# TS Inter Maths 1A &1B Practice papers (Reduced Syllabus)

###### TS Inter Maths 1A and 1B Practice papers as per reduced syllabus were designed by the ‘Basics in Maths‘ team.

These Practice papers to do help the intermediate First-year Maths students.

# TS Inter Maths 1A and 1B Practice papers as per reduced syllabus are very useful in IPE examinations.

## MATHS 1A PRACTICE PAPER – 1

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## MATHS 1A PRACTICE PAPER – 2

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## MATHS 1A PRACTICE PAPER – 3

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## MATHS 1A PRACTICE PAPER – 4

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## MATHS 1B PRACTICE PAPER – 1

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## MATHS 1B PRACTICE PAPER – 2

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## MATHS 1B PRACTICE PAPER – 3

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## MATHS 1B PRACTICE PAPER – 4

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## TS inter 1st year

TS inter 1st year: These blueprints were designed by ‘Basics in Maths’ team. These to-do’s help the TS intermediate first-year Maths students fall in love with mathematics and overcome their fear.

These blueprints cover all the topics of the TS I.P.E first-year maths syllabus and help in I.P.E exams.

TS inter 1st year

## TS Inter second year

TS Inter second year: This note is designed by the ‘Basics in Maths’ team. These notes to do help the TS intermediate second-year Maths students fall in love with mathematics and overcome the fear.

These notes cover all the topics covered in the TS I.P.E second year maths 2B syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations.

### TS Inter second year

#### 1. CIRCLES

Circle: In a plane, the set of points that are at a constant distance from a fixed point is called a circle.

∗ The fixed point is called the centre (C) of the circle and the constant distance is called the radius(r) of the circle

Unit circle: If the radius of the circle is 1 unit, then that circle is called the unit circle.

Point Circle: A circle is said to be a point circle if its radius is zero. A point circle contains only one point in the centre of the circle.  •

∗ The equation of the circle with centre (h, k) and radius r is

(x – h)2 + (y – k)2 = r2

∗ The equation of the circle with centre origin and radius r is

x2 + y2 = r2

⇒ x2 + y2 = r2 is called standard form of the circle.

The general equation of the second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, where a, b, f, g, h and c are real numbers, represent a circle iff (i) a = b ≠ 0 (ii) h = 0 and (iii) g2 + f2 + c ≥ 0

∗ The general equation of the circle is x2 + y2 + 2gx + 2fy + c = 0

It’s centre c = (– g, – f) and radius

∗ The equation of the circle passing through origin is x2 + y2 + 2gx + 2fy = 0.

∗ The equation of the circle whose centre on the x-axis is x2 + y2 + 2gx + c = 0.

∗ The equation of the circle having centre on y-axis is x2 + y2 + 2fy + c = 0.

∗ The circles which have the same centre are called concentric circles.

∗ The equation of the circle concentric with the circle x2 + y2 + 2gx + 2fy + c = 0 is

x2 + y2 + 2gx + 2fy + k = 0.

∗ The length of the intercept made by a circle x2 + y2 + 2gx + 2fy + c = 0 on

• x -axis is   if g2 – c > 0
• y -axis is if f2 – c > 0

Note: –

(a) if g2 – c = 0, then A1 A2 = 0 ⇒ the circle touches the x- axis at only one point.

(b)  if f2 – c = 0, then B1 B2 = 0 ⇒ the circle touches the y- axis at only one point.

(c) if g2 – c < 0, then the circle does not meet the x- axis.

(d) if f2 – c < 0, then the circle does not meet the y- axis.

∗ The equation of the circle having the line segment joining A (x1, y1) and B (x2, y2) as a diameter is

(x – x1) (x – x2) + (y – y1) (y – y2) = 0.

∗ Let A, B be any two points on a circle then,

• The line is called the secant line of the circle.
• The line segment is called the chard of the circle.
• AB is called the length of the chord.

∗ A chord passing through the centre is called the diameter of the circle.

∗ The angle subtended by a chord on the circumference of at any point is equal.

The perpendicular bisector of a chord of a circle is asses through the centre of the circle.

∗ The angle in a semicircle is 900.

∗ The equation of the circle passing through three non-collinear points A (x1, y1), B (x2, y2), C (x3, y3) is

Where ci = − (x2 + y2) and i = 1,2,3

∗ centre of the circle is

#### Parametric form:

If P (x, y) is a point on the circle with centre (h, k) and radius r, then

X = h + r cosθ, y = k + r sinθ  0 ≤ θ ≤ 2π.

⇒ A point n the circle x2 + y2 = r2 is taken as (r cosθ, r sinθ) and simply denoted by θ.

Note:

1.  If the centre of the circle is the origin, then the parametric equations are x = r cosθ, y = r, 0 ≤ θ ≤ 2π.
2. The point (h + rcosθ1, k + r sin θ1) is referred to as the point θ1 on the circle having the centre (h, k) and radius r.

Notations:

S = x2 + y2 + 2gx + 2fy + c

S1 = xx1 + yy1 + g(x +x1) + f (y +y1) + c

S11 = x12 + y12 + 2gx1 +2fy1 + c

S12 = x1x2 + y1y2 + g(x1 + x2 ) + f (y1 + y2) + c

Position of a point with respect to the circle:

A circle divides the plane into three parts.
1. The interior of the circle

2. The circumference which is the circular curve.

3. The exterior of the circle.

Power of point:

Les S = 0 be a circle with radius ‘r’ and centre ‘C’ and P (x1, y1) be a point on the circle, then CP – r2 is called the power of point ‘P’ concerning S = 0.

• The power of point P (x1, y1) w.r.t. S = 0 is S11.

•Let S = 0 be a circle in a plane and P (x1, y1) be any point in the same plane. then

1. P lies in the interior of the circle ⇔ S11 < 0.
2. P lies on the circle ⇔ S11 = 0.
3. Plies in the exterior of the circle ⇔ S11 > 0.

Secant and tangent of a circle:

Let P be any point on the circle and Q be neighbourhood point of P lying on the circle. join P and Q, then the line PQ is the secant line.

The limiting position of the secant line PQ when Q is approached to the point P along the circle is called a tangent to the circle at P.

Length of the tangent:

If P is any point on the circle S = 0 and T is any exterior point of the circle, then PT is called the length of the tangent.

∗ If S = 0 is a circle and P (x1, y1) is an exterior point with respect o S = 0, then the length of the tangent from P (x1, y1) to S =0 is

Condition for a line to be a tangent:

• A straight-line y = mx + c (i) meet the circle x2 + y 2 = r2 in two distinct points if
• Touch the circle x2 + y 2 = rif
• Does not touch the circle x2 + y 2 = r2 in two distinct points if

Note:

1. For all real values of m, the straight line is a tangent to the circle x2 + y2 = r2 and the slope of the line is m.
2. A straight-line y = mx + c is a tangent to the circle x2 + y2 = r2 if c = .
3. The equation of a tangent to the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 having the slope m is  where r is the radius of the circle.
4. The circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 touches (i) x – axis if g2 = c (ii) y – axis if f2= c.

Chord joining two points on a circle:

If P (x1, y1) and Q (x2, y2) are two points on the circle S = 0 then the equation of secant line PQ is S1 + S2 = S12.

Equation of tangent at a point on the circle:

The equation of the tangent at the point (x1, y1) to the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 is S1 = 0.

The equation of the tangent at the point (x1, y1) to the circle x2 + y2 = r2 is xx1 + yy1 – r2 = 0.

Point of contact:

If a straight-line lx + my + n = 0 touches the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 at P (x1, y1), then this line is the tangent to the circle S = 0. And the equation of the tangent is

(x1 + g) x + (y1 +f) y + (gx1 + fy1 + c) = 0.

∗ The equation of the chord joining two points θ1, θ2 on the circle x2 + y2 + 2gx + 2fy + c = 0 is

∗ The equation of the chord joining the points θ1, θ2 on the circle x2 + y2 = r2 is

∗ The equation of the tangent at P(θ) on the circle x2 + y2 + 2gx + 2fy + c = 0 is

∗ The equation of the tangent at P(θ) on the circle x2 + y2 = r2 is

x cosθ + y sinθ = r.

Normal:

The normal at any point P of the circle is the line which is passing through P and is perpendicular to the tangent at P.

• The equation of the normal at P (x1, y1) of the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 is

(x – x1) (y1 + g) – (y – y1) (x1 + g) = 0.

• The equation of the normal at P (x1, y1) of the circle x2 + y 2 = r2 is xy1 – yx1 = 0.

Chord of contact and Polar:

∗ If P (x1, y1) is an external point of the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0, then there exists two tangents from P to the circle S = 0.

Chord of contact: –

If the tangents are drawn through P (x1, y1)

to a circle S = 0 touch the circle at points A and B then the secant line AB is called the chord of contact of P with respect to S = 0

∗ If P (x1, y1) is an external point of the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0, then the equation of the chord of contact of P with respect to S =0 is S1 = 0.

Note:

1. If the point P (x1, y1) is on the circle S = 0, then the tangent itself can be defined as the chord of contact.
2. If the point P (x1, y1) s an interior point of the circle S = 0, then the chord of contact does not exist.

Pole and Polar: –

Let S = 0 be a circle and P be any point if any line is drawn through the point Pin the plane other than the centre of S = 0. then the points of intersection meet the circle in two points A and B, of tangents drawn at A and B lie on a line called polar of P and P, is called Pole of polar.

∗ The equation of the polar of P (x1, y1) with respect to the circle S = 0 is S1 = 0.

Note: –

1. If Plies outside the circle S = 0, then the polar of P meets the circle in two points and the polar becomes the chord f contact of P.
2. If P lies on the circle S = 0, then the polar P becomes the tangent at P o the circle.
3. If P lies inside the circle S = 0, then the polar of P does not meet the circle.
4. If P is the centre of the circle S = 0, then the polar of P does not exist.
5. The pole of the line lx + my + n = 0 with respect to the circle x2 + y2 = r2 is
6. The pole of the line lx + my + n = 0 with respect to the circle x2 + y2 + 2gx + 2fy + c = 0 is
7. The polar of P (x1, y1) with respect to the circle S = 0 passes through Q (x2, y2)  ⟺ polar of Q passes through P.

Conjugate points: Two P and Q are said to be conjugate points with respect to the circle S = 0, if the polar of P with respect to S = 0 passes through Q.

⇒ The condition for the points P (x1, y1), Q (x2, y2) to be conjugate with respect to the circle S = 0 is S12 = 0.

Conjugate lines: Two lines L1 = 0 and L2 = 0 are said to conjugate lines with respect to the circle S = 0 if the pole of L1 = 0 is lies on L2 = 0.

The condition for the lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 to be conjugate with respect to the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 is

r2 (l1l2 + m1m2) = (l1g + m1f – n1) (l2g + m2f – n2)

⟹ The condition for the lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 to be conjugate with respect to the circle S ≡ x2 + y2 = r2 is  r2 (l1l2 + m1m2) = n1n2

Inverse points: Let S = 0 be a circle with centre C and radius r. two points P and Q are said to be inverse points with respect to the circle S = 0 if

1. C, P, Q are collinear.
2. P, Q lies on the same side of C.
3. CQ = r2.

⟹  If lies inside of the circle S = 0, then Q lies outside of the circle.

⟹  If P lies on the circle S = 0, then P =Q.

⟹  Let S = 0 be a circle with centre C and radius r. The polar of P meets the line CP in Q iff  P, Q is inverse points.

⟹  f P, Q are inverse points with respect to S = 0, then P, Q are conjugate points with respect to the circle S = 0.

⟹  If P, Q are inverse points with respect to S = 0, then Q is the foot of the perpendicular from P on the polar of P with respect to the circle S = 0.

Equation of the chord with the given middle point:

The equation of the chord of the circle S = 0 having P (x1, y1) as its midpoint is S1 = S11.

Common tangents to the circle:

⟹ A straight line L = 0 is said to be a common tangent to the circle S = 0 and S= 0 if it is a tangent to both S = 0 and S’ = 0.

Two circles are said to touch each other if they have only one common tangent.

The relative position of two circles:

Let C1, C2 centres and r1, r2 be the radii of two circles S = 0 and S’ = 0respectively.

1.If C1C2 > r1+ r2, then two circles do not intersect.

⟹2 direct common tangents and

2 transverse common tangents

Total 4 common tangents

⟹P divides C1C2 in the ratio r1: r2 internally

Here P is called the internal centre of similitude (I.C.S)

⟹ Q divides C1C2 in the ratio r1: r2 externally

Here Q is called the external centre of similitude (E.C.S)

2.If C1C2 = r1+ r2, two circles touch each other

⟹ Q divides C1C2 in the ratio r1: r2 externally

⟹ two direct common tangents and one common tangent. Total 3 tangents

Here Q is called the external centre of similitude (E.C.S)

3.

⟹ two direct common tangents

Here Q is called the external centre of similitude (E.C.S)
Q divides C1C2 in the ratio r1: r2 externally

⟹ internal centre of similitude does not exist.

4.

⟹ only one common tangent

internal centre of similitude does not exist.

5.

no. of common tangents zero.

Note: the combined equation of the pair of tangents drawn from an external point P (x1, y1) to the circle S = 0 is S S11 = S12.

## 2.SYSTEM OF CIRCLES

A set of circles is said to be a system of circles if it contains at least two circles.

The angle between two intersecting circles:

If two circles S = 0 and S’ = 0 intersect at P then the angle between the tangents of two circles at P is called angle between the circles at P.

⟹ If two circles S = 0 and S’ = 0 intersect at P and Q then the angle between the tangents of two circles at P and Q are equal.

⟹ If d is the distance between the centres of the two intersecting circles with radii r1, r2 and θ is the angle between the circles then.

⟹ If θ is the angle between the circles x2 + y2 + 2gx + 2fy + c = 0 and x2 + y2 + 2g’x + 2f’y + c’ = 0 then

⟹ Two intersecting lines are said to be Orthogonal if the angle between the circles is a right angle.

Condition for the orthogonality:

⟹ The condition that the two circles x2 + y2 + 2gx + 2fy + c = 0 and x2 + y2 + 2g’x + 2f’y + c’ = 0 cut each other orthogonally is 2gg’ + 2ff’ = c + c’.

⟹ If d is the distance between the centres of the two intersecting circles with radii r1, r2. Two circles cut orthogonally if d2 = r12 + r22.

∎  If S = 0, S’ = 0 are two circles intersecting at two distinct points, then S – S’ = 0 represents a common chord of these two circles.

∎ If S = 0, S’ = 0 are two circles touch each other, then S – S’ = 0 represents a common tangent of these two circles.

∎ If S ≡ x2 + y2 + 2gx + 2fy + c =  0 and L ≡ lx + my + n = 0 are the equation of the circle and  a line respectively intersecting each other, then S + λ L = 0 represent a circle passing through the point intersection of  S = 0 and L = 0 ∀ λ ∈ ℛ.

Radical axes:

The radical axis of two circles s defined as the locus of the point which moves so that its powers with respect to the two circles are equal.

(OR)
The locus of a point, for which the powers with respect to given non-concentric circles are equal, is a straight line is called Radical axis of the given circles.

∎ The equation of Radical axis f the circles S = 0 and S’ = 0 is S – S’ = 0.

The radical axis of any two circles is perpendicular to the line joining their centres.

The lengths of tangents from a point on the radical axis of two circles are equal if exist.

Radical axis of two circles bisects all common tangents of the two circles.

∎ If the centres of any three circles are non-collinear then the radical axis of each pair of circles chosen from these three circles re concurrent.

Radical centre:  The point of concurrence of radical axes of each pair of three circles is called radical centre (see above figure).

∎ If the circle S = 0 cuts the each of the two circle S’ = 0 and S’’ =0 orthogonally then the centre of S =0 lies on the radical axis of S’ = 0 and S’’ = 0.

∎ Radical axis of two circles is

• The ’common chord’ if the two circles intersect at two distinct points.
• The ‘common tangent’ at the point of contact if the two circles touch each other.

The radical axis of any two circles bisects the line joining the points of contact of common tangents to the circles.

Let S = 0, S’ = 0 and S’’ =0 be three circles whose centres are non- collinear and no two circles of these are intersecting then the circles having

• Radical centre of these circles as the centre of the circle.
• Length of the tangent from the radical centre to any one of these three circles cuts the given three circles orthogonally.

### CONIC SECTIONS

Conic: The locus of a point moving on a plane such that its distance from a fixed point and a fixed straight line is in the constant ratio is called Conic.

OR

The locus of a point moving on a plane such that its distance from a fixed point and a fixed line on the plane are in a constant ratio ‘e’, is called a Conic.

Focus: The fixed point is called focus and it is denoted by S.

Directrix: The fixed straight line is called the directrix.

Eccentricity: The constant ratio is called eccentricity and it is denoted by ‘e’.

Conic is the locus of a point P moving on a plane such that SP/PM = e, PM is the perpendicular distance from P to directrix at M.

If e = 1, then the conic is parabola.

if 0 < e < 1, then the conic is Ellipse.

if e > 1, then the conic is Hyperbola.

## 3.PARABOLA

If e = 1, then the conic is parabola.

• The standard form of parabola is y2 = 4ax.
• Focus S = (a. 0).
• Equation of directrix is x + a = 0.
• Vertex A = (0, 0) and A is the mid-point of SZ.

• Equation of the parabola with focus (α, β) and directrix lx + my + n = 0 is

• If the focus is situated on the left side of the directrix, the equation of the parabola with vertex as the origin and the axis is X-axis is y2 = – 4ax.
• The vertex being the origin, if the axis of the parabola is taken as Y – axis, equation of the parabola is x2 = 4 ay or x2 = – 4 ay according to the focus is above or below the X-axis.

Nature of the curve:

The nature of the parabola f the equation y2 = 4 ax (a>0)

• F y = 0, then 4 ax = 0 and x = 0

∴ the curve passes through the origin.

• If x = 0, then y2 = 0. Which gives y = 0. Y – axis is the tangent to the parabola at origin.
• Let P(x, y) be any point on the parabola (a>0) and y2 = 4 ax, we have x ≥ 0 and

∴ for any positive real value of x, we obtain two value of y of equal magnitude but opposite in sign. This shows that the curve is symmetric about X-axis and lies in the first and fourth quadrants.

The curve does not exist on the left side of the Y-axis since x ≥ 0 for any point (x, y) on the parabola.

Chord: The line segment joining two points on a parabola is called a chord.

Focal chord: A chord which is passing through focus is called Focal Chord.

Double ordinate: A chord through a point P on the parabola, which is perpendicular to the axis of the parabola is called Double ordinate.

Latus rectum: The double ordinate passing through the focus is called Latus rectum.

⟹ Length of Latus rectum = 4a.

Various forms of the parabola

1. y2 = 4ax

focus: (a, 0)

equation of directrix: x + a = 0

axis of parabola: y = 0

vertex: (0. 0)

2. y2 = −4ax

equation of directrix: x − a = 0

focus: (−a, 0)

axis of parabola: y = 0

vertex: (0. 0)

3. x2 = 4ay

focus: (0, a)

equation of directrix: y + a = 0

axis of parabola: x = 0

vertex: (0. 0)

4. x2 = −4ax

focus: (0, −a)

equation of directrix: y − a = 0

vertex:  (0. 0)    axis of parabola: x = 0

5. (y – k) 2 = 4a (x – h)

focus: (h + a, k)

equation of directrix: x – h + a = 0

axis of parabola: y – k = 0

vertex: (h. k)

6. (y – k) 2 = −4a (x – h)

focus: (h – a, k)
equation of directrix: x – h – a = 0

axis of parabola: y – k = 0

vertex: (h. k)

7. (x – h) 2 = 4a (y – k)

focus: (h, k + a)

equation of directrix: y – k + a = 0

axis of parabola: x – h = 0

vertex: (h. k)

8. (x – h)2 = −4a (y – k)

focus: (h, k – a)

equation of directrix: y – k – a = 0

vertex: (h. k)
axis of parabola: x – h =0

9.

focus: (α, β)

equation of directrix: lx + my + n = 0

axis of parabola: m (x – α) – l (y – β) = 0

vertex: A

Note:

1. Equation of the parabola whose axis parallel to X – axis is x = ly2 + my + n.
2. Equation of the parabola whose axis parallel to Y – axis is y = lx2 + mx + n.

Focal distance:  The distance of a point on the parabola from its focus is called Focal distance.

⟹ Focal distance of parabola s x1 + a

Parametric equations of a parabola:

The point (at2, 2at) satisfy the equation y2 = 4ax, the parametric equations of parabola are  x = at2, y = 2at. The point P(at2, 2at) is generally denoted by the point ‘t’ or P(t).

Notation:

1. S ≡ y2 – 4 ax
2. S1 ≡yy1 – 2a (x + x1)
3. S12 ≡ y1y2 – 2a (x1 + x2)
4. S11 ≡ y12 – 4 ax1

Equation of a tangent and normal at a point on the parabola:

∎ y = mx + c is a tangent to the parabola y2 = 4ax, then c = a/m or a =cm, and the point of contact is (a/m2, 2a/m).

∎ if m = 0, the line y = c is parallel to the axis of the parabola (i.e., x – axis)

y = c ⟹ c2 = 4ax ⟹ x = c2 /4a

∴ point of contact is (c2/4a, c).

∎ if m ≠ 0 and c = 0, then

Y = mx ⟹ x = 4a/m2 and y = 4a/m

∴ point of contact is (4a/m2, 4a/m).

∎ The equation of the chord joining the points (x1, y1) and (x2, y2) is S1 + S2 = S12.

∎ The equation of the tangent at P (x1, y1) to the parabola S = 0 is S1= 0.

∎ The equation of the normal at P (x1, y1) is (y – y1) = – y1/2a (x – x1).

Parametric form:

∎ The equation of the tangent at a point ‘t’ on the parabola y2 = 4ax is x – yt + at2 = 0.

∎ Equation of the normal at a point ‘t’ on the parabola y2 = 4ax is y + xt = 2at + at3.

∎ The condition for the straight-line lx + my + n = 0 to be a tangent to the parabola y2 = 4 ax is

am2 = nl and point of contact is (n/l, –2 am/l).

∎ common tangent to the parabolas y2 = 4 ax and x2 = 4 by is x a1/3 + y b1/3 + a2/3 b2/3 = 0.

∎ The equation of the chord of contact of the external point (x1, y1) w.r. t parabola S = 0 is S1 = 0.

∎ The equation of the polar of the point (x1, y1) w.r. t parabola S = 0 is S1 = 0.

∎ The pole of the line lx + my + n = 0 w.r.t. parabola y2 = 4ax is (n/l, -2am/l).

∎ If two points P (x1, y1), Q (x2, y2) are conjugate points w.r.t. parabola S = 0, then S12 = 0.

∎ The lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 are conjugate lines with respect to the parabola y2 = 4 ax, then l1n2 + l2n1 = 2a m1m2.

## 4.ELLIPSE

Ellipse: A conic with eccentricity less than unity s called Ellipse.

∎ Equation of Ellipse in standard form is

⇒b2 = a2 (1 – e2) ⇒ e2 – 1 = -b2/a2

Major and Minor axis:

⟹ The line segment AA’ and BB’ of length 2a and 2b respectively are axes of the Ellipse.

⟹ If a > b AA’ is called Major axis and BB’ is called Minor axis and vice-versa if a<b.

Various form of Ellipse:

1.

Major axis: along the x-axis

Length of Major axis:2a

Minor axis: along y – axis

Length of Minor Axis:2b

Centre: (0, 0)

Foci: S = (ae, 0) and S’ = (–ae, 0)

Equation of directrices: x = a/e and x = –a/e

Eccentricty:

2.

Major axis: along y – axis

Length of Major axis:2b

Minor axis: along x – axis

Length of Minor Axis:2a

Centre: (0, 0)

Foci: S = (0, be) and S’ = (0, –be)

Equation of directrices: x = b/e and x = –b/e

Eccentricty:

Centre not at the origin

3.

Major axis: along with y = k

Length of Major axis:2a

Minor axis: along x = h

Length of Minor Axis:2b

Centre: (h, k)

Foci: S = (h +ae, k) and S’ = (h – ae, k)

Equation of directrices: x = h + a/e and x = h – a/e

Eccentricty:

4.

Major axis: along x = h

Length of Major axis:2b

Minor axis: along with y = k

Length of Minor Axis:2a

Centre: (h, k)

Foci: S = (h, k + be) and S’ = (h, k – be)

Equation of directrices: xy = k + b/e and y = k – b/e

Eccentricty:

Chord: The line segment joining two points on a parabola is called a  chord of Ellipse.

Focal chord: A chord which is passing through one of the foci is called Focal Chord.

Latus rectum: A focal chord perpendicular to the major axis of the Ellipse is called Latus Rectum. Ellipse has two latera recta.

Length of the Latus rectum:

1.The coordinates of the four ends of the latera recta of the ellipse

L = (ae, b2/a), L’ = (ae, -b2/a) and L1 = (-ae, b2/a), L1’= (-ae, -b2/a).

length of the Latus rectum = 2b2/a.

2.length of the Latus rectum of an ellipse   is 2a2/b and the coordinates of the four ends of the latera recta are

L = (a2/b, be), L’ = (-a2/b, be) and L1 = (a2/b, -be), L1’ = (-a2/b, -be).

3. The equation of the Latus rectum of the Ellipse   through S is x = ae and through S’ is x = -ae.

4. The equation of the Latus rectum of the Ellipse   through S is y = be and through S’ is y = -be.

5. If P (x, y) is any point on the Ellipse  whose foci are S and S’, then SP +S’P is constant.

Auxiliary circle: The circle described on the major axis of an Ellipse as the diameter is called the Auxiliary circle of the Ellipse. The Auxiliary circle of the Ellipse is x2 + y2 = a2.

Parametric equations:  The parametric equations of the Ellipse  are x = a cosθ and y = b sinθ.

Notation:

Equation of Tangent and Normal

The equation of any tangent to the Ellipse can be written as

The condition for a straight-line y = mx + c to be a tangent to the Ellipse       is  c2 = am2 + b2.

∎ The point of contact of two parallel tangents to the Ellipse are (-a2m/c, b2/c) and (a2m/c, -b2/c)

∎ The equation of the chord joining two points (x1, y1) and (x2, y2) on the Ellipse S = 0 is S1 + S2 = S12.

∎ The equation of the Normal at P (x1, y1) to the Ellipse is

∎ Equation of the tangent at P(θ) on the Ellipse

∎ Equation of the normal at P(θ) on the Ellipse S = 0 is

∎ When θ = 0, π; equation of Normal is y =0.

∎ When θ = π/2, 3π/2; equation of Normal is x =0.

∎ The condition for the line lx + my + n = 0 to be a tangent the Ellipse S = 0 is a2l2 + b2m2 = n2.

∎ The condition for the line x cosα + y sinα = p to be a tangent the Ellipse S = 0 is a2 cos2 α + b2 sin2 α = p2.

∎ The pole of the line lx + my + n = 0 with respect to the Ellipse S = 0 is (-a2l/n, -b2m/n).

∎ The condition for the two lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 to be conjugate with respect to the Ellipse S = 0 is a2l1l2 + b2m1m2 = n1n2. a2l1l2.

## 5.HYPERBOLA

Hyperbola: Hyperbola is a conic in which the eccentricity is greater than the unity.
Standard form of Hyperbola:

Equation of Hyperbola in standard form is

Centre: C (0, 0)

Foci: (± ae, 0)

Directrix: x = ± a/e.

Ecdntricity:

Notation:

Rectangular Hyperbola:

If in a Hyperbola the length of the transverse axis (2a) is equal to the length of the conjugate axis(2b), then the hyperbola is called rectangular hyperbola.

Its equation is x2 – y2 = a2 and eccentricity is .

Auxiliary circle: The circle described on the transverse axis of hyperbola as diameter is called the auxiliary circle of the hyperbola.

The equation of the auxiliary circle of S = 0 is x2 + y2 = a2.

Parametric equations:  The parametric equations of the Parabola S = 0 are x = a secθ and y = b tanθ.

Conjugate Hyperbola:

The hyperbola whose transverse and conjugate axis are respectively the conjugate and transverse axis of a given hyperbola is called a conjugate hyperbola.

The equation of hyperbola conjugate to   S ≡  is S’ ≡

∎ For

The transverse axis lies on along X-axis and its length is 2a.

The conjugate axis lies on along Y-axis and its length is 2b.
∎ For

The transverse axis lies on along Y-axis and its length is 2b.

The conjugate axis lies on along X-axis and its length is 2a.

Various form of Hyperbola:

Let  and

1.Hyperbola

The transverse axis along X-axis: y =0

Length of the transverse axis:2a

The conjugate axis along Y-axis: x = 0

Length of the conjugate axis: 2b

Centre: (0, 0)

Foci: (± ae, 0)

Equation of the directrices: x = ± a/e

Eccentricity:

2.Conjugate Hyperbola

The transverse axis along Y-axis: x = 0

Length of the transverse axis:2b

The conjugate axis along X-axis: y = 0

Length of the conjugate axis: 2a

Centre: (0, 0)

Foci: (0, ± be)

Equation of the directrices: y = ± b/e

Eccentricity:

Centre not at the origin:

3.Hyperbola

The transverse axis along X-axis: y = k

Length of the transverse axis:2a

The conjugate axis along Y-axis: x = h

Length of the conjugate axis:2b

Centre: (h, k)

Foci: (h± ae, k)

Equation of the directrices: x = h± a/e

Eccentricity:

4.Hyperbola

The transverse axis along Y-axis: x = h

Length of the transverse axis:2b

The conjugate axis along X-axis: y = k

Length of the conjugate axis: 2a

Centre: (h, k)

Foci: (h, k ± be)

Equation of the directrices: y = k± b/e

Eccentricity:

Equation of tangent and normal at a point on the Hyperbola:

∎ The equation of the tangent at P (x1, y1) to the hyperbola S = 0 is S1 = 0.

∎ The equation of the tangent at P(θ) on the hyperbola S = 0 is

∎ The equation of the Normal at P (x1, y1) to the hyperbola S = 0 is

∎ Equation of the normal at P(θ) on the Hyperbola S = 0 is

∎ The condition for a straight-line y = mx + c to be a tangent to the hyperbola S = 0 is c2 = am2 − b2.

Asymptotes of a hyperbola:

The equations of asymptotes of hyperbola S = 0 are      and the joint equation of asymptotes is

## 6. INTEGRATION

∎ Let E be a subset of R such that E contains a right or left the neighbourhood of each of its points and let f: E → R be a function. If there is a function F on E such that F’(x) = f(x) ∀ x ∈ E, then we call F an anti-derivative of f or a primitive of f.

Indefinite integral: Let f: I→R. Suppose that f has an antiderivative F on I. Then we say that f has an integral on I and for any real constant c, we call F + c an indefinite integral of over I, denote it by    ∫f(x) dx and read it as ‘integral’ f(x) dx.

∫f = ∫f(x) dx = F(x) + c. here c is called constant of integration.

In the indefinite integral ∫f(x)dx, f is called ‘integrand’ and x is called the variable of integration.

⟹

⟹ if f: I⟶R is differentiable on I, then ∫f’(x)dx = f(x) + c.

Standard forms:

Properties of integrals:

∎ ∫ (f ±g) (x) dx = ∫f(x) dx ± ∫g(x) dx + c

∎ ∫(af) (x) dx = a ∫f(x) dx + c

∎ ∫ (f1 + f2 + … + fn) (x) dx = ∫f1(x) dx + ∫f2(x) dx + … +∫fn(x) dx +c

∎ ∫f(g(x)) g’(x) dx = F(g(x)) + c

∎ ∫f(ax + b) dx = 1/a F(ax +b) + c

Some important formulae:

1. ∫eax dx = 1/a eax + c
2. ∫sin (ax + b) dx = -1/a cos (ax + b) + c
3. ∫cos (ax + b) dx = 1/a sin (ax + b) + c
4. ∫sec2 (ax + b) dx = 1/a tan (ax + b) + c
5. ∫ cosec2 (ax + b) dx = 1/a cot (ax + b) + c
6. ∫cosec (ax + b) cot (ax + b) dx = -1/a cosec (ax + b) + c
7. ∫sec (ax + b) tan (ax + b) dx = 1/a sec (ax + b) + c

Integration by parts:

Let u, v real valued differentiable functions in I. Suppose that u,v has an integral on I, then uv’ has an integral on I and

∫(uv’) (x) dx = (uv) – ∫(u’v) (x) dx + c or ∫(uv) dx = u ∫v dx – ∫ [u’ ∫v dx] dx + c

Integration of exponential functions:

∫ex dx = ex + c; ∫x ex dx = (x – 1) ex + c

∫ ex [f(x) +f’(x)] dx = ex f(x) + c

Integration of logarithmic functions:

∫log x dx = x log x – x + c

Integration of inverse trigonometric functions:

Evaluation of integrals form  :

Working rule:  reduce ax2 + bx + c to the form of a[(x + α)2 + β] and then integrate using the substitution t = x + α.

Evaluation of integrals form

Working rule:

Case(i) if a >0 and b2 – 4ac < 0, then reduce ax2 + bx + c to the form of a[(x + α)2 + β] and then integrate.

Case(ii) if a <0 and b2 – 4ac >0, then reduce ax2 + bx + c to the form of (-a) [ β – (x + α)2 +] and then integrate.

Evaluation of integrals form

Working rule:  write px + q in the form of A (ax2 + bx +c)’ + B, then integrate.

Evaluation of integrals form

Working rule:  write cos x =cos2(x/2) – sin2(x/2) and sin x = 2 sin(x/2) cos (x/2)

Put t = tan(x/2), then dt = ½ sec2 (x/2) dx

Cos x = 1 – t2 / 1 + t2, sin x = 2t/1 + t2 then integrate.

Evaluation of integrals form

Working rule:  t = sqrt. (px + q and then integrate.

Evaluation of integrals form

Working rule:  we find real numbers A, B and C such that

(a cos x + b sin x + c) = A(d cos x + e sin x +f)’ + B(d cos x + e sin x +f) + C then by substituting this expression in the  integrand, evaluate the integral.

Integration – partial fraction method:

Let R(x) = f(x) / g(x), g(x) ≠ 0 where f, g are polynomials. If degree of f(x) ≥degree of g(x), then divide f(x) by g(x) by synthetic division method and find polynomials.

Q(x) and h(x) such that f(x) = Q (x) g(x) + h(x) here h = 0 or h ≠ 0 and degree h(x) < degree of g(x). Then R(x) =Q(x) + h(x)/g(x)

We get solution of h(x) / g(x) using partial fractions and then integrate.

Partial fractions:

∎ If R(x) = f(x) / g(x) is proper fraction, then

Case(i): – For every factor of g(x) of the form (ax + b) n, there will be a sum of n partial fractions of the form:

Case(ii): – For every factor of g(x) of the form (ax2 + bx + c) n, there will be a sum of n partial fractions of the form:

∎ If R(x) = f(x) / g(x) is improper fraction, then

Case (i): – If degree f(x) = degree of g(x), then f(x)/g(x) = k + h(x)/g(x) where k is the quotient of the highest degree term of f(x) and g(x).

Case (ii): – If f(x) > g(x)

R(x) =f(x) /g(x) = Q(x) + h(x)/g(x)

Reduction formulas:

## 7.DEFINITE INTEGRATION

Partition: Let a, b∈ R be such that a < b. Then, a finite set P = {x0, x1, …, x i- 1, xi, xi + 1, …, xn} of elements of [a, b] is called to be a partition of [a, b] if a =  x0 <  x1 < … < x i- 1 <  xi <  xi + 1 < … < xn = b.

Norm: if {x0, x1, …, xn} is a partition of [a, b], then the norm of the partition P, denoted by ∥P∥, is defined by ∥P∥ = max {x1 – x0, x2 – x1, …, xn – xn-1}. We donate the set of all partitions of [a, b] by 𝒫 ([a, b]).

Definite integral:
Riemann sum:
Let f: [a, b] → R be a bounded function for all x in [a, b]. Let P = {x0, x1, …, x i- 1, xi, xi + 1, …, xn} be partition of [a, b], and t ∈ [xi-1, xi], for I = 1, 2, …, n. A sum of the form is called Riemann sum of f relative to P.

Let f is Riemann integrable on [a, b]. if there exists a real number A such that S (P, f) approaches A as ∥P∥ approaches to ‘0’. In other words, given ϵ > 0, there is a δ > 0 such that  for any partition P of [a, b] with ∥P∥ < δ irrespective of the choice of ti in [xi-1, xi]. Such an A, if exists, is unique and is denoted by , it is read as the definite integral of f from a to b. an a is called the lower limit and b is called the upper limit. The function f in  is called ‘integrand’.

if f: [a, b] → R is continuous, then is exists.

∎ If f is continuous on [0, p] where p is a positive integer then

The fundamental theorem of integral calculus:

If f is integrable on [a, b] and if there is a differentiable function F on [a, b] such that F = f, then

we write

properties of definite integrals:

∎ Let f: [a, b] → R be bounded. Let c ∈ (a, b). then f is integrable on [a, b] if and only if it is integrable on [a, c] as well as on [c, b] and in this case

Method of substitution:  Let g: [c, d] → R have continuous derivative on [c, d]. Let f: g([c, d]) → R be continuous. Then (fog) g’ is integrable on [c, d] and

∎ Let f be integrable on [a, b]. Then the function h, defined on [a, b] as h(x) = f (a + b – x)

for all x in [a, b] and

∎ Let f be integrable on [0, a]. Then the function h, defined on [0, a] as h(x) = f (a – x)

for all x in [a, b] and

∎ Let f: [-a, a] → R be integrable on [0, a]. Suppose that f is either odd or even. Then f is integrable on  [-a, a] and

∎ Let f: [0,2 a] → R be integrable on [0, a].

• If f (2a – x) = f(x) for all x in [a, 2a] then f is integrable on [0, 2a] and
• If f (2a – x) = – f(x) for all x in [a, 2a] then f is integrable on [0, 2a] and

∎ If f and g are integrable on [a, b], then their product fig is integrable on [a, b].

Integration by parts:

∎ Let f: R→ R be a continuous periodic function and T be the period of it. Then any positive integer n

Reduction formulae:

∎ Let n≥2 be an integer, then

∎ Let m and n be positive integers, then

Areas under curves:
(i)If f: [a, b] → [0, ∞) is continuous, then the area A of the region bounded by the curve y = f(x), the X-axis and the line x = a and x =b is given by

A =

(ii) If f: [a, b] → (−∞, 0] is continuous, then the graphs of y = f(x)and y = − f(x) on [a, b] are symmetric about the X-axis. So, the area bounded by the graph of y = f(x), the X-axis and the lines x = a, x =b is same as the area bounded by the graph of y = – f(x), the X-axis and the lines x = a and y = b which is given by A =

From (i) and (ii) A =

(iii) Let f: [a, b] → R be continuous and f(x) ≥ 0 ∀ x ∈ [a, c] and f(x) ≤ 0 ∀ x∈ [c, b] where a < c < b. Then the area of the region bounded by the curve y= f(x), the X – axis, and the lines x = a and x = b is given by

Area of region =A =

(iv) Let f: [a, b] → R and g: [a, b] → R be continuous f(x) ≤ g(x) ∀ x∈ [a, b]. Then the area f the region bounded by the curve y = f(x), y = g(x) and the lines x = a, x = b is given by

(v) Let f and g be wo continuous real value functions on [a, b] and c ∈ (a, b) such that f(x) < g(x) ∀ x∈ [a, c) and g(x) < f(x) ∀ x∈ (c, b] with f (c) = g (c). area of the region bounded by y = f (x), y = g(x), and the lines x = a, x = b is given by

(vi) Let f: [a, b] → R and g: [a, b] → R be continuous functions. Suppose that, there exist points x1, x2 ∈ (a, b) with x1< x2 such that f(x1) = g(x1) and f(x2) = g(x2) and f(x) ≥ g(x) ∀ x ∈ (x1, x2). Then the area of the region bounded by the curves by y = f (x), y = g(x), and the lines x = x1, x = x2 is given by and if f(x) ≤ g(x) ∀ x ∈ (x1, x2). ThenIn either case, area is .

## 8. DIFFERENTIAL EQUATIONS

Differential equation: An equation involving one dependent variable and its derivative with respect to one or more independent variables is called a ‘Differential equation’.

If a differential equation contains only one independent variable, then it is called ‘an ordinary differential equation and if it contains more than one independent variable, then it is called ‘a partial differential equation’.

Degree of the differential equation:  If a differential can be expressed as a polynomial equation in the derivatives occurring in it using the algebraic operations such that the exponent of each of the derivatives is the least, then the large exponent of the highest order derivative in the equation is called the degree of the differential equation.

Otherwise, the degree is not defined for a differential equation.

Order of differential equation: The order of the differential equation is the order of the highest derivative occurring in it.

Note: The general form of an ordinary differential equation of nth order is

Formation of the differential equation: suppose that an equation y = ϕ (x, α1, α2, …, αn) where α1, α2, …, αn are parameters, representing a family of curves is given. Then successively differentiating the above equation, a differential equation of the form

We know that y = mx is a straight line passing through the origin

m = dy/ dx ⟹

Solving differential equations:

1. Variable separable method:

If a given differential equation can be put in the form of f(x) dx + g(y) dy = 0 then its solution can be obtained by integrating each of them. This method is called the variable separable method.

Ex: xdy – y dx = 0 can be written as dx/x =dy/y

By integrating we get ∫dx/x = ∫dy/y

⇒ logx = logy + logc

⇒ logx = log yc

∴ x = yc is the required solution

2. Homogeneous Differential Equation:

Homogeneous function: – A function f (x, y) of two variables x and y is said to be a homogeneous function of degree n, if f(kx, ky) = kn f(x, y)  for all values of k for which both sides of the above are meaningful.

Homogeneous Differential Equation: – A differential equation of the form where f (x, y) and g (x, y) are homogeneous functions of x and y of the same degree is called a homogeneous differential equation.

Method of solving the homogeneous differential equation: –

Consider the homogeneous equation   …… (1)

where f (x, y) and g (x, y) are homogeneous functions of x and y of the same degree.

f (x, y) = xn ϕ (y/x) and g(x) xn ψ (y/x)

eqn (1) becomes  ……. (2)

put y = vx. Then  ……. (3)

from (2) and (3)

This can be solved by the variable separable method.

3. Non-Homogeneous Differential Equations:

The differential equation of the form where a, b, c, a’, b’, c’ are constants and c and c’ are not both zero are called non-homogeneous equations. Reduce the above equation to a homogeneous equation by suitable substitution for x and y.

Case(i): –

Suppose that b = – a’. then    becomes

⇒ (a’x + b’y + c’) dy – (ax – a’y + c) dx = 0

⇒ a’(x dy + y dx) + b’ y dy – ax dx + c’ dy – cdx = 0

By integrating we get

a’ xy + b’ y2/2 – a x2/2 + c’y – cx = k

which is a required solution.

Case(ii): –

Suppose that  Then  becomes

Put ax + by = v, then

this can be solved by the variable separable method.

Case(iii): –

Suppose that b ≠ – a’ and a/a’ ≠ b/b’, then taking x = X + h, y = Y + k, where X and Y are variables and h, k are constants. We get  . Hence…..(i)    becomes

Now choose constants h and k such that

ah + bk + c = 0

a’h + b’k +c’ = 0

by solving above equations, we get h. k values

Hence, equation (1) becomes

This is the homogeneous equation in X and Y and then solve by the homogeneous method by putting Y = VX.

3. Linear Differential Equations:

A differential equation of the form   = R,  where P1, P2, …, Pn and R are constants or functions x only, is said to be a linear differentiable equation of nth order.

Method of solving the linear differentiable equation of 1st order: –

The linear differentiable equation of the first order is

Multiplying both sides of (1) by, we get

My App:

## TS Inter Second Year Maths 2A concept

TS Inter Second Year Maths : This note is designed by ‘Basics in Maths’ team. These notes to do help the TS intermediate second year Maths students fall in love with mathematics and overcome the fear.

These notes cover all the topics covered in the TS I.P.E second year maths 2A syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations.

## 1. COMPLEX NUMBERS

•  The equation x2 + 1 = 0 has no roots in real number system.

∴ scientists imagined a number ‘i’ such that i2 = − 1.

Complex number:  if x, y are any two real numbers then the general form of the complex number is

z = x + i y;  where x real part and y is imaginary part.

∗ z =  x + iy can be written as (x, y)

∗If z1 = x1 + i y1, z2 = x2 + i y2, then

∗ z1 + z2 = (x1 + x2, y1 + y2) = (x1 + x2) + i (y1 + y2)

∗ z1 − z2 = (x1 − x2, y1 − y2) = (x1 − x2) + i (y1 − y2)

∗ z1∙   z2 = (x1 x2 −y1 y2, x1y2 + x2y1) = (x1x2 −y1 y2) + i (x1y2 +x2 y1)

∗ z1/ z2 = (x1x2 + y1 y2/x22 +y22, x2 y1 – x1y2/ x22 +y22)

= (x1x2 + y1 y2/x22 +y22) + i (x2 y1 – x1y2/ x22 +y22)

Multiplicative inverse of complex number:

Multiplicative inverse of complex number z is 1/z.

z = x + i y then 1/z = x – i y/ x2 + y2

Conjugate complex number:

• The complex numbers x + iy, x – iy are called conjugate complex numbers.
• The sum and product of two conjugate complex numbers are real.
• If z1, z2 are two complex numbers then

Modulus and amplitude of complex number:

Modulus: – If z = x + iy, then the non-negative real number  is called the modulus of z and it is denoted by or ‘r’.

Amplitude: – The complex number z = x + i y is represented by the point P (x, y) on the XOY plane. ∠XOP = θ is called amplitude of z or argument of z.

∗ x = r cosθ, y = r sinθ

⇒ x2 + y2 = r2 cos2θ + r2 sin2θ = r2 (cos2θ + sin2θ) = r2(1)

⇒ x2 + y2 = r2

⇒ r =   and  = r.

∗ Arg (z) = tan−1(y/x)

∗ Arg (z1.z2) = Arg (z1) + Arg (z2) + nπ for some n ∈ { −1, 0, 1}

∗ Arg(z1/z2) = Arg (z1) − Arg (z2) + nπ for some n ∈ { −1, 0, 1}

Argand plane: The plane containing all complex numbers is called the Argand plane. This was introduced by the mathematician Gauss (1777-1855), who first thought that complex numbers can be represented as a two-dimensional plane.

The square root of a complex number:

## 2.DE- MOIVER’S THEOREM

De- Moiver’s theorem: For any integer n and real number θ, (cosθ + i sinθ) n = cos nθ + i sin nθ.

cos α + i sin α can be written as cis α

cis α.cis β= cis (α + β)

1/cisα = cis(-α)

cisα/cisβ = cis (α – β)

(cosθ + i sinθ) -n = cos nθ – i sin nθ

(cosθ + i sin θ) (cosθ – i sin θ) = cos2θ – i2 sin2θ = cos2θ + sin2θ = 1.

cosθ + i sin θ = 1/ cosθ – i sin θ and cosθ – i sin θ = 1/ cosθ + i sin θ

(cosθ – i sin θ) n = (1/ (cosθ –+i sin θ)) n = (cosθ + i sin θ)-n = cos nθ – i sin nθ

nth root of a complex number: let n be a positive integer and z0 ≠ 0 be a given complex number. Any complex number z satisfying z n = z0 is called an nth root of z0. It is denoted by z01/n or

let z = r (cosθ + i sin θ) ≠ 0 and n be a positive integer. For k∈ {0, 1, 2, 3…, (n – 1)}   let . Then a0, a1, a2, …, an-1 are all n distinct nth roots of z and any nth root of z is coincide with one of them.

nth root of unity:  Let n be a positive integer greater than 1 and

Note:

• The sum of the nth roots of unity is zero.
• The product of nth roots of unity is (– 1) n – 1.
• The nth roots of unity 1, ω, ω2, …, ωn-1 are in geometric progression with common ratio ω.

Cube root of unity:

x3 – 1 = 0 ⇒ x3 = 1

x =11/3

## 3.QUADRATIC EXPRESSIONS

Quadratic Expression: If a, b, c are real or complex numbers and a ≠ 0, then the expression ax2 + bx + c is called a quadratic expression in variable ‘x’.

∎ A complex number α is said to be a zero of the quadratic expression ax2 + bx + c if aα2 + bα + c = 0.

Quadratic Equation:  If a, b, c are real or complex numbers and a ≠ 0, then ax2 + bx + c = 0 is called a quadratic equation in variable ‘x’.

∎ A complex number α is said to be root or solution of the quadratic equation ax2 + bx + c = o if aα2 + bα + c = 0.

The roots of a quadratic equation:

∎ The zeroes of the quadratic expression ax2 + bx + c are same as the roots of quadratic equation ax2 + bx + c = o.

∎ The roots of the quadratic equation are

If α, β are the roots of the quadratic equation ax2 bx +c= 0, then α + β = -b/a and αβ = c/a.

Discriminate:

If ax2 + bx + c = 0 is a quadratic equation, then b2 – 4ac is called the discriminant of quadratic equation. b2 – 4ac is also the discriminant of quadratic expression ax2 + bx + c. It is denoted by ∆

∴ ∆ = b2 – 4 ac.

Nature of the roots:

The nature of the roots of the quadratic equation as follows:

1. If ∆ > 0, then roots are real and distinct.
2. If ∆ = 0, then roots are real and equal.
3. If < 0, then roots are imaginary.

Note:

• If ∆ > 0 and b2 – 4 ac is a perfect square, then the roots are rational and distinct.
• If ∆ < 0 and b2 – 4 ac is not a perfect square, then the roots are irrational and distinct. Further, the roots are conjugate surds.

If α, β are the roots of the quadratic equation ax2 + bx +c= 0, then ax2 + bx + c = a (x – α) (x – β)

The quadratic equation whose roots are α, β is (x – α) (x – β) = 0 ⇒ x2 – (α + β) x + αβ = 0.

∎ The necessary and sufficient condition for the quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 to have common root is (c1a2 – c2a1)2 = (a1b2 – a2b1) (b1c2 – b2c1)

the common root is (c1a2 – c2a1)/ (a1b2 – a2b1).

If f(x) = ax2 + bx +c= 0 is a quadratic equation then

1. The quadratic equation whose roots are the reciprocals of the roots of f(x) = 0 is f(1/x) = 0.
2. Whose roots are greater than by ‘k’ then those of f(x) = 0 is f (x – k) = 0.
• Whose roots are smaller by ‘k’ than those of f(x) = 0 is f (x + k) = 0.
1. Whose roots are multiplied by ‘k’ of hose f(x) = 0 is f (x/k) = 0.

Sign of quadratic Expressions – change in signs:

∎ If the roots of quadratic equation ax2 + bx + c = 0 are complex roots, then for x ∈ R, ax2 + bx + c and ‘a’ have the same sign.

∎ If the roots of quadratic equation ax2 + bx + c = 0 are real and equal then for x ∈ R – {-b/2a}, ax2 + bx + c and ‘a’ have the same sign.

∎ Let α, β are the roots of the quadratic equation ax2 + bx +c= 0and α < β, then

• x ∈ R, α < x < β ⇒ ax2 + bx + c and ‘a’ have the opposite sign.
• x ∈ R, x < α or x> β ⇒ ax2 + bx + c and ‘a’ have the same sign.

Maximum and Minimum values:

Let f(x) = ax2 + bx +c is a quadratic expression then

• if a < 0, then f(x) has maximum value at x = -2b/a and maximum value is 4ac – b2/4a.
• if a > 0, then f(x) has minimum value at x = -2b/a and minimum value is 4ac – b2/4a.

Quadratic inequation:

A quadratic in equation in one variable is of the form ax2 + bx +c > 0 or ax2 + bx +c ≥ 0 or  ax2 + bx +c< 0 or ax2 + bx +c ≤ 0 where a, b, c are real numbers and a ≠ 0. The values of x which satisfy given inequation are called the solution of the in equations.

⟹ Quadratic inequations are solved by two methods (i) Algebraic method (ii) Graphical method.

## 4.THEORY OF EQUATIONS

Polynomial: If n is non- negative integer and a0, a1, a2. …, an are real or complex numbers and a0 ≠ 0, then an expression

f(x) = a0 xn + a1 xn – 1+a2xn – 2 + … + an is called polynomial in x of degree n.

a0, a1, a2. …, an are called the coefficients of the polynomial f(x), a0 is called leading coefficient and an is called constant term.

Monic polynomial: A polynomial with leading coefficient is 1 is called a monic polynomial.

Remainder theorem: let f(x) be a polynomial of degree n> 0. Let a ∈C. Then there exist a polynomial q(x) of degree n – 1 such that

f(x) = (x – a) q(x) + f(a).

Factor theorem: let f(x) be a polynomial of degree n> 0. Let a ∈ C. We say that (x – a) is factor of f(x) if there exist a polynomial q(x) such that f(x) = (x – a) q(x).

⟹ let f(x) be a polynomial of degree n> 0, then (x – a) is factor of f(x) iff f(a) = 0.

The fundamental theorem of algebra: Every non-constant polynomial equation has at least one root.

⟹ The set of all roots of a polynomial f(x) = 0 of degree n> 0 is non-empty and has at most n elements. Also there exist α1, α 2. …, α n in C such that f(x) = a (x – α1) (x – α2) (x – α3) … (x – αn) where a is the leading coefficient of f(x).

The relations between the roots and the coefficients:

Let xn + p1 xn – 1+ p2xn – 2 + … + pn = 0 be a polynomial equation of degree n

Let α1, α 2. …, α n be roots.

xn + p1 xn – 1+ p2xn – 2 + … + pn = (x – α1) (x – α2) (x – α3) … (x – αn)

= xn – (α1+ α 2+ …+ α n) xn – 1+( α1 α 2+ α2 α 3 +…+ α n-1 α n) xn-2 – … (-1)n α1. α 2…. α n

These equalities the relation between the roots and coefficient of the polynomial equation whose leading coefficient is 1.

Note:

1.For the quadratic equation: Let α, β be the roots of the quadratic equation ax2 + bx + c = 0, then

Sum of the roots = α + β = -b/a

Product of the roots = αβ = c/a

2.For the cubic equation: Let α, β and γ be the roots of the cubic equation ax3+ bx2 + cx + d = 0, then

Sum of the roots = α + β + γ = -b/a

Product of the roots taken two at a time= αβ + β γ + γ α = c/a

Product of the roots = αβ γ = -d/a

Notation: Let α, β and γ be the roots of a cubic polynomial, then

α + β + γ is denoted by ∑α,  αβ + β γ + γ α is denoted by ∑ αβ , 1/α + 1/β + 1/γ is denoted by ∑1/α and  α2 β + β2 α  + α2 γ + γ2 β + β2 γ + γ2 α = ∑ α2 β + ∑ α β2

synthetic division: This method has two types

∗ Finding the quotient and remainder, when

a0 xn + a1 xn – 1+a2xn – 2 + … + an (a0 ≠ 0) is divided by (x – a).

∗ Finding the quotient and remainder, when

a0 xn + a1 xn – 1+a2xn – 2 + … + an (a0 ≠ 0) is divided by x2 – px – q.

Method of finding the quotient and remainder, when a0 xn + a1 xn – 1+a2xn – 2 + … + an (a0 ≠ 0) is divided by (x – a)- (Horner’s Method):

The procedure of the above method:

• First write down the coefficients of xn, xn-1, …x, x0. If any term with xk (0 ≤ k < 1) is missing, take the coefficient of it as zero.
• Draw a vertical line to the left of ‘a0’ and write ‘a’ to the left of the vertical line on the same horizontal level as that of ‘a0’.
• Under a0 write 0 and draw the horizontal line below it. Below the horizontal line and below 0, write the sum a + 0 as the first term of the 3rd row, which is equal to b0 with a and write this product below a1 in the second row. The sum a1 + ab0 is b­1. Write this in the 3rd row next to b0. Continue this process until the terms of the second and the third rows are filled.
• From the table, the quotient is b0 xn-1 + b1 xn – 2+ … + bn-1 and the remainder is R = an + abn-1.

Note:

1. If the divisor is (x + a) then the above method can be used by replacing a with – a.
2. If the divisor is ax – b, then replace a by b/a.

Method of finding the quotient and remainder, when  a0 xn + a1 xn – 1+a2xn – 2 + … + an (a0 ≠ 0) is divided by x2 – px – q:

The procedure of the above method:

• First write down the coefficients of xn, xn-1, …x, x0. If any term with xk (0 ≤ k < 1) is missing, take the coefficient of it as zero.
• Draw a vertical line to the left of ‘a0’ and write p, q as column figures to the left of the vertical line in the second and third rows respectively. These are the negatives of the coefficient of x and the constant term in the divisor. Draw a horizontal line below the third row.
• Put 0 in two rows underneath a0 write this sum a + 0 + 0 as the first term of the 4th row, which is equal to b0. Next, multiply b0 with p and write this product below a1 and write the next column entry as 0. The sum a1 + pb0 + 0 is b­1. Write this in the 4th row underneath a1 multiply b1 with p and b0 with q and write this product underneath a2. let the sum of a2, pb1 and qb0 be b2. continue this process until the terms an-1 are obtained. Name the 4th row under an-1 R1. below an put 0 and qbn-2 in the second and third rows respectively. Let the sum as an, 0 qbn-2 be R2. write it in the 4th row below an.

Trial and Error method: To find a root of f(x) = 0, we have to find out a value of x, for which f(x) = 0. Some times we can do this by inspection. This method is called trial and error method.

Multiple roots or repeated roots: let f(x) be a polynomial of degree n > 0. Let α1, α 2. …, α n be the roots of f(x) = 0 so that f(x) = a0 (x – α1) (x – α2) (x – α3) … (x – αn). A complex number α is said to be a root of f(x) = 0 of multiplicity m, if α = αk for exactly m values of k among 1,2, 3…, n. Roots of multiplicity m>1 are called multiple roots or repeated roots.

Roots of multiplicity 1 are called simple roots.

∎ Let f(x) be a polynomial of degree n> 0. Let α be a root of f(x) = 0 of multiplicity m. If m>1, then α is a root of the equation f’(x) = 0 of multiplicity m – 1. If m = 1, then f’(α) ≠ 0

∎ Let f(x) be a polynomial of degree n > 0. Let α be a root of f(x) = 0 of multiplicity m, then α is a root of the equation f(k)(x) = 0 of multiplicity m –k (k = 1, 2, 3, …, m – 1).

∎ Let f(x) be a polynomial of degree n> 0. Let α be a root of f(x) = 0 of multiplicity m Iff f(α) = f’(α) = … = f (m – 1) (α) and f(m)(α) ≠ 0.

Procedure to find multiple roots:  Let f(x) be a polynomial. First, we find f’(x) and then find the HCF of f(x) and f’(x). Now we note that, if α is a root of the HCF of multiplicity k, then α is a multiple order of (k + 1) of f(x) =0.

∎ let f(x) be a polynomial with real coefficients. Let α ∈ C, then

∎ Let f(x) be a polynomial of degree n > 0, with real coefficients. Let a0 be the leading coefficient of f(x).

1. If the equation f(x) = 0 has no real roots, then n is even and f(α), a0 have the same sign for all real values of α
2. If n is odd, then the equation f(x) = 0 has at least one real root.

∎ Let f(x) be a polynomial of degree n > 0, with real coefficients. Let a and b be rational numbers, b > 0 and irrational. Then   is a root of f(x) = 0 if and only if another root is a .

Roots with the change of sign:

If α1, α 2. …, α n are the roots of f(x) = 0, then -α1, – α 2. …, – α n are the roots of f(-x) = 0.

Roots multiplied by a given number:

If α1, α 2. …, α n are the roots of f(x) = 0, then for any non-zero complex number k, the roots of f(x/k) = 0 are kα1, k α 2. …, kα n.

Roots subtracted by a given number:

If α1, α 2. …, α n are the roots of f(x) = 0, then α1-h, α 2-h. …, – α n-h are the roots of f (x +h) = 0.

Roots added by a given number:

If α1, α 2. …, α n are the roots of f(x) = 0, then α1+h, α 2+h. …, – α n+h are the roots of f (x -h) = 0.

Reciprocal roots:

Let α1, α 2. …, α n are the roots of f(x) = 0. Suppose none of them non- zero, then 1/α1,1/ α 2. …,1/ α n are the roots of xn f (1/x) = 0.

∎ if α is a root of f(x) = 0, then α2 is a root of

Reciprocal equation:

Let f(x) be a polynomial of degree n > 0 is said to be reciprocal if f(0) ≠ 0 and ∀x ∈C-{0}, where a0 is the leading coefficient of f(x).

If f(x) is a reciprocal polynomial, then the equation f(x) = 0 is reciprocal equation.

∎ If f(x) = a0 xn + a1 xn – 1+a2xn – 2 + … + an  be a polynomial of degree n > 0, then f(x) is reciprocal iff an – k = ak for k = 0, 1, 2,… , n or  an – k = – ak for k = 0, 1, 2,… , n .

The reciprocal polynomial of class one and class two:

A reciprocal polynomial f(x) of degree n with leading coefficient a0 is said to be class one or class two according to as f(0) = a0 or  –a0.

If f(x) is a reciprocal polynomial then the equation f(x) = 0 is said to be the reciprocal equation of class one or class two according to as f(x) is a reciprocal polynomial of class one or class two.

Note:

∎ For an odd degree, the reciprocal equation of class one -1 is the root and for an odd degree reciprocal equation of class two, 1 is root.

∎ For an even degree, the reciprocal equation of class two, – 1 and 1 roots.

∎ To solve the reciprocal equation of order 2m, divide the equation by xm and put x + 1/x = y or x – 1/x = y according to the equation of class one or class two.  The degree of the transformed equation is m.

∎ For an odd degree reciprocal equation. To find the roots of it, divide f(x) by (x + 1) or (x – 1) according as the equation of class one or class two. Let Q(x) be the quotient obtained, then f(x) = (x+1) Q(x) or f(x) = (x – 1) Q(x) according as the equation of class one or class two and Q(x) is even degree reciprocal polynomial. The roots of Q(x) = 0 can be obtained by above procedure.

## 5.PERMUTATIONS AND COMBINATIONS

The fundamental principle of counting: if a work w1 can be performed in ‘m’ different ways and a second work w2 can be performed in ‘n ‘different ways, then the two works can be performed in ‘mn’ ways.

Permutation: From a given finite set of elements selecting some or all of them and arranging them in a line is called a ‘linear permutation’ or ‘permutation’.

Circular permutation

Permutations of ‘n’ dissimilar thing taken ‘r’ at a time:

∎ If n, r are positive integers and r ≤ n, then the no. of permutations of n dissimilar things taken as ‘r’ at a time is n (n – 1) (n – 2) (n – 3) … (n – r + 1).

Notation:

The number of permutations of n dissimilar things taken as r at a time is denoted by   nPr or P (n, r) (1≤r≤n).

nPr  =  n (n – 1) (n – 2) (n – 3) … (n – r + 1)

∎ If n ≥1 and 0≤r≤n, then

nPn = n! and nP0 = 1.

∎ For 1≤r≤n , nPr = n. (n – 1) P (r – 1)

∎ If n, r are positive integers and 1 ≤r < n, then nPr = (n – 1) Pr + r. (n – 1) P (r – 1).

∎ The sum of all r-digit numbers that can be formed using the given ‘n’ non-zero digits (1 ≤r ≤n≤9) is

(n – 1) P (r – 1) × [ sum of the given digits × 1111… 1(r times)]

∎If ‘0’ is one digit among the given ‘digits, then we get that the sum of all r-digit numbers that can be formed using the given ‘n’ digits including ‘0’ is

{ (n – 1) P (r – 1) × [ sum of the given digits × 1111… 1(r times)]} – { (n – 2) P (r – 2) × [ sum of the given digits × 1111… 1((r-1) times)]}.

Note: If a set A has m elements and the set B has n elements, then the no. of injections into A to B is nPm if m ≤n and 0 if m> n.

Permutations when repetitions are allowed:

∎ Let n and r be positive integers. If the repetition of things is allowed, then the no. of permutations of ‘n’ dissimilar things taken ‘r’ at a time is nr.

Palindrome: A number or a word which reads the same either from left to right or right to left is called a palindrome.

Ex:  121, 1331, ATTA, AMMA etc.

Note: The no. of palindromes with r distinct letters that can be formed using given n distinct letters is

(i) nr/2 if r is even (ii) nr+1/2 if r is odd.

Circular permutation: From a given finite set of elements selecting some or all of them and arranging them around a circle is called a ‘circular permutation’.

The no. of circular permutations of ‘n’ dissimilar things (taken all at a time) is (n – 1)!

∎ In case of the garlands of flowers, chains of beads etc, no. of circular permutations = ½ (n – 1)!

Permutations with constraint repetitions:

∎ The no. of linear permutations of ‘n’ things n which ‘p’ things are alike and the rest are different is

∎ The no. of linear permutations of ‘n’ things n which ‘p’ like things of one kind, q like things of the second kind, r like things of the third kind and the rest are different is

Combinations:

A combination is only a selection. There is no importance to the order or arrangement of things in a combination.

∎ The no. of combinations of ‘n’ dissimilar things taken ‘r’ at a time is denoted by nCr or C (n, r)

∎ For 0≤r≤n, nCr = n C n – r

∎ If m, n are distinct positive integers, then the no, of ways of dividing (m + n) things into two groups containing m things and ‘n’ things is

∎ If m, n, p are distinct positive integers, then the no, of ways of dividing (m + n + p) things into three groups containing m things, ‘n’ things and ‘p’ things is

∎ The no. of ways of dividing 2n dissimilar things into two equal groups containing ‘n’ things in each case is

∎ The no. of ways of dividing ‘mn’ dissimilar things into m equal groups containing ‘n’ things in each case is

∎ The no. of ways of distributing ‘mn’ dissimilar things equally among  m  persons is

• For 0 ≤ r, s ≤ n, if nCr = nCs then r =s or n = r + s.

• If 1 ≤ r ≤ n, then nCr-1 + nCr = (n+1) Cr.

• If 2 ≤ r ≤ n, then nCr-2 +2 nCr-1 = (n+2) Cr.

• If p things are alike of one kind, q things are alike of the second kind and r things are alike of the third kind, then the number of ways of selecting any no, of things out of these (p + q +r) things is (p + 1) (q+1)(r+1) – 1.

• The number of ways of selecting one or more things out of ‘n’ dissimilar things is 2n – 1.

• If p1, p2,…, pn are distinct primes and α1, α 2,…, α n are positive integers, then the number of positive divisors of is (α1+1)( α2+1) … (αk + 1).

Exponents of a prime in n! (n ∈ z+): Exponents of a prime number ‘p’ in n! is the largest integer ‘k’ such that pk divides n!

## 6.BINOMIAL THEOREM

Binomial: Binomial means two terms connected by either ‘+’ or ‘– ‘.

Binomial expansions:

(x + y)1 = x + y

(x + y)2 = x2 + 2xy + y2

(x + y)3 = x3 + 3x2 y + 3xy2 + y3 and so on, are called binomial expansions.

Binomial coefficients:

Coefficients of expansion (x + y) are 1, 1.

Coefficients of expansion (x + y)2 are 1, 2, 1

Coefficients of expansion (x + y) are 1, 3, 3, 1

And so on, are called binomial coefficients.

Pascal triangle:

Binomial theorem: Let n be a positive integer and x, a be real numbers, then

(x + a) n = nC0 xn a0 + nC1xn – 1 a1 + nC2 x n – 2 a2 +… + nCr xn – r ar + … + nCn x0 an

Note: –

Let n be a positive integer and x, a be real numbers, then

(i) (x + a) n = ∑ nCr xn – r ar

(ii) The expansion of (x + a) n has (n + 1) terms.

(iii) The rth term in the expansion of (x + a) n, which is denoted by Tr, is given by Tr = nCr-1 xn – r +1 ar-1 for 1≤ r≤ n + 1.

The general term of the binomial expansion:

In the expansion of (x + a)n, the (r + 1)th term is called the general term of the binomial expansion and it is given by Tr+1 = nCr xn –r  ar for 0≤ r≤ n.

(x – a) n = nC0 xn (-a)0 + nC1xn – 1 (-a)1 + nC2 x n – 2 (-a)2 +… + nCr xn – r (-a) r + … + nCn x0 (-a) n

= nC0 xn a0nC1xn – 1 a1 + nC2 x n – 2 a2 –… +(–1) r nCr xn – r ar + … + (–1) n nCn x0 an

And the general term is Tr+1 = (–1) r nCr xn –r ar for 0≤ r≤ n.

Trinomial Expansion: Let n ∈ N and a, b, c ∈ R, then (a + b + c) n can be expand using the binomial theorem taking a as the first term and (b + c) as the second term

(a + b + c) n = (a + (b+ c)) n = ∑ nC0 an-r (b + c) r (0≤ r≤ n)

⟹ no. of terms in the expansion of (a + b + c) n =

Middle terms in (x + a) n:

∎ if n is even then  term is the middle term.

∎ if n is odd then terms are the middle terms.

Binomial coefficients: The coefficients in the binomial expansion (x + a) n are nC0, nC1, …, nCr, …, nCn these coefficients are called binomial coefficients. When n is fixed these coefficients are denoted by C0, C1, …, Cr, …, Cn. respectively.

Note:

• The binomial expansion of (1 + x) n = nC0 + nC1x + nC2 x 2 +… + nCn xn. This expansion is called the standard binomial expansion.

With the standard notation, if n is a positive integer, then

• C0 + C1 + C2 …+ Cn = 2n
• C0 + C2 + C4 …+ Cn = 2n-1 if n is even
• C0 + C2 + C4 …+ Cn-1 = 2n-1 if n is odd
• C1 + C3 + C5 …+ Cn-1 = 2n-1 if n is even
• C1 + C3 + C5 …+ Cn = 2n-1 if n is odd

Integral part and Fractional part: If x is any real number, then there exist an integer n such that n ≤ x < n+ 1. This integer n is called an integral part of the real number x and it is denoted by [x]. The real number x – [x] is called fractional part of x and it is denoted by {x}.

Numerically greatest term:  In the binomial expansion of (1 + x) n, the rth term Tr is called numerically greatest term if,

⟹ if  = p, where p is a positive integer then, pth and (p + 1)th are the numerically greatest terms.

⟹ if = p + F, where p is a positive integer and 0 < F < 1 then, (p + 1)th  is the numerically greatest term.

⟹ To find the numerically greatest term(s) in the binomial expansion of (a + x)n we write (a + x)n = an(1 + x/a)n and then find the numerically greatest term(s) by using above rules.

Largest Binomial coefficient:

The largest binomial coefficient(s) among nC0, nC1, …, nCr, …, nCn is (are)

(i) if n is even integer.

(ii) if n is an odd integer

Binomial theorem for Rational Index:

If m is a rational number and x is a real number such that – 1 < x < 1, then

Rational Index: –

## 7. PARTIAL FRACTIONS

Rational fraction:  If f(x) and g(x) are two polynomials and g(x) is a non-zero polynomial, then is called a rational fraction or polynomial fraction or simply a fraction.

Ex:

Proper and Improper Fractions: A rational fraction is called a Proper fraction if the degree of f(x) is less than the degree of g(x). Otherwise, it is called an improper fraction.

Ex: is a proper fraction and is an Improper fraction.

Irreducible Polynomial:  A polynomial f(x) is said to be irreducible if it can not be express as a product of two polynomials g(x) and h(x) such that the degree of each polynomial is less than the degree of f(x). If f(x)is not irreducible then we say that f(x) is reducible.

Ex: 3x – 1, x2 + x + 1 are irreducible polynomials.

Division Algorithm for Polynomials: If f(x) and g(x) are two polynomials with g(x) ≠ 0, then there exist unique polynomials q(x) and r(x) such that f(x) = q(x) g(x) + r(x) , where either r(x) = 0 or the degree of r(x) is less than the degree of g(x).

Partial Fraction:  If a proper fraction is expressed as the sum of two or more proper fractions, wherein the power of the denominator of irreducible polynomials, then each proper fraction in the sum is called a partial fraction of the given fraction.

Partial Fraction of  when g(x) contains linear factors:

Rule – 1:   Let  be a proper fraction. To each non- repeated factor of g(x), there will be a partial fraction of the form where A is a non-zero real number, to be determined.

Rule – 2:   Let  be a proper fraction. To each factor (ax + b)n, a ≠ 0 where ‘n’ is a positive integer, of g(x) there will be a partial fraction of the form where A1, A2, …, An are to be determined constants. Note that An ≠ 0 and Rule –1 is a particular case of Rule-2 for n = 1.

Partial Fraction of  when g(x) contains irreducible factors:

Rule – 3: Let  be a proper fraction. To each non- repeated quadratic factor (ax2 + bx + c), a ≠ 0 of g(x) there will be a partial fraction of the form where A, B are real numbers, to be determined.

Rule – 4: Let  be a proper fraction. If n (>1)∈ N is the largest exponent so that (ax2 + bx + c)n, a ≠ 0) factor of g(x) there will be a partial fraction of the form where A1, A2, …, An and B1, B2, …, Bn are real numbers, to be determined.

Partial Fraction of when  is an Improper fraction:

Case (1): If degree f(x) = degree of g(x) then by                                                                                                division algorithm there exist a unique constant k and r(x) such that f(x) = k g(x) + r(x), where either r(x) = 0 or the degree of r(x) is less than the degree of g(x) and the constant k is the quotient of the coefficient of the highest degree terms of f(x) and g(x).

can be expressed as k +   where is a proper fraction which can be resolved into a partial fraction using the above rules.

Case (2): If degree f(x) > degree of g(x) then by division algorithm can be expressed as q(x) +   where q(x) is a non-zero polynomial and is a proper fraction which can be resolved into a partial fraction using above rules.

## 8. MEASURES OF DISPERSION

The measure of dispersion: In a measure of central tendency, we have to know a measure to describe the variability. This method is called a measure of dispersion.

Measuring dispersion of a data is significant because it determines the reliability of an average by pointing out as to how far an average is representative of the entire data.

Some measures of dispersion are: (i) Range (ii) Mean deviation (iii) Standard deviation

Range:

For ungrouped data, the range is the difference between the maximum and minimum value of the series of observations.

For grouped data range is approximated as the difference between the upper limit of the largest class and the lower limit of the smallest class.

Mean deviation:

To find the dispersion of values of x from a central value ‘a’ we find the deviation about ‘a’. They are   (x – a)’s. To find the mean deviation we have to sum up all such deviations.

Mean deviation from the mean for ungrouped data:

Let x1, x2, …, xn be n observations of discrete data.

Steps for finding the Mean deviation from the mean for ungrouped data:

1. First, we have to find the mean ( ) of the n observations. Let it be ‘a’
2. Find the deviations of each xi from ‘a’, i.e., x1 – a, x2 – a, …, xn – a.
3. Find the absolute values of i.e., of these deviations by ignoring the negative sign, if any, in the deviation computed in step 2.
4. Find the arithmetic mean of the absolute values of the deviations.

M.D from the mean =

Mean deviation from the median for ungrouped data:

Let x1, x2, …, xn be n observations of discrete data.

Steps for finding the Mean deviation from the median for ungrouped data:

1. First, we have to find the median of the n observations. Let it be ‘a’
2. Find the deviations of each xi from ‘a’, i.e., x1 – a, x2 – a, …, xn – a.
3. Find the absolute values of i.e., of these deviations by ignoring the negative sign, if any, in the deviation computed in step 2.
4. Find their arithmetic mean as M.D from median =

Mean deviation for a grouped data:

A data can be arranged or grouped as a frequency distribution in two ways: (i) Discrete frequency distribution and (ii) Continuous frequency distribution.

(i) Discrete frequency distribution:

If x1, x2, …, xn are of ‘n’ observations occurring with frequencies f1, f2, …, fn Then we can represent this data in the following manner:

 xi x1 x2 x3 … xn fi f1 f2 f3 … fn

This form is called the discrete frequency distribution

Mean deviation about the mean and median:

Mean =

M.D(mean) =

M.D(median) =

Where N is total frequency.

(ii) Continuous frequency distribution: –

Continuous distribution is a series in which the data is classified into different class-intervals along with their respective frequencies.

Mean deviation about the means and median:

Mean =

M.D(mean) =

Median =

M.D(median) =

Where N is total frequency.

Step- Deviation method: If the midpoints of the class intervals xi as well as their associated frequencies are very large then we use this method.

Arithmetic mean =

Where

Variance and Standard Deviation of un grouped data:

If x1, x2, …, xn are n observations and is their mean, then

We have the following cases:

Case(i): if = 0, then each= 0 which implies all observations are equal to the mean  and there is no dispersion.

Case(ii): if  is small, then it indicates that each observation xi is very close to the mean  and hence the degree of dispersion is low.

Case(iii): if  is larger, then it indicates the higher degree of dispersion of the observations from the mean .

Variance = σ2=

Standard deviation

∎ The coefficient of variation of a distribution (C.V.) =

## 9. PROBABILITY

Random Experiment: If the result of an experiment is not certain and is any one of the several possible outcomes, then the experiment is called ‘random experiment.

Sample space: The set of all possible outcomes of an experiment is called ample space when ever the experiment conducted and is denoted by ‘S’.

Event: Any subset of the sample space is called an event.

Complimentary of an event: The complementary of an event E , is denoted by Ec , is the event given by Ec = S – E which is called the complimentary event of E.

Equally likely events: two events are said to be equally likely events when chance of occurrence of one event is equal to that of other.

Exhaustive events: A set of events is said to be exhaustive if the performance of the experiment always result in the occurrence of the at least one of them.

The events E1, E2, …, En are said to be exhaustive if E1∪ E2∪…, ∪ En = S.

Mutually Exclusive events: A set of events is said to be mutually exclusive if happening of one of them prevents the happening of any one of remaining events.

The events E1, E2, …, En are said to be exhaustive if Ei ∩ Ej =∅ for i ≠ j, 1 ≤i, j≤ n.

Classical definition of Probability: In a random experiment, let there be n mutually exclusive, exhaustive and equally likely events.

E be the event of the experiment. ‘m’ elementary events are favourable to an event E, then the probability of E is defined as P (E) =

For any event E, 0 ≤ P(E) ≤1.

∎ If Ec is the non-occurrence of E, then the probability of on-occurrence of E is P (Ec)

P (Ec) = 1 – P (E) ⇒ P (Ec) + P (E) = 1

Limitations of the Classical definition of the probability:

1. If the out comes of the random experiment are not equally likely, then the probability of an event in such experiment is not defined.
2. If the random experiment contains infinitely many out comes, then his definition cannot be applied to find the probability of an event in such an experiment.

Relative frequency (Statistical or Emperical) definition probability:

Suppose a random experiment is repeated n times, out of which an event E occurs m(n) times, then the ratio  is called the nth relative frequency of the event E.

Let r1, r2, …, rn be the sequence. If rn tends to a definite limit, , l is defined to be the probability of the event E and we write P (E) =

Deficiencies of the relative frequency definition of probability:

1. Repeating a random experiment infinitely many times is practically impossible.
2. The sequence of relative frequencies is assumed to tend to a definite limit, which may not exist.
3. The values r1, r2, …, rn are not real variables. Therefore, it is not possible to prove the existence and the uniqueness of the limit of rn as n → ∞, by applying methods used in calculus.

Probability Function:

Let S be the sample space of a random experiment, which is finite. Then a function P: S → R satisfying the following axioms is called a Probability function.

(i) P (E) ≥ 0 ∀ E ∈ S (axiom of non-negativity)

(ii) P (S) = 1 (axiom of certainty).

(iii) If E1, E2 ∈ S and E1 ∩ E2 =∅, then P (E1 ∪ E2) = P (E1) + P (E2) (axiom of additivity).

For each E ∈ S, the real number P (E) s called the probability of the event E. If E = {a}, then we write p(a) instead of P ({a}).

Note:

1. P (∅) = 0 for any sample space S, S ∅ = S and S∩ ∅ = ∅. P (S) = (S ∅) = P (S) + P (∅) = P (S) (∵ P (∅) = 0).
1. If S is countably infinite, then axiom (iii) of the above definition is to be replaced by (iii)*: if is a sequence of pairwise mutually exclusive events, then
2. Suppose S be a sample space of a random experiment. Let P be a probability function. If E1, E2, …, En are finitely many pairwise mutually exclusive events, then P (E1∪ E2∪…, ∪ En) = P (E1) + P (E2) + … + P (En)

⟹ If E1, E2 are any two events in a sample space S, then
Addition theorem on probability:
If E1, E2 are any two events in a sample space S and P is a probability function, then P (E1∪ E2) = P (E1) + P (E2) – P (E1∩ E2)

P (E1 – E2) = P (E1) – P (E1∩ E2)

P (E2 – E1) = P (E2) – P (E1∩ E2)

Set – theoretic descriptions:

 Event Set-theoretic description Event A or Event B to occur A∪B Both event A and B occur A∩B Neither A nor B occur (A∪B) c = Ac ∩ Bc A occurs but B does not occur A ∩ Bc or A\B Exactly one of the event A, B to occur (A∩B) c ∪ (Ac ∩ B) or (A – B) ∪ (B – A)                 or (A∪B) – (A ∩ B) Not more than one of the events A, B occurs (A∩B) c ∪ (Ac ∩ B) ∪ (Ac ∩ Bc) Event B occurs whenever event A occurs A ⊆ B

Conditional event: If A, B are two events of random experiment, then the event of happening (occurring) B after the event A happens(occurs) is called conditional event. It is denoted by B\A.

Conditional probability:  If A, B are two events and P (A) ≠ 0, then the probability of B after the event A has occurred is called conditional probability. It is denoted by P (B/A) and is defined by                  P(B/A) =

Multiplication theorem of probability: If A, B are two events of random experiment with P (A) > 0 and P (B) > 0, then P (A∩B) = P (A) P (B/A) = P (B) P (A/B).

Two events A and B said to be independent if P (B/A) = P(B) or P (A/B) = P (A)

Two events A and B said to be independent if P (A∩B) = P (A). P (B)

The events A1, A2, …, An are   said to be independent if P (A1∩ A2∩ …∩ An) = P (A1). PA2). …. P(An).

Bayes Theorem:

If A1, A2, …, An are mutually exclusive and exhaustive events in a sample space S such that P (Ai) > 0 for i = 1, 2, 3, …, n and E is an event with P (E) > 0, then

## 10. RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS

Random variable: Let S be a sample space of a random experiment. A real valued function X: S → R is called random variable.

∎ A set A is said to be countable if there exist a bijection from A into a subset of N.

Probability distribution Function:  If X: S → R is a random variable connected with a random experiment and P is a probability function associate with it. The unction F: R → R defined by F(x) = P (X ≤ x) is called probability distribution function of the random variable X.

Discrete or discontinuous Random variable:   Let S be a sample space, a random variable X: S → R is said to be Discrete or discontinuous if the range of X is countable.

I.POBABILITY DISTRIBUTION:

If X: S → R is a discrete random variable with range {x1, x2, x3………}, then {P(X = xr; r = 1, 2, 3, …, n} is called probability distribution of X.

The table for the probability distribution of the discrete random variable X is:

 X = xi x1 x2 x3 … xn P (X = xi) P (x1) P (x2) P (x3) … P (xn)

Mean (μ) =∑xi P (X = xi)

Variance = σ2 = ∑ (xi – μ)2 P (X = xi) = ∑ xi2 P (X = xi) – μ2

Standard deviation is σ.

II.BINOMIAL or BERNOULLI DISTRIBUTION:

Let n be a positive integer and p be the random number such that 0 < p < 1. A random variable X with range {0, 1, 2, 3, …, n} is said to have a Binomial distribution with parameters n and p, if

P (X = x) = nCx px qn – x for x = 0, 1, 2, …, n and q = 1 – p.

Mean = np

Variance = npq.

III. POISSON DISTRIBUTION:

Let λ > 0 be a real number. A random variable X with range {1, 2, 3, …, n} is said to be Poisson distribution with parameter λ if,

Mean = λ

Variance = λ

Poisson distribution as a limiting form of Binomial distribution:

Poisson distribution can be derived as the limiting case of binomial distribution in the following case.

If λ > 0 for each positive integer n > λ, let Xn be the Binomial variable B (n, λ/n). using the fact that we can prove that for every non-negative integer k,

My App:

## TS Inter Maths 1B Concept

Ts Inter Maths 1B Concept:  designed by the ‘Basics in Maths’ team. These notes to do help the TS intermediate first-year Maths students fall in love with mathematics and overcome the fear.

These notes cover all the topics covered in the TS I.P.E  first year maths 1B syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations.

## 0.COORDINATE GEOMETRY( BASICS)

• Distance between two points A(x1, y1), B(x2, y2) is

• distance between a point A(x1, y1) to the origin is

• The midpoint of two points A(x1, y1), B(x2, y2) is

•     If P divides the line segment joining the points A(x1, y1), B(x2, y2) in the ratio m:n then the coordinates of P are

• Area of the triangle formed by the vertices A (x1, y1), B (x2, y2) and C (x3, y3) is

### 1. LOCUS

Locus: The set of points that are satisfying a given condition or property is called the locus of the point.

Ex:- If a point P is equidistant from the points A and B, then AP =BP

Ex 2: – set of points that are at a constant distance from a fixed point.

here the locus of a point is a circle.

• In a right-angled triangle PAB, the right angle at P and P is the locus of the point, then

AB2 = PA2 + PB2

•Area of the triangle formed by the vertices A (x1, y1), B (x2, y2), and C (x3, y3) is

### 2.CHANGE OF AXES

Transformation of axes:

When  the origin is shifted to  (h, k), without changing the direction of axes then

•To remove the first degree terms of the equation ax2  + 2hxy + by2 +2gx +2fy+ c = 0, origin should be shifted to the point

•If the equation ax2 + by2 +2gx +2fy+ c = 0, origin should be shifted to the point

Rotation of axes:

When the  axes are rotated through an angle θ then

•To remove the xy term of the equation ax2 + 2hxy + by2  = 0, axes should be rotated through an angle θ is given by

### 3.STRAIGHT LINES

Slope:-  A-line makes an angle θ with the positive direction of the X-axis, then tan θ is called the slope of the line.

It is denoted by “m”.

m= tan θ

• The slope of the x-axis is zero.

• Slope of any line parallel to the x-axis is zero.

• The y-axis slope is undefined.

• The slope of any line parallel to the y-axis is also undefined.

• The slope of the line joining the points A (x1, y1) and B (x2, y2) is

Slope of the line ax + by + c = 0 is

### Types of the equation of a straight line:

• Equation of x- axis is y = 0.
• Equation of any line parallel to the x-axis is y = k, where k is the distance from above or below the x-axis.
• Equation of y- axis is x = 0.
• Equation of any line parallel to y-axis is x = k, where k is the distance from the left or right side of the y-axis.

Slope- intercept form

The equation of the line with slope m and y-intercept c is y = mx + c.

Slope point form:

The equation of the line passing through the point (x1, y1) with slope m is

y – y1 = m (x – x1)

Two points form:

The equation of the line passing through the points (x1, y1) and (x2, y2) ’ is

Intercept form:

The equation of the line with x-intercept a, y-intercept b is

• The equation of the line ∥ el    to ax +by + c = 0 is ax +by + k = 0.

• The equation of the line ⊥ler   to ax +by + c = 0 is bx −ay + k = 0.

Note: –

1. If two lines are parallel then their slopes are equal

m1 = m2

1. If two lines are perpendicular then product of their slopes is – 1

m1 × m2 = – 1

1. The area of the triangle formed by the line ax + by + c = 0 with the coordinate axes is
2. The area of the triangle formed by the line   with the coordinate axes is

Perpendicular distance (Length of the perpendicular):

The perpendicular distance from a point P (x1, y1) to the line ax + by + c = 0 is

• The perpendicular distance from origin to the line ax + by + c = 0 is

Distance between two parallel lines:

•The distance between the parallel lines ax1 + by1 + c1 = 0 and ax2 + by2 + c2 = 0 is

Perpendicular form or Normal form:

The equation of the line which is at a distance of ‘p’ from the origin and α (0≤ α ≤ 3600) is the angle made by the perpendicular with the positive direction of the x-axis is x cosα + y sinα = p.

Symmetric form:

The equation of the line passing through point P (x1, y1) and having inclination θ is

Parametric form:

if P (x, y) is any point on the line passing through A (x1, y1) and

making inclination θ, then

x = x1 + r cos θ, y = y1 + r sin θ

where ‘r’  is the distance from P to A.

• The ratio in which the line L ≡ ax + by + c = 0 divide the line segment joining the points A (x1, y1), B (x2, y2) is – L11: L22.

Where L11 = ax1 + by1 + c and L22 = ax2 + by2 + c.

Note: – the points A (x1, y1), B (x2, y2) lie on the same side or opposite side of line L = 0 according to L11 and L22 have the same sign or opposite sign.

∗  x-axis divides the line segment joining the points A (x1, y1), B (x2, y2) in the ratio – y1: y2.

∗  y-axis divides the line segment joining the points A (x1, y1), B (x2, y2) in the ratio – x1: x2.

Point of intersection of two lines:

the point of intersection of two lines a1x + b1y + c = 0 and a2x + b2y + c = 0 is

#### Concurrent Lines:

Three or more lines are said to be concurrent lines if they have a point in common.

The common point is called the point of concurrence.

∗  The condition that the lines a1x + b1y + c1 = 0, a2x + b2y + c2 = 0 and a3x + b3y + c3 = 0 to be concurrent is

a3(b1c2 – b2c1) + b3(c1a2 – c2a1) + c3(a1b2 – a2b1).

∗ The condition that the lines ax + hy +g = 0, hx + by + f = 0 and gx +fy + c = 0 is

abc + 2fgh – af2 – bg2 – ch2 = o.

Note: – if two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 said to be identical (same) if

#### Family of a straight line:

Family of straight lines: – A set of straight lines having a common property is called a family of straight lines.

Let L1 ≡ a1x + b1y + c1 = 0 and L2 ≡ a2x + b2y + c2 =0 represent two intersecting lines, theThe equation λ1 L1 + λ2 L2 = 0 represent a family of straight lines passing through the point of intersection of the lines L1 = 0 and L2 = 0.

∗  The equation of the straight line passing through the point of intersection of the lines L1 = 0 and L2 = 0 is L1 + λL2 = 0.

The angle between two lines:

If θ is the angle between the lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 then

∗  If θ is an acute angle then

∗ If θ is the angle between two lines, then (π – θ) is another angle between two lines.

∗ If θ≠π/2 is angle between the lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, then

∗ If m1, m2 are the slopes of two lines then

Note: – The lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are

∗ Parallel iff

∗ Perpendicular iff a1a2 + b1b2 = 0

The foot of the perpendicular:

If Q (h. k) is the foot of the perpendicular from a point P (x1, y1)to the line ax + by +c = 0 then

Image of the point:

If Q (h. k) is the image of point P (x1, y1)   with respect to the line ax + by +c = 0 then

Collinear Points:

If three points are said to be collinear, then they lie on the same line.

∗ If A, B, and C are collinear, then

Slope of AB = Slope of BC (or) Slope of BC = Slope of AC (or) Slope of AB = Slope of AC

### 4. PAIR OF STRAIGHT LINES

∎ ax2 + 2hxy + by2 = 0 is called the second-degree homogeneous equation in two variable x and y.

This equation always represents a pair of straight lines which are passing through the origin.

∎ If l1x + m1y = 0 and l2x + m2y = 0 are two lines represented by the equation ax2 + 2hxy + by2 = 0, then ax2 + 2hxy + by2 = (l1x + m1y) (l2x + m2y)

⇒ a = l1l2; 2h = l1m2 + l2m1; b = m1m2

∎ If m1, m2 are the slopes of the lines represented by the equation ax2 + 2hxy + by2 = 0, then

m1+ m2 = – 2h/b and m1 m2 = a/b

∎ The lines represented by the equation ax2 + 2hxy + by2 = 0 are

∎ If h2 = ab, then the lines represented by the equation ax2 + 2hxy + by2 = 0 are coincident.

∎ If two lines represented by the equation ax2 + 2hxy + by2 = 0 are equally inclined to the coordinate axes then h = 0 and ab < 0.

∎ The equation of the pair of lines passing through the point (h, k) and

(i) Parallel to the lines represented by the equation ax2 + 2hxy + by2 = 0 is

a (x – h)2 + 2h (x – h) (y – k) + b (y – k)2 = 0

(ii) Perpendicular to the lines represented by the equation ax2 + 2hxy + by2 = 0 is

b (x – h)2 – 2h (x – h) (y – k) + a (y – k)2 = 0

Angle between the lines:

If θ is the angle between the lines represented by the equation ax2 + 2hxy + by2 = 0, then

∎ If a + b = 0, then two lines are perpendicular.

Area of the triangle:

The area of the triangle formed by the lines ax2 + 2hxy + by2 = 0 and the line lx + my + n = 0 is

Angular Bisectors:

⇒ the angle between angular bisectors is always 900
L1 = o, L2 = o are two non-parallel lines the locus of the point P such that the perpendicular distance from P to the first lie is equal to the perpendicular distance from P to second line is called the angular bisector of two lines.

⇒ If two lines are a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, then the angular bisectors are

∎ The equation of the pair of angular bisectors of ax2 + 2hxy + by2 = 0 is

h (x2 – y2) = (a – b =) xy.

∎ If ax2 + 2hxy + by2 + 2gx 2fy + c= 0 represents a pair of straight lines then

(i) abc + 2fgh – af2 – bg2 – ch2 = 0

(ii) h2 ≥ ab, g2 ≥ ac and f2 ≥ bc

If two lines represented by ax2 + 2hxy + by2 + 2gx 2fy + c= 0 are

l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0, then

ax2 + 2hxy + by2 + 2gx 2fy + c = (l1x + m1y + n1) (l2x + m2y + n2)

a = l1l2; 2h = l1m2 + l2m1; b = m1m2 ; 2g = l1n2 + l2n1;  2f =  m1n2 + m2n1 and c = n1n2

The point of intersection of the lines represented by ax2 + 2hxy + by2 + 2gx 2fy + c= 0 is

∎ If two line have same homogeneous path then the lines represented by the first pair is parallel to the lines represented by the second pair.

ax2 + 2hxy + by2 + 2gx 2fy + c= 0 …………… (2)

ax2 + 2hxy + by2 = 0 …………… (1)

equation (1) and equation (2) form a parallelogram, one of the diagonals of parallelogram which is not passing through origin is 2gx + 2fy + c = 0.

∎ If two lines represented by ax2 + 2hxy + by2 + 2gx 2fy + c= 0 are parallel then

• h2 = ab (ii) af2 = bg2 (iii) hf = bg, gh = ab

Distance between parallel lines is

### 5. THREE DIMENSIONAL COORDINATES

• Let X’OX, Y’OY be two mutually perpendicular lines passing through a fixed point ‘O’. These two lines determine the XOY – plane (XY- plane). Draw the line Z’OZ perpendicular to XY – plane and passing through ‘O’.

The fixed point ‘O’ is called origin and three mutually perpendicular lines X’OX, Y’OY, Z’OZ are called Rectangular coordinate axes.

Three coordinate axes taken two at a time determine three planes namely XOY- plane, YOZ-plane, ZOY-plane or XY-plane, YZ-plane, ZX-plane respectively.

For every point P in space, we can associate an ordered triad (x, y, z) of real numbers formed by its coordinates.

The set of points in space is referred to as ‘Three-Dimensional Space’ or R3– Space.

∗ If P (x, y, z) is a point in a space, then

x is called x-coordinate of P

y is called y-coordinate of P

z is called z-coordinate of P

Distance between two points in space:

∗ Distance between the points A (x1, y1, z1) and B (x2, y2, z2) is

∗ Distance between the point P (x, y, z) to the origin is

Translation of axes:

When the origin is shifted to the point (h, k, l), then

X = x – h; Y = y – k; Z = z – l and x = X + h; y = Y + k; z = Z + l

∗ The foot of the perpendicular from P (x, y, z) to X-axis is A (x, 0, 0).

The perpendicular distance of P from X-axis is

Similarly,

The perpendicular distance of P from Y-axis is

The perpendicular distance of P from Z-axis is

Collinear points: If three or more points lie on the same line are called collinear points.

Section formula:

The point dividing the line segment joining the points A (x1, y1, z1) and B (x2, y2, z2) in the ratio m : n is given by

The mid-point of the line segment joining the points A (x1, y1, z1) and B (x2, y2, z2) is

The centroid of the triangle whose vertices are A (x1, y1, z1), B (x2, y2, z2) and C (x3, y3, z3) is

Tetrahedron:

→ It has 4 vertices and 6 edges.
→ A Tetrahedron is a closed figure formed by four planes not all passing through the same point.

→ Each edge arises as the line of intersection of two of the four planes.

→ The line segment joining the vertices to the centroid of opposite face. The point of concurrence is        called centroid of Tetrahedron.

→ Centroid divides the line segment in the ratio 3:1.

→ The centroid of the Tetrahedron whose vertices are A (x1, y1, z1), B (x2, y2, z2), C (x3, y3, z3) and C (x4, y4, z4) is

The line segment joining the points (x1, y1, z1), (x2, y2, z2) is divided by

XY – plane in the ratio – z1: z2

YZ – plane in the ratio – x1: x2

XZ – plane in the ratio – y1: y2

### 6. DIRECTION COSINES AND RATIOS

Consider a ray OP passing through origin ‘O’ and making angles α, β, γ respectively with the positive direction of X, Y, Z axes.

Cos α, Cos β, Cos γ are called Direction Cosines (dc’s) of the ray OP.

Dc’s are denoted by (l, m, n), where l = Cos α, m = Cos β, n = Cos γ

• A line in a space has two directions, it has two sets of dc’s, one for each direction. If (l, m, n) is one set of dc’s, then (-l, -m, -n) is the other set.

• Suppose P (x, y, z) is any point in space such that OP = r. If (l, m, n) are dc’s of a ray OP then x = lr, y= mr, z = nr.

• If OP = r and dc’s of OP are (l, m, n) then the coordinates of P are (lr, mr, nr).

• If P (x, y, z) is a point in the space, then dc’s of OP are

• If (l, m, n) are dc’s of a line then l2 + m2 + n2 = 1.

⇒ cos2α + cos2β + cos2γ = 1.

Direction Ratios:

Any three real numbers which are proportional to the dc’s of a line are called direction ratios (dr’s) of that line.

• Let (a, b, c) be dr’s of a line whose dc’s are (l, m, n). Then (a, b, c) are proportional to (l, m, n)

and a2 + b2 + c2 ≠ 1.

• Dr’s of the line joining the points (x1, y1, z1), (x2, y2, z2) are (x2 – x1, y2 – y1, z2 – z1)

• If (a, b, c) are dr’s of a line then its dc’s are

• If (l1, m1, n1), (l2, m2, n2) are dc’s of two lines and θ is angle between them then Cos θ = l1l2 + m1m2 + n1n2

If two lines perpendicular then l1l2 + m1m2 + n1n2 = 0.

• If (a1, b1, c1), (a2, b2, c2) are dr’s of two lines and θ is the angle between them then

If two line are perpendicular then a1a2 + b1b2 + c1c2 = 0.

## 7. THE PLANE

Plane: A plane is a proper subset of R3 which has at least three non-collinear points and any two points in it.

∎ Equation of the plane passing through a given point A (x1, y1, z1), and perpendicular to the line whose dr’s (a, b, c) is a(x – x1) + a(y – y1)  + a(z – z1) = 0.

∎ The equation of the plane hose dc’s of the normal to the plane (l, m, n) and perpendicular distance from the origin to the pane p is lx + my + nz = p

∎ The equation of the plane passing through three non-collinear points A (x1, y1, z1), B (x2, y2, z2) and C (x3, y3, z3) is

∎ The general equation of the plane is ax + by + cz + d = 0, where (a, b, c) are Dr’s of the normal to the plane.

Normal form:

The equation of the plane ax + by + cz + d = 0 in the normal form is

Perpendicular distance:

The perpendicular distance from (x1, y1, z1) to the plane ax + by + cz + d = 0 is

The perpendicular distance from the origin to the plane ax + by + cz + d = 0 is

Intercepts:

X- intercept = aIf a plane cuts X –axis at (a, 0, 0), Y-axis at (0, b, 0) and Z-axis at (0, 0, c) then

Y-intercept = b

Z-intercept = c

The equation of the plane in the intercept form is

The intercepts of the plane ax + by + cz + d =0 is -d/a, -d/b, -d/c

∎ The equation of the plane parallel to ax + by + cz + d = 0 is ax + by + cz + k = 0.

∎ The equation of XY – plane is z = 0.

∎ The equation of YZ – plane is x = 0.

∎ The equation of XZ – plane is y = 0.

∎ Distance between the two parallel planes ax + by + cz + d1 =0 and ax + by + cz + d2 =0 is

The angle between two planes:

The angle between the normal to two planes is called the angle between the planes.

If θ is the angle between the planes a1 x + b1 y + c1 z + d1 =0 and a2 x + b2 y + c2 z + d2 =0 then

If two line are perpendicular then a1a2 + b1b2 + c1c2 = 0.

∎ The distance of the point P (x, y, z) from

### 8. LIMITS AND CONTINUITY

Intervals:

Let (a, b) ∈ R such that a ≤ b, then the set

• {x ∈ R: a ≤ x ≤ b}, is denoted by [a, b] and it is called as closed interval
• {x ∈ R: a < x < b}, is denoted by (a, b) and it is called as open interval
• {x ∈ R: a < x ≤ b}, is denoted by (a, b] and it is called as open closed interval
• {x ∈ R: a ≤ x < b}, is denoted by [a, b) and it is called as closed open interval
• {x ∈ R: x ≥ a}, is denoted by [a, ∞)
• {x ∈ R: x > a}, is denoted by (a, ∞)
• {x ∈ R: x ≤ a}, is denoted by (- ∞, a]
• x ∈ R: x < a}, is denoted by (- ∞, a)

#### Limit:

If f(x) is a function of x such that if x approaches to a constant value ‘a’, then the value of f(x) also approaches to ‘l’. Then the constant ‘I’ is called a limit of f(x) at x = a

Or

A real number l is called the limit of the function f, if for all ϵ> 0 there exist δ > 0 such that    whenever  ⟹

###### Properties of Limits:

Sand witch theorem:( Squeez Principle):

f, g, and h are functions such that f(x) ≤ g(x) ≤ h(x), then  and if

###### Left- hand and Right-hand Limits:

If x < a, then  is called left-hand limit

If x > a, then  is called right-hand limit

Note:

###### In Determinate forms:

if a function f(x) any of the following forms at x = a:

Then f(x) is said to be indeterminate at x = a.

Continuity:

Condition 1:  If the condition is like x = a and x ≠ a, then we use following property.

If then f(x) is continuous at x = a, otherwise f(x) is not continuous.

Condition 2: If the condition is like x ≤ a and x >a, or x < a and x ≥a then we use following property.

If     then f(x) is continuous at x = a, otherwise f(x) is not continuous.

## 9.DIFFERENTIATION

Let f be a function defined on a neighbourhood of a real number ‘a’ if exist then we say that f is differentiable at x a and it is denoted by f'(a).

∴ f’(a) =

∎ If right hand derivative = left hand derivative, then f is differentiable at ‘a’.

i.e.,

### First principle in derivative:

The first principle of the derivative of f at any real number ‘x’ is f’(x) =

∎ The differentiation of f(x) is denoted by

means differentiation of ‘y’ with respect to ‘x’

The derivative of constant function is zero i.e., f’(a) = 0 where ‘a’ is any constant.

∎ Let I be an interval in R u and v are real valued functions on I and x ∈ I. Suppose that u and v are differentiable at ‘x’, then

• (u ± v) is also differentiable at ‘x’ and (u ± v)’(x) = u’ (x) ± v’(x).
• ‘uv’ is also differentiable at ‘x’ and (uv)’(x) = u(x) v’(x) + v(x) u’(x).
• αu + βv is also differentiable at ‘x’ and (αu + βv)’(x) = αu’(x) + βv’(x), α, β are constants.
• is also differentiable at ‘x’ and

∎ (f o g)’ (x) = f’(g(x)). g’(x).

### Formulae:

#### Parametric Differentiation:

If x = f(t) and y = g(t) then the procedure of finding  in terms of the parameter ‘t’ is called parametric equations.

Implicitly differentiation:

An equation involving two or more variables is called an implicit equation.

ax2 + 2hxy + b y2 = 0 is an implicit equation in terms of x and y.

The process of finding    from an implicit equation is called implicitly differentiation.

Derivative of one function w.r.t.  another function:

The derivative of f(x) w.r.t g(x) is

Second order derivative:

Let y = f(x) be a function, if y is differentiable then the derivative of f is f’(x). If ‘(x) is again differentiable then the derivative of f’(x) is called second order derivative. And it is denoted by f” (x) or

## 10. ERRORS AND APPROXIMATIONS

Approximations:

Let y f(x) be a function defined an interval I and x ∈ I. If ∆x is any change in x, then ∆y be the corresponding change in y thus ∆y = f (x + ∆x) – f (x).

Let

where ϵ is very small

For ‘ϵ.∆x’ is very small and hence,

Approximate value is f (x + ∆x) = f(x) + f’(x). ∆x

Differential:

Let y f(x) be a function defined an interval I and x ∈ I. If ∆x is any change in x, then called differential of y = f(x) and it is denoted by df.

∴ dy = f’(x). ∆x

Errors:

Let y f(x) be a function defined an interval I and x ∈ I. If ∆x is any change in x, then ∆y be the corresponding change in y.

### The Following formulae will be used in Solving problems

CIRCLE:

If ‘r’ is radius, ‘d’ is diameter ‘P’ is the perimeter or circumference and A is area of the circle then

d= 2r, P = 2πr = πd and A = πr2sq.u

SECTOR:

If ‘r’ is the radius, ‘l’ is the length of arc and θ is of the sector then

Area = ½ l r = ½ r2θsq.u.

Perimeter = l + 2r = r (θ + 2) u.

CYLINDER:

Length of the Arc ‘l’ = rθ (θ must be in radians).

If ‘r is the radius of the base of cylinder and ‘h’ is the height of the cylinder, then

Area of base = πr2 sq.units.

Lateral surface area = 2πrh units.

Total surface area = 2πr (h + r) units.

Volume = πr2 h cubic units.

CONE:

If ‘r’ is the radius of base, ‘h’ is the height of cone and ‘l’ is slant height then

l 2 + r2 = h2

Lateral surface area = πrl units.

Total surface area = πr (l + r) sq. units.

Volume =  cubic units.

SPHERE:

If ‘r’ is the radius of the Sphere then

Surface area = πr2 sq. units.

Volume =  πr3 cubic units.

## 11. TANGENTS AND NORMALS

Tangent of a Curve:
If the secant line PQ approaches to the same position as Q moves along the curve and approaches to either side then limiting position is called a ‘Tangent line’ to the curve at P. The point P is called point of contact

Let y = f(x) be a curve, P a point on the curve. If Q(≠P) is another point on the curve then the line PD is called secant line.

Gradient of a curve:

Let y = f(x) be a curve and P (x, y) be a point on the curve. The slope of the tangent to the curve y = f(x) at P is called gradient of the curve.

Slope of the tangent to the curve y = f(x) at P (x, y) is m =

∎ The equation of the tangent at P (x1, y1) to the curve is y – y1 = m (x – x1) where m =

Normal of a curve:

Let y = f(x) be a curve and P (x, y) be a point on the curve. The line passing through P and perpendicular to the tangent of the curve y = f(x) at P is called Normal of the curve.

∎ The equation of the tangent at P (x1, y1) to the curve is y – y1 = -1/m (x – x1).
Slope of the normal is -1/m. where m =

Lengths of tangent, normal, subtangent and subnormal:

PT → Normal; QN → subnormal
PN → Tangent; QT → subtangent

∎ if m =  then

Angle between two curves:

If two curves intersect at a point P., then the angle between the tangents of the curves at P is called the angle between the curves at P.

∎ If m1, m2 are the slopes of two tangents of the two curves and θ is the angle between the curves then

Tanθ =

Note:

• If m1= m2, then two corves are touch each other.
• if m1× m2 = –1, then two curves intersect orthogonally.

## 12. RATE MEASURE

Average rate of change:

if y = f(x) then the average rate of change in y between x = x1 and x = x2 is defined as

Instantaneous rate of change:

if y = f(x), then the instantaneous rate of change of a unction f at x = x0 is defined as

Rectilinear Motion:

A motion of a particle in a line is called Rectilinear motion. The rectilinear motion is denoted by s = f(t) where f(t) is the rule connecting ‘s’ and ‘t’.

Velocity, Acceleration:

A particle starts from a fixed point and moves a distance ‘S’ along a straight-line during time ‘t’ then

Velocity =

Acceleration =

Note:

(i) If v> 0, then the particle s moving away from the straight point.

(ii) If v < 0, then particle s moving away towards the straight point.

(iii) If v = 0, then the particle comes rest.

## 13.ROLLE’S & LANGRANGEE’S THEOREM

Rolle’s Theorem:

Suppose a, b (a < b) are two real numbers. Let f: [a, b] → R be a function satisfying the following conditions:

(i) f is continuous on [a, b]

(ii) f is differentiable on (a, b) and

(iii) f(a) = f(b)

then there exists at least one c ∈ (a, b) such that f’(c)= 0.

Lagrange’s Theorem:

Suppose a, b (a < b) are two real numbers. Let f: [a, b] → R be a function satisfying the following conditions:

(i) f is continuous on [a, b]

(ii) f is differentiable on (a, b) and

then there exists at least one c ∈ (a, b) such that f’(c)=

## 14.INCREASING & DECREASING FUNCTIONS

Let f be a real function on an interval I then f is said to be

(i) an increasing function on I if

x1 < x2 ⇒ f (x1) ≤ f (x2) ∀ x1, x2 ∈ I

(ii) decreasing function on I if

x1 < x2 ⇒ f (x1) ≥ f (x2) ∀ x1, x2 ∈ I

Let f be a real function on an interval I then f is said to be

(i) strictly increasing function on I if

x1 < x2 ⇒ f (x1) < f (x2) ∀ x1, x2 ∈ I

(ii) strictly decreasing function on I if

x1 < x2 ⇒ f (x1) > f (x2) ∀ x1, x2 ∈ I

Let f(x) be a real valued function defined on I = (a, b) or [a, b) or (a, b] or [a, b]. Suppose f is continuous on I and differentiable in (a, b). If

(i) f’ (c) > 0 ∀ c ∈ (a, b), then f is strictly increasing on I

(ii) f’ (c) < 0 ∀ c ∈ (a, b), then f is strictly decreasing on I

(iii) f’ (c) ≥ 0 ∀ c ∈ (a, b), then f is increasing on I

(iv) f’ (c) ≤ 0 ∀ c ∈ (a, b), then f is decreasing on I

Critical point:

A point x = c in the domain of the function said to be ‘critical point’ of the function f if either f’ (c) = 0 or f’ (c) does not exists.

Stationary point:

A point x = c in the domain of the function said to be ‘stationary point’ of the function f if  f’ (c) = 0.

MAXIMA & MINIMA

Global maxima – Global minima:

Let D be an interval in R and f: D → R be a real function and c ∈ D. Then f is said to be

(i) a global maximum on D if f(c) ≥ f(x)

(ii) a global minimum on D if f(c) ≤ f(x)

Relative maximum:

Let D be an interval in R and f: D → R be a real function and c ∈ D. Then f is said to be relative maximum at c if there exist δ > 0 such that f(c) ≥ f(x) ∀ x ∈ (c – δ, c + δ).

Here, f (c) is called relative maximum value of f(x) at x = c and the point x = c is called point of relative maximum.

Relative minimum:

Let D be an interval in R and f: D → R be a real function and c ∈ D. Then f is said to be relative maximum at c if there exist δ > 0 such that f(c) ≤ f(x) ∀ x ∈ (c – δ, c + δ).

Here, f (c) is called relative maximum value of f(x) at x = c and the point x = c is called point of relative minimum.

The relative maximum and minimum value of f are called extreme values.

If f is either minima or maxima f’ (α) = 0.

Let f be a continuous function om [a, b] and α ∈ (a, b)

(i) If f’ (α) = 0 and f’’ (α) >0, then f(α) is relative minimum.

(ii) if f’ (α) = 0 and f’’ (α) <0, then f(α) is relative maximum

My App:

## 1.Functions

Set: A collection of well-defined objects is called a set.

Ordered pair: Two elements a and b listed in a specific order form. An ordered pair denoted by (a, b).

Cartesian product: Let A and B are two non-empty sets. The Cartesian product of A and B is denoted by A × B and is defined as a set of all ordered pairs (a, b) where a ϵ A and b ϵB

Relation: Let A and B are two non-empty sets the relation R from A to B is a subset of A×B.

⇒ R: A→B is a relation if  R⊂ A × B

#### Function:

A relation f: A → B is said to be a function if ∀ aϵ A there exists a unique element b such that (a, b) ϵ f.                                            (Or)

A relation f: A → B is said to be a function if

(i) x ϵ A ⇒ f(x) ϵ B

(ii)  x1 , x2 ϵ A , x1 = x2 in A  ⇒ f(x1) = f(x2) in B.

Note:   If A, B are two finite sets then the no. of   functions that can be defined from A to B is  n(B)n(A)

VARIOUS TYPES OF FUNCTIONS

One– one Function (Injective):- A function f: A→ B is said to be a one-one function or injective if different elements in A have different images in B.

(Or)

A function f: A→ B is said to be one-one function if f(x1) = f(x2) in B ⇒ x1 = x2 in A.

Note: No. of one-one functions that can be defined from A into B is n(B) p n(A)   if  n(A) ≤ n(B)

On to Function (Surjection): – A function f: A→ B is said to be onto function or surjection if for each yϵ B ∃ x ϵ A such that f(x) =y

Note: if n(A) = m and n(B) = 2 then no. of onto functions = 2m – 2

Bijection: – A function f: A→ B is said to be Bijection if it is both ‘one-one and ‘onto’.

Constant function:  A function f: A→ B is said to be constant function if f(x) = k ∀ xϵA

Identity function:  Let A be a non-empty set, then the function defined by I: A → A, I(x)=x is called identity function on A.

Equal function:  Two functions f and g are said to be equal if

(i)   They have same domain (D)

(ii)  f(x) = g(x) ∀ xϵ D

Even function:  A function f: A→ B is said to be even function if f (- x) = f(x) ∀ xϵ A

Odd function:   A function f: A→ B is said to be odd function if f (- x) = – f(x) ∀ xϵ A

Composite function:  If f: A→B, g: B→C are two functions then the composite relation is a function from A to C.

gof: A→C is a composite function and is defined by gof(x) = g(f(x)).

Step function:  A number x = I + F

I → integral part    = [x]

F → fractional part = {x}

∴ x = [x] + {x}

If y = [x] then domain = R and

Range = Z

0 ≤ x ≤ 1, [x] = 0

1≤ x ≤ 2, [x] = 1

-1 ≤ x ≤ 0, [x] = -1

If k is any integer [ x + k] = k + [x]

The value of [x] is lies in x – 1 < [x] ≤ 1.

Inverse function: If f: A → B is bijection then f -1  is exists

f-1: B → A is an inverse function of f.

### SOME IMPORTANT POINTS

of subsets of a set of n elements is 2n

of proper subsets of a set of n elements is 2n – 1

Let A and B are two non-empty finite sets and f: A → B is a function. This function will

One-one if n(A) ≤ n(B)

On to if n(A) ≥ n(B)

Bijection   if n(A) = n(B).

## 3. MATRICES

Matrix: An ordered rectangular array of elements is called a matrix

• Matrices are generally enclosed by brackets like
• Matrices are denoted by capital letters A, B, C and so on
• Elements in a matrix are real or complex numbers; real or complex real-valued functions.

Oder of Matrix: A matrix having rows and ‘n’ columns is said to be of order m x n. Read as m by n.

### Square Matrix: A matrix in which the no. of rows is equal to the no. of columns is called a square matrix.

Principal diagonal ( diagonal)  Matrix: If A  = [aij] is a square matrix of order ‘n’ the elements  a11 , a22 , a33 , ………. ann is said to constitute its principal diagonal.

Trace Matrix: The sum of the elements of the principal diagonal of a square matrix A is called the trace of the matrix. It is denoted by Tr (A).

Ex:-

Diagonal Matrix: If each non-diagonal element of a square matrix is ‘zero’ then the matrix is called a diagonal matrix.

Scalar Matrix: If each non-diagonal elements of a square matrix are ‘zero’ and all diagonal elements are equal to each other, then it is called a scalar matrix.

Identity Matrix or Unit Matrix: If each of the non-diagonal elements of a square matrix is ‘zero’ and all diagonal elements are equal to ‘1’, then that matrix is called a unit matrix.

Null Matrix or Zero Matrix: If each element of a matrix is zero, then it is called a null matrix.

Row matrix & column Matrix: A matrix with only one row s called a row matrix and a matrix with only one column is called a column matrix.

Triangular matrices:

A square matrix A = [aij] is said to be upper triangular if aij = 0   ∀ i > j

A square matrix A = [aij] is said to be lower triangular matrix aij = 0  ∀ i < j

Equality of matrices: matrices A and B are said to be equal if A and B of the same order and the corresponding elements of A and B are equal.

### Product of Matrices:

Let A = [aik]mxn and B = [bkj]nxp be two matrices ,then the matrix C = [cij]mxp  where

Note: Matrix multiplication of two matrices is possible when no. of columns of the first matrix is equal to no. of rows of the second matrix.

Transpose of Matrix: If A = [aij] is an m x n matrix, then the matrix obtained by interchanging the rows and columns is called the transpose of A. It is denoted by AI or AT.

Note: (i) (AI)I = A (ii) (k AI) = k . AI    (iii)  (A + B )T = AT + BT  (iv)  (AB)T = BTAT

Symmetric Matrix: A square matrix A is said to be symmetric if AT =A

If A is a symmetric matrix, then A + AT is symmetric.

Skew-Symmetric Matrix: A square matrix A is said to be skew-symmetric if AT = -A

If A is a skew-symmetric matrix, then A – AT is skew-symmetric

Minor of an element: Consider a square matrix

the minor an element in this matrix is defined as the determinant of the 2×2 matrix obtained after deleting the rows and the columns in which the element is present.

Cofactor of an element: The cofactor of an element in i th row and j th column of A3×3 matrix is defined as it’s minor multiplied by (- 1 ) i+j .

### Properties of determinants:

• If each element of a row (column) of a square matrix is zero, then the determinant of that matrix is zero.

• If A is a square matrix of order 3 and k is scalar then.
• If two rows (columns) of a square matrix are identical (same), then Det. Of that matrix is zero.

• If each element in a row (column) of a square matrix is the sum of two numbers then its determinant can be expressed as the sum of the determinants.

• If each element of a square matrix are polynomials in x and its determinant is zero when x = a, then (x-a) is a factor of that matrix.
• For any square matrix A  Det(A) =  Det (AI).
• Det(AB) = Det(A) . Det(B).
• For any positive integer n Det(An) = (DetA)n.

Singular and non-singular matrices: A Square matrix is said to be singular if its determinant is zero, otherwise it is said to be the non-singular matrix.

Ad joint of a matrix: The transpose of the matrix formed by replacing the elements of a square matrix A with the corresponding cofactors is called the adjoint of A.

Invertible matrix: Let A be a square matrix, we say that A is invertible if there exists a matrix B such that AB =BA = I, where I is the unit matrix of the same order as A and B.

Augmented matrix: The coefficient matrix (A) augmented with the constant column matrix (D) is called the augmented matrix. It is denoted by [AD].

Sub matrix: A matrix obtained by deleting some rows and columns (or both) of a matrix is called the submatrix of the given matrix.

Let A be a non-zero matrix. The rank of A is defined as the maximum of the order of the non-singular submatrices of A.

• Note: If A is a non-zero matrix of order 3 then the rank of A is:
• 1, if every 2×2 submatrix is singular
• 2, if A is singular and at least one of its 2×2 sub-matrices is non-singular

(iii)  3, if A is non – singular.

Consistent and Inconsistent: The system of linear equations is consistent if it has a solution, in-consistent if it has no solution.

• Note: The system of three equations in three unknowns AX = D has
• A unique solution if rank(A) = rank ([AD]) = 3
• Infinitely many solutions if rank (A) = ([AD]) < 3
• No solution if rank (A) ≠ rank ([AD])

### Solutions of a homogeneous system of linear equations:

The system of equations AX = 0 has

• The trivial solution only if rank(A) = 3
• An infinite no. of solutions if rank(A) < 3

## 4.ADDITION OF VECTORS

Directed line: If A and B are two distinct points in the space, the ordered pair (A, B) denoted by AB is called a directed line segment with initial point A and terminal point B.

⇒ A directed line passes through three characteristics: (i) length (ii) support (iii) direction

Scalar: A quantity having magnitude only is called a scalar. We identify real numbers as a scalar.

Ex: – mass, length, temperature, etc.

Vector: A quantity having length and direction is called a vector.

Ex: – velocity, acceleration, force, etc.

⇒ If is a vector then its length is denoted by

Position of vector: If P (x, y, z) is any point in the space, then is called the position vector of the point P with respect to origin (O). This is denoted by

Like and unlike vectors:  If two vectors are parallel and having the same direction then they are called like vectors.

If two vectors are parallel and having opposite direction then they are called, unlike vectors.

Coplanar vectors:
Vectors whose supports are in the same plane or parallel to the same plane are called coplanar vectors.

### VECTOR ADDITION

Triangle law: If are two vectors, there exist three points A, B, and C in a space such that   defined by

Parallelogram law: If two vectors and represented by two adjacent sides of a parallelogram in magnitude and direction then their sum is represented in magnitude and direction by the diagonal of the parallelogram through their common point.

Scalar multiplication: Let be a vector and λ be a scalar then we define vector λ  to be the vector if either is zero vector or λ is the scalar zero; otherwise λ is the vector in the direction of with the magnitude if λ>0 and λ  = (−λ)(− ) if λ<0.

The angle between two non-zero vectors:   Let be two non-zero vectors, let  then ∠AOB has two values. The value of ∠AOB, which does not exceed 1800 is called the angle between the vectors and , it is denoted by ( ).

Section formula: Let be two position vectors of the points A and B with respect to the origin if a point P divides the line segment AB in the ratio m:n then

Linear combination of vectors:  let  be vectors x1, x2, x3…. xn be scalars, then the vector is called the linear combination of vectors.

Components: Consider the ordered triad (a, b, c) of non-coplanar vectors If r is any vector then there exist a unique triad (x, y, z) of scalars such that  . These scalars x, y, z are called the components of with respect to the ordered triad   (a, b, c).

• i, j, k are unit vectors along the X, Y and Z axes respectively and P(x, y, z) is any point in the space then = r = x i + y j +z k   and

Regular polygon: A polygon is said to be regular if all the sides, as well as all the interior angles, are equal.

• If a polygon has sides then the no. of diagonals of a polygon is
• The unit vector bisecting the angle between  is

### Vector equation of a line and plane

⇒The vector equation of the line passing through point A () and ∥el to the vector  is

Proof:-

Then AP,  are collinear vector proof: let P ( ) be any point on the line a

the equation of the line passing through origin and parallel to the vectoris

• the  vector equation of the line passing through the points A( )  and B(  )  is
• Cartesian equation of the line passing through A ( x1, y1, z1) and  B ( x2, y2, z2) is
• The vector equation of the plane passing through point A( ) and parallel to the vectors and is
• The vector equation of the plane passing through the point A( ), B( ) and parallel to the vector is
• The vector equation of the plane passing through the points A( ), B( ) and C( ) is

$large&space;bar{r}=&space;(1-t)bar{a}&space;+&space;t&space;bar{b}$

## 5.PRODUCT OF VECTORS

Dot product (Scalar product): Let are two vectors. The dot product or direct product of and  is denoted byand is defined as

• If = 0, = 0 ⟹  = 0.
• If ≠0, ≠ 0 then
• The dot product of two vectors is a scalar
• If are two vectors, then

• If θ is the angle between the vectors then.

⟹

⟹ If   > 0, then θ is an acute angle

⟹ If    < 0, then θ is obtuse angle 0

⟹ If    = 0, then  is perpendicular to

• If is any vector then

Component and Orthogonal Projection:

Let=,=  be two non-zero vectors. Let the plane passing through B ( ) and perpendicular to intersects

In M, then is called the component of on

• The component (projection) vector of  on is
• Length of the projection (component) =
• Component of perpendicular to =

If ,,    form a right-handed system of an orthonormal triad, then

• If then = a1b1 + a2b2 + a3b3
• If  then

Parallelogram law:

In a parallelogram, the sum of the squares of the lengths of the diagonals is equal to the sum of the squares of the lengths of its sides.

In ∆ABC, the length of the median through vertex A is

Vector equation of a plane:

The vector equation of the plane whose perpendicular distance from the origin is p and unit normal drawn from the origin towards the plane is,

•The vector equation of the plane passing through point A ( ) and perpendicular to the is

•If θ is the angle between the planes then

Cross product (vector product): Let and be two non-zero collinear vectors. The cross product of   and  is denoted by ×  (read as a cross ) and is defined as

are orthogonal triad then

• The vector × is perpendicular to both  and and also perpendicular to the plane containing them

• The unit vector perpendicular to both and  is

• Let then

• If and  are two sides of a triangle then the area of the triangle =

• If A ( ), B ()and C ( )are the vertices of a ∆ABC, then its area

• The area of the parallelogram whose adjacent sides and    is

• The area of the parallelogram whose diagonals  and    is

• If A ( ), B ( )and C ( )are three points then the perpendicular distance from A to the line passing through B, C is

Let,and be three vectors, then () . is called the scalar triple product of,andand it is denoted by

Ifthen

•In determinant rows(columns) are equal then the det. Value is zero.
•In a determinant, if we interchange any two rows or columns, then the sign of det. Is change.
•Four distinct points A, B, C, and D are said to be coplanar iff
The volume of parallelepiped:
If ,andare edges of a parallelepiped then its volume is
The volume of parallelepiped:
The volume of Tetrahedron with, and are coterminous edges is
The volume of Tetrahedron whose vertices are A, B, C and D is
Vector equation of a plane:
The vector equation of the plane passing through point A () and parallel to the vectors and is
The vector equation of the plane passing through the points A ( ) and B( ) and parallel to the vector is
The vector equation of the plane passing through the points A (), B( ) and C( ) is
Skew lines:
The lines which are neither intersecting nor parallel are called Skew lines

The shortest distance between the Skew lines:
If are two skew lines, then the shortest distance between them is

If A, B, C and D are four points, then the shortest distance between the line joining the points AB and CD is

•The plane passing through the intersection of the planes is
the perpendicular distance from point A (a ̅) to the plane is

Let ,and be three vectors, then is called the vector triple product of, and.

Scalar product of four vectors:

Vector product of four vectors:

## 6. TRIGONOMETRY UPTO TRANSFORMATIONS

The word ’trigonometry’ derived from the Greek words ‘trigonon’ and ‘metron’. The word ‘trigonon’ means a triangle and the word ‘metron’ means a measure.

Angle: An angle is a union of two rays having a common endpoint in a plane.

There are three systems of measurement of the angles.

• Sexagesimal system (British system)
• Centesimal system (French system)
• Circular measure (Radian system)

Sexagesimal system: – In this system, a circle can be divided into 360 equal parts. Each part is called one degree (0). One circle = 3600

Further, each degree can be divided into 60 equal parts. Each part is called one minute (‘).

and each minute can be divided into 60 equal parts. Each part is called one second (“)

Sexagesimal system: – In this system, a circle can be divided into 400 equal parts. Each part is called one grade (g). One circle = 400g

Further, each grade can be divided into 100 equal parts. Each part is called one minute (‘).

and each minute can be divided into 100 equal parts. Each part is called one second (“)

Circular measure: Radian is defined as the amount of the angle subtended by an arc of length ’r’ of a circle of radius ‘r’.

One radian is denoted by 1c. One circle = 2πc

Relation between the three measures:

3600 = 400g = 2 πc

1800 = 200g = πc

Trigonometric Ratios:

Trigonometric identities: –

∗ sin2θ + cos2θ = 1

1 – cos2θ = sin2θ

1 – sin2θ = cos2θ

∗ sec2θ − tan2θ = 1

sec2θ = 1 + tan2θ

tan2θ = sec2θ – 1

(secθ − tanθ) (secθ + tanθ) = 1

∗  cosec2θ − cot2θ = 1

co sec2θ = 1 + cot2θ

cot2θ = cosec2θ – 1

(cosec θ – cot θ) (cosec θ + cot θ) = 1

• sin θ. cosec θ = 1

sec θ. cos θ = 1

tan θ. cot θ = 1

All Silver Tea Cups Rule:

Note: If 900 ±θ or 2700 ±θ then

‘sin’ changes to ‘cos’; ‘tan’ changes to ‘cot’; ‘sec’ changes to ‘cosec’

‘cos’ changes to ‘sin’; ‘cot’ changes to ‘tan’; ‘cosec’ changes to ‘sec’.

If 1800 ±θ or 3600 ±θ then, no change in ratios.

Values of Trigonometric Ratios:

Complementary angles: Two angles A and B are said to be complementary angles, if A + B = 900.

supplementary angles: Two angles A and B are said to be supplementary angles, if A + B = 1800.

Let E ⊆ R and f: E → R be a function, then f is called periodic function if there exists a positive real number ‘p’ such that

• (x + p) ∈ E ∀ x∈ E
• F (x+ p) = f(x) ∀ x∈ E

If such a positive real number ‘p’ exists, then it is called a period of f.

The algebraic sum of two or more angles is called a ‘compound angle’.

For any two real numbers A and B

sin (A + B) = sin A cos B + cos A Cos B

sin (A − B) = sin A cos B − cos A Cos B

cos (A + B) = cos A cos B − sin A sin B

cos (A − B) = cos A cos B + sin A sin B

tan (A + B) =

tan (A − B) =

cot (A + B) =

⋇ cot (A − B) =

sin (A + B + C) = ∑sin A cos B cos C − sin A sin B sin C

cos (A + B + C) = cos A cos B cos C− ∑cos A sin B sin C

tan (A + B + C) =

⋇ cot (A + B + C) =

⋇ sin (A + B) sin (A – B) = sin2 A – sin2 B = cos2 B – cos2 A

⋇ cos (A + B) cos (A – B) = cos2 A – sin2 B = cos2 B – sin2 A

Extreme values of trigonometric functions:

If a, b, c ∈ R such that a2 + b2 ≠ 0, then

Maximum value =

Minimum value =

If A is an angle, then its integral multiples 2A, 3A, 4A, … are called ‘multiple angles ‘of A and the multiple of A by fraction likeare called ‘submultiple angles.

⋇ sin 2A = 2 sin A cos A =

⋇ cos 2A = cos2 A – sin2 A

= 2 cos2 A – 1

= 1 – 2sin2 A

=

⋇ tan 2A =

⋇ cot 2A =

∎ If   is not an add multiple of

⋇ sin A = 2 sin  cos  =

⋇ cos A = cos2  – sin2

= 2 cos2    – 1

= 1 – 2sin2

=

⋇ tan A =

⋇ cot A =

⋇ sin3A = 3 sin A −4 sin3 A

⋇ cos 3A = 4 cos3 A – 3 cos A

⋇ tan 3A =

⋇ cot 3A =

⋇ tan A + cot A = 2 cosec 2A

⋇ cot A – tan A = 2 cot 2A

For A, B∈ R

⋇ sin (A + B) + sin (A – B) = 2sin A cos B

⋇ sin (A + B) −sin (A – B) = 2cos A sin B

⋇ cos (A + B) + cos (A – B) = 2 cos A cos B

⋇ cos (A + B) − cos (A – B) = − 2sin A sin B

For any two real numbers C and D

⋇ sin C + sin D = 2sin cos

⋇ sin C −sin D= 2cos  sin

⋇ cos C + cos D = 2 coscos

⋇ cos C − cos D = − 2sin   sin

If A + B + C = π or 1800, then

⋇ sin (A + B) = sin C; sin (B + C) = sin A; sin (A + C) = sin B

⋇ cos (A + B) = − cos C; cos (B + C) = −cos A; cos (A + C) = − cos B

If A + B + C = 900 or  then

⋇ sin   = cos  ; sin    = cos  ; sin    = cos

⋇ cos    = sin ; cos    = sin ; cos    = sin

If then

⋇ sin (A + B) = cos C; sin (B + C) = cos A; sin (A + C) = cos B

⋇ cos (A + B) = sin C; cos (B + C) = sin A; cos (A + C) = sin B

## 7. TRIGONOMETRIC EQUATIONS

Trigonometric equation: An equation consisting of the trigonometric functions of a variable angle θ ∈ R is called a ‘trigonometric equation’.

The solution of the equation: The values of the variable angle θ, satisfying the given trigonometric equation is called a ‘solution’ of the equation.

The set of all solutions of the trigonometric equation is called the solution set’ of the equation. A ‘general solution’ is an expression of the form θ0 + f(n) where θ0 is a particular solution and f(n) is a function of n ∈ Z involving π.

If k ∈ [− 1, 1] then the principle solution of θ of sin x = k lies in

General solution of sin x = sin θ is x = nπ + (−1) n θ, n ∈ Z

If k ∈ [− 1, 1] then the principle solution of θ of cos x = k lies in

General solution of cos x = cos θ is x = 2nπ ± θ, n ∈ Z

If k ∈R then the principle solution of θ of tan x = k lies in

General solution of tan x = tan θ is x = nπ + θ n ∈ Z

If sin θ = 0, then the general solution is θ = nπ, n ∈ Z

If tan θ = 0, then the general solution is θ = nπ, n ∈ Z

If cos θ = 0, then the general solution is θ = (2n + 1) , n ∈ Z

If sin2 θ = sin2 𝛂, cos2 θ = cos2 𝛂 or tan2 θ = tann2 𝛂 then the general solution is 𝛉 = nπ ± θ, n ∈ Z

## 8.INVERSE TRIGONOMETRIC FUNCTIONS

If A, B are two sets and f: A→ B is a bijection, then f-1 is existing and f-1: B → A is an inverse function.

The function Sin-1: [−1, 1] → is defined by Sin-1 x = θ ⇔ θ∈  and sin θ = x

The function Cos-1: [−1, 1] → [0, π] is defined by Sin-1 x = θ ⇔ θ∈ [0, π] and cos θ = x

The function Tan-1: R →  is defined by Tan-1 x = θ ⇔ θ∈  and tan θ = x

The function Sec-1: [−∞, −1] ∪ [1, ∞] → is defined by Sin-1 x = θ ⇔ θ∈ and sec θ= x

The function Cosec-1: [−∞, −1] ∪ [1, ∞] →   is defined by cosec-1 x = θ ⇔ θ∈ and Cosec θ= x

The function Cot-1: R → (0, π) is defined by Cot-1 x = θ ⇔ θ ∈ (0, π) and cot θ = x

Properties of Inverse Trigonometric functions:

Sin-1 x = Cosec-1(1/x) ∀ x ∈ [−1, 1] – {0}

Cos-1x = Sec-1(1/x) ∀ x ∈ [−1, 1] – {0}

Tan-1 x = Cot-1(1/x), if x > 0