some useful algebraic formulae -...

Wisdom institute of mathematics Formulae for XII class/D.K. Rohaj

Wisdom institute of mathematics//Narela, Delhi – 40//D. k. Rohaj//mo: 9811469438

Some useful algebraic formulae

ab4baba )12

ba2baba )11

acbcabcbacbaabc3cba )10

babababa )9

abbababa )8

abbababa 7)

bababa 6)

)acbcab(2cbacba 5)

baab3baba 4)

baab3baba 3)

ab2baba 2)

ab2baba 1)

22

2222

222333

2244

2233

2233

22

2222

333

333

222

222



RELATIONS AND FUNCTIONS

1) A relation R in a set A is called reflexive, if (a, a) R, a A

2) A relation R in a set A is called symmetric, if:

(a1, a2) R (a2 , a1) R, a1, a2 A

3) A relation R in a set A is called transitive, if:

(a1, a2) R & (a2, a3) R (a1 , a3) R, a1, a2, a3 A

4) A relation R in a set A is called equivalence, if R is reflexive, symmetric and

transitive.

5) A function f: X → Y is called one-one (injective), if:

f (x1) = f (x2) x1 = x2, x1, x2 X

6) A function f: X → Y is called onto (surjective), if there exists an element x in X,

for every y Y, such that f (x) = y

7) A function f: X → Y is called one-one and onto (bijective), if f is both one-one

and onto.

8) Composition of functions:

gof (x) = g (f (x)), fog (x) = f (g (x)), fof (x) = f (f (x))

9) If f is one-one and onto, then f must be invertible.

10) If a function f: X → Y is invertible, then:

a) f must be one-one and onto,

b) there exists a function g: Y → X such that fog = Iy and gof = Ix.

c) g is called the inverse of f, i. e., f – 1 = g.

11) A binary operation * on a set A is a function * : A × A → A. We denote:

* (a, b) by a * b.

12) A binary operation * on the set X is called commutative, if:

a * b = b * a, a, b X.

13) A binary operation * : A × A → A is said to be associative if:

(a * b) * c = a * (b * c), a, b, c A

14) e is called the identity element of binary operation * : A × A → A, if:

e є A and a * e = a = e * a, a A

15) If a * b = e = b * a, then a is invertible and b = a– 1, where e is the identity element.



INVERSE TRIGONOMETRY

03

113

223

21

13

222

313

10

02

1

2

1

2

31

12

3

2

1

2

10

θcot

θ sec

cosecθ

tanθ

cosθ

sinθ

2

π

3

π

4

π

6

π0

Functions Domain Range of principal branch

Sin-1 [– 1, 1]

2,

2

Cosec-1 R – (– 1, 1)

2,

2– {0}

Tan-1 R

2,

2

Cot-1 R (0, π)

Cos-1 [– 1, 1] [0, π]

Sec-1 R – (– 1, 1) [0, π] –

2

1. Sin-1x should not be confused with (sin x) -1. In fact (sin x) -1 = xsin

1 and

similarly other trigonometric functions.

2. Sin (sin -1 x) = x if – 1 ≤ x ≤ 1 and sin-1 (sin x) = x if2

x2

. In other

words if y = sin -1x, then sin y = x.



3. Principal value of inverse trigonometric functions lies in the range of

principal branch.

4. sin -1 (-x) = – sin -1 x

5. cosec -1 (-x) = – cosec -1 x

6. tan -1 (-x) = – tan -1 x

7. cos -1 (-x) = π – cos -1 x

8. sec -1 (-x) = π – sec -1 x

9. cot -1 (-x) = π – cot -1 x

10. Sin -1 x + Cos -1 x = 2

11. sec -1 x + cosec -1 x = 2

12. tan -1 x + cot -1 x = 2

13. tan -1 x + tan -1 y = tan -1

xy1

yx

14. tan -1 x – tan -1 y = tan -1

xy1

yx

15. 2tan -1 x = tan -1 2x1

x2

16. 2tan -1 x = sin -1 2x1

x2

17. 2tan -1 x = cos -1 2

2

x1

x1

18. 2 Sin -1 x = sin -1 212 x21cos)x1x2(

19. sin -1 x = cos -1 2x1

20. sin -1 x = tan -1

2x1

x

21. sin -1 x = cosec -1

x

1

22. cos -1 x = sin -1 2x1



23. cos -1 x = tan -1

x

x1 2

24. cos -1 x = sec -1

x

1

25. tan -1 x = sin -1

2x1

x

26. tan -1 x = cos -1

2x1

1

27. tan -1 x = cot -1

x

1

28. sin -1 x ± sin -1 y = sin -1 (x 2y1 ± y 2x1 )

29. cos -1 x ± cos -1 y = cos -1 (xy 2y1 2x1 )

Some important substitutions:

Expression substitution

1). a2 – x2 x = a sin θ or a cos θ

2). a2 + x2 x = a tan θ or a cot θ

3). x2 – a2 x = a sec θ or a cosec θ

4). xa

xa

x = a cos 2θ

6). 22

22

xa

xa

x2 = a2 cos 2θ

TRIGONOMETRY

1. sin x = hypotenuse

larperpendicu; cos x =

hupotenuse

base; tan x =

base

larperpendicu

2. sin x = x eccos

1; cosec x =

xsin

1

3. cos x = xcos

1 x sec ;

xsec

1

4. tan x = tan x

1 cot x ;

x cot

1



5. tan x = sin x

xcos cot x ;

xcos

xsin

6. sin 2 x + cos2 x = 1; sin2 x = 1 – cos2 x ; cos2 x = 1 – sin2 x

7. sec2 x – tan2 x = 1; tan2 x = sec2 x – 1; sec2 x = 1 + tan2 x

8. cosec2 x – cot2 x = 1; cosec2 x = 1 + cot2x; cot2 x = cosec2 x – 1

9. tan (A+B) = B tan Atan1

BtanAtan

10. tan (A – B) = B tan Atan1

BtanAtan

11. cot (A+B) = BcotAcot

1Bcot Acot

12. cot (A - B) = AcotBcot

1Bcot Acot

13. sin (A + B) = sin A cos B + cos A sin B

14. sin (A – B) = sin A cos B – cos A sin B

15. cos (A + B) = cos A cos B – sin A sin B

16. cos (A - B) = cos A cos B + sin A sin B

17. sin A + sin B = 2 sin2

BA cos

2

BA

18. sin A – sin B = 2 cos 2

BA sin

2

BA

19. cos A + cos B = 2 cos 2

BA cos

2

BA

20. cos A – cos B = – 2 sin 2

BA sin

2

BA

21. 2sin A cos B = sin (A+B) + sin (A-B)

22. 2cos A sin B = sin (A+B) – sin (A-B)

23. 2cos A cos B = cos (A+B) + cos (A-B)

24. 2sin A sin B = cos (A-B) – cos (A+B)

25. cos 2x = 2 cos2 x – 1 )2/x(cos2xcos1 2

26. cos 2x = 1 – 2sin2 x )2/x(sin2xcos1 2

27. cos 2x = xtan1

xtan1

2

2

28. cos 2x = cos2 x – sin2 x

29. tan 2x = xtan1

xtan22

30. sin 2x = 2 sin x cos x )2/xcos()2/xsin(2xsin



31. sin 2x = xtan1

xtan22

32. sin 3x = 3sin x – 4sin3 x

33. cos 3x = 4cos3 x – 3cos x

34. tan 3x = xtan31

xtanxtan3

2

3

35. sin

x2

= cos x ; cos

x2

= sin x

36. tan

x2

= cot x ; cot

x2

= tan x ---first quadrant

37. sec

x2

= cosec x ; cosec

x2

= sec x

38. sin

x2

= cos x ; cos

x2

= – sin x

39. tan

x2

= - cot x ; cot

x2

= – tan x --second quadrant

40. sec

x2

= – cosec x ; cosec

x2

= sec x

41. sin (π – x) = sin x ; cos (π – x) = – cos x

42. tan (π – x) = – tan x ; cot (π – x) = – cot x --second quadrant

43. sec (π – x) = – sec x ; cosec(π – x) = cosec x

44. sin (π + x) = – sin x ; cos (π + x) = – cos x

45. tan (π + x) = tan x ; cot (π + x) = cot x --third quadrant

46. sec (π + x) = – sec x ; cosec(π + x) = – cosec x

47. sin

x2

3= – cos x; cos

x2

3= – sin x

48. tan

x2

3= cot x ; cot

x2

3= tan x --third quadrant

49. sec

x2

3= – cosec x ; cosec

x2

3= – sec x

50. sin (2π – x) = – sin x ; cos (2π – x) = cos x

51. tan (2π – x) = – tan x ; cot (2π – x) = – cot x --fourth quadrant

52. sec (2π – x) = sec x ; cosec(2π – x) = – cosec x



53. sin

x2

3= – cos x; cos

x2

3= sin x

54. tan

x2

3= – cot x ; cot

x2

3= – tan x --fourth quadrant

55. sec

x2

3= cosec x; cosec

x2

3= – sec x

2

π

2

2

SECOND FIRST

QUADRANT QUADRANT

S A 2

π 0, 2π

T C 2

THIRD FOURTH

QUADRANT QUADRANT

2

3

2

3

2

3π

fourth

third

ondsec

First

x cotxsecx eccosxtanx cosxsinQuadrant



DETERMINANTS

Some useful properties and formulae based on determinants:

1. Sum of product of elements of a row (or column) with corresponding cofactors is

equal to | A | and otherwise zero.

2. | A | = | transpose of A |

3. If any two rows (or columns) of a determinant are interchanged, then sign of

determinant changes.

4. If any two rows (or columns) of determinant are identical (all corresponding

elements are same), then value of determinant is zero.

5. If A is a matrix of order n and k is any constant then we can write | kA | = kn | A |

6. If (a, b), (c, d), (e, f) are vertices of a triangle then,

1fe

1dc

1ba

2

1 triangleof Area

7. The area of triangle formed by three collinear points is always equals to zero.

8. A (adj A) = (adj A) A = | A | I, where I is the identity matrix.

9. A square matrix A is said to be singular matrix if | A | = 0 otherwise it is known as

non-singular matrix.

10. | AB| = | A | | B |

11. If n is the order of matrix A, then | adj A | = | A | n-1.

12. A system of equations is said to consistent if its solution (one or more) exists

otherwise it is known as inconsistent system.

13. A square matrix A is invertible if and only if it is non-singular matrix i.e. | A | ≠ 0.

14. A-1 = A

1.(adj A)

15. AA-1 = I and | A-1|= A

1

16. If | A| 0, then the system is consistent and has a unique solution.



17. If | A| = 0 and (adj A) B = 0, then the system is consistent and has infinitely many

solutions.

18. If |A| = 0 and (adj A) B 0, then the system is inconsistent.

19. (AB) -1 = A-1B-1

CONTINIUTY AND DIFFERENTIABILITY

1. Suppose f is a real function on a subset of the real numbers and c is a point in the

domain of f. Then f is continuous at c if limit of f(x) at x = c is equal to f(c)

otherwise it is discontinuous at x = c.

2. Every polynomial function is continuous at every real number.

3. Every rational function is continuous at every real number.

4. If f and g are two continuous functions at a real number ‘c’ then (f + g), (f - g),

(f ×g) and (f/g) are also continuous at ‘c’.

5. If g is continuous at c and f is continuous at g(c), then (fog) is continuous at c.

6. If L. H. L = R. H. L = f (C) then f is continuous function at C.

7. L.H.D is the limit of h

)x(f)hx(f

at h tends to zero.

8. R.H.D is the limit of h

)x(f)hx(f at h tends to zero.

9. If L.H.D = R.H.D, then function is differentiable.

Function Derivative of function w.r.t x

1. xn n x n-1

2. Sin x cos x

3. Cos x – sin x

4. tan x sec 2 x

5. Cosec x – cosec x cot x

6. Sec x sec x tan x

7. Cot x – cosec 2 x

8. x x2

1



9. x

1

2x

1

10. ex ex

11. log x x

1

12. ax ax log x (where is a is any constant)

Some important rules to find the derivative of two or more functions:

There are u and v are two functions and k is any constant

13. (k × u)′ = k × u′

14. (u ± v) ′ = u′ ± v′

15. (u × v) ′ = (u × v′) + (v × u′) (product rule)

16. 2v

)vu( )uv(

v

u

(quotient rule)

Some important properties of logarithm:

2) log(ab) = b × ( log a )

3) log

b

a = log(a) – log(b)

4) log(a × b) = log(a) + log(b)

5) logba = b log

a log

6) e log x = x or alogax

= x

7) If logbx = a, then x = ba

Some important substitutions:

Expression substitution

1). a2 – x2 x = a sin θ or a cos θ

2). a2 + x2 x = a tan θ or a cot θ



3). x2 – a2 x = a sec θ or a cosec θ

4). xa

xa

x = a cos θ

6). 22

22

xa

xa

x2 = a2 cos θ

Derivative of inverse trigonometric functions:

Function Derivative of function

1. sin-1 x 2x1

1

2. cos-1 x 2x1

1

3. tan-1 x 2x1

1

4. cot-1 x 2x1

1

5. sec-1 x 1xx

12

6. cosec-1 x 1xx

12

Rolle’s Theorem:

Let f: [a, b] → R be a continuous on [a, b] and differentiable on (a, b), such that

f (a) = f (b), where a and b are some real numbers.

Then there exists some c in (a, b) such that f ′ (c) = 0.

Mean value theorem:

Let f: [a, b] → R be a continuous function on [a, b] and differentiable on (a, b). Then

there exists some c in (a, b) such that f ′ (c) =ab

)a(f)b(f

.

Some useful results on continuous and differentiable functions:

1. A polynomial function is everywhere continuous and differentiable.



2. The exponential function, sine and cosine functions are every where continuous

and differentiable.

3. Logarithmic function is continuous and differentiable in its domain.

4. The sum, difference, product and quotient of continuous functions is continuous.

5. The sum, difference, product and quotient of differentiable functions is

differentiable.

6. If a function is differentiable at a point, it is necessarily continuous at that point.

But the converse is not necessarily true.

APPLICATION OF DARIVATIVES

Rate of change of quantities:

1. Rate of change of y with respect to x isdx

dy.

2. dx

dyis positive if y increases as x increases and is negative if y decreases as x

increases.

Some useful formulae of mensuration:

1. Area of circle = πr2

2. Circumference of circle = 2πr

3. Area of rectangle = l × b

4. Perimeter of rectangle = 2 × (l + b)

5. Area of square = side × side

6. Perimeter of square = 4 × side

7. Total surface area of cube = 6 × (side)2

8. Curved surface area of cube = 4 × (side)2

9. volume of cube = (side)3

10. Total surface area of cuboids = 2(lb + bh + lh)

11. Volume of cuboids = lbh

12. Total surface area of cylinder = 2πrh + 2πr2

13. Curved surface area of cylinder = 2πrh

14. Volume of cylinder = πr2h

15. Total surface area of cone = πrl + πr2

16. Curved surface area of cone = πrl



17. Volume of cone =3

hr 2

18. Surface area of sphere = 4πr2

19. Volume of sphere = 3

r 4 3

20. Total surface area of hemisphere = 3πr2

21. Curved surface area of hemisphere = 2πr2

22. Volume of hemisphere = 3

r 2 3

Increasing and decreasing functions

Some definitions of increasing and decreasing functions:

Without using derivative:

1. f is said to be increasing on I if x1 < x2 f (x1) ≤ f (x2) for all x1, x2 I

2. f is said to be strictly increasing on I if x1 < x2 f (x1) < f (x2) for all x1, x2 I

3. f is said to be decreasing on I if x1 < x2 f (x1) ≥ f (x2) for all x1, x2 I

4. f is said to be strictly decreasing on I if x1 < x2 f (x1) > f (x2) for all x1, x2 I

Using derivative:

Let f be continuous on [a, b] and differentiable on (a, b). Then

1. f is increasing in [a, b] if f ′ (x) > 0 for each x (a, b).

2. f is decreasing in [a, b] if f ′ (x) < 0 for each x (a, b).

3. f is a constant function in [a, b] if f ′ (x) = 0 for each x (a, b).

4. f is strictly increasing in (a, b) if f ′ (x) > 0 for each x (a, b).

5. f is strictly decreasing in (a, b) if f ′ (x) < 0 for each x (a, b).

6. f is a increasing (decreasing) in R if it is so in every interval of R.

Tangent and normal

Slope (gradient) of a line:

1. If θ is the angle of a line with the positive direction of x-axis, then the slope of

that line is tan θ. 2. If (x1, y1) and (x2, y2) are two points lies on a line, then the slope of that line

is12

12

xx

yy

.

3. If the equation of a straight line is ax + by + c = 0, then the slope of that line is

(b

a ).



4. Let y = f (x) be a continuous curve and let P(x1, y1) be a point on it, then pdx

dy

is the slope of tangent to the curve y = f(x) at point P and the slope of normal is

pdx

dy

1

5. Slope of x-axis is equal to zero.

6. Slope of y-axis is equal to ∞.

7. If m1 and m2 are slopes of two parallel lines, then m1 = m2.

8. If m1 and m2 are slopes of two perpendicular lines, then m1× m2 = -1.

Equation of a straight line: If a straight line passing through a point (x1, y1) and having slope m, then the

equation of that line is (y – y1)= m (x – x1).

Angle between two lines: The angle θ between two lines having slopes m1 and m2 is given by

21

21

mm1

mm tan

Angle of intersection of two curves:

The angle of intersection of two curves is defined to be the angle between the

tangents to the two curves at their point of intersection.

Approximations

1. If y = f (x), ∆x is increment in x and ∆y is increment in y, then

∆y = f (x + ∆x) – f (x)

2. The differential of x is dx and dx = ∆x.

3. The differential of y is dy and dy ≈ ∆y.

4. ∆y =dx

dy × ∆x.

Maxima and minima:

1. First derivative test for local maxima and local minima:

i. x = c is a point of local maxima and f (c) is the local maximum value of

f(x) if f ′ (x) > 0 at every point in the left of c and f ′ (x) < 0 at every point

in the right of c.

ii. x = c is a point of local minima and f (c) is the local minimum value of f(x)

if f ′ (x) < 0 at every point in the left of c and f ′ (x) > 0 at every point in the

right of c.

iii. If f ′ (x) does not change sign as x increases through c, then c is neither a

point of local maxima nor a point of local minima. Such a point is called a

point of inflection.



2. Second derivative test for local maxima and local minima:

i. x = c is a point of local maxima and f ( c ) is local maximum value of

function if f ′ (x) = 0 and f ′′ (x) < 0

ii. x = c is a point of local minima and f ( c ) is local minimum value of

function if f ′ (x) = 0 and f ′′ (x) > 0

iii. The test fails if f ′ (c) = 0 and f ′′ (c) = 0. In this case, we go back to the

first derivative test.

INTEGRATION

Very important formulae of integration:

1. ∫k dx = k x + c, where k is a constant.

2. ∫ xn dx = 1n

x 1n

+ c

3. ∫ ax dx = alog

ax

+ c

4. ∫ ex dx = ex + c

5. ∫ x

1 dx = log |x | + c

6. ∫ sin x dx = – cos x + c

7. ∫ cos x dx = sin x + c

8. ∫ tan x dx = log |sec x | + c = – log |cos x | + c

9. ∫ cosec x dx = log |cosec x – cot x | + c= log |tan 2

x| +c

10. ∫ sec x dx = log | sec x + tan x | + c = log |tan

2

x

2| + c

11. ∫ cot x dx = log |sin x | + c

12. ∫ sec2 x dx = tan x + c

13. ∫ cosec2 x dx = – cot x + c

14. ∫ sec x tan x dx = sec x + c

15. ∫ cosec x cot x dx = – cosec x + c

16. 22 xa

1dx = log | x +

22 xa | + c



17. 22

1

axdx = log | x +

22 ax | + c

18. 22 xa

1dx = sin -1

a

x + c

19. 22

1

axdx = log

a x

a x

+ c

20. 22 a x

1dx =

a

1 tan -1

a

x + c

21. 22 x a

1dx = log

x a

x a

+ c

22. 22 x a dx = 222

22 x a x log2

a x a

2

x + c

23. 22 a x dx = 222

22 a x x log2

aa x

2

x + c

24 . 22 x a dx = a

xsin

2

a x a

2

x 1 2

22 + c

Some useful formulae of trigonometry to find integration:

1. ∫ sin2 x dx =

2

2cos1 xdx

2. ∫ cos2 x dx =

2

2cos1 xdx

3. ∫ tan2 x dx = ∫ sec2 x – 1 dx

4. ∫ cot2 x dx = ∫ cosec2 x – 1 dx

5. ∫ sin3 x dx =

4

sin3xsin x 3 dx

6. ∫ cos3 x dx =

4

3coscos3 xxdx

Some standard results on integration:

1. ∫k f (x) dx = k ∫f (x) dx, where k is a constant.

2. ∫{f (x) ± g (x)} dx = ∫f (x) dx ± ∫g (x) dx

3. If ∫f (x) dx = g (x) + c, then ∫f (ax + b) dx = a

b) (ax g + c

Forms Solutions

1. )x(f)x(f ndx or

dx

)x(f

)x(fn

Put f (x) = t



2. )x(f)x(f n dx or

dx

)x(f

)x(fn

Put f (x) = t

3. If degree of numerator is greater

then the degree of denominator

divide nr by dr

4. linear

lineardx

Make the nr same as dr

5. linearlinear dx Make the outer function same as inner

function

6. linearlinear

1dx

Rationalize of denominator

7. quadraticlinear dx,

quadratic

lineardx, quadratic

lineardx

Make the linear function same as the

derivative of quadratic function

8. linear

quadratic dx

Make the quadratic function same as linear

function using factorization

9. quadratic dx,

quadratic

1dx,

quadratic

1 dx

Make perfect square

Forms Solutions

1. ∫ sin mx cos nx dx,

∫ cos mx cos nx dx,

∫ sin mx sin nx dx,

∫ cos mx sin nx dx

Use these identities:

2sin A cos B = sin (A+B) + sin (A – B)

2cos A sin B = sin (A+B) – sin (A – B)

2cos A cos B = cos (A+B) + cos (A – B)

2sin A sin B = cos (A – B) – cos (A+B)

2. ∫sinn x cosn x dx Multiply and divide by 2n and then using

formula {2 sin x cos x = sin 2x}

3. ∫sinm x cosn x If power of cos x is even then put cos x = t

If power of sin x is even, then put sin x = t

4. )1x(x

1n

dx, where n R Put xn ± 1 = t



5. )(

)(

xg

xf dx

Remove the square root of nr

6. xba 2sin

1 dx, xba 2cos

1dx,

xbxa 22 cossin

1 dx,

xxba cossin

1 dx, etc.

Divide nr and dr by cos2 x, and then

put tan x = t

7. xba sin

1 dx, xba cos

1 dx,

xbxa cossin

1dx, etc.

Using: sin x =

2tan1

2tan2

2 x

x

,

Cos x =

2tan1

2tan1

2

2

x

x

8.

xdxc

xbxa

cossin

cossin dx Make nr = A ×

dx

d (dr) + B (dr)

9.

rxqxp

cxbxa

cossin

cossin

Make nr = A ×dx

d (dr)) + B (dr) + C

1.

1

124

2

axx

x dx,

1

124

2

axx

xdx,

Divide nr and dr by x2

2. quadraticlinear

1dx Put linear =

t

1

3. quadratic pure quadratic pure

1dx Put x =

t

1

4. linearquadratic

1dx Put linear= t

5. linearlinear

1dx Put linear= t

Integration by parts:



1. If u and v are two functions then:

dx dx v u dx v)(u dxvudx

d

2. Proper choice of first and second function:

Use the word I L A T E :

I stands for sin-1 x, cos-1 x, tan-1 x, cosec-1 x, sec-1 x, cot-1 x

L stands for the logarithm functions.

A stands for the algebraic function.

T stands for the trigonometric functions.

E stands for exponential functions.

FORM SOLUTION

1. ∫log x dx, ∫sin-1 x dx, ∫cos-1 x dx,

∫tan-1 x dx, ∫sec-1 x dx, etc.

Take unity as second function using by

parts

2. ∫ex {f(x) + f ′ (x)} dx Open the bracket and evaluate ∫ex f(x) dx

Integration by Partial fraction:

If the degree of nr is less then the degree of dr and dr is expressible as the product of

factors, then we can use partial fraction.

FORM OF THE RATIONAL FUNCTIONS FORM OF PARTIAL FRACTION

1. ))(( bxax

qpx

)()( bx

B

ax

A

2. 2)( ax

qpx

2)()( ax

B

ax

A

3. ))()((

2

cxbxax

rqxpx

)()()( cx

C

bx

B

ax

A

4. )()( 2

2

bxax

rqxpx

)()()( 2 bx

C

ax

B

ax

A

5. ))(( 2

2

cbxxax

rqxpx

)()( 2 cbxx

CBx

ax

A

Definite integrals:



If ∫f (x) dx = g (x), then b

a

dxxf )( = g (b) – g (a)

Integration from first principle or ab-initio method or as the limit of a sum:

b

a

xf dx )( = 0

limh

h [f (a) + f (a +h) + f (a + 2h) + f (a + 3h) + + f {a + (n – 1) h}],

where h = n

ab

The following results will be helpful in evaluating definite integrals as the limits of a sum

1. 1 + 2 + 3 + 4 + 5 + 6 + ………+ (n – 1) = 2

)1( nn

2. 12 + 22 + 32 + 42 + 52 + 62 + ………+ (n – 1)2 = 6

)2)(1( nnn

3. 13 + 23 + 33 + 43 + 53 + 63 + ………+ (n – 1)3 =

2

2

)1(

nn

4. a + ar + ar2 + ar3 + ……… + ar n-1 = 1

)1(

r

ra n

, where r > 1

Important properties of definite integral:

1. b

a

dxxf )( = b

a

tf )( dt, i. e., integration is independent of the change of variable.

2. b

a

dxxf )( = – a

b

dxxf )( , i. e., if the limits of definite integral are interchanged then its

value changes by minus sign only.

3. b

a

xf )( dx =

b

a

xaf )( dx (most important property)

4.

a

a

xf )( dx = 2 a

xf

0

)( dx, if f (x) is an even function. i. e., f (-x) = f (x)

5.

a

a

xf )( dx = 0, if f (x) is an odd function. i. e., f (-x) = - f (x)

6. a

xf

2

0

)( dx = 2 a

xf

0

)( dx, if f(2a – x) = f (x)

7. a

xf

2

0

)( dx = 0, if f (2a – x) = – f (x)



8. b

a

dxxf )( =

b

a

dxxbaf )(

APPLICATIONS OF INTEGRATION IN FINDING AREAS:

CURVES GENERAL EQUATIONS

Straight line a x + b y + c = 0

x – axis y = 0

y – axis x = 0

Circle (x – x1)2 + (y – y1)

2 = r2

Standard ellipse 1

2

2

2

2

b

y

a

x

Parabola (y – y1)2 = 4 a (x – x1) or (x – x1)

2 = 4 a (y – y1)

Straight line y = x Parabola x2 = 4ay



Circle x2 + y2 = r2 Ellipse 1b

y

a

x2

2

2

2

DIFFRENTIAL EQUATIONS

Order of a differential equation: The order of a differential equation is the order of

highest order derivative appearing in the equation.

Degree of a differential equation: The degree of a differential equation is the degree

of the highest order derivative, when differential coefficients are made free from

radicals and fractions.

General solution of a differential equation: The solution which contains as many as

arbitrary constants as the order of differential equation.

Particular solutions of a differential equation: Solutions obtained by giving

particular values to the arbitrary constants in the general solution.

Some special types of differential equations and their solutions:

1. dx

dy= f (x) dy 1 = C dx )(xf

2. dx

dy = f (y) C dy

f(y)

1 dx 1

3. dx

dy = f (x, y) C dy (y) f dx )(xf (variable separable form)

4. dx

dy = f (ax + by + c)

Put ax + by + c = v

5. ),(

),(

yxg

yxf

dy

dx , where f (x, y)

and g (x, y) are functions of

same degree

Put y

x = v



6. ),(

),(

yxg

yxf

dx

dy , where f (x, y)

and g (x, y) are functions of

same degree

Put x

y = v

7. dx

dy + P y = Q, where P and

Q are functions of x or

constants

C dx F) I.(F) (I. Qy ,

where I. F = integrating factor = e∫P dx

8. dy

dx+ P x = Q, where P and

Q are functions of y or

constants

C dy F) I.(F) (I. Qx ,

where I. F = integrating factor = e∫P dy

VECTOR ALGEBRA

Some basic concepts:

1. AB is the symbol of AB vector which has magnitude as well as direction from A

to B.

2. In this vector point A is called the initial point and point B is called the terminal

point.

3. | AB | is the length of

AB , it is also known as the magnitude of

AB .

4. Unit vector in the direction of a is

a

aa

5. Magnitude of a unit vector is always equals to one.

6.

CB AC AB ,

BC AB AC and

AC BA BC are known as triangle

law of vector addition.

7.

BA AB

8.

a k k

a , where k is constant.

9. If k z jy i x a , then scalar components and direction ratios of

a are x, y, z

and vector component of a are k z and jy ,ix , where î, ĵ, k are unit vectors in

the direction of x-axis, y-axis and z-axis respectively.

10. If a = x1î + y1ĵ + z1 k and

b = x2î + y2ĵ + z2 k then,



a)

ba = (x1 + x2) î + (y1 + y2) ĵ + (z1 + z2) k , it is known as resultant

of the vectors a and

b .

b)

ba = (x1 – x2) î + (y1 – y2) ĵ + (z1 – z2) k

c) If

ba then, x1 = x2, y1 = y2, z1 = z2.

d) If a and

b are collinear, then

2

1

2

1

2

1

z

z

y

y

x

x .

11. If 21

21

21

111 a a of magnitude then ,ˆˆˆ zyxkzjyixa

12. If A (x1, y1, z1) and B (x2, y2, z2) are two points then, AB = (x2 – x1) î + (y2 – y1) ĵ + (z2 – z1) k

Scalar (or dot) product of two vectors:

If a = x1î + y1ĵ + z1 k and

b = x2î + y2ĵ + z2 k then,

1.

ba = (x1 × x2) + (y1 × y2) + (z1 × z2).

2.

ba = ba cos θ,

3. 1kk 1,jj ,1ii

4. Ifa

b , then

ba = 0.

5. 0ki 0,kj ,0ji

6. cos

b a

ba

7 . Projection of

b

babona

Vector (or cross) product of two vectors:



If a = x1î + y1ĵ + z1 k and b = x2î + y2ĵ + z2 k then,

1.

ba =

ˆˆˆ

222

111

zyx

zyx

kji

2.

ˆ sinbaba , therefore

ˆ andb ,a where,

b a

ba

sin are in the

form of right handed system.

3. 0kk,0jj,0ii

4. Unit vector perpendicular to

ba

babanda

5. If banda represent the adjacent sides of a triangle, then:

Area of the triangle = ba 2

1

6. If banda represent the adjacent sides of a parallelogram:

i) Area of the parallelogram = ba

ii)

ba represent the longer diagonal of parallelogram.

THREE DIMENSIONAL GEOMETRY

1. Direction cosines of a line: If a directed line passing through the origin makes angles

α, β, and γ with x, y and z-axes, respectively, called direction angles, then cosine of

these angles, namely, cos α, cos β and cos γ are called direction cosines of the

directed line. If a line in space does not pass through the origin, then, we draw a line

through the origin and parallel to the given, and then we can define the direction

cosine of the given line.

2. Direction ratios of a line: Any three numbers which are proportional to the direction

cosines of a line are called the direction ratios of a line.



3. cos2 α + cos2 β + cos2 γ = 1, i. e., l2 + m2 + n2 = 1, where l, m, n are the direction

cosines of a line.

4. If a line passes through two points (x1, y1, z1) and (x2, y2, z2), then direction ratios of

the line are (x2 – x1), (y2 – y1), (z2 – z1) or (x1 – x2), (y1 – y2), (z1 – z2)

5. If a line passes through two points P(x1, y1, z1) and Q(x2, y2, z2), then direction

cosines of the line are

PQ

zz,

PQ

yy,

PQ

xxor

PQ

zz,

PQ

yy,

PQ

xx 212121121212 ,

where PQ = 212

212

212 )zz()yy()xx(

6. If the direction ratios of two lines are 111 c,b,a and 222 c,b,a ,

then22

22

22

21

21

21

212121

cbacba

ccbbaacos

, where, is the acute angle between these

lines.

7. If two lines having direction ratios 111 c,b,a and 222 c,b,a , are collinear (or parallel),

then their direction ratios are proportional i. e.,2

1

2

1

2

1

c

c

b

b

a

a

8. If two lines having direction cosines 111 n,m,l and 222 n,m,l ,

then 212121 nnmmllcos , where, is the angle between these two lines.

9. If two lines having direction ratios 111 c,b,a and 222 c,b,a , are perpendicular, then

0ccbbaa 212121

10. If two lines having direction cosines 111 n,m,l and 222 n,m,l , are perpendicular, then

0nnmmll 212121

FORM OF EQUATION OF STRAIGHT LINE IN THREE

DIMENTIANOL GEOMETRY:

1) Equation of a line through a given point

a and parallel to a given vector

b :

i. Vector form:

bar

ii. Cartesian form:c

zz

b

yy

a

xx 111

, where ( 111 z,y,x ) is a passing

through point and (a, b, c) are direction ratios of parallel vector

b .



2) Equation of a line passing through two given points

a and

b :

i. Vector form:

abar

ii. Cartesian form: 12

1

12

1

12

1

zz

zz

yy

yy

xx

xx

, where ( 111 z,y,x ) and

( 222 z,y,x ) are passing through points.

Some important formulas based on the equation of lines

1) If the equations of two given lines are

2211 bar andbar , then:

i. Angle between these lines in vector form:

b b

bb

21

21

cos

ii. If these lines are perpendicular to each other, then: 0bb 21

iii. If these lines are parallel, then:

21 bb

iv. If these lines are skew lines, then shortest distance between them =

21

2112

bb

bbaa

v. Distance between two parallel lines

bar andbar 21 , is

b

aa b 12

2) If equations of the lines are1

1

1

1

1

1

c

zz

b

yy

a

xx

&

2

2

2

2

2

2

c

zz

b

yy

a

xx

,then:

i. Angle between these lines is given by: cbacba

ccbbaa cos

22

22

22

21

21

21

212121

ii. If these lines are perpendicular to each other, then: 212121 ccbbaa =0

iii. If these lines are parallel, then: 2

1

2

1

2

1

c

c

b

b

a

a



3) Condition of coplanarity of two lines:

i. Vector form: if the equations are

2211 barandbar

then: 0bbaa 2112

ii. Cartesian form: if the equations are 1

1

1

1

1

1

c

zz

b

yy

a

xx

and

2

2

2

2

2

2

c

zz

b

yy

a

xx

then:

222

111

121212

cba

cba

zzyyxx

= 0

Equations of a plane

1) Equation of a plane in normal form:

i. Vector form: dnr

ˆ , where n is the unit normal vector along the normal from the

origin to the plane, and d is the perpendicular distance from the origin to the plane.

ii. Cartesian form lx + my + nz = d, where l, m, n are the direction cosines of the

normal vector, d is the perpendicular distance from the origin to the plane.

2) Equation of a plane passing through a given point

a and perpendicular to a

given vector

N :

i. Vector form is 0Nar

ii. Cartesian form: A (x – x1) + B (y – y1) + C (z – z1) = 0, where A, B and C are

direction ratios of

N , and ( 111 z,y,x ) is a given point.

3) Equation of a plane passing through three non-collinear points

a , ,

b and

c :

i. vector form: 0acabar

ii. Cartesian form:

131313

121212

111

zzyyxx

zzyyxx

zzyyxx

= 0

4) Intercept form of the equation of a plane: 1c

z

b

y

a

x , where a, b, and c are

intercepts on x, y, and z-axes, respectively.

5) Equation of a plane which passing through the intersection of two given planes:



i. Vector form : if two given planes are 11 dnr

ˆ and 22 dnr

ˆ , then, the equation of

required plane is given by 2121 ddnnr

ˆˆ

ii. Cartesian form: if two given planes are A1x + B1y + C1z = D1 and A2x + B2y + C2z = D2

then, the equation of required plane is given by

(A1x + B1y + C1z – D1) + λ (A2x + B2y + C2z – D2) = 0

Some important formulae based on a plane and a line in three

dimensional geometry

1) Angle between two planes:

i. Vector form: if the equations of planes are 11 dnr

and 22 dnr

then,

21

21

nn

nncos

ii. Cartesian form: if the equations of planes are A1x + B1y + C1z = D1 and

A2x + B2y + C2z = D2, then, 22

22

22

21

21

21

212121

CBACBA

CCBBAAcos

2) Perpendicular distance of a point from a plane:

i. Vector form: if point is P with position vector

a and equation of plane is

dNr

, then, distance =

N

dNa

ii. Cartesian form: if point is P (x1, y1, z1) and equation of plane is Ax + By + Cz =

D, then, distance = 222

111

CBA

DCzByAx

3) Angle between a line and a plane:



i. Vector form: if equation of line is

bar and the equation of plane is

dNr

then,

Nb

Nbsin

ii. Cartesian form: if equation of line is c

zz

b

yy

a

xx 111

and equation of

plane is Ax + By + Cz = D, then, 222222 cbaCBA

CcBbAasin

PROBABILITY

CBACBACBAand Co of A, B Exactly tw

CBAd Cof A, B anAll three

BABA Be of A andExactly on

CBA or Cne of A, BAt least o

BABAnor B Neither A

BABA but not

BAA and B

BA) of A or B least oneA or B (at

ANot A

Notation TheoreticSet EquivalentEvent of nDescriptio verbal

Some useful formulae of probability:

1) )BA(P)B(P)A(P)BA(P

2) )A(P1)A(P

3) )BA(P)A(P)BA(P

4) )BA(P)B(P)BA(P

5) )BA(P1)BA(P



6) )BA(P1)BA(P

Mutually exclusive events:

1) If P (A B) = 0, then A and B are known as mutually exclusive events.

Conditional probability:

1) Probability of the event A given that B has already occurred = P (A | B) = )B(P

)BA(P

2) Probability of the event B given that A has already occurred = P (B | A) = )A(P

)BA(P

3) C|)BA(P)C|B(P)C|A(PC|)BA(P

4) )B|A(P1)B|A(P

Multiplication rule of probability:

1) P (A B) = ( )B|A(P)B(P)BA(P

2) P (A B) = )A|B(P)A(P)BA(P

Independent events:

1) If )B(P)A(P)BA(P , then A and B are independent events.

2) If )B(P)A(P)BA(P , then A and B are not independent and they are known

as dependent events

Total probability:

)E|A(P)E(P..............................)E|A(P)E(P)E|A(P)E(P)A(P nn2211

Bayes’ Theorem:

.., n.........., ........4, 3, 2, 1 any i forn

)E|A(P)E(P

)E|A(P)E(P)A|E(P

1jjj

iii

Random variable: A random variable is real valued function whose domain is

the sample space of a random experiment.

Mean of a random variable: The mean or Expectation of a random variable X is

the sum of the products of all possible values of X by their respective probabilities.

i. e.,

n

1iiixp)X(EX



Variance of a random variable: 2

x = Var (X) = E (X2) – [E (X)]2.

2x = Var (X) =

n

1i

2n

1iii

2ii xpxp

Standard deviation of a random variable:

S. D = )Xvar(

some useful algebraic formulae -...

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