Download Calculus and Analytical Geometry: Tangents, Normals, and Integration and more Cheat Sheet Mathematics in PDF only on Docsity!
S.No. Topic Page No.
INDEX MATHEMATICS
- Straight Line 2 –
- Circle
- Parabola
- Ellips 5 –
- Hyperbola 6 –
- Limit of Function 8 –
- Method of Differentiation 9 –
- Application of Derivatves 11 –
- Indefinite Intedration 14 –
- Definite Integration 17 –
- Fundamental of Mathematics 19 –
- Quadratic Equation 22 –
- Sequence & Series 24 –
- Binomial Theorem 26 –
- Permutation & Combinnation 28 –
- Probability 29 –
- Complex Number 31 –
- Vectors 32 –
- Dimension 35 –
- Solution of Triangle 41 –
- Inverse Trigonometric Functions 44 –
- Statistics 47 –
- Mathematical Reasoning 49 –
- Sets and Relation 50 –
MATHEMATICS
FORMULA BOOKLET - GYAAN SUTRA
STRAIGHT LINE
1. Distance Formula:
2 2 d (x – x ) 1 2 (y – y ) 1 2.
2. Section Formula :
x = m n
mx 2 n x 1
; y = m n
my 2 n y 1
.
3. Centroid, Incentre & Excentre:
Centroid G
y y y , 3
x 1 x 2 x 3 1 2 3 ,
Incentre I (^)
a b c
ay by cy , a b c
ax 1 bx 2 cx 3 1 2 3
Excentre I 1
a b c
ay by cy , a b c
ax 1 bx 2 cx 3 1 2 3
4. Area of a Triangle:
ABC =
x y 1
x y 1
x y 1
2
1
3 3
2 2
1 1
5. Slope Formula:
Line Joining two points (x 1 y 1 ) & (x 2 y 2 ), m = 1 2
1 2 x x
y y
6. Condition of collinearity of three points:
x y 1
x y 1
x y 1
3 3
2 2
1 1 = 0
7. Angle between two straight lines :
tan = 1 2
1 2 1 m m
m m
.
CIRCLE
1. Intercepts made by Circle x 2 + y 2 + 2gx + 2fy + c = 0 on the Axes:
(a) 2 g c 2 on x -axis (b) 2 f c 2 on y - aixs
2. Parametric Equations of a Circle:
x = h + r cos ; y = k + r sin
3. Tangent :
(a) Slope form : y = mx ± (^) a 1 m^2
(b) Point form : xx 1 + yy 1 = a^2 or T = o (c) Parametric form : x cos + y sin = a.
4. Pair of Tangents from a Point: SS 1 = T². 5. Length of a Tangent : Length of tangent is S 1 6. Director Circle: x^2 + y^2 = 2a^2 for x^2 + y^2 = a^2 7. Chord of Contact: T = 0 1. Length of chord of contact = 2 2 R L
2 L R
- Area of the triangle formed by the pair of the tangents & its chord of
contact = (^22)
3
R L
R L
- Tangent of the angle between the pair of tangents from (x 1 , y 1 )
=
2 2 L R
2 R L
- Equation of the circle circumscribing the triangle PT 1 T 2 is : (x x 1 ) (x + g) + (y y 1 ) (y + f) = 0. 8. Condition of orthogonality of Two Circles: 2 g 1 g 2 + 2 f 1 f 2 = c 1 + c 2. 9. Radical Axis : S 1 S 2 = 0 i.e. 2 (g 1 g 2 ) x + 2 (f 1 f 2 ) y + (c 1 c 2 ) = 0. 10. Family of Circles: S 1 + K S 2 = 0, S + KL = 0.
PARABOLA
1. Equation of standard parabola :
y^2 = 4ax, Vertex is (0, 0), focus is (a, 0), Directrix is x + a = 0 and Axis is y = 0. Length of the latus rectum = 4a, ends of the latus rectum are L(a, 2a) & L’ (a, 2a).
2. Parametric Representation: x = at² & y = 2at 3. Tangents to the Parabola y² = 4ax: 1. Slope form y = mx + m
a (m 0) 2. Parametric form ty = x + at^2
- Point form T = 0 4. Normals to the parabola y² = 4ax :
y y 1 = 2 a
y 1 (^) (x x 1 ) at (x1, y 1 )^ ; y = mx^ ^ 2am^ ^ am
(^3) at (am (^2) 2am) ;
y + tx = 2at + at^3 at (at^2 , 2at).
ELLIPSE
1. Standard Equation : 2
2
2
2
b
y
a
x = 1, where a > b & b² = a² (1 e²).
Eccentricity: e = 2
2
a
b 1 , (0 < e < 1),^ Directrices :^ x = ±^ e
a .
Focii : S (± a e, 0). Length of, major axes = 2a and minor axes = 2b Vertices : A ( a, 0) & A (a, 0).
Latus Rectum : = ^
2
2 2 a 1 e a
2 b
2. Auxiliary Circle : x² + y² = a² 3. Parametric Representation : x = a cos & y = b sin 4. Position of a Point w.r.t. an Ellipse:
The point P(x1, y 1 ) lies outside, inside or on the ellipse according as;
b
y
a
x 2
2 1 2
2 1 > < or = 0.
5. Position of A Point 'P' w.r.t. A Hyperbola :
S 1 1
b
y
a
x 2
2 1 2
2 1 >, = or < 0 according as the point (x1, y 1 ) lies inside, on
or outside the curve.
6. Tangents :
(i) Slope Form : y = m x (^) a 2 m^2 b^2
(ii) Point Form : at the point (x 1, y 1 ) is (^1) b
y y
a
x x 2
1 2
1 .
(iii) Parametric Form : (^) 1 b
yt an
a
x sec
.
7. Normals :
(a) at the point P (x 1 , y 1 ) is 1
2
1
2
y
b y
x
a x = a^2 + b^2 = a^2 e^2.
(b) at the point P (a sec , b tan ) is
tan
b y
sec
a x = a^2 + b^2 = a^2 e^2.
(c) Equation of normals in terms of its slope 'm' are y
= mx
2 2 2
2 2
a b m
a b m
.
8. Asymptotes :^0 b
y
a
x and 0 b
y
a
x .
Pair of asymptotes :^0 b
y
a
x 2
2
2
2 .
9. Rectangular Or Equilateral Hyperbola : xy = c^2 , eccentricity is 2.
Vertices : (± c,^ ±c) ; Focii : (^) 2 c , 2 c. Directrices : x + y = (^2) c
Latus Rectum ( l ) : = 2 2 c = T.A. = C.A.
Parametric equation x = ct, y = c/t, t R – {0}
Equation of the tangent at P (x 1 ,^ y 1 ) is 1 y 1
y
x
x (^) = 2 & at P (t) is t
x
Equation of the normal at P (t) is x t^3 y t = c (t^4 1). Chord with a given middle point as (h, k) is kx + hy = 2hk.
LIMIT OF FUNCTION
1. Limit of a function f(x) is said to exist as x a when,
^ h 0
Limit f (a h) = h 0
Limit (^) f (a + h) = some finite value M.
(Left hand limit) (Right hand limit)
2. Indeterminant Forms:
,
, 0 , º, 0º,and 1 .
3. Standard Limits :
x 0
Limit (^) x
sin x = (^) x 0
Limit (^) x
tan x = (^) x 0
Limit (^) x
tan x
1
= (^) x 0
Limit (^) x
sin x
1
= (^) x 0
Limit (^) x
e 1
x = (^) x 0
Limit x
n( 1 x ) = 1
x 0
Limit (1 + x)
1/x (^) = x
Limit
x
x
(^) = e, x 0
Limit x
a 1 x = logea, a > 0,
x a
Limit x a
x a
n n
= nan – 1.
4. Limits Using Expansion
(i) .........a^0 3!
x ln a
2!
xln a
1!
xln a a 1
2 2 3 3 x
(ii) ...... 3!
x
2!
x
1!
x e 1
2 3 x
(iii) ln (1+x) = .........for^1 x^1 4
x
3
x
2
x x
2 3 4
(iv) ..... 7!
x
5!
x
3!
x sinx x
3 5 7
2. Basic Theorems
1. dx
d (f ± g) = f(x) ± g(x) 2. dx
d (k f(x)) = k dx
d f(x)
3. dx
d (f(x). g(x)) = f(x) g(x) + g(x) f(x)
4. dx
d
g(x )
f (x )
g(x)
g(x)f(x) f(x)g(x ) 2
5. dx
d (f(g(x))) = f(g(x)) g(x)
Derivative Of Inverse Trigonometric Functions.
dx
d sin x
1 x
, dx
d cos x
, for – 1 < x < 1.
dx
d tan x
, dx
d cot x
(x R)
dx
d sec x
2
, dx
d cosec x
= – |x| x 1
2
, for x (– , – 1) (1, )
3. Differentiation using substitution
Following substitutions are normally used to simplify these expression.
(i) (^) x 2 a^2 by substituting x = a tan , where – 2
< 2
(ii) (^) a 2 x^2 by substituting x = a sin , where – 2
2
(iii) (^) x 2 a^2 by substituting x = a sec , where [0, ], 2
(iv) a x
x a
by substituting x = a cos , where (0, ].
4. Parametric Differentiation
If y = f() & x = g() where is a parameter, then
dx/d
dy/ d
dx
dy .
5. Derivative of one function with respect to another
Let y = f(x); z = g(x) then g'(x)
f'(x )
dz/d x
dy/d x
d z
d y .
6. If F(x) =
u(x) v(x) w(x )
l(x) m(x) n(x )
f(x) g(x) h(x )
, where f, g, h, l, m, n, u, v, w are differentiable
functions of x then F (x) = u(x) v(x) w(x )
l(x) m(x) n(x )
f'(x) g'(x) h'(x )
u(x) v(x) w(x )
l'(x) m'(x) n'(x )
f(x) g(x) h(x )
u'(x) v'(x) w'(x )
l(x) m(x) n(x )
f(x) g(x) h(x )
APPLICATION OF DERIVATIVES
1. Equation of tangent and normal
Tangent at (x 1 , y 1 ) is given by (y – y 1 ) = f(x 1 ) (x – x 1 ) ; when, f(x 1 ) is real.
And normal at (x 1 , y 1 ) is (y – y 1 ) = – f(x)
(x – x 1 ), when f(x 1 ) is nonzero
real.
2. Tangent from an external point
Given a point P(a, b) which does not lie on the curve y = f(x), then the equation of possible tangents to the curve y = f(x), passing through (a, b) can be found by solving for the point of contact Q.
f(h) = h a
f(h) b
7. Lagrange’s Mean Value Theorem (LMVT) :
If a function f defined on [a, b] is
(i) continuous on [a, b] and (ii) derivable on (a, b) then there exists at least one real numbers between a and b (a < c < b) such
that b a
f(b) f(a )
= f(c)
8. Useful Formulae of Mensuration to Remember : 1. Volume of a cuboid = bh. 2. Surface area of cuboid = 2(b + bh + h). 3. Volume of cube = a^3 4. Surface area of cube = 6a^2 5. Volume of a cone = 3
r 2 h.
6. Curved surface area of cone = r ( = slant height) 7. Curved surface area of a cylinder = 2rh. 8. Total surface area of a cylinder = 2rh + 2r^2. 9. Volume of a sphere = 3
r^3.
10. Surface area of a sphere = 4r^2. 11. Area of a circular sector = 2
r^2 , when is in radians.
12. Volume of a prism = (area of the base) × (height). 13. Lateral surface area of a prism = (perimeter of the base) × (height). 14. Total surface area of a prism = (lateral surface area) + 2 (area of the base) (Note that lateral surfaces of a prism are all rectangle). 15. Volume of a pyramid = 3
(area of the base) × (height).
16. Curved surface area of a pyramid = 2
(perimeter of the base) ×
(slant height). (Note that slant surfaces of a pyramid are triangles).
INDEFINITE INTEGRATION
1. If f & g are functions of x such that g(x) = f(x) then,
f(x)^ dx = g(x)^ + c^ ^
d
d x
{g(x)+c} = f(x), where c is called the constant of
integration.
2. Standard Formula:
(i) (^) (ax + b)n^ dx =
ax b
a n
n
1
(ii) (^)
dx
ax b
=
a
n (ax + b) + c
(iii) (^) eax+b^ dx =
a
eax+b^ + c
(iv) (^) apx+q^ dx =
p
a
n a
p x q
(v) (^) sin (ax + b) dx =
a
cos (ax + b) + c
(vi) (^) cos (ax + b) dx =
a
sin (ax + b) + c
(vii) (^) tan(ax + b) dx =
a
n sec (ax + b) + c
(viii) (^) cot(ax + b) dx =
a
n sin(ax + b)+ c
(ix) (^) sec² (ax + b) dx =
1
a
tan(ax + b) + c
(x) (^) cosec²(ax + b) dx = (^)
1
a
cot(ax + b)+ c
3. Integration by Subsitutions
If we subsitute f(x) = t, then f (x) dx = dt
4. Integration by Part :
f (x)g(x) dx = f(x)^ ^
g( x)dx – ^ f(x)^ ^ g(x)^ dx dx dx
d
5. Integration of type
2 (^2 )
dx dx , , ax bx c dx ax bx c (^) ax bx c
Make the substitution
b x t 2a
6. Integration of type
2 (^2 )
px q px q dx, dx, (px q) ax bx c dx ax bx c (^) ax bx c
Make the substitution x +
b
2a
= t , then split the integral as some of two
integrals one containing the linear term and the other containing constant term.
7. Integration of trigonometric functions
(i) (^) 2
dx
a b sin x
OR^ 2
dx
a bcos x
OR (^) 2 2
dx
a sin x b sin x cos x c cos x
put tan x = t.
(ii)
dx
a b sin x
OR
dx
a bcos x
OR
dx
a b sin x c cos x
put tan^
x
2
= t
(iii)
a.cos x b.sin x c
.cos x m.sin x n
dx. Express Nr A(Dr) + B
d
dx
(Dr) + c & proceed.
8. Integration of type
2
4 2
x 1 dx x Kx 1
where K is any constant.
Divide Nr & Dr by x² & put x
x
= t.
9. Integration of type
dx
(ax b) px q
OR (^) 2
dx
(ax bx c) px q
; put px + q = t^2.
10. Integration of type
2
dx
(ax b) px qx r
(^) , put ax + b =
t
;
2 2
dx
(ax b) px q
(^) , put x =
t
DEFINITE INTEGRATION
Properties of definite integral
1.
b
a
f (x )dx =
b
a
f (t )dt 2.
b
a
f (x )dx = –^
a
b
f( x )dx
3. (^)
b
a
f (x )dx =
c
a
f (x )dx +
b
c
f (x )dx
4.
a
a
f (x )dx =
a
0
(f (x) f( x ))dx =
0 , f(–x) –f(x )
2 f(x)dx , f(–x) f(x )
a
0
5.
b
a
f( x )dx =
b
a
f( a b x )dx
FUNDAMENTAL OF MATHEMATICS
Intervals :
Intervals are basically subsets of R and are commonly used in solving inequalities or in finding domains. If there are two numbers a, b R such that a < b, we can define four types of intervals as follows :
Symbols Used (i) Open interval : (a, b) = {x : a < x < b} i.e. end points are not included. ( ) or ] [ (ii) Closed interval : [a, b] = {x : a x b} i.e. end points are also included. [ ] This is possible only when both a and b are finite. (iii) Open-closed interval : (a, b] = {x : a < x b} ( ] or ] ] (iv) Closed - open interval : [a, b) = x : a x < b} [ ) or [ [
The infinite intervals are defined as follows : (i) (a, ) = {x : x > a} (ii) [a, ) = {x : x a} (iii) (– , b) = {x : x < b} (iv) (, b] = {x : x b} (v) (– ) = {x : x R}
Properties of Modulus :
For any a, b R |a| 0, |a| = |–a|, |a| a, |a| –a, |ab| = |a| |b|,
b
a
|b|
| a | , |a + b| |a| + |b|, |a – b| ||a| – |b||
Trigonometric Functions of Sum or Difference of Two Angles:
(a) sin (A ± B) = sinA cosB ± cosA sinB 2 sinA cosB = sin(A+B) + sin(AB) and and 2 cosA sinB = sin(A+B) sin(AB)
(b) cos (A ± B) = cosA cosB sinA sinB 2 cosA cosB = cos(A+B) + cos(AB) and 2sinA sinB = cos(AB) cos(A+B) (c) sin²A sin²B = cos²B cos²A = sin (A+B). sin (A B) (d) cos²A sin²B = cos²B sin²A = cos (A+B). cos (A B)
(e) cot (A ± B) = cotB cotA
cotAcotB 1
(f) tan (A + B + C) = 1 tanAtanB tanBtanC tanCtanA
tanA tanB tanC tanAtanBtan C
.
Factorisation of the Sum or Difference of Two Sines or Cosines:
(a) sinC + sinD = 2 sin 2
C D
cos 2
C D
(b) sinC sinD = 2 cos 2
C D
sin 2
C D
(c) cosC + cosD = 2 cos 2
C D
cos 2
C D
(d) cosC cosD = 2 sin 2
C D
sin 2
C D
Multiple and Sub-multiple Angles :
(a) cos 2A = cos²A sin²A = 2cos²A 1 = 1 2 sin²A; 2 cos² 2
= 1 + cos , 2 sin² 2
= 1 cos .
(b) sin 2A = 1 tan A
2 tan A 2
, cos 2A = 1 tanA
1 tan A 2
2
(c) sin 3A = 3 sinA 4 sin^3 A (d) cos 3A = 4 cos^3 A 3 cosA
(e) tan 3A = 1 3 tan A
3 tanA tan A 2
3
Important Trigonometric Ratios:
(a) sin n = 0 ; cos n = 1 ; tan n = 0, where n
(b) sin 15° or sin 12
= 2 2
= cos 75° or cos 12
;
cos 15° or cos 12
= 2 2
= sin 75° or sin 12
;
tan 15° = 31
= (^2) 3 = cot 75° ; tan 75°
= 31
= (^2) 3 = cot 15°
(c) sin 10
or sin 18° = 4
& cos 36° or cos 5
= 4
