MATHEMATICS

Prepare By: RANGIN KHAN

Web:www.ranginissb.blogspot.com

Also Pdf Available

GUIDELINES

Quadratic Equation. An equation of second degree is called a quadratic equation. The equation ax2+bx+c=0, where a, b, c ε R and ≠0 is called the standard form of Quadratic equation in one variable.

Solution by Factorization. The method of solving quadratic equations by using factorization is based upon the fact that

If a b = 0, then either a = 0 or b = 0 or both a =0 and b = 0

Solution by Quadratic Formula. The solution of the quadratic equation ax2+bx+c=0 is given by

-b + √ b2 -4ac

2a

Sum and Product of roots of equation ax2+bx+c=0. If α, β are the roots of quadratic equation ax2+bx+c = 0, then

(A) α + β = - b (B) αβ = c

a a

(C) α - β = √ b2 -4ac

a

Nature of the roots of equation ax2+bx+c=0. if a ax2+bx+c=0 where a, b, c ε Q and

a ≠ 0, then b2–4ac is called its discriminant, and if

b2–4ac = 0, then the roots are real and equal.

b2–4ac > 0 and a perfect square, then the roots are rational and unequal.

b2–4ac > 0 and not a perfect square, then the roots are irrational and if one root is p + √q, then the other will be p=√q and vice-versa.

b2–4ac <0, then the roots are complex and if one root is p + iq, then the other will be p-iq and vice-versa.

How to write Quadratic Equation when roots are known. If α, β are the roots of quadratic equation, then quadratic equation is

x2 – (α+β) + αβ = 0 {i.e. x2 – (sum of the roots)x + (product of roots) = 0}

Important Formulae.: Some facts of multiplication are very important. They are known as formulae and should be learnt by heart. They can be verified by actual multiplication.

(a + b)2 = a2 + 2ab + b2

(a b)2 = a2 2ab + b2

(a + b)2 + (a b)2 = 2(a2 + b2)

(a + b)2 (a b)2 = 4ab

(a + b +c)2 = a2 + b2 +c2 + 2ab + 2bc + 2ac

= a2 + b2 +c2 + 2(ab + bc + ac)

a2 b2 = (a +b)(a b)

(a + b)3 = a3 + b3 + 3ab(a + b) = a3 + b3 + 3a2b + 3ab2

(a b)3 = a3 – b3 – 3ab(a – b) = a3 – b3 – 3a2b + 3ab2

a3 + b3 = (a + b)(a2 ab + b2)

a3 b3 = (a b)(a2 + ab + b2)

a3 + b3 + c3 -3abc = (a + b + c) (a2 + b2 + c2 ab bc ac)

EXERCISE

Q.1 Prove that sum of cube roots of unit is zero and their product is 1.

Q. 2 Find the cube roots of -1 and hence find their sum and product.

Q. 3 Find the cube roots of -64 & 125.

Q. 4 Prove that 2 + w2 = 3 .

2 + w

7 7

Q. 5 Prove that 1+ √ -3 1- √-3

2 2

Q. 5 Solve the following equation:

(x – 1)2 -3(x+1) = 0

x x

Q. 7 Solve y + y +1 = 13

y+1 y 6

Q. 8 Solve 1 + x+1 = 2 1

x+1 2

Q. 9 Solve 3x2–2x-√3x2–2x+4 = 16

Q. 10 Solve the given equation: √x+8 √x+3 = √12x +13

Q. 11 Solve by using quadratic formula. 4.22x+1 – 9.2x + 1 = 0

Q. 12 Solve (3x2 – 5x + 4)2 -5(3x2 – 5x) = 26

Q. 13 If α, β are roots of px2+qx+r=0, p≠0 form an equation whose roots are -1 & -1

α3 β3

Q. 14 If α, β are roots of the equation px2+qx+r=0, prove that

α β q

β α p

Q. 15 If α, β are roots of ax2+ bx+ c=0, a=0 find the equation whose roots are -1 & -1

α3 β3

Q. 16 If α, β are roots of 2x2+ 3x+ 4=0, a=0 find the equation whose roots are α & β

β α

Q. 17 If α, β are roots of x2 – px+q=0, find the value of

(i) α2 + β2 (ii) α3 + β3

Q. 18 Find the equation whose roots are reciprocals of the roots of 2x212x+16=0

Q. 19 Area of a square is numerically less than twice its diagonal by 2. Find its perimeter.

Q. 20 The sum of six times a certain positive integer and its square is 91. Find the integer.

Q. 21 The product of a negative number less one and three times this number less two is 14. Determine the number.

Q. 22 The diagonal of a rectangle is 26m and its perimeter is 68m. Find out its dimensions.

Q. 23 Find the solution set of the equation 3 + 2 = 2 and x + y = 5

x y

Q. 24 The sum of the squares of two numbers is 925 and the difference of their squares is 875. What are the numbers?

GUIDELINES

Sequence. A sequence is a function whose domain is the set of natural numbers.

Series. Sum of the terms of a sequence is known as series.

Arithmetic Progression (A.P.). A sequence every terms of which after the first term,is obtained from the previous term by adding a fixed number is called an Arithmetic progression. If the sequence a1, a2, a3, ., an is an A.P, then

an – an-1 = d, a constant ` n ε N. (d is called the common difference of A.P)

If a is the first term and d is the common difference of an A.P. then its nth term denoted by an is given by an = a + (n-1)d.

sum upto n terms is denoted by Sn is given by Sn = n [2a + (n-1)d]

2

Arithmetic Mean (A.M.). If A is the arithmetic mean between a and b, then A = a + b

2

Geometric Progression (G.P.). A sequence every term of which after the first term, is obtained from the previous term by multiplying it by a fixed number is called a Geometric Progression.

If the sequence a1, a2, a3, ., an is a G.P., then an = r, a constant n ε N,

an-1

r is called the common ratio of G.P. If a is the first term and r is the common ration of a G.P. then its nth term denoted by an is given by an = a.rn-1

Sum upto n terms is denoted by Sn is given by

Sn = a(1 – rn) , r ≠1

1 – r

If r < 1, the sum to the infinite G.P. = a .

1 - r

Geometric Mean (A.M.). If G is the geometric mean between a and b, then

G = +√ab, a> 0, b >0

Harmonic Progression (G.P.). Reciprocal of the terms of an arithmetic sequence is known as Harmonic sequence. (i.e. if the sequence a1, a2, a3, ., an is an A.P., then

1 , 1 , 1 , .., 1 is a H.P.

a1 a2 a3 an

Harmonic Mean (A.M.). If H is the Harmonic mean between a and b, then H = 2ab

a + b

EXERCISE

Q. 1 Which term of the sequence -3, 3, 9, 15, . is 75?

Q. 2 Which term of the following sequence is 125?

5, 10, 15, 20, 25,

Q. 3 How many terms are there in the A.P 2 , 1 , 1 , -17

3 2 3 6

Q. 4 If pth term of an arithmetic progression is q and the qth term is P. Find the (p+q) term.

Q. 5 If b,c,p,q,r are in A P, prove that b+r=r+q=2p.

Q. 6 Insert three arithmetic means between 2 & 11.

Q. 7 Find the arithmetic progression, whose 8th term is 23 and 102nd term is 305.

Q. 8 The sum of three numbers in an A.P is 30 and the product of the extremes is 96. Find the number.

Q. 9 The sum of the first n terms of two A.Ps are in the ration 3n+31: 5n 3. Show that their 9th terms are equal.

Q. 10 The path of an A.P is q and qth term is p. Find the (p + q)th term.

Q. 11 Find the 9th term of 18, -12, 8, .

Q. 12 For what value of n is an+1+bn+1 the Geometric mean between a & b?

an + bn

Q. 13 Find the sum to n terms, the series 2+22+222+ to n terms.

Q. 14 Sum to 100 terms the series, 0.9 + 0.09 + 0.009 + 0.0009 + ..

Q. 15 Find the sum of the following series upto n terms.

1 + 3.2 + 5.22 + 7.23 + is n term.

Q. 16 A boy has 231 marbles. He arranges them in rows so that each row contains one marble less than the preceding. The last row consists of one marble only, which forms the vertex of a triangle. How many marbles are there in the base of the triangle?

Q. 17 Find the three numbers in G.P. whose sum is 19 and whose product is 216.

Q. 18 Find the first term of a G.P whose second term is 2 and the sum to infinity is 8.

Q. 19 The sum of first six terms of a G. P. is 9 times the sum of the first 3 terms. Find the common ration and first term.

GUIDELINES

Binomial Theorem for a positive index. If n is a positive integer and a, b are real numbers, then

(a + b)n = nc0 an + nc1 an-1 b + nc2 an-2 b2 + + ncr an-r br + ..+ ncn-1 abn-1 + ncn bn where nc0 , nc1, nc2, .., ncr, , ncn are called Binomial coefficients.

General Term. Tr = ncr an-1br Coefficients of terms equidistant from the beginning and end are equal.

Binomial Theorem for any index.

(1 + x)n = 1 + nx + n(n-1) x2+ n(n-1)(n-2) x3+ . + n(n-1)(n-2)(n-r+1) xr + .....to ∞.

2! 3! R!

General Term. Tr = n(n-1)(n-2)……(n-r+1) xr

r!

EXERCISE

Q. 1 Evaluate: ( 1 + 2 √x)5 +(1 - 2√x)5

Q. 2 Find the middle term of (x - 1 )2n+1

x

Q. 3 Find sixth term in the expansion of 2x _ 3 10

3 2x

x4 _ 1 15

Q. 4 Write in the simplified form the term involving x-17 in the expansion of x5

Q. 5 Find the term independent of x in of 2x + 1 9

3x2

Q. 6 Find the coefficient of x6 in (a3 + 3bx2)-6.

Q. 7 Find the coefficient of the term involving x3 in the expansion of (2x½)7

Q. 8 Evaluate 5√31 to five places of decimals.

Q. 9 Expand (1.02)7 by binomial theorem.

Q. 10 Identify the following series as a binomial expansion and find its sum:

1 + 2. 1 + 2.5 . 1 + 2.5.8 . 1 + ……..

32 1.2 34 1.2.3 36

Q. 11 Shown that the middle term is the expansion of (1+x)2n is

1.3.5………….(2n – 1) 2nxn

n!

GUIDELINES

Standard Results

1. ∫xn dx = xn+1 + c

n+1

2. ∫1 dx = log IxI +c

x

3. ∫ 1 dx = 1 log Iax+bI +c

ax+b a

4. ∫(ax + b)n dx = (ax+b)n+1 + c

(n+1) a

5. ∫ex dx = ex + c

6. ∫eax dx = eax + c

a

7. ∫ax dx = ax + c

ln a

8. ∫abx dx = abx + c

b ln a

9. ∫ sin x dx = - cos x +c

10. ∫ sin (ax+b) dx = - cos(ax+b) +c

a

11. ∫ cos x dx = sin x +c

12. ∫ cos (ax+b) dx = sin(ax+b) +c

a

13. ∫ sec2x dx = tan x +c

14. ∫ sec2 (ax+b) dx = tan(ax+b) +c

a

15. ∫ cosec2x dx = -cot x +c

16. ∫ cosec2 (ax+b) dx = - cot(ax+b) +c

a

17. ∫ sec x tan x dx = sec x +c

18. ∫ sec(ax+b) tan(ax+b) dx = sec(ax+b) +c

a

19. ∫ cosec x cot x dx = -cosec x +c

20. ∫ cosec(ax+b) cot(ax+b) dx = - cosec(ax+b) +c

a

21. ∫ sec x dx = log Isec x + tan xI + c

22. ∫ sec (ax+b) dx = 1 log Isec (ax+b) + tan (ax+b)I + c

a

23. ∫ cosec x dx = log Icosec x – cot xI + c

24. ∫ cosec (ax+b) dx = 1 log Icosec (ax+b) -cot (ax+b)I + c

a

25. ∫ tab x dx = - log Icos xI + c

26. ∫ tan (ax+b) dx = - 1 log Icos (ax+b)I + c

a

27. ∫ cot x dx = log Isin xI + c

28. ∫ cot (ax+b) dx = 1 log Isin (ax+b)I + c

a

29. ∫ dx = 1 tan-1 x +c

x2 + a2 a a

30. ∫ dx = 1 log x -a +c

x2 - a2 2a x+a

31. ∫ dx = 1 log x +a +c

a2 - x2 2a x-a

32. ∫ dx = sin-1 x +c

√x2 - a2 a

33. ∫ dx = log Ix + √x2 + a2I + c

√x2 + a2

34. ∫ dx = log Ix + √x2 - a2 I + c

√x2 - a2

35. ∫√x2 + a2 dx = x √x2+a2 + a2 log I x + √x2 + a2 + c

2 2

36. ∫√x2 - a2 dx = x √x2-a2 + a2 log I x + √x2 - a2 + c

2 2

37. ∫√a2 - x2 dx = x √a2-x2 + a2 sin-1 x + c

2 2 a

38. ∫ [f(x)]n . f'(x) dx = [f(x)]n+1 + c

n + 1

39. ∫ f'(x) dx = log I f (x) I + c

f(x)

EXERCISE

Q. 1 Resolve into partial fractions -2x + 4

(x2 + 1) (x-1)2

Q. 2 Resolve the following into partial fractions:

x-4 .

(x-1)(x-2)(x2+4)

Q. 3 Resolve the 2 – 3x2 + x .into Partial fraction.

x(x – 1)(x2 + 1)

Q. 4 Resolve x2 into partial fraction.

(1 – x) (1 + x2)2

Q. 5 Evaluate the definite integral ∫ (Sin22xCos22x)dx.

Q. 6 Evaluate ∫ dx .

(2x+3)2/3

Q. 7 Evaluate ∫ √ 4 – x2 dx

Q. 8 Find ∫x3(x2-1)4/3 dx

Q. 9 Evaluate ∫eax Sinbx dx.

Q. 10 Evaluate the following integrals:

(i) ∫x3(x2-1)3/4 dx (ii) ∫(lnx)2dx

Q. 11 Evaluate ∫x2exdx

Q. 12 Evaluate ∫ dy where a≠0

√ay + b

Q. 13 Evaluate ∫Sin2x Cos x dx.

Q. 14 Evaluate ∫ xsin xdx

Q. 15 ∫ x2+3x+4 dx

x – 2

Q. 16 ∫ dx . ∫ √a2 – x2 dx

((a + x2)2

Q. 17 Evaluate ∫ x(x3+ 1)2. dx

Q. 18 Determine ∫ x. Cot x2. dx

Q. 19 Evaluate ∫ du . (a ≠ b)

√u + a + √u + b

GUIDELINES

Trigonometric Ratios. There are six possible ratios of a, b and c.

We defined:

Sin α = a Cosine α = b

c c

B

Tangent α = a β

b

c a

Cotangent α = b = 1 .

a tan α α Y

A b C

Secant α = c = 1 .

b cos α

Cosecant α = c = 1 .

a sin α

Identities.

Sin2θ + Cos2θ = 1 Sin2θ = 1 – Cos2θ

2

Sec2θ = 1 + tan2θ Cos2θ = 1 + Cos2θ

2

Cosec2θ = 1 + Cot2θ

Arc Length

S = rθ (where S = arc length, r = radius, θ = angle).

1800 = π radius.

Addition Formulae

1. sin (α+β) = sinα cosβ + cos α sinβ

2. cos (α+β) = cosα cosβ – sin α sinβ

3. tan (α+β) = tan α + tan β

1 - tan α tan β

Subtraction Formulae

1. sin (α-β) = sinα cosβ - cos α sinβ

2. cos (α-β) = cosα cosβ + sin α sinβ

3. tan (α-β) = tan α – tan β

1 + tan α tan β

Double Angle Identities

1. sin 2 θ = 2sin θ cos θ

2. cos 2 θ = cos2 θ – sin2 θ = 2cos2 θ – 1 = 1 – 2sin2 θ

3. tan 2 θ = 2 tan α .

1 – tan2 α

Half Angle Formulae

1. sinθ= + 1 – cos θ

2 2

2. cos θ= + 1+ cos θ

2 2

3. tan θ= + 1- cos θ

2 1 + cos θ

4. tan θ = sin θ = 1 - cos θ

2 1 + cos θ sin θ

Product of Sum Formulae

1. 2sin α cos β = sin (α+β) + sin (α-β)

2. 2cos α sin β = sin (α+β) - sin (α-β)

3. 2cos α cos β = cos (α+β) + cos (α-β)

4. - 2sin α sin β = cos (α+β) - cos (α-β)

Sum to Product Formulae.

1. sin α + sin β = 2 sin α+β cos α-β

2 2

2. sin α - sin β = 2 cos α+β sin α-β

2 2

3. cos α + cos β = 2 cos α+β cos α-β

2 2

4. cos α - cos β = -2 sin α+β sin α-β

2 2

Law of Sines. In any triangle ABC

a = b = c .

sin A sin B sin C

Law of Cosines. In any triangle ABC

cos A = b2 + c2 – a2

2bc

cos B = a2 + c2 – b2

2ac

cos C = a2 + b2 – c2

2ab

LAWS OF AREA OF TRIANGLE:

Area= √s(s-a)(s-b)(s-c) where S = a + b +c

2

Area= a2 Sin B Sin C ,

2 Sin A

Area= b2 Sin A Sin C ,

2 Sin B

Area= c2 Sin B Sin A ,

Sin C

VALUES OF TRIGONOMETRIC FUNCTIONS FOR SELECTED ANGLES

Degree

-180

-135

-90

-45

0

30

45

60

90

135

180

Radian

-π

-3 π

4

- π

2

- π

4

0

π

6

π

4

π

3

π

2

3π

4

π

Sin θ

0

___

-1/√2

-1

___

-1/√2

0

1/2

___

1/√2

___

√3/ 2

1

___

1/√2

0

Cos θ

-1

___

-1/√2

0

___

1/√2

1

___

√3/ 2

___

1/√2

1/2

0

___

-1/√2

-1

Tan θ

0

1

∞

-1

0

___

1/√3

1

___

√3

∞

-1

0

EXERCISE

Q. 1 Prove 1 + sec θ = tan θ + sin θ, cos θ ≠ 1

1 – sec θ sin θ – tan θ

Q. 2 Prove that 1 + Cosec x = 1 + Sinx

Cosec x – 1 1 – Sinx

Q. 3 Prove that (Cosec Ø - Cot Ø)2 = 1 – Cos Ø

1 + Cos Ø

Q. 4 Prove that 1 + Cos Ø = Sin Ø

Sin Ø 1 – Cos Ø

Q. 5 Prove that 1 – Sec θ = Cos θ – 1

1 + Sec θ Cos θ + 1

Q. 6 Show that Sin Ø = Cot Ø

– Cos 2 Ø

Q. 7 Show that Sin50 – Sin30 + Sin20 = 4SinθCos(3 θ)Cos(5 θ)

2 2

Q. 8 Prove, 1 + Cos 2θ = 2 .

1 + tan2θ

Q. 9 Prove that: Sin (x+y) = tan x + tan y, when cos x Cos y = 0

Cos x Cos y

Q. 10 Cot x + Cosec x = Cot x Cosec x

Sin x + Tan x

Q. 11 Prove that Sin (A+B) = TanA + TanB

CosACosB

Q. 12 If tanθ = 3 and θ is the first quadrant. Find the values of other trigonometric functions. 4

Q. 13 If tanθ = 3 and θ is the second quadrant. Find the values of other trigonometric functions. 4

Q. 14 If Cos α = 1 , Cos β = √ 3 and α, β are the first quadrant, Find Cos (α – β).

2 2

Q. 15 Find the remaining trigonometric functions when Tan θ = -1 and R(θ) is in 2nd quadrant. 3

Q. 16 Solve 2sin2θ + 2√2 sin θ – 3 = 0

Q. 17 Solve the trigonometric equation: 3Sin2θ – sinθ = 1

4

Q. 18 Show that Sin-1 3 + Sin-1 4 = π

5 5 2

Q. 19 Prove that tan-1 1 + 1 tan-1 1 = π

3 2 7 8

Q. 20 Find the area of triangle ABC, with α=5.2cm, β=300 & ү = 400

Q. 21 Solve the triangle ABC, in which α=300, β=400 & a = 10 cm.

Q. 22 Solve the triangle ABC in which α=40.7, β=800 & c = 50.4 cm.

Q. 23 While flying at a height of 1200 meters, a pilot observes the measure of the angle of depression of an airport to be 600 and that of a town to be 300. Find the distance between the town and the airport.

Q. 24 A man observes that the angle of elevation of the top of a mountain measures 450 from a point on the ground. On walking 100 meters away from the point, the angle of elevation measures 43.45o. Find the height of the mountain.

Q. 25 From the top of tower, if the angle of depression of a ships water line is 45o. Find the distance ship and the foot of the tower, if height of tower is 60 meters.

Q. 27 While flying at a height of 1200 meters, a pilot observes the measure of the angle of depression of an airport to be 300 and that of a town to be 200. Find the distance between the town and the airport.

GUIDELINES

Distance. The length of the segment joining the points P (x1, y1) and Q (x2, y2) given by PQ = √( x2- x1)2 + ( y2- y1)2

Internal Division. The coordinates of the point R which divides the join of P(x1, y1) and Q (x2, y2) internally in the ratio m:n are given by

mx2 + nx1 , my2 +ny1

m + n m + n

Mid Point. If R is the mid point of the segment joining the point P(x1, y1) and Q (x2, y2) then its coordinates are

x2 + x1 , y2 + y1

2 2

Centroid of Triangle. If A (x1, y1), B (x2, y2) and C (x3, y3) are the vertices of a triangle ABC, then coordinates of its centroid are given by

x1 + x2 + x3 , y1 + y2 +y3

3 3

Incentre of Triangle. If A (x1, y1), B (x2, y2) and C (x3, y3) are the vertices of a triangle ABC, then coordinates of incentre are given by

ax1 + bx2 +c x3 , ay1 + by2 +cy3

a + b + c a + b + c

Area of Triangle. The area of the triangle ABC whose vertices are A(x1, y1), B(x2, y2), C(x3, y3) is given by

x1 y1 1

= 1 x2 y2 1

2 x3 y3 1

In case the area is zero, the points are said to be collinear.

Slope of the Line. If P (x1, y1) and Q (x2, y2) are any two pints on a line, then the slope of line denoted by m is defined as m = y2 – y1 = tan θ

x2 – x1

Where θ is angle of inclination of the line with the positive direction of x-axis

and 0< θ < 1800, except θ = 900, when θ = 900, m does not exist, then the line is said to be parallel to y-axis. When θ = 0, then m = 0, the line is said to be parallel to x-axis.

EQUATIONS OF LINE

Slope Pint Form. Line with given slope m and a given point (x1, y1) on it is

y-y1 = m (x – x1)

Two Point Form. Line passing through two given points (x1, y1) and (x2, y2) is

y y1 y2 y1

x x1 x2 x1

Intercept Form. Line making intercepts a, b on the x-axis and y-axis is x + y = 1

a b

Perpendicular Form. Line with perpendicular length from origin and perpendicular making an angle α with positive direction x-axis is x cos α + y sin α = p

General Form of the Line. General form of the line is ax+by+c = 0, where a, b, c being real numbers and a, b cannot be simultaneously equal to zero.

Results

1. The length of perpendicular from the point (x1, y1) on the line ax+by + c= 0 denoted by d is given by

d = I ax1 + by1 + cI

√a2 + b2

2. If α is the acute angle between the two intersecting lines a'x + b'y +c' = 0 and

ax + by + c = 0, then

Tan α= ab' – a'b

aa' + bb'

3. A family of lines through the point of intersection of two given lines ax+by+c = 0 and a'x + b'y + c' = 0 is

(ax + by +c) + k(a'x + b'y +c') = 0 where k is any real number.

Condition of Concurrency of three Straight Lines

a1 b1 c1

a2 b2 c2 = 0

a3 c3 c3

Circle. The path traced by a moving point at a fixed distance from a fixed point, is called a circle.

Diameter of a Circle. A chord of a circle passing through its centre is called a diameter of a circle. A diameter divides the circle into two equal parts, each part being a semi circle.

Tangent of Circle. A straight line which meets a circle at one and only one point is called a tangent to the circle.

1. The equation of a circle with centre (h, k) and radius r is (x h)2 + (y k)2 = r2. In case h = 0, k = 0, the centre of the circle is at origin and then the above equation become x2 + y2 = r2.

2. The general equation of the circle is x2 + y2 + 2gx + 2fy + c = 0 where g, f, c are constants and g2 + f2 c > 0. The centre of the circle is (g, f) and radius is √g2 + f2 – c.

3. We observe that g = -coefficient of x and f = coefficient of y

2 2

4. The equation of the circle when the coordinate of the end points of the diameter are (x1, y1) and (x2, y2) is (x x1)(x x2) + (y y1)(y y2) = 0

5. The equation of the tangent to the circle x2 + y2 = a2 at the point (x1, y1) on the circle is xx1 yx1 = a2. The equation of the normal at same point is xy1 yx1 = 0.

6. The equation of the tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 at the point (x1, y1) is xx1 + yy1 + g(x + x1) + f(y+y1) + c = 0. The equation of the normal to the circle at the same point is (x x1)(y1 + f) + (y y1)(x1 g) = 0

7. The condition that the line y = mx +c may be tangent to the circle x2 + y2 = a2 is

c = + a √1 + m2 so that y = mx + a √1 + m2 is the equation of general tangent to the circle x2 + y2 = a2

THE PARABOLA

Main Facts about standard forms of Parabola

Equation

y2 = 4ax

a>0

y2 = 4ax

a>0

x2 = 4ay

a>0

x2 = 4ay

a>0

Vertex

(0, 0)

(0, 0)

(0, 0)

(0, 0)

Focus

(a, 0)

( a, 0)

(0, a)

(0, a)

Directrix

x + a = 0

x – a = 0

y + a = 0

y a = 0

Axis

y = 0

y = 0

x = 0

x = 0

Length of Latusrectum

4a

4a

4a

4a

Equation of latusrectum

x a = 0

x + a = 0

y a = 0

y +a = 0

The Ellipse. An ellipse is defined as the locus of a point P which moves so that its distance from a fixed point F always bears a constant ratio e (0 < e <1) to its perpendicular from a fixed line L. The standard equation of ellipse is:

x2 + y2 = 1 where b2 = a2 (1 – e2)

a2 b2

Some of the Elementary Properties of the Ellipse.

1. The eccentricity of the ellipse (0 < e <1) is given by e2 = a2 – b2

a2

2. The equation of directrices are x = a and x = a

e e

3. Length of latus rectum = 2b2

a

The Hyperbola. A hyperbola is defined as the locus of a point which moves so that its distance from a fixed point “F” always bears a constant ratio e (e > 1) to its perpendicular from a fixed line L.

x2 y2 = 1 where b2 = a2 (e2 1)

a2 b2

1. Length of the transverse axis = 2a

2. Length of the conjugate axis = 2a

3. The equation of directrices are x = a and x = a

e e

4. Eccentricity e is given by e2 = a2 + b2

a2

5. Length of latus rectum is 2b2

a

Tangent to the Conics.

1. Equation of tangent to the parabola y2 = 4ax at the point

(x1 , y1) is given by yy1 = 2a(x + x1)

2. Equation of tangent to the ellipse x2 + y2 = 1at the point (x1 , y1) is given by

a2 b2

xx1 + yy1 = 1

a2 b2

3. Equation of tangent to the hyperbola x2 y2 = 1 at the point (x1 , y1) is given by

a2 b2

xx1 yy1 = 1

a2 b2

EXERCISE

Q. 1 Find the distance between the parallel lines 3x+4y+10=0 and 6x+8y-9=0

Q. 2 Find the equation of a straight line parallel to x-axis and at a distance of 6 units below it.

Q. 3 Find the distance between the parallel lines x+y-2=0, 2x+2y-1=0

Q. 4 Find the combined equation of two straight, lines through the origin and perpendicular to the pair given by ax2+2hxy+by2 = 0

Q. 5 Find the distance (6, -2) from the line 3x – 4y + 4 = 0

Q. 6 The distance of a point (1, 4) from a line passing through the intersection of the lines x-2y+3=0 and x-y-5=0 is 4 units. Find its equation.

Q. 7 Find the equation of the straight line passing through (3, -4) and parallel to 2x+3y= 4

Q. 8 Find the equation of a line passing through the intersection of the line 2x+5y-8=0, 3x+2y+5=0 and making an angle of 450 with the line 2x+3y-7=0

Q. 9 Find the equation of the straight line passing through the point (3,1) and parallel to the line 2x+5-4=0

Q. 10 Find the equation of the circle passing through points (-1,-2) and (6,-1) and touching x-axis.

Q. 11 Find the equation of a circle having centre at (2, 3) with radius 7.

Q. 12 Find the circle passing through the three points (0,3), (2,1), (1,0)

Q. 13 Find the equation of which is perpendicular to 2x+3y+4 = and passes through

(2, -1).

Q. 14 Find the parametric coordinates for the point (-8,8) on the circle x2+y2 = 100.

Q. 15 Find the equation of the circle which contains the points of intersection of the circles with equations:

X2+y2+2x–4y–11=0 and x2+y2–2x–3=0 and which contains the point (3, -2) also.

Q. 16 Find the equation of tangent and normal at (7, -7) to y2 = 7x

Q. 17 Find the area of the triangle whose vertices are A (3, 8), B(7,2) & C(-1,-1).

Q. 18 If the vertices of a triangle are A(2,-1), B(-2, 0), C(3,2). Find the tangent of the angle at A.

Q. 19 Obtain the coordinates of the centroid of the triangle whose vertices are

(-2,5), (4,-1) and (5,4)

Q. 20 Find the area above the x-axis, under the curve x2 + y2 = 1between the ordinates x = -1 and in y -1. 4 9

Q. 21 Find the centroid of the triangle; the equation of whose sides are

12x2 – 20xy +7y2 = 0 and 2x – 3y + 4 = 0

Q. 22 The points A(2, -1), B(-1, 4) and N(-2, 2) are the mid points of the sides of a triangle. Find its vertices.

Q. 23 Find the area of the triangle whose vertices are A (4, 5), B(-3, 2) & C(1, 4).

Q. 24 Show that (-4, 0), (4, 4√3) form the vertices of an equilateral triangle.

Q. 25 Find the vertex, focus and axis of the parabola x2+4x-y+5=0 (a≠0, b=0).

Q. 26 Find equation of the parabola whose focus is (1, 2) and vertex is (1, -3)

Q. 27 Find the equation of the circle whose diameter is the latus rectum of the Parabola x2 = -36y.

Q. 28 Find the equation of the parabola whose focus is (3, 4) and the directrix is the line x+y-1=0

Q. 29 Find the centre the vertices and the Foci of the ellipse given by the following equation and also draw graph.

4x2-16x+25y2+200y+316=0

Q. 30 Find the equation of the tangents and normal of he circle, x2+y2+6x+4y = 132 at point (6, 6).

Q. 31 Find the slope of the line through the mid point of the segment from A (-4,4) to B(2,2) and the point which is three fifth the way from C(5, 3) to D(-3, -2)

Q. 32 Find the equation of the tangents and normal to the hyperbola 2x2 – 3y2 = 1 at the point (2, 1)

Q. 33 Find the centre and radius of the circle 4x2+4y2-12x+4y-15=0

Q. 34 Find the circles through the points (3, 0), (0, -2), (-3, 4).

Q. 35 Find the point of divisions of the line joining (1, -2) to (-3, 4) with the ratio 2 : 3 an 3 : 5 respectively.

GUIDELINES

Definition. Let we have a function f(x), then a number ‘l ‘ is said to be the limit of the function f(x), when x→a, f(x) → l

Where 1 is unique and definite. We write it as Lim f(x) = l

x→a

Note (1): x→a means (i) x ≠ a (ii) x assumes values nearer and nearer to ‘a’ or we can say that x is very close to ‘a’.

Note (2): x→a should be read as

1. x approaches a

2. x tends to a

3. x assumes values nearer and nearer to a

4. x is very close to a

Evaluation of Limits. We give below some of the typical methods of calculating the limits of a function f(x) as x tends to a finite quantity.

Method I. Method of Factorization

If f(x) is of the form g(x) factorize g(x) and h(x) and cancel the common factors, then put the value of x. h(x)

Method II. Methods of Substitution

For evaluating Lim g(x), we follow the following steps:

x→a Ф(x)

(i) Put x = a + h, where h is small [≠0]

... as x →a, h→0

(ii) Simplify numerator and denominator and cancel h throughout [h≠0]

Method III. Methods of Rationalization

Here we rationalize the factor containing the square root and simplify and we put the value of x.

Some Standard Limits

1. Lim xm – am = mxm-1

x→a x – a

2. Lim sin x = 0

x→o

3. Lim cos x = 1

x→o

4. Lim sin x = 1

x→o x

5. Lim tan x = 1

x→o x

6. Lim (1 + x)1/x = e

x→o

7. Lim (1 + 1)x = e

x→o x

8. Lim ex – 1 = 1

x→o x

9. Lim ax – 1 = lna

x→o x

Abinitio Method or First Principle

Let y = f(x) be a function of x. Let x be a small change is x and y the corresponding change in y. Then

Step-1 y+ y = f(x + x)

Step-2 y = f(x + x) – f(x)

Step-3 y = f(x + x) – f(x)

x x

Step-4 Now applying the limit as x→0, we get

Lim y = Lim f(x + x) – f(x)

x→0 x x→0 x

If this limit exists, it is called derivative or differential coefficient of y with respect to x and is denoted by dy or f'(x)

dx

Thus dy or f'(x) = Lim f(x + x) – f(x)

dx x→0 x

Fundamental Theorems

(A) Differential coefficient of a constant is zero. i.e., d (e)= 0

dx

Let u, v, w be derivable functions. Then

(B) d (u + v) = du + dy

dx dx dx

(C) d (u.v) = u. dv + v. du

dx dx dx

v. du – u. dv

d u dx dx .

dx v v2

Differential Coefficient of a Function of a Function. If y is function of u and u is a function of x, then

dy dy du

dx du dx

Parametric Equations

If x = f(t), y = g(t) then

dy

dt . g'(t)

dx f'(t)

dt

Standard Results

1. d (xn) = nxn-1

dx

2. d (un) = nun-1 du

dx dx

Standard Results

1. d (sin x) = cos x

dx

2. d (cos x) = -sin x

dx

3. d (tan x) = Sec2 x

dx

4. d (cosec x) = cosec x. cot x

dx

5. d (sec x) = sec x. tan x

dx

6. d (cot x) = cosec2 x

dx

7. d (sin-1 x) = 1 .

dx √1 – x2

8. d (cos-1 x) = 1 .

dx √1 – x2

9. d (tan-1 x) = 1 .

dx 1 + x2

10. d (cosec-1 x) = 1 .

dx IxI √ x2 1

11. d (sec-1 x) = 1 .

dx IxI √ x2 1

12. d (cot x) = 1 .

dx 1 + x2

13. d (ex) = ex

dx

14. d (log x) = 1 .

dx x

15. d (ax) = ax . log a

dx

16. d (Log ax) = 1 .. Log ae

dx x

To find local maximum or local minimum. Second derivative Test (Working rule)

1. Find f'(x)

2. Solve f'(x) = 0, Let x0 be one of he critical point.

3. Find f'' (x)

4. If f''(x0) < 0, then f has a local maximum value at x = x0

If f''(x0) > 0, then f has a local minimum value at x = x0

EXERCISE

Q. 1 Evaluate Lim Cosec x – Cot x

x→0 x

Q. 2 Evaluate Lim 1 - Cosx

x→0 x2

Q. 3 Evaluate Lim Sin4x Lim Sin5x a ≠ 0

x→0 x x→a x

Q. 4 Evaluate Lim x2 - 9

x→3 x-3

Q. 5 Evaluate Lim Tanx – Sin x

x→0 Sin3x

Q. 6 Find the limit lim √1 + x -1 .

x→0 x

Q. 7 Evaluate the limit lim xn – an

x→0 x – a

Q. 8 Prove that lim ax – 1 If a ε R+, show that lim ax – 1 = lna

x→0 x x→0 x

Q. 9 Find Lim tanx

x→π/2 tan3x

Q. 10 Find Lim x2 – 5x + 2

x→∞ 5x2 + 6x – 4

Q. 11 Find the limit of the sequence {an} where an = 4n2 .

n2-3n+2

Q. 12 Find the derivative by the First Principle method of f(x) = 3x3-x.

Q. 13 Find dy where x = lnt + sin and y = et + cos t

dx

Q. 14 Differentiate Tan2x w.r.t Cot3x

Q. 15 If y = (a + bx)ex then show that d2y - 2 dy + y = 0

dx2 dx

Q. 16 Find dy from the equation ax2 + 2hxy + by2 + 2gx + 2fy + c = 0

dx

where a, h, b, g, f, c ε R and hx + by + f ≠ 0.

Q. 17 Find dy from the equation ax2 + 2hxy + by2 + 2gn + 2fy + c = 0

dx

where a, c, f, g, & h are real numbers

Q. 18 Find dy , if x = Sint3 + Cost3 & y = Sint +2Cos-1t

dx

Q. 19 Find dy , at the point t. If x = logt + sint y = et + cost

dx

Q. 20 Find dy/dx when x = lnt + Sin t and y = et + cost where it exists.

Q. 21 If y = esinxcosx, find dy If f(x) = esinxcosx. Find f' (x)

dx

Q. 22 Evaluate Cos5xdx

Q. 23 If Sin y = x Sin(a+y), show that dy = Sin2 (a+y)

dx Sin a

Q. 24 Find the f'(x), of followings:

f(x) = xSin x f(x) = ½ tan x + ln cos x

Q. 25 Let f : R+ → R be defined by f (x) = xSin x ε R+. Find f' (x).

Q. 26 Find the points of extreme values (i.e. points of maximum & minimum values) for the function f(x) = 2x3-15x2+36x+10.

GUIDELINES

Set. A set is a collection of well defined distinct objects, each object being the element of the set. No element is repeated in a set.

Equal Set. Two sets are said to be equal if they contain the same elements. Symbolically, A = B

Complement of a Set. If A is a subset of U then the set U \ A is called the Complement of A with respect to U and is denoted by Ac.

Union of two Sets. The Union of two sets A and B, written as A U B, is the set of all elements which belong either to A or to B or to both A and B.

Intersection of two Sets. The Intersection of two sets A and B, written as A ∩ B, is the set of all elements which are common to both A and B.

Fundamental Properties of Union and Intersection Commutative Property of Union

A U B = B U A

Complex Number. If Z is a complex number denoted by Z = a + ib, then a is called a real part and b is called the imaginary prt.

Re (Z) = a Im (Z) = b

Addition. If Z1 = (a, b), Z2 = (c, d) then

Z1 + Z2 = (a + c, b + d)

Multiplication. Z1 Z2 = (ac – bd, ad + bc)

Additive Identity. (0, 0) Multiplicative Identity. (1, 0)

Additive Inverse. Z = -a, -ib

Multiplicative Inverse. a , b .

a2 + b2 a2 + b2

__

Conjugate. Z = a – ib

Modulus. |Z| = r = √a2 + b2

EXERCISE

Q. 1 If A = {1,2,3}, B = {3,4,5} and U = {1,2,3,4,5} verify De Morgan’s Law.

Q. 2 Separate (2x – 3yi)4 in its real and imaginary parts.

Q. 3 Simplify √2 + i

√2 – i

Q. 4 Find the multiplicative inverse of (-3, 8).

Q. 5 Find the modulus of complex number (4, -3).

Q. 6 Verify that: (-3,-4)+(2, 5) = (2,5)+(-3,-4)

4 2 0 2 1 -1

Q. 7 If A = 5 6 7 and B = 3 -2 -1 find AB and BA

-3 1 9 2 -5 -1

1 2 3 2 3 4

Q. 8 If A = 4 6 8 and B = -4 6 -8 Show that (A+B) and A'+B'

-1 2 -3 1 2 5

Q. 9 By using the properties of the determinants and without expanding it show that:

1 1 1

a b c = O

b+c c+a a+b

Q. 10 Apply Carmer’s rule to solve the following equations:

9x+7y+3z=6

5x – y+4z=1

6x+8y+2z=4

Notes Available :

www.ranginissb.blogspoth.com

RANGIN KHAN√

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