First year

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TS Inter Maths 1B Concept

 This note is 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 two points

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

distance between a point and origin

  • 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

section formula

  • 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

locus 1Ex 2: – set of points that are at a constant distance from a fixed point. circle for locus

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

right- angled triangle

•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


transformation of axes

•To remove the first degree terms of the equation ax2  + 2hxy + by2 +2gx +2fy+ c = 0, origin should be shifted to the point    formula 1

•If the equation ax2 + by2 +2gx +2fy+ c = 0, origin should be shifted to the point  TS inter 1B change of axes 1

Rotation of axes:

When the  axes are rotated through an angle θ then

rotation-of-axes-diagram.j

rotation of axes

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

formula 2


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”.  slope - diagram

 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 formula 1Slope of the line ax + by + c = 0 is  s inter slope of the straight line

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

equation of the line two

Intercept form:

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

equation of the line in the intercept form

intercept form

• 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 TS inter 1B Straight lines 2
  2. The area of the triangle formed by the line  TS inter 1B Straight lines 4 with the coordinate axes is  TS inter 1B Straight lines 3

Perpendicular distance (Length of the perpendicular):

The perpendicular distance from a point P (x1, y1) to the line ax + by + c = 0 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  TS inter 1B Straight lines 5

Distance between two parallel lines:

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

distance between parallel lines

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 normal formthe 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

symmetric form

Parametric form:

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

making inclination θ, then                                                                                              parametric form

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

poin of intersection of two lines

 

Concurrent Lines:                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 TS inter 1B Straight lines 6

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

angle between two lines

∗  If θ is an acute angle then

acute angle between two lines

∗ 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

angle between two lines using tanѳ

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

angle between two lines when slopes are given

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

∗ Parallel iff  TS inter 1B Straight lines 7

∗ 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    oot of the perpendicular diagram

foot of the perpendicular

 

 

 

 

 

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         image of the point diagram

image of the point 1

 

 

 

 

 

 

 

 

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

TS inter 1B pair of straight lines 1

∎ The lines represented by the equation ax2 + 2hxy + by2 = 0 are TS inter 1B pair of straight lines 2

∎ 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 TS inter 1B pair of straight lines 3

∎ 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

TS inter 1B pair of straight lines 4

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 areTS inter 1B pair of straight lines 6

∎ 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 TS inter 1B pair of straight lines 7

∎ 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)

TS inter 1B pair of straight lines 8

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 TS inter 1B pair of straight lines 9


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’.

TS inter 1B 3D coordinates 1

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 TS inter 1B 3D coordinates 2

∗ Distance between the point P (x, y, z) to the origin is TS inter 1B 3D coordinates 3

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 TS inter 1B 3D coordinates 4

Similarly,

The perpendicular distance of P from Y-axis is TS inter 1B 3D coordinates 5

The perpendicular distance of P from Z-axis is TS inter 1B 3D coordinates 6

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 TS inter 1B 3D coordinates 7

The mid-point of the line segment joining the points A (x1, y1, z1) and B (x2, y2, z2) is TS inter 1B 3D coordinates 8

The centroid of the triangle whose vertices are A (x1, y1, z1), B (x2, y2, z2) and C (x3, y3, z3) is TS inter 1B 3D coordinates 8

Tetrahedron:TS inter 1B 3D coordinates 11

→ 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 TS inter 1B 3D coordinates 10

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.TS inter 1B DC's and DR's 1

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 TS inter 1B DC's and DR's 2

• 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) TS inter 1B DC's and DR's 3

  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 TS inter 1B DC's and DR's 4

• 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

TS inter 1B DC's and DR's 5

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.TS inter The Plane 1

∎ 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

TS inter The Plane 2

∎ 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

TS inter The Plane 3

Perpendicular distance:

The perpendicular distance from (x1, y1, z1) to the plane ax + by + cz + d = 0 is TS inter The Plane 4

The perpendicular distance from the origin to the plane ax + by + cz + d = 0 isTS inter The Plane 5

Intercepts:TS inter The Plane 8

 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 TS inter The Plane 6                                                                      

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 TS inter The Plane 7

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

TS inter 1B DC's and DR's 5

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

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

TS inter The Plane 9

 


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)

Neighbourhood:
Let a ∈ R. If δ > 0, then the open interval (a – δ, a + δ) is called the δ – neighbourhood of ‘a’

TS inter Limits and continuity 1

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

TS inter Limits and continuity 2

Or

A real number l is called the limit of the function f, if for all ϵ> 0 there exist δ > 0 such that TS inter Limits and continuity 3   wheneverTS inter Limits and continuity 4  ⟹ TS inter Limits and continuity 2

Properties of Limits:

TS inter Limits and continuity 5

Sand witch theorem:( Squeez Principle):

f, g, and h are functions such that f(x) ≤ g(x) ≤ h(x), then TS inter Limits and continuity 6 and if TS inter Limits and continuity 7

Left- hand and Right-hand Limits:

If x < a, then  is TS inter Limits and continuity 8called left-hand limit

If x > a, then TS inter Limits and continuity 9 is called right-hand limit

Note:

 TS inter Limits and continuity 10

In Determinate forms:

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

TS inter Limits and continuity 11

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

TS inter Limits and continuity 12

Continuity:

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

If TS inter Limits and continuity 13 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   TS inter Limits and continuity 14  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 TS inter 1B Differentiation 1exist then we say that f is differentiable at x a and it is denoted by f'(a).

∴ f’(a) = TS inter 1B Differentiation 1

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

i.e.,  TS inter 1B Differentiation 2

First principle in derivative:

The first principle of the derivative of f at any real number ‘x’ is f’(x) = TS inter 1B Differentiation 3

∎ The differentiation of f(x) is denoted by TS inter 1B Differentiation 4

  TS inter 1B Differentiation 5   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 TS inter 1B Differentiation 7

∎ (f o g)’ (x) = f’(g(x)). g’(x).

TS inter 1B Differentiation 8

Formulae:

TS inter 1B Differentiation 9

Derivative of Trigonometric & Inverse trigonometric functions:

TS-inter-1B-Differentiation-10

Derivative of Hyperbolic & Inverse Hyperbolic functions:

TS inter 1B Differentiation 11

 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. TS inter 1B Differentiation 12

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 TS inter 1B Differentiation 5   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 TS inter 1B Differentiation 13

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 TS inter 1B Differentiation 14


 APPLICATION OF DERIVATIVES

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 TS-inter-1B-Errors and Approximations1

TS-inter-1B-Errors and Approximations2 - Copy   where ϵ is very small

TS-inter-1B-Errors and Approximations3

For ‘ϵ.∆x’ is very small and hence, TS-inter-1B-Errors and Approximations4

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, thenTS-inter-1B-Errors and Approximations5 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.

TS-inter-1B-Errors and Approximations 6


The Following formulae will be used in Solving problems

CIRCLE: TS inter1B application of derivative 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:TS inter1B application of derivative 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:TS inter1B application of derivative 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: TS inter1B application of derivative 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 = TS inter1B application of derivative geometry 5 cubic units.

SPHERE: TS inter1B application of derivative Sphere

If ‘r’ is the radius of the Sphere then

Surface area = πr2 sq. units.

Volume = TS inter1B application of derivative fraction 7 πr3 cubic units.


11. TANGENTS AND NORMALS

Tangent of a Curve:TS inter1B Tangents & Norma's 1
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 = TS inter1B Tangents & Normal 2

∎ The equation of the tangent at P (x1, y1) to the curve is y – y1 = m (x – x1) where m =TS inter1B Tangents & Normal 2

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 =TS inter1B Tangents & Normal 2

Lengths of tangent, normal, subtangent and subnormal: TS inter1B Tangents & Normals 4

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

∎ if m =TS inter1B Tangents & Normal 2  then

TS inter1B Tangents & Normals 5

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 = TS inter1B Tangents & Normals 6

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 TS inter1B Rate measure 1

Instantaneous rate of change:

if y = f(x), then the instantaneous rate of change of a unction f at x = x0 is defined as TS inter1B Rate measure 2TS inter1B Rate measure 2TS inter1B Rate measure 2

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 = TS inter1B Rate measure 3

Acceleration = TS inter1B Rate measure 4

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)= TS inter 1B Rolle's and Lagrange's theorem 1


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


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maths IA concept feature image

Ts Inter Maths IA Concept


This note is designed by ‘Basics in Maths’ team. These notes to do help the TS intermediate first year Math 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 1A 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.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

                           cartesion product              

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

onto function

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’.

bijection

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.

composite function

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}

step functionIf 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.

domain and range

in equations

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).


2. MATHEMATICAL INDUCTION


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.

matrix

Types of Matrices

 Square Matrix: A matrix in which the no. of rows is equal to the no. of columns is called a square matrix.

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.

principle diagonal matrix

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:-

trace of matrix

Diagonal Matrix: If each non-diagonal element of a square matrix is ‘zero’ then the matrix is called a diagonal matrix.

diagonal of 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.

scallar 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.

identity matrix

Null Matrix or Zero Matrix: If each element of a matrix is zero, then it is called a null matrix.

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.

row and column matrices

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

triangular matrices

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.

equality of matrices

Product of Matrices:

 Let A = [aik]mxn and B = [bkj]nxp be two matrices ,then the matrix C = [cij]mxp  where

product of matrices

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.  

transpose of matrix

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   

minor of an elemen

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.

minor of an element example

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.

det-1

  • 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.

det-2

  • 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.

         det-3

  • 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.

singular and non-singular matrices

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.

adjoint of matrix 2

 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.

invertible matrix

Augmented matrix: The coefficient matrix (A) augmented with the constant column matrix (D) is called the augmented matrix. It is denoted by [AD].

augmented matrix

Sub matrix: A matrix obtained by deleting some rows and columns (or both) of a matrix is called the submatrix of the given matrix.

sub 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:

hogenious 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.

directed line

⇒ 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 TS inter addition of vectors 4 is a vector then its length is denoted by TS inter addition of vectors 28

Position of vector: If P (x, y, z) is any point in the space, then TS inter addition of vectors 1 is called the position vector of the point P with respect to origin (O). This is denoted by TS inter addition of vectors 2

Like and unlike vectors:  If two vectors are parallel and having the same direction then they are called like vectors.

like vectors

 

If two vectors are parallel and having opposite direction then they are called, unlike vectors.


un like 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 TS inter 1A product of vectors 2 are two vectors, there exist three points A, B, and C in a space such that   defined by TS inter addition of vectors 7

triangle law

Parallelogram law: If two vectorsTS inter 1A vector a and TS inter addition of vectors 5 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.

parallelogram law 2                                                                    parallelogram law

Scalar multiplication: LetTS inter 1A vector a be a vector and λ be a scalar then we define vector λTS inter 1A vector a  to be the vectorTS inter addition of vectors 29 if eitherTS inter 1A vector a is zero vector or λ is the scalar zero; otherwise λTS inter 1A vector a is the vector in the direction of TS inter 1A vector awith the magnitude TS inter addition of vectors 9if λ>0 and λTS inter 1A vector a  = (−λ)(−TS inter 1A vector a ) if λ<0.

add. vectors notes

The angle between two non-zero vectors:   LetTS inter 1A product of vectors 2 be two non-zero vectors, let TS inter addition of vectors 10  then ∠AOB has two values. The value of ∠AOB, which does not exceed 1800 is called the angle between the vectorsTS inter 1A vector a and TS inter addition of vectors 5, it is denoted by (TS inter 1A product of vectors 2 ).

TS inter addition of vectors 12

Section formula: LetTS inter 1A product of vectors 2 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

section formula

Linear combination of vectors:  let TS inter addition of vectors 13 be vectors x1, x2, x3…. xn be scalars, then the vectorTS inter addition of vectors 14 is called the linear combination of vectors.

Components: Consider the ordered triad (a, b, c) of non-coplanar vectorsTS inter addition of vectors 15 If r is any vector then there exist a unique triad (x, y, z) of scalars such that TS inter addition of vectors 16 . These scalars x, y, z are called the components of TS inter addition of vectors 2with 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 thenTS inter addition of vectors 1 = r = x i + y j +z k   andTS inter addition of vectors 17

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 TS inter addition of vectors 18 
  • The unit vector bisecting the angle between  is  TS inter addition of vectors 19

Vector equation of a line and plane

⇒The vector equation of the line passing through point A (TS inter 1A vector a) and ∥el to the vector TS inter addition of vectors 5 is

vector equation of a line

Proof:-

vector equation opf a line 2

 Then AP,  are collinear vector proof: let P (TS inter addition of vectors 2 ) be any point on the line a

TS inter addition of vectors 20      

   the equation of the line passing through origin and parallel to the vectorTS inter addition of vectors 5isTS inter addition of vectors 21      

  • the  vector equation of the line passing through the points A(TS inter 1A vector a )  and B( TS inter addition of vectors 5 )  is TS inter addition of vectors 23
  • Cartesian equation of the line passing through A ( x1, y1, z1) and  B ( x2, y2, z2) is TS inter addition of vectors 22
  • The vector equation of the plane passing through point A(TS inter 1A vector a ) and parallel to the vectors TS inter addition of vectors 5andTS inter 1A vector c is  TS inter addition of vectors 24
  • The vector equation of the plane passing through the point A(TS inter 1A vector a ), B(TS inter addition of vectors 5 ) and parallel to the vector TS inter 1A vector c is TS inter addition of vectors 25
  • The vector equation of the plane passing through the points A(TS inter 1A vector a ), B(TS inter addition of vectors 5 ) and C( TS inter 1A vector c) isTS inter addition of vectors 26

large bar{r}= (1-t)bar{a} + t bar{b}

5.PRODUCT OF VECTORS

TS inter 1A vectors dotproduct title

Dot product (Scalar product): LetTS inter 1A product of vectors 2 are two vectors. The dot product or direct product of TS inter 1A vector a and TS inter 1A vector b  is denoted byTS inter 1A product of vectors 3and is defined as 

  • IfTS inter 1A vector a = 0, TS inter 1A vector b = 0 ⟹ TS inter 1A product of vectors 3  = 0.
  • If TS inter 1A vector a≠0,TS inter 1A vector b ≠ 0 thenTS inter 1A product of vectors 4
  • The dot product of two vectors is a scalar
  • If TS inter 1A product of vectors 2 are two vectors, then

     TS inter 1A product of vectors 1

  • If θ is the angle between the vectorsTS inter 1A product of vectors 2 then. TS inter 1A product of vectors 4

         ⟹    TS inter 1A product of vectors 5

         ⟹ IfTS inter 1A product of vectors 3   > 0, then θ is an acute angle

         ⟹ If  TS inter 1A product of vectors 3  < 0, then θ is obtuse angle 0

          ⟹ If  TS inter 1A product of vectors 3  = 0, thenTS inter 1A vector a  is perpendicular toTS inter 1A vector b

  • IfTS inter 1A vector a is any vector then  TS inter 1A product of vectors 6

Component and Orthogonal Projection:

LetTS inter 1A vector a=TS inter 1A vector OA,TS inter 1A vector b=TS inter 1A vector OB  be two non-zero vectors. Let the plane passing through B (TS inter 1A vector b ) and perpendicular to TS inter 1A vector aintersectsTS inter 1A vector OA

TS inter 1A product of vectors 7

In M, then TS inter 1A vector OM is called the component of TS inter 1A vector b on TS inter 1A vector a

  • The component (projection) vector of TS inter 1A vector b on TS inter 1A vector a is TS inter 1A product of vectors 8
  • Length of the projection (component) =TS inter 1A product of vectors 9
  • Component ofTS inter 1A vector b perpendicular toTS inter 1A vector a = TS inter multiplication of vectors 1

If TS inter 1A vector i,TS inter 1A vector j, TS inter 1A vector k   form a right-handed system of an orthonormal triad, then 

TS inter 1A product of vectors 10

  • If TS inter 1A product of vectors 11 then TS inter 1A product of vectors 3 = a1b1 + a2b2 + a3b3
  • IfTS inter 1A product of vectors 11  then TS inter 1A product of vectors 12

Parallelogram law:TS inter multiplication of vectors 3

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.

TS inter multiplication of vectors 2

In ∆ABC, the length of the median through vertex A is TS inter multiplication of vectors 4

Vector equation of a plane:TS inter multiplication of vectors 10

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,TS inter multiplication of vectors 5

•The vector equation of the plane passing through point A (TS inter 1A vector a ) and perpendicular to theTS inter multiplication of vectors 6 isTS inter multiplication of vectors 7

•If θ is the angle between the planes TS inter multiplication of vectors 8 then TS inter multiplication of vectors 9


TS inter VECTORS Cross product 1

Cross product (vector product): Let TS inter 1A vector aandTS inter addition of vectors 5 be two non-zero collinear vectors. The cross product of TS inter 1A vector a  and TS inter addition of vectors 5  is denoted by TS inter 1A vector a×TS inter addition of vectors 5  (read as a cross ) and is defined as TS inter VECTORS Cross product 2

TS inter VECTORS Cross product 3

TS inter VECTORS Cross product 4are orthogonal triad then

TS inter VECTORS Cross product 5

• The vectorTS inter 1A vector a ×TS inter addition of vectors 5 is perpendicular to both TS inter 1A vector a and TS inter addition of vectors 5 and also perpendicular to the plane containing themTS inter VECTORS Cross product 6

• The unit vector perpendicular to bothTS inter 1A vector a and TS inter addition of vectors 5  isTS inter VECTORS Cross product 7

• LetTS inter VECTORS Cross product 8 then TS inter VECTORS Cross product 9

• If TS inter 1A vector aand TS inter addition of vectors 5 are two sides of a triangle then the area of the triangle =TS inter VECTORS Cross product 10

• If A (TS inter 1A vector a ), B ()and C (TS inter 1A vector c )are the vertices of a ∆ABC, then its areaTS inter VECTORS Cross product 12

TS inter VECTORS Cross product 11

• The area of the parallelogram whose adjacent sidesTS inter 1A vector a and TS inter addition of vectors 5   is TS inter VECTORS Cross product 13

• The area of the parallelogram whose diagonals TS inter 1A vector a and TS inter addition of vectors 5   is   TS inter VECTORS Cross product 10

• If A (TS inter 1A vector a ), B (TS inter addition of vectors 5 )and C (TS inter 1A vector c )are three points then the perpendicular distance from A to the line passing through B, C is

TS inter VECTORS Cross product 14


TS inter scallar tripple product 1

LetTS inter 1A vector a,TS inter addition of vectors 5andTS inter 1A vector c be three vectors, then (TS inter scallar tripple product 3) . TS inter 1A vector c is called the scalar triple product ofTS inter 1A vector a,TS inter addition of vectors 5andTS inter 1A vector cand it is denoted byTS inter scallar tripple product 2
TS inter scallar tripple product 4
IfTS inter VECTORS Cross product 8TS inter scallar tripple product 21then
TS inter scallar tripple product 5
•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 TS inter scallar tripple product 6
The volume of parallelepiped:
If TS inter 1A vector a,TS inter addition of vectors 5andTS inter 1A vector care edges of a parallelepiped then its volume is TS inter scallar tripple product 7
The volume of parallelepiped:
The volume of Tetrahedron withTS inter 1A vector a,TS inter addition of vectors 5 andTS inter 1A vector c are coterminous edges isTS inter scallar tripple product 8
The volume of Tetrahedron whose vertices are A, B, C and D is  TS inter scallar tripple product 9
Vector equation of a plane:
The vector equation of the plane passing through point A (TS inter 1A vector a) and parallel to the vectorsTS inter addition of vectors 5 and TS inter 1A vector cis TS inter scallar tripple product 10
The vector equation of the plane passing through the points A ( TS inter 1A vector a) and B( TS inter addition of vectors 5) and parallel to the vectorTS inter 1A vector c isTS inter scallar tripple product 11
The vector equation of the plane passing through the points A (TS inter 1A vector a), B( TS inter addition of vectors 5) and C(TS inter 1A vector c ) is TS inter scallar tripple product 12
Skew lines:TS inter scallar tripple product 13
The lines which are neither intersecting nor parallel are called Skew lines

The shortest distance between the Skew lines:
If TS inter scallar tripple product 16 are two skew lines, then the shortest distance between them is TS inter scallar tripple product 14

 
If A, B, C and D are four points, then the shortest distance between the line joining the points AB and CD is TS inter scallar tripple product 15

•The plane passing through the intersection of the planesTS inter scallar tripple product 17 is TS inter scallar tripple product 18
the perpendicular distance from point A (a ̅) to the plane TS inter scallar tripple product 19 is TS inter scallar tripple product 20

TS inter 1A vector tripple product 1

Let TS inter 1A vector a,TS inter addition of vectors 5andTS inter 1A vector c be three vectors, thenTS inter 1A vector tripple product 2 is called the vector triple product ofTS inter 1A vector a,TS inter addition of vectors 5 andTS inter 1A vector c.

TS inter 1A product of four vectors 1

Scalar product of four vectors:

TS inter 1A scalar product of four vectors 1

Vector product of four vectors:

TS inter 1A vector product of four vectors 1


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

TS inrer relation betwee the measurements

Trigonometric Ratios:TS inter trigonometric ratios1

TS inter 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

TS inter trigonometric identities 1

 ∗  cosec2θ − cot2θ = 1

         co sec2θ = 1 + cot2θ

cot2θ = cosec2θ – 1

(cosec θ – cot θ) (cosec θ + cot θ) = 1

TS inter trigonometric identities 2

• sin θ. cosec θ = 1

sec θ. cos θ = 1

tan θ. cot θ = 1

All Silver Tea Cups Rule:

TS inrer trigonometry all silver tea cups

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:

TS inrer trigonometry ratios

TS inter trigonometric ratios values

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.

TS inrer trigonometry periodic fu

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.

TS inrer trigonometry periodic functions1

TS inrer trigonometry COMPOUND AANGLES 1

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) =ts inter ttrriggonomertty compound angles 2

cot (A + B) =ts inter ttrriggonomertty compound angles 3

⋇ cot (A − B) = ts inter ttrriggonomertty compound angles 4

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) =ts inter ttrriggonomertty compound angles 5

⋇ cot (A + B + C) =ts inter ttrriggonomertty compound angles 6

⋇ 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

ts inter ttrriggonomertty compound angles 11

Extreme values of trigonometric functions:

If a, b, c ∈ R such that a2 + b2 ≠ 0, then

Maximum value = ts inter ttrriggonomertty compound angles 12

Minimum value =ts inter ttrriggonomertty compound angles 13

ts inter trigonometry Multiple and submultiple angles 1

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 likets inter trigonometry Multiple and submultiple angles 2are called ‘submultiple angles.

⋇ sin 2A = 2 sin A cos A =ts inter trigonometry Multiple and submultiple angles 5

⋇ cos 2A = cos2 A – sin2 A

                 = 2 cos2 A – 1

                 = 1 – 2sin2 A

                =ts inter trigonometry Multiple and submultiple angles 6

⋇ tan 2A =ts inter trigonometry Multiple and submultiple angles 3

⋇ cot 2A =ts inter trigonometry Multiple and submultiple angles 4

∎ If ts inter trigonometry Multiple and submultiple angles 7  is not an add multiple of ts inter trigonometry Multiple and submultiple angles 8

⋇ sin A = 2 sints inter trigonometry Multiple and submultiple angles 7  costs inter trigonometry Multiple and submultiple angles 7  =ts inter trigonometry Multiple and submultiple angles 10

⋇ cos A = cos2 ts inter trigonometry Multiple and submultiple angles 7  – sin2 ts inter trigonometry Multiple and submultiple angles 7

                 = 2 cos2 ts inter trigonometry Multiple and submultiple angles 7   – 1

                 = 1 – 2sin2 ts inter trigonometry Multiple and submultiple angles 7

                  =ts inter trigonometry Multiple and submultiple angles 9

⋇ tan A =ts inter trigonometry Multiple and submultiple angles 11

⋇ cot A =ts inter trigonometry Multiple and submultiple angles 12

ts inter trigonometry Multiple and submultiple angles 13

⋇ sin3A = 3 sin A −4 sin3 A

⋇ cos 3A = 4 cos3 A – 3 cos A

⋇ tan 3A =ts inter trigonometry Multiple and submultiple angles 14

⋇ cot 3A =ts inter trigonometry Multiple and submultiple angles 15

⋇ tan A + cot A = 2 cosec 2A

⋇ cot A – tan A = 2 cot 2A

ts inter trigonometry Multiple and submultiple angles 16

TS inter tranformations10

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 = 2sinTS inter tranformations1 cosTS inter tranformations2

⋇ sin C −sin D= 2cosTS inter tranformations1  sinTS inter tranformations2

⋇ cos C + cos D = 2 cosTS inter tranformations1cos TS inter tranformations2

⋇ cos C − cos D = − 2sinTS inter tranformations1   sinTS inter tranformations2   

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 orts inter trigonometry Multiple and submultiple angles 8  then

⋇ sin TS inter tranformations4  = cosTS inter tranformations5  ; sinTS inter tranformations6    = cosTS inter tranformations7  ; sinTS inter tranformations8    = cosTS inter tranformations9

 

⋇ cos TS inter tranformations4   = sinTS inter tranformations5 ; cosTS inter tranformations6    = sinTS inter tranformations7 ; cos TS inter tranformations8   = sinTS inter tranformations9

If TS inter tranformations3 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 TS inter trigonometric equations1  

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   TS inter trigonometric equations2

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 TS inter trigonometric equations3  

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)ts inter trigonometry Multiple and submultiple angles 8 , 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] →TS inter inverse trigonometric functions1 is defined by Sin-1 x = θ ⇔ θ∈ TS inter inverse trigonometric functions1 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 →TS inter inverse trigonometric functions2  is defined by Tan-1 x = θ ⇔ θ∈TS inter inverse trigonometric functions2  and tan θ = x

The function Sec-1: [−∞, −1] ∪ [1, ∞] →TS inter inverse trigonometric functions5 is defined by Sin-1 x = θ ⇔ θ∈TS inter inverse trigonometric functions5 and sec θ= x

The function Cosec-1: [−∞, −1] ∪ [1, ∞] →TS inter inverse trigonometric functions6   is defined by cosec-1 x = θ ⇔ θ∈TS inter inverse trigonometric functions6 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

TS inter domain and range of inverse trigonometric functions

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

Tan-1 x = Cot-1(1/x) −π, if x < 0

Sin-1 (−x) = − Sin-1(x) ∀ x ∈ [−1, 1]

Cos-1 (−x) = π − Cos-1(x) ∀ x ∈ [−1, 1]

Tan-1 (−x) = − Tan-1(x) ∀ x ∈ R

Cosec-1 (−x) = − Cose-1(x) ∀ x ∈ (− ∞, − 1] ∪ [1, ∞)

Sec-1 (−x) = π − Sec-1(x) ∀ x ∈ (− ∞, − 1] ∪ [1, ∞)

Cot-1 (−x) =π − Cot-1(x) ∀ x ∈ R 

 (i) If θ∈TS inter inverse trigonometric functions1, then Sin−1(sin θ) = θ and if x ∈ [−1, 1], then sin (Sin−1x) = x

 (ii) If θ∈ [0, π], then Cos−1(cos θ) = θ and if x ∈ [−1, 1], then cos (Cos−1x) = x

 (iii) If θ∈TS inter inverse trigonometric functions2 , then tan−1(tann θ) = θ and if x ∈ R, then tan (Tan−1x) = x

 (iv) If θ∈ (0, π), then Cot−1(cot θ) = θ and if x ∈ R, then cot (Cot−1x) = x

 (v) If θ∈ [0, TS inter inverse trigonometric functions12) ∪ (TS inter inverse trigonometric functions12 , π], then Sec−1(sec θ) = θ and

 if x ∈ (− ∞, − 1] ∪ [1, ∞), then sec (Sec−1x) = x

 (vi) If θ∈ TS inter inverse trigonometric functions6 , then Cosec−1(cosec θ) = θ and

if x ∈ (− ∞, − 1] ∪ [1, ∞), then cosec (Cosec−1x) = x

(i) If θ∈TS inter inverse trigonometric functions1 , then Cos−1(sin θ) = TS inter inverse trigonometric functions7

 (ii) If θ∈ [0, π], then Sin−1(cos θ) =TS inter inverse trigonometric functions7

 (iii) If θ∈TS inter inverse trigonometric functions2 , then Cot−1(tan θ) =TS inter inverse trigonometric functions7

 (iv) If θ∈ (0, π), then Tan−1(cot θ) =TS inter inverse trigonometric functions7

 (v) If θ∈ TS inter inverse trigonometric functions5, then Cosec−1(sec θ) =TS inter inverse trigonometric functions7

 (vi) If θ∈TS inter inverse trigonometric functions6 , then Sec−1(cosec θ) =TS inter inverse trigonometric functions7

  1. Sin1x = Cos( TS inter inverse trigonometric functions8)if 0 ≤ x ≤ 1 and Sin1x =− Cos1 ( TS inter inverse trigonometric functions8) if −1 ≤ x ≤ 0
  2. Sin1x = Tan1TS inter inverse trigonometric functions9 if x ∈ (−1, 1)
  3. Cos1x = Sin1 (TS inter inverse trigonometric functions13) if x ∈ [0, 1] and Cos1x = π − Sin1 (TS inter inverse trigonometric functions13)  if x ∈ [−1, 0]
  1. Tan1x = Sin1TS inter inverse trigonometric functions10 = Cos−1 TS inter inverse trigonometric functions11or x > 0

Cos−1 x + Sin−1x = TS inter inverse trigonometric functions12  ∀ x ∈ [−1, 1]

Tan−1 x + Cot−1x =TS inter inverse trigonometric functions12  ∀ x ∈ R

Sec−1 x + Cosec−1x = TS inter inverse trigonometric functions12 ∀ x ∈ (−∞, −1] ∪ [1, ∞) 

Sin−1 x + Sin−1y = Sin−1(x TS inter inverse trigonometric functions14 + yTS inter inverse trigonometric functions13  ) if 0 ≤x ≤ 1, 0 ≤y ≤ 1and x2 + y2 ≤ 1

                                    =π− Sin−1(x TS inter inverse trigonometric functions14 + y TS inter inverse trigonometric functions13 ) if 0 ≤x ≤ 1, 0 ≤y ≤ 1and x2 + y2 > 1

Cos−1 x + Cos−1y = Cos−1(x y −TS inter inverse trigonometric functions13  TS inter inverse trigonometric functions14 ) if 0 ≤x, y ≤ 1and x2 + y2 ≥ 1

                                    =π− Cos−1(x y −TS inter inverse trigonometric functions13 TS inter inverse trigonometric functions14  ) if 0 ≤x ≤ 1, 0 ≤y ≤ 1and x2 + y2 < 1

Tan−1 x + Tan−1y = Tan−1TS inter inverse trigonometric functions15  if x > 0, y> 0 and xy < 1

                                    =π + Tan−1 TS inter inverse trigonometric functions15 if x > 0, y> 0 and xy > 1

                                    =   Tan−1 TS inter inverse trigonometric functions15if x < 0, y< 0 and xy > 1

                                  = −π + Tan−1TS inter inverse trigonometric functions15  if x < 0, y< 0 and xy < 1

Tan−1 x − Tan−1y = Tan−1 TS inter inverse trigonometric functions16 if x > 0, y> 0 or x < 0, y< 0

2 Sin−1 x = Sin−1 (2x ) if x≤TS inter inverse trigonometric functions17

                       = π− Sin−1 (2x ) if x >TS inter inverse trigonometric functions17

2 Cos−1 x = Cos−1(2x2 – 1) if x ≥TS inter inverse trigonometric functions17

                        =Cos−1(1–2x2) if x <TS inter inverse trigonometric functions17

2 Tan−1 x = Tan−1 TS inter inverse trigonometric functions23 ifTS inter inverse trigonometric functions18 < 1

                         = π + Tan−1 TS inter inverse trigonometric functions23 ifTS inter inverse trigonometric functions18 ≥ 1

                         = Sin−1 TS inter inverse trigonometric functions19 if x ≥ 0

                         = Cos−1 TS inter inverse trigonometric functions20 if x ≥ 0

3Sin−1x = Sin−1(3x – 4x3)

3Cos−1x = Cos−1(4x3 – 3x)

3Tan−1x = tan−1TS inter inverse trigonometric functions21


9.HYPERBOLIC FUNCTIONS

TS inter Hyperbolic functions 1

The function f: R→R defined by f(x) =  ∀ x ∈ R is called the ‘hyperbolic sin’ function. It is denoted by sinh x.

∴ sinh x =TS inter Hyperbolic functions 2

Similarly,

cosh x = TS inter Hyperbolic functions 3 ∀ x ∈ R 

tanh x = TS inter Hyperbolic functions 4 ∀ x ∈ R 

coth x =TS inter Hyperbolic functions 5  ∀ x ∈ R

sech x = TS inter Hyperbolic functions 6  ∀ x ∈ R

cosech x = TS inter Hyperbolic functions 7  ∀ x ∈ R

Identities:

cosh2x – sinh2 x = 1

    cosh2x = 1 + sinh2 x

    sinh2 x = cosh2 x – 1

sech2 x = 1 – tanh2 x

    tanh2 x = 1 – sesh2 x

cosech2 x = coth2 x – 1

     coth2 x = 1 + coth2 x

Addition formulas of hyperbolic functions:

sinh (x + y) = sinh x cosh y + cosh x sinh y

sinh (x − y) = sinh x cosh y − cosh x sinh y

cosh (x + y) = cosh x cosh y + sinh x sinh y  

cosh (x − y) = cosh x cosh y − sinh x sinh y  

tanh (x + y) = TS inter Hyperbolic functions 8

tanh (x − y) = TS inter Hyperbolic functions 9

coth (x + y) =TS inter Hyperbolic functions 10  

sinh 2x = 2 sinh x cosh 2x = TS inter Hyperbolic functions 11

cosh 2x = cosh2x + sinh2 x = 2 cosh2x – 1 = 1 + 2 sinh2x =TS inter Hyperbolic functions 12

tanh 2x =TS inter Hyperbolic functions 13

sinh 3x = 3 sinh x + 4 sinh3x

cosh 3x = 4 cosh3 x – 3 cosh x

tanh 3x = TS inter Hyperbolic functions 14

Inverse hyperbolic functions:

Sinh−1x =TS inter Hyperbolic functions 15  ∀ x ∈ R

Cosh−1x = TS inter Hyperbolic functions 21  ∀ x ∈ (1, ∞)

Tanh−1x = TS inter Hyperbolic functions 16   ∀ TS inter inverse trigonometric functions18< 1

Coth−1x = TS inter Hyperbolic functions 17   ∀ TS inter inverse trigonometric functions18> 1

Sech−1x = TS inter Hyperbolic functions 18   ∀ x ∈ (0, 1]

Cosech−1x = TS inter Hyperbolic functions 19   if x < 0 and x ∈ (−∞, 0)

                         = TS inter Hyperbolic functions 18  if x > 0

TS inter Hyperbolic functions 20


10. PROPERTIES OF TRIANGLES

 In ∆ABC,TS inter Properties of triangles 1

Lengths AB = c; BC = a; AC =b

Area of the tringle is denoted by ∆.

Perimeter of the triangle = 2s = a + b + c

A = ∠CAB; B = ∠ABC; C = ∠BCA.

R is circumradius.

Sine rule:

In ∆ABC,

TS inter Properties of triangles 2

 ⟹ a = 2R sin A; b = 2R sinB; c = 2R sin C

Where R is the circumradius and a, b, c, are lengths of the sides of ∆ABC.

Cosine rule:

In ∆ABC,

a2 = b2 + c2 – 2bc cos A    ⟹cos A = TS inter Properties of triangles 3

b2 = a2 + c2 – 2ac cos B    ⟹ cos B = TS inter Properties of triangles 4

c2 = a2 + b2 – 2ab cos C    ⟹ cos A = TS inter Properties of triangles 5

projection rule:

In ∆ABC,

a = b cos C + c cos B

b = a cos C + c cos A

c = a cos B + b cos A

Tangent rule (Napier’s analogy):

In ∆ABC,

TS inter Properties of triangles 6

Half angle formulae and Area of the triangle:TS inter Properties of triangles 8

In ∆ABC, a, b, and c are sides

TS inter Properties of triangles 7   and area of the triangle TS inter Properties of triangles 13

1.Half angle formulae: –

TS inter Properties of triangles 9

TS inter Properties of triangles 10

TS inter Properties of triangles 11

TS inter Properties of triangles 12

2.Formulae for ∆: – 

∆ = ½ ab sinC= ½ bc sin A=½ ac sin B

  TS inter Properties of triangles 13  where TS inter Properties of triangles 7

   = 2R2sin A sin B sinC

   = r.s

   =TS inter Properties of triangles 14

  =TS inter Properties of triangles 15

In circle and Excircles of a triangle:TS inter Properties of triangles 16

⋇The circle that touches the three sides of an ∆ABC internally is called ‘incircle’. The centre of the incircle is ‘I’ and the radius is ‘r’.

Formulae for ‘r’: –

r = TS inter Properties of triangles 27

 = (s – a) tanTS inter Properties of triangles 17  = (s – b) tanTS inter Properties of triangles 18  = (s – c) tanTS inter Properties of triangles 19

 = 4R sinTS inter Properties of triangles 17sinTS inter Properties of triangles 18 sinTS inter Properties of triangles 19

 =TS inter Properties of triangles 20

The circle that touches the side BC internally and the other two sides AB and AC externally is called the ‘Excircle’ opposite to the angle A. Its centre is I1 and the radius is r1. A TS inter Properties of triangles 21triangle has three ex circles. The remaining circles centre and radius are respectively I2, r2 and I3, r3.

  Formulae for ‘r1’: –

r1 = TS inter Properties of triangles 22

 = s tan TS inter Properties of triangles 17 

= (s – b) cotTS inter Properties of triangles 19  = (s – c) cotTS inter Properties of triangles 18

 = 4R sinTS inter Properties of triangles 17  cosTS inter Properties of triangles 18  cosTS inter Properties of triangles 19

 =TS inter Properties of triangles 23

Formulae for ‘r2’: –

r2 = TS inter Properties of triangles 24

= s tanTS inter Properties of triangles 18   

 = (s – c) cotTS inter Properties of triangles 17  = (s – a) cotTS inter Properties of triangles 19

= 4R cosTS inter Properties of triangles 17  sinTS inter Properties of triangles 18 cosTS inter Properties of triangles 19

 =TS inter Properties of triangles 25 

Formulae for ‘r3’: –

r3 = TS inter Properties of triangles 28

 = s tanTS inter Properties of triangles 19

= (s – a) cot TS inter Properties of triangles 18 = (s – b) cotTS inter Properties of triangles 17

 = 4R cosTS inter Properties of triangles 17  cosTS inter Properties of triangles 18  sinTS inter Properties of triangles 19

 =TS inter Properties of triangles 26


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