Wisdom institute of mathematics Formulae for XII class/D.K. Rohaj
Wisdom institute of mathematics//Narela, Delhi – 40//D. k. Rohaj//mo: 9811469438 Page 1
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
Wisdom institute of mathematics Formulae for XII class/D.K. Rohaj
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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.
Wisdom institute of mathematics Formulae for XII class/D.K. Rohaj
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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.
Wisdom institute of mathematics Formulae for XII class/D.K. Rohaj
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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
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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
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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
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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
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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
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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.
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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
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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
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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.
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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
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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 ).
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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.
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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
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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
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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
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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:
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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:
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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)
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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
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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
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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,
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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:
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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.
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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 .
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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
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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:
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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:
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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
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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
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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(