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Inter Second Year Accountancy Material

This note is designed by the ‘Basics in Maths’ team. These notes to do help the TS intermediate second-year Accountancy students.
These notes cover all the topics covered in the TS I.P.E second year Accountancy syllabus and concept to help you solve all the types of Inter Accountancy problems asked in the I.P.E and entrance examinations.


1. DEPRECIATION

DEPRECIATION: It is a permanent, continuous and gradual shrinking in the book value of a fixed asset.

DEPLETION: It refers to the physical deterioration by the exhaustion of natural resources. (e.g. ore deposits in mines, oil wells, quarries, etc.,)

FIXED INSTAL MENT METHOD: A method under which the depreciation provided annually on the fixed asset remains the same throughout the life span of the asset.

NONCASH EXPENSE: Expenses that may be operational in nature but that do not affect the payment of cash (e.g. depreciation).

OBSOLESCENCE: diminution in the value of fixed assets due to new inventions, new improvement, change in fashions, change in customer’s tastes and preferences.

RESIDUAL VALUE (scrap value): The realisable value of fixed assets after the expiry of its estimated economic life.

SINKING FUND: A fund created for the repayment of along-term liability or the replacement of an asset at a set date in the future.

DEPRECIATIONFUND: A fund created for the replacement of an asset at a set date in the future.

WRITTEN DOWN VALUE: The value of a fixed asset after depreciation.

WRITTEN DOWN VALUE METHOD: A method under which depreciation is calculated at a fixed percentage on the original cost of the asset in the first year and on written down value in the subsequent year.

DOUBLE ENTRY SYSTEM: The accounting system of recording both the receiving (debit) and giving (credit) aspects of a business transaction is called a double-entry system.

SINGLE ENTRY SYSTEM: It is a mixture of a double-entry, single entry and no entry. It is an incomplete double-entry system.

STATEMENT AFFAIRS: To find out capital, a statement showing various assets and liabilities of a business concern is prepared on a particular date, which is called a statement of affairs. It is similar to a balance sheet.

CAPITAL COMPARISON METHOD: Under this method, the profit/ loss for a particular period is ascertained by comparing the closing capital of the opening capital.

Meaning of depreciation

Depreciation means a fall in the value or quality of an asset. The word depreciation is derived from the Latin word “depretium”. ‘De’ means decline and ‘pretium’ means price.  it is the decline in the price or value of the fixed assets. Depreciation is described as a permanent, continuing and gradual shrinkage in the value of fixed assets. It is based on the cost of the asset consumed in a business and not on its market value.

The depreciation is that part of the original cost of a fixed asset that is consumed during its period of use by the business. Thus, depreciation is the loss of value of a fixed asset arising from use, effluxion of time or obsolescence. Depreciation sometimes restricted to fixed tangible assets but in the UK, it also usually includes the amortization of intangible assets. However, the tangible fixed assets lose their value over a period of time as they are used in the business operation and they do not last forever. If any amount is received on the sale of the fixed asset is deducted from the cost of it. Then the remaining value of the fixed asset is said to have” depreciated value” by that amount over its period of usefulness to the business.

For example, if a motor vehicle was bought for Rs 100000 and it was sold 5 years later for Rs 10000. It means the motor van value has depreciated over the period of its use by Rs 90000. Therefore, the depreciation for each year will be estimated while the fixed asset continues to be useful and be charged(debited) to the profit and loss account. Since depreciation is treated as a business expense and charged to the profit and loss account, it reduces the net profit of the business and the value of the asset.

DEFINITIONS OF DEPRECIATION

According to the American institute of certified public accountants (AICPA-USA),” depreciation accounting is a system of accounting while aims to distribute the cost or other basic value of a tangible capital asset, less salvage value (if any) over the estimated useful life of the asset (which may be a group of assets) in a systematic and rational manner .it is a process of allocation, not of valuation.”

SIGNIFICATION OR NEED FOR PROVIDING DEPRECIATION
  1. To ascertain true results of business operations: – As depreciation is treated as expenditure, it must be charged to profit and loss account against income to assets the real profit /loss of the business.
  2. To show the fixed asset at their original worth in the balance sheet: -If depreciation is not provided, the value of the asset shown in the balance sheet is not correct. Hence, the real value of the asset must be shown in the balance sheet after deducting depreciation from its book value.

For example, machinery is purchased for ₹10,000, and its estimated working life is 10 years.

Then, the depreciation to be charged to the asset every year will be 10,000/10 years =1000. Though the value of the asset is depreciated by ₹1000 every year, if it is shown as ₹ 10,000 in the balance sheet, it will not reflect the real financial position of the business.

  1. To provide funds for the replacement of asset: – The amount charged as depreciation is nothing but setting aside a profit and this amount accumulates and will be available for replacement of the asset at the end of its useful life.
  2. To ascertain the true cost of production: -for calculating the cost of production, it is necessary to charge depreciation as an item of cost of production. Otherwise, it would not present true cost of production.
  3. To follow the legal provisions: – in the case of a joint-stock company, it is compulsory to provide depreciation on fixed assets and without providing for depreciation on fixed assets and without providing for depreciation, dividends cannot be declared to the shareholders.

CAUSES OF DEPRECIATION

1. Wear and tear: – The decrease in the value of the fixed asset due to the constant use is said to be wear and tear. It is a deterioration in an asset value, arising from its use in business operations for earnings income.

Example: Machinery, Furniture and fixtures are depreciated by wear and tear.

2 . Depletion: – It is the loss of natural resources mineral wealth due to the constant extraction of raw material from mines, quarries, oil wells etc.

3. Accidents: – The assets lose their value if it is met with an accident. An accident means the breakdown of an asset which decreases the value of assets. It is not a gradual decrease but causes a permanent loss in asset value.

4. Obsolescence: – it is the processing out of date. due to new inventions, technological improvement, availability of the better type of asset, or change in market demand the existing asset becomes obsolete or outmoded. This results in a decrease in the value of the asset.

5. Fluctuations: – the asset value may get decreased due to the fluctuations/charges in the market conditions.


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TS Inter second year Maths 2B Concept

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.


1. CIRCLES

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

circle for second year maths

∗ 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.

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            equation of the circle with centre origin and radius r

(x – h)2 + (y – k)2 = r2                 

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

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 TS inter circle radius 1

∗ 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 onlength of the intercepts made by circle

  • x -axis is TS inter circle length of intercept by x axis  if g2 – c > 0
  • y -axis is TS inter circle length of intercept by y axisif 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

equation of the circle passing through end points of diameter

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

 

∗ Let A, B be any two points on a circle then,secantline and chord of the circle

  • 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.

angle sustended by chord at any point

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

perpendicular bisector of the chord assing through centre

∗ The angle in a semicircle is 900.

angle in a semicircle

 

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

equatin of the circle passing through three points

Where ci = − (x2 + y2) and i = 1,2,3

∗ centre of the circle is

centre of the circle lassing through 3 points

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: ts inter 2B position of point

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.

power of point

  • 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. thents inter 2B position of the point 2

  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.ts inter 2B secant and tangent to a circle

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:ts inter 2B length of 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 TS inter circle length of tangent

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 ts inter 2B condition for a line to be a tangent 1
  • Touch the circle x2 + y 2 = rif ts inter 2B condition for a line to be a tangent 2
  • Does not touch the circle x2 + y 2 = r2 in two distinct points if ts inter 2B condition for a line to be a tangent 3

Note:

  1. For all real values of m, the straight line ts inter 2B condition for a line to be a tangent 4 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 = ts inter 2B condition for a line to be a tangent 5 .
  3. The equation of a tangent to the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 having the slope m is  ts inter 2B condition for a line to be a tangent 6where 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

ts inter 2B equation of chord in parametric form1

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

ts inter 2B equation of chord in parametric form2

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

ts inter 2B equation of tangent in parametric form

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

x cosθ + y sinθ = r.

Normal:ts inter 2B normal to the circle

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.ts inter 2B angents from external point to the circle

 

 

 

 

Chord of contact: –

If the tangents are drawn through P (x1, y1)ts inter 2B chord of contact diagram

 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 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: –ts inter 2B pole and polar of circle

Let S = 0 be a circle and P be any point if any line drawn through the point Pin the plane other than the centre of S = 0.then the points of intersection meets 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 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 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 are 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:ts inter maths 2B common tangent to the circle

⟹ A straight line L = 0 is said to be 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 be 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.ts inter maths 2B does not meet the circles

⟹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 otherts inter maths 2B the circles touch each other externally

   ⟹ 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. ts inter maths 2B the circles intersects at distinct points1

⟹ two direct common tangentsts inter maths 2B the circles intersects at distinct points

    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. ts inter maths 2B the circles touch internally1ts inter maths 2B the circles touch internally

⟹ only one common tangent

internal centre of similitude does not exist.

 

5. ts inter maths 2B one circle lies inside the other circle1     ts inter maths 2B one circle liec inside the other circle

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:ts inter 2B angle between two circles diagram

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.

ts inter 2B angle between two circles when d r1 r2 given

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

ts inter 2B angle between two circles when gg' ff' cc' known

⟹ 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:ts inter 2B radical axis diagram

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.

ts inter 2B radical axes of three circles are 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 SECTIONSts inter 2B conic sections diagram

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.TS inter 2B parabola diagram

  • 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

ts inter 2B equation of the parabola 1

  • 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 ts inter 2B equation of the parabola 2

∴ 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 = 4axTS inter 2B parabola diagram1

    focus: (a, 0)

equation of directrix: x + a = 0

axis of parabola: y = 0

    vertex: (0. 0)

2. y2 = −4axTS inter 2B parabola diagram2

equation of directrix: x − a = 0                                 

  focus: (−a, 0)

axis of parabola: y = 0

vertex: (0. 0)

3. x2 = 4ayTS inter 2B parabola diagram3

    focus: (0, a)

equation of directrix: y + a = 0

    axis of parabola: x = 0

vertex: (0. 0)

4. x2 = −4axTS inter 2B parabola diagram4

    focus: (0, −a)

equation of directrix: y − a = 0

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

5. (y – k) 2 = 4a (x – h)TS inter 2B parabola diagram5

    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)TS inter 2B parabola diagram6

  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)TS inter 2B parabola diagram7

    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) TS inter 2B parabola diagram8

    focus: (h, k – a)

equation of directrix: y – k – a = 0

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

9. TS inter 2B parabola equation focus is in quadrantTS inter 2B parabola diagram9

    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.TS inter 2B ellipse diagram

∎ Equation of Ellipse in standard form is TS inter 2B standard form of ellipse equation

⇒b2 = a2 (1 – e2) ⇒ e2 – 1 = -b2/a2

TS inter 2B standard form of ellipse equation2

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.TS inter 2B ellipse form equation1TS inter 2B ellipse diagram1

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: TS inter 2B eccentricity of ellipse form equation1

2.TS inter 2B ellipse form equation2     TS inter 2B ellipse diagram2

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:TS inter 2B eccentricity of ellipse form equation2

Centre not at the origin

3.TS inter 2B ellipse form equation3TS inter 2B ellipse diagram3

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:TS inter 2B eccentricity of ellipse form equation3

4.TS inter 2B eccentricity of ellipse form equation4TS inter 2B ellipse diagram4

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: TS inter 2B eccentricity of ellipse form equation4

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 TS inter 2B eccentricity of ellipse form equation5

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  TS inter 2B eccentricity of ellipse form equation6 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  TS inter 2B eccentricity of ellipse form equation5 through S is x = ae and through S’ is x = -ae.

4. The equation of the Latus rectum of the Ellipse  TS inter 2B eccentricity of ellipse form equation6 through S is y = be and through S’ is y = -be.

5. If P (x, y) is any point on the Ellipse TS inter 2B ellipse equation without condition  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 TS inter 2B eccentricity of ellipse form equation5 is x2 + y2 = a2.

Parametric equations:  The parametric equations of the Ellipse TS inter 2B ellipse equation without condition are x = a cosθ and y = b sinθ.

Notation:

TS inter 2B ellipse notation

Equation of Tangent and Normal

 The equation of any tangent to the Ellipse can be written as TS inter 2B quation of the tangent to ellipse

 The condition for a straight-line y = mx + c to be a tangent to the Ellipse    TS inter 2B ellipse equation without condition   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 TS inter2B equation of the normal to the ellipse

∎ Equation of the tangent at P(θ) on the Ellipse   TS inter2B equation of the tangentto the ellipse2 

∎ Equation of the normal at P(θ) on the Ellipse S = 0 isTS inter2B equation of the normal to the ellipse2

∎ 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.TS inter 2B Hyperbola diagram
Standard form of Hyperbola:

Equation of Hyperbola in standard form is TS inter 2B equation of standard form of Hyperbola

 Centre: C (0, 0)

Foci: (± ae, 0)

Directrix: x = ± a/e.

Ecdntricity: TS inter 2B eccentricity of Hyperbola

Notation:    TS inter 2B Hyperbola 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 sqrt 2 .

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 TS inter 2B equation of Hyperbola2

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  TS inter 2B equation of conjugate Hyperbola

 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:TS inter 2B various forms of Hyperbola diagram

Let TS inter 2B equation of Hyperbola2 and      TS inter 2B equation of conjugate Hyperbola

1.Hyperbola TS inter 2B equation of Hyperbola2

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:TS inter 2B eccentricity of Hyperbola

2.Conjugate Hyperbola TS inter 2B equation of 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:TS inter 2B eccentricity of conjugate Hyperbola

Centre not at the origin:

3.Hyperbola TS inter 2B equation of standard form of Hyperbola1

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:TS inter 2B eccentricity of Hyperbola

4.Hyperbola TS inter 2B equation of conjugate Hyperbola2

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: TS inter 2B eccentricity of conjugate Hyperbola

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 TS inter 2B equation of tangent to the Hyperbola

∎ The equation of the Normal at P (x1, y1) to the hyperbola S = 0 is TS inter 2B equation of normal to the Hyperbola

∎ Equation of the normal at P(θ) on the Hyperbola S = 0 is TS inter 2B equation of normal to the Hyperbola2

∎ 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    TS inter 2B asymptotes of Hyperbola  and the joint equation of asymptotes is TS inter 2B asymptotes of Hyperbola2


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.

⟹  TS inte 2B integration formula 1

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

Standard forms:

TS inte 2B integration 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. ∫ cossec2 (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

TS inte 2B integration formula 3

TS inte 2B integration formula 4

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:

TS inte 2B integration of inverse trigonometric functions

Evaluation of integrals form  : TS inte 2B evaluation of integration form1

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 TS inte 2B evaluation of integration form2

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 TS inte 2B evaluation of integration form3

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

Evaluation of integrals form  TS inte 2B evaluation of integration form4

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   TS inte 2B evaluation of integration form5

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

Evaluation of integrals form    TS inte 2B evaluation of integration form6

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:

TS inte 2B integration partial fractions 1

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:

TS inte 2B integration partial fractions 2

∎ 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:

TS inte 2B integration reduction formulae


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 TS inter equation of Riemann sum 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 TS inter definite integration equation 1 for any partition P of [a, b] with ∥P∥ < δ irrespective of choice of ti in [xi-1, xi]. Such an A, if exists, is unique and is denoted by TS inter definite integration 2, it is read as the definite integral of f from a to b. an a is called lower limit and b is called upper limit. The function f inTS inter definite integration 2  is called ‘integrand’.    

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

∎ If f is continuous on [0, p] where p is a positive integer then TS inter definite integration 3

The fundamental theorem of the 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 TS inter definite integration 1           

we write TS inter definite integration 2          

properties of definite integrals:

TS inter properties of definite integration

∎ 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

TS inter definite integration 3

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 TS inter definite integration 4                                                      

∎ 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 TS inter definite integration 5

∎ 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 TS inter definite integration 6

∎ 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 TS inter definite integration 7

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

  • If f (2a – x) = f(x) for all x in [a, 2a] hen f is integrable on [0, 2a] and TS inter definite integration 8
  • If f (2a – x) = – f(x) for all x in [a, 2a] hen f is integrable on [0, 2a] and TS inter definite integration 9

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

Integration by parts:

TS inter definite integration by parts

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

TS inter definite integration 10

Reduction formulae:

∎ Let n≥2 be an integer, then TS inter definite integration reduction 1

∎ Let m and n be positive integers, then

TS inter definite integration reduction 2

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 =TS inter definite integration 2TS inter definite integration area of curves1

(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 = TS inter definite integration area of curves3TS inter definite integration area of curves2

From (i) and (ii) A = TS inter definite integration area of curves4

(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 TS inter definite integration area of curves5

TS inter definite integration area of curves6

Area of region =A = TS inter definite integration area of curves7

(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 TS inter definite integration area of curves9the lines x = a, x = b is given by

TS inter definite integration area of curves8

 

(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 TS inter definite integration 12(c). area of the region bounded by y = f (x), y = g(x), and the lines x = a, x = b is given by TS inter definite integration 11

 

(vi) Let f: [a, b] → R and g: [a, b] → R be continuous functions. Suppose that, there exist points x1, x2 ∈ (a, b) TS inter definite integration 16with 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 byTS inter definite integration 13 and if f(x) ≤ g(x) ∀ x ∈ (x1, x2). ThenTS inter definite integration 14In either case, area is TS inter definite integration 15.


8.DIFFERENTIAL EQUATIONS

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

If a differential equation contains only one independent variable, then it is called ‘ordinary differential equation and if it contains more than one independent variable, then it is called ‘ 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 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 TS inter differential equation 1

 

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 above equation, a differential equation of the form  TS inter differential equation 2

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

m = dy/ dx ⟹ TS inter differential equation 3

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 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 unction: – A function f (x, y) of two variable 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 side of the above are meaningful.

Homogeneous Differential Equation: – A differential equation of the formTS inter differential equation 4 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 TS inter differential equation 4  …… (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 TS inter differential equation 5 ……. (2)

put y = vx. Then TS inter differential equation 6 ……. (3)

from (2) and (3)

TS inter differential equation 7

This can be solved by the variable separable method.

3.Non-Homogeneous Differential Equations:

The differential equation of the formTS inter differential equation 8 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 TS inter differential equation 8   becomes TS inter differential equation 9

⇒ (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 required solution.

Case(ii): –

Suppose that TS inter differential equation 10 Then TS inter differential equation 8 becomesTS inter differential equation 11

Put ax + by = v, then

TS inter differential equation 12

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 TS inter differential equation 13 . HenceTS inter differential equation 8…..(i)    becomes

TS inter differential equation 14

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 TS inter differential equation 15

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 formTS inter differential equation 16   = 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 first order isTS inter differential equation 17

Multiplying both sides of (1) byTS inter differential equation 18, we get

TS inter differential equation 19


 

maths ii a concept feature image

TS Inter Second Year Maths 2A concept

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 =  + 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

conjugate complex numbers properties

Modulus and amplitude of complex number:

Modulus: – If z = x + iy, then the non-negative real numberTS inter complex numbers 1 is called modulus of z and it is denoted by or ‘r’.

Amplitude: – The complex number z = x + i y 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 = TS inter complex numbers 1  and TS inter complex numbers 2 = 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 contains all complex numbers is called 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.

rgand plane

The square root of a complex number:

square root of complex numbers

 

 


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 orts inter n th root of complex number

let z = r (cosθ + i sin θ) ≠ 0 and n be a positive integer. For k∈ {0, 1, 2, 3…, (n – 1)}   let ts inter n th root of complex number2 . 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

    ts inter n th root of unity

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

cube roots of unity

 

 

 


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  TS inter 2A quadratic equation formula

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

TS inter 2A relaton between zeros and coefficients

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):

TS inter 2A synthetic division method 1

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:

TS inter 2A synthetic division method 2

  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 TS inter 2A bar of f of alpha

∎ 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 TS inter 2A sqrt of b irrational. Then TS inter 2A a plus sqrt of b   is a root of f(x) = 0 if and only if another root is a inter 2A a minus sqrt of b.

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 TS inter 2A f of sqrt of x

Reciprocal equation:

Let f(x) be a polynomial of degree n > 0 is said to be reciprocal if f(0) ≠ 0 and TS inter 2A reciprocal equation ∀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  TS inter 2A permutations formula 1

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 TS inter 2A combinations forula8

∎ 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 TS inter 2A combinations forula9

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)

TS inter 2A combinations forula1

∎ 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 TS inter 2A combinations forula2

∎ 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 TS inter 2A combinations forula3

∎ The no. of ways of dividing 2n dissimilar things into two equal groups containing ‘n’ things in each case is TS inter 2A combinations forula4

∎ The no. of ways of dividing ‘mn’ dissimilar things into m equal groups containing ‘n’ things in each case is TS inter 2A combinations forula5

∎ The no. of ways of distributing ‘mn’ dissimilar things equally among  m  persons is TS inter 2A combinations forula6

• 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 TS inter 2A combinations forula7 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:

TS inter 2A 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 = TS inter 2A no. of terms in a trinomial expansions

 Middle terms in (x + a) n:

 ∎ if n is even then TS inter 2A middle terms in a binomial expansion1 term is the middle term.

∎ if n is odd then TS inter 2A middle terms in a binomial expansion2 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, TS inter 2A numerically greatest term in a binomial theorem

⟹ if TS inter 2A numerically greatest term in a binomial theorem1 = p, where p is a positive integer then, pth and (p + 1)th are the numerically greatest terms.

⟹ if TS inter 2A numerically greatest term in a binomial theorem1 = 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) TS inter 2A largest binomial coefficient1 if n is even integer.

(ii) TS inter 2A largest binomial coefficient2 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

TS inter 2A binomial theorem for rational index1

Rational Index: –

TS inter 2A binomial theorem for rational index2

TS inter 2A binomial theorem for rational index3


7. PARTIAL FRACTIONS

Rational fraction:  If f(x) and g(x) are two polynomials and g(x) is a non-zero polynomial, thenTS inter 2A Partial Fractions 2 is called a rational fraction or polynomial fraction or simply a fraction.

Ex:TS inter 2A Partial Fractions 3

Proper and Improper Fractions: A rational fractionTS inter 2A Partial Fractions 2 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: TS inter 2A Partial Fractions 4 is a proper fraction andTS inter 2A Partial Fractions 5 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 TS inter 2A Partial Fractions 1 when g(x) contains linear factors:

Rule – 1:   Let TS inter 2A Partial Fractions 2  be a proper fraction. To each non- repeated factor of g(x), there will be a partial fraction of the formTS inter 2A Partial Fractions 6 where A is a non-zero real number, to be determined.

Rule – 2:   Let TS inter 2A Partial Fractions 2  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 formTS inter 2A Partial Fractions 7 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 TS inter 2A Partial Fractions 1 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 formTS inter 2A Partial Fractions 8 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 formTS inter 2A Partial Fractions 9 where A1, A2, …, An and B1, B2, …, Bn are real numbers, to be determined.

Partial Fraction of TS inter 2A Partial Fractions 1when TS inter 2A Partial Fractions 1  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).

TS inter 2A Partial Fractions 2  can be expressed as k + TS inter 2A Partial Fractions 10  whereTS inter 2A Partial Fractions 10 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 algorithmTS inter 2A Partial Fractions 2 can be expressed as q(x) + TS inter 2A Partial Fractions 10   where q(x) is a non-zero polynomial and TS inter 2A Partial Fractions 10 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.

TS inter 2A Measures of dispersion 1

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 (TS inter 2A Measures of dispersion ) 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 =TS inter 2A Measures of dispersion 4

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.,TS inter 2A Measures of dispersion 3 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 =TS inter 2A Measures of dispersion 4

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 = TS inter 2A Measures of dspersion 5

M.D(mean) =TS inter 2A Measures of dspersion 6

M.D(median) =TS inter 2A Measures of dspersion 7

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 = TS inter 2A Measures of dspersion 5

M.D(mean) =TS inter 2A Measures of dspersion 6

Median = TS inter 2A Measures of dispersion 10

M.D(median) =TS inter 2A Measures of dspersion 7

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 = TS inter 2A Measures of dispersion 8

Where  TS inter 2A Measures of dispersion 9

Variance and Standard Deviation of un grouped data:

If x1, x2, …, xn are n observations and is their mean, thenTS inter 2A Measures of dispersion 13

We have the following cases:

Case(i): if TS inter 2A Measures of dispersion 14= 0, then eachTS inter 2A Measures of dspersion 18= 0 which implies all observations are equal to the mean  and there is no dispersion.

Case(ii): if TS inter 2A Measures of dispersion 14 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 TS inter 2A Measures of dispersion 14 is larger, then it indicates the higher degree of dispersion of the observations from the mean .

Variance = σ2= TS inter 2A Measures of dispersion 15

Standard deviation

TS inter 2A Measures of dispersion 16

∎ The coefficient of variation of a distribution (C.V.) =TS inter 2A Measures of dispersion 17


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) =TS inter 2A Probability 1

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 TS inter 2A Probability 2  is called the nth relative frequency of the event E.

Let r1, r2, …, rn be the sequence. If rn tends to a definite limit,TS inter 2A Probability 3 , l is defined to be the probability of the event E and we write P (E) = TS inter 2A Probability 3

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)*: ifTS inter 2A Probability 4 is a sequence of pairwise mutually exclusive events, then TS inter 2A Probability 5
  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)TS inter 2A Probability 6

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) = TS inter 2A Probability 7

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

TS inter 2A Probability 8


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,

TS inter 2A Probability 9

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 TS inter 2A Probability 10 we can prove that for every non-negative integer k,TS inter 2A Probability 11


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