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Mathematics Study Notes: Relations, Functions, Calculus, and Geometry, Schemes and Mind Maps of Mathematics

Mathematics

This comprehensive set of study notes covers essential mathematical concepts, including relations and functions, inverse trigonometric functions, calculus, matrices, and three-dimensional geometry. It provides detailed explanations, examples, and exercises to enhance understanding and problem-solving skills. The notes are particularly useful for students preparing for high school or university mathematics courses.

Typology: Schemes and Mind Maps

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Vidyarthi Tutorials H10 Raghu Nagar, Pankha Road, opp.Janakpuri. Ph 9818084221 1

Brilliant Academy, 303, 3rd Flr, Aggarwal Tower, sec5 Market, Dwarka, Ph 9015048802

POINTS TO

REMEMBER IN CLASS

XII MATHEMATICS

Balraj Khurana

INDEX

1. Relations and functions - Pg 2

2. Inverse trigonometric functions - Pg 5

3. Calculus identities - Pg 6

4. Continuity - Pg 7

5. Differentiation - Pg 8

6. Application of derivative - Pg 9

7. Indefinite integral - Pg 11

8. Definite integral - Pg 14

9. Matrices - Pg 16

10. Determinants - Pg 19

11. Solution of system of linear equations - Pg 21

Partial preview of the text

Download Mathematics Study Notes: Relations, Functions, Calculus, and Geometry and more Schemes and Mind Maps Mathematics in PDF only on Docsity!

POINTS TO

REMEMBER IN CLASS

XII MATHEMATICS

By

Balraj Khurana

INDEX

1. Relations and functions - Pg 2

2. Inverse trigonometric functions - Pg 5

3. Calculus identities - Pg 6

4. Continuity - Pg 7

5. Differentiation - Pg 8

6. Application of derivative - Pg 9

7. Indefinite integral - Pg 11

8. Definite integral - Pg 14

9. Matrices - Pg 16

10. Determinants - Pg 19

11. Solution of system of linear equations - Pg 21

RELATIONS AND FUNCTIONS

I. RELATION

i. Let A and B be two sets. A relation between A and B is a collection of ordered pairs (a, b) such that a A and b B ii. If 𝑅: 𝐴 → 𝐵 is a relation from A to B, then 𝑅 ⊆ 𝐴 × 𝐵 iii. If n(A) = m, n(B) = n ,then total number of relations from A to B is 2mn. iv. Domain of R = {𝑎: (𝑎, 𝑏) ∈ 𝑅} v. Range of R = {𝑏: (𝑎, 𝑏) ∈ 𝑅} vi. Co-domain of R = 𝐵

II. Equivalence Relation

Let S be a set and R a relation between S and itself. We call R an equivalence relation on S if R has the following three properties:

 Reflexivity : Every element of S is related to itself ⟹ (𝑎, 𝑎) ∈ 𝑅 ∀ 𝑎 ∈ 𝑆.  Symmetry : If a is related to b then b is related to a. (𝑎, 𝑏) ∈ 𝑅 ⟹ (𝑏, 𝑎) ∈ 𝑅 ∀ 𝑎, 𝑏 ∈ 𝑆.  Transitivity : If a is related to b and b is related to c , then a is related to c. (𝑎, 𝑏) ∈ 𝑅 , (𝑏, 𝑐) ∈ 𝑅 ⟹ (𝑎, 𝑐) ∈ 𝑅 ∀ 𝑎, 𝑏, 𝑐 ∈ 𝑆.

Antisymmetric - A relation is antisymmetric if a R b and b R a⟹ a = b for all values a and b.

III. FUNCTIONS :  Definition - Any relation on A x B in which i. No two second elements have a common first element and ii. Every first element has a corresponding second element is called a function. It is also called mapping. A function is said to map an element x in its domain to an element y in its range. 𝑓: 𝐴 → 𝐵 𝑜𝑟 𝑓: 𝑥 → 𝑓(𝑥) 𝑡ℎ𝑒𝑛 𝑓(𝑥) = 𝑦 where y is a function of x.  DOMAIN - The set of all the first elements of the ordered pairs of a function is called the domain  RANGE - The set of all the second elements of the ordered pairs of a function is called the range  CODOMAIN - If (a, b) is an ordered pair of the function 𝑓: 𝐴 → 𝐵 then the set B is called the Co-Domain. The range is a subset of the co-domain.

IV. Some important facts about a function from A to B:

Ceiling function x = Least integer that is greater than or equal to x.domain= R; range = Z; discontinuous
Reciprocal function f(x) = 1 𝑥 ; domain = R - {o};range = R - {o} continuous in R+ and R-
Modulus function f(x) = |𝑥| = {

; Domain = R; Range = R + ; continuous.

Signum function f(x) = {

|𝑥| 𝑥 , ∀𝑥 ≠ 0 0 , 𝑥 = 0

; domain = R ;range = {-1 , 0 ,1}; discontinuous.

VII. COMPOSITION OF FUNCTIONS - function composition is the application of one function to the results of another. For instance, the functions f : X → Y and g : Y → Z can be composed by computing the output of g when it has an input of f(x) instead of x. A function g ∘ f : X → Z defined by ( g ∘ f )( x ) = g ( f ( x )) for all x in X.

 The composition of functions is always associative. That is, if f , g , and h are three functions with suitably chosen domains and codomains, then f ∘ ( g ∘ h ) = ( f ∘ g ) ∘ h ,  The functions g and f are said to commute with each other if g ∘ f = f ∘ g.

VIII. INVERSE OF A FUNCTION - Let ƒ be a bijective function whose domain is the set X , and whose range is the set Y. Then, if it exists, the inverse of ƒ is the function ƒ–^1 with domain Y and range X , defined by the following rule:

 A function with a codomain is invertible if and only if it is both one-to-one and onto or a bijection and has the property that every element y ∈ Y corresponds to exactly one element x ∈ X.  Domain (f) = range(f-1) and range (f) = domain (f-1)

 Inverses and composition - If ƒ is an invertible function with domain X and range Y , then

 There is a symmetry between a function and its inverse. Specifically, if the inverse of ƒ is ƒ–^1 , then the inverse of ƒ–^1 is the original function ƒ. i.e. If 𝑓−1^ ∘ 𝑓(𝑥) = 𝐼𝑋 then 𝑓 ∘ 𝑓−1(𝑦) = 𝐼𝑌  Only one-to-one functions have a unique inverse.  If the function is not one-to-one, the domain of the function must be restricted so that a portion of the graph is one-to-one. You can find a unique inverse over that portion of the restricted domain.  The domain of the function is equal to the range of the inverse. The range of the function is equal to the domain of the inverse.

IX. Inverse of a composition

The inverse of g o ƒ is ƒ–^1 o g –^1.

The inverse of a composition of functions is given by the formula

X. BINARY OPERATION on a set – Let A be a non-empty set.A binary operation * on the set A is a function ∗: 𝐴 × 𝐴 → 𝐴 such that a*b ∈ 𝐴∀ (𝑎, 𝑏) ∈ 𝐴 × 𝐴

 Commutative property - A binary operation * on the set A is said to be commutative if ab = ba** ∀ 𝑎, 𝑏 ∈ 𝐴.  Associative property - A binary operation * on the set A is said to be associative if a(bc) = (a* b)c* ∀ 𝑎, 𝑏, 𝑐 ∈ 𝐴  Identity element of a binary operation – Given a binary operation ∗: 𝐴 × 𝐴 → 𝐴, a unique element e ∈ 𝐴, if it exists , is called the identity element for * if ae = a = ea** ∀ 𝑎 ∈ 𝐴.  Inverse of an element - Given a binary operation ∗: 𝐴 × 𝐴 → 𝐴, the identity element e ∈ 𝐴, an element a is called invertible w.r.t.* if ∃𝑏 ∈ 𝐴 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 𝐚 ∗ 𝐛 = 𝐞 = 𝐛 ∗ 𝐚 .Then b is called the inverse of a and is denoted by a-1^ i.e. a * a-1= e = a-1^ *** a.**

INVERSE TRIGONOMETRIC FUNCTIONS

INVERSE TRIGONOMETRIC FUNCTIONS or cyclometric functions - are the so-called inverse functions of the trigonometric functions, when their domain are restricted to principal value branch to make the trigonometric functions bijectiveThe principal inverses are listed in the following table.

Name Usual notation Definition Domain of x for real result

Range of usual principal value (radians)

Range of usual principal value (degrees)

arcsine y = sin-^1 x x = sin y −1 ≤ x ≤ 1 −π/2 ≤ y ≤ π/2 −90° ≤ y ≤ 90°

arccosine y = cos-^1 x x = cos y −1 ≤ x ≤ 1 0 ≤ y ≤ π 0° ≤ y ≤ 180°

 Use 𝑠𝑖𝑛−1^ ( 𝑝 ℎ) = 𝑐𝑜𝑠

V. SUM FORMULA

 𝑠𝑖𝑛−1𝑥 ± 𝑠𝑖𝑛−1𝑦 = 𝑠𝑖𝑛−1[𝑥√1 − 𝑦^2 ± 𝑦√1 − 𝑥^2 ].

 𝑐𝑜𝑠−1𝑥 ± 𝑐𝑜𝑠−1𝑦 = 𝑐𝑜𝑠−1[𝑥𝑦 ∓ √1 − 𝑥^2 √1 − 𝑦^2 ]

 𝑡𝑎𝑛−1𝑥 ± 𝑡𝑎𝑛−1𝑦 = 𝑡𝑎𝑛−1^ (

𝑥±𝑦 1∓𝑥𝑦)  𝑡𝑎𝑛−1𝑥 + 𝑡𝑎𝑛−1𝑦 + 𝑡𝑎𝑛−1𝑧 = 𝑡𝑎𝑛−1^ ( 𝑥+𝑦+𝑧−𝑥𝑦𝑧 1−𝑥𝑦−𝑦𝑧−𝑧𝑥)

VI. MULTIPLE FORMULA

 2𝑠𝑖𝑛−1𝑥 = 𝑠𝑖𝑛−1[2𝑥√1 − 𝑥^2 ]

 2𝑐𝑜𝑠−1𝑥 = 𝑐𝑜𝑠−1[2𝑥^2 − 1]

 2 𝑡𝑎𝑛−1𝑥 = 𝑡𝑎𝑛−^

2𝑥 1−𝑥^2 = 𝑠𝑖𝑛

−1 2𝑥 1+𝑥^2 = 𝑐𝑜𝑠

−1 1−𝑥^2 1+𝑥^2  3𝑠𝑖𝑛−1𝑥 = 𝑠𝑖𝑛−1[3𝑥 − 4𝑥^3 ]  3𝑐𝑜𝑠−1𝑥 = 𝑐𝑜𝑠−1[4𝑥^3 − 3𝑥]  3 𝑡𝑎𝑛−1𝑥 = 𝑡𝑎𝑛− 3𝑥−𝑥^3 1−3𝑥^2

CALCULUS

I. ALGEBRAIC AND TRIGONOMETRICIDENTITIES

a^3 + b^3 = (a+b)(a^2 – ab + b^2 )
a^3 - b^3 = (a - b)(a^2 + ab + b^2 )
sin^ cos (^2) x  2 x  1 4.^1  tan^2 x  sec^2 x
1  cot^2 x  csc^2 x
Sin (u±𝑣) = sin^ u^ ^ cos^ v^ ^ cos^ u^ sin v
1. cos (u±𝑣) = cos^ u^ ^ cos^ v^ sin^ u^ sin v
2. tan(u±𝑣) =

tan tan tan tan

u v u v

2𝑡𝑎𝑛𝑢 1+𝑡𝑎𝑛^2 𝑢

cos2u = cos^2 u – sin^2 u = 2 cos^2 u – 1 = 1 – 2sin^2 u = 1−𝑡𝑎𝑛^2 𝑢 1+𝑡𝑎𝑛^2 𝑢 11.tan( 2 u ) 

tan tan

u  u

Sin3u= 3sinu – 4sin^3 u
Cos3u = 4cos^3 u – 3cosu
Tan3u = 3𝑡𝑎𝑛𝑢−𝑡𝑎𝑛^3 𝑢 1−3𝑡𝑎𝑛^2 𝑢 15.sin^2 u 

 cos u

16.cos (^2) u ^1 2

 cos u

tan^2 u 

cos cos

u u

Sin^3 u = 3𝑠𝑖𝑛𝑢−𝑠𝑖𝑛3𝑢 4
cos^3 u =

3𝑐𝑜𝑠𝑢+𝑐𝑜𝑠3𝑢 4

sinu.sinv = 1 2

[𝑐𝑜𝑠(𝑢 − 𝑣) − 𝑐𝑜𝑠(𝑢 + 𝑣)]

cosu.cosv = 1 2

[𝑐𝑜𝑠(𝑢 + 𝑣) + 𝑐𝑜𝑠(𝑢 − 𝑣)]

Sinu.cosv = 1 2

[𝑠𝑖𝑛(𝑢 + 𝑣) + 𝑠𝑖𝑛(𝑢 − 𝑣)]

cosu.sinv = 1 2 [𝑠𝑖𝑛(𝑢 + 𝑣) − 𝑐𝑜𝑠(𝑢 − 𝑣)]
sinu + sinv = 2𝑠𝑖𝑛 (𝑢+𝑣) 2 𝑐𝑜𝑠^

(𝑢−𝑣) 2

sinu - sinv = 2𝑐𝑜𝑠 (𝑢+𝑣) 2 𝑠𝑖𝑛^

(𝑢−𝑣) 2

cosu + cosv = 2𝑐𝑜𝑠 (𝑢+𝑣) 2 𝑐𝑜𝑠^

(𝑢−𝑣) 2

27. cosu - cosv = 2𝑠𝑖𝑛 (𝑢+𝑣) 2 𝑐𝑜𝑠^

(𝑣−𝑢) 2

law of sines: a A

b B

c sin sin sin C

  law of cosines: c^2^  a^2  b^2^  2 ab cos C

area of triangle using trig.

Area sin 2

 ac B

II. CONIC SECTION FORMULA

1. Circle formula: ^ ^ ^ 

(^2 2 ) x  h  y  k  r

2. Parabola formula: ^ ^ ^ 

2 x  h  4 p y  k

Ellipse formula:

x a

y b

c a b

2 2

  1  2 ^2

Hyperbola formula:

x a

y b

c a b

2 2

  1  2 ^2

eccentricity:

c a

parameterization of ellipse:

2 2 2 2 1 becomes^ cos^ ,^ sin

x y x a t y b t a b

III. FORMULAS OF LIMITS

a. Change of base rule for logs: log^

ln

a x^

x

a

b. lim

sin x

x  (^0) x

c. lim

sin x

x  x

d. lim 𝑥→𝑎

𝑥𝑛−𝑎𝑛 𝑥−𝑎 = 𝑛𝑎

𝑛−

e. lim 𝑥→

𝑒𝑥− 𝑥 = 1 f. lim 𝑥→

𝑎𝑥− 𝑥 = 𝑙𝑜𝑔𝑒𝑎 g. lim 𝑥→

log(1+𝑥) 𝑥 = 1

IV. CONTINUITY

 DEFINITION - Continuity of a function(x) at a point – A function f(x) is said to be continuous at the point x = a if lim 𝑥→𝑎

 Continuity of a function f(x) at x = a means i. f(x) is defined at a i.e. the point a lies in the domain of f ii. lim 𝑥→𝑎

𝑓(𝑥)𝑒𝑥𝑖𝑠𝑡𝑠 𝑖. 𝑒. lim 𝑥→𝑎−^

𝑓(𝑥) = lim 𝑥→𝑎+^

III. RULES OF DIFFERENTIATION

 Chain rule : if y = f(u) and u = g(x) then 𝑑𝑦 𝑑𝑥 =^

𝑑𝑓 𝑑𝑢.^

𝑑𝑢 𝑑𝑥  Product rule : If u and v are two functions of x then 𝑑(𝑢.𝑣) 𝑑𝑥 = 𝑢.^

𝑑𝑣 𝑑𝑥 + 𝑣.^

𝑑𝑢 𝑑𝑥 = 𝑢𝑣

 Quotient rule :If u and v are two functions of x then 𝑑 𝑑𝑥 (

𝑢 𝑣) =^

𝑣𝑢′−𝑢𝑣′ 𝑣^2

 Parametric differentiation : if y =f(t), x= g(t) then , dy dx

dy dt dx dt

 Derivative formula for inverses

df dx df dx

x f a x a



 

( )

 Logarithmic differentiation : If y = f(x)g(x)^ then take log on both the sides. Write logy = g(x) log[f(x)]. Differentiate by applying suitable rule for differentiation.  If y is sum of two different exponential function u and v, i.e. y = u + v. Find 𝑑𝑢 𝑑𝑥 𝑎𝑛𝑑^

𝑑𝑣 𝑑𝑥 by logarithmic differentiation separately then evaluate 𝑑𝑦 𝑑𝑥 as^

𝑑𝑦 𝑑𝑥 =^

𝑑𝑢 𝑑𝑥 +^

𝑑𝑣 𝑑𝑥

 Intermediate Value Theorem : If a function is continuous between a and b , then it takes on every value

between f ( ) a and f b ( ).

 Extreme Value Theorem :If f is continuous over a closed interval, then f has a maximum and minimum value over that interval.

 Mean Value Theorem(for derivatives) : If f ( ) x is a continuous function over a b , , and f(x) is

differentiable in ( a,b )then at some point c between a and b :

f b f a b a

f c

^  (the tangent at x = c is

parallel to the chord joining (a, f(a)) and (b , f(b)) ) 

 Rolle’s Theorem If (i) f ( ) x is a continuous function over a b , , (ii) f(x) is differentiable in ( a,b ) (iii) f( a )

= f(b)then there exists some point c between a and b such that f’(c) = 0 ( the tangent at x = c is parallel

to x axis )

VI. APPLICATION OF DERIVATIVE

I. APPROXIMATIONS, DIFFERENTIALS AND ERRORS

 Absolute error - The increment ∆𝑥 in x is called the absolute error in x.

 Relative error - If ∆𝑥 is an error in x , then Δ𝑥 x is called the relative error in x.  Percentage error - If ∆𝑥 is an error in x , then Δ𝑥 x × 100^ is called the percentage error in x  Approximation -

1. Take the quantity given in the question as y + ∆𝑦= f(x + ∆𝑥) 2. Take a suitable value of x nearest to the given value. Calculate ∆𝒙 3. Calculate y= f(x) at the assumed value of x.]

calculate 𝑑𝑦𝑑𝑥 at the assumed value of x 5. Using differential calculate ∆𝑦 = 𝑑𝑦𝑑𝑥 × ∆𝑥 6. find the approximate value of the quantity asked in the question as y + ∆𝑦, from the values of y and ∆𝑦 evaluated in step 3 and 5. II. Tangents and normals –  Slope of the tangent to the curve y = f(x) at the point (x 0 ,y 0 ) is given by 𝑑𝑦 𝑑𝑥}(𝑥 0 ,𝑦 0 )  Equation of the tangent to the curve y = f(x) at the point (x 0 ,y 0 ) is (y - y 0 ) = 𝑑𝑦 𝑑𝑥}(𝑥 0 ,𝑦 0 ) (x − x^0 ).  Slope of the normal to the curve y = f(x) at the point (x 0 ,y 0 ) is given by − 𝑑𝑥 𝑑𝑦}(𝑥 0 ,𝑦 0 )  Equation of the normal to the curve y = f(x) at the point (x 0 ,y 0 ) is (y - y 0 ) = − 𝑑𝑥 𝑑𝑦}(𝑥 0 ,𝑦 0 )^ (x − x^0 )  To curves y = f(x) and y = g(x) are orthogonal means their tangents are perpendicular to each other at the point of contact  𝑡ℎ𝑒 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 𝑜𝑟𝑡ℎ𝑜𝑔𝑜𝑛𝑎𝑙𝑖𝑡𝑦 𝑜𝑓 𝑡𝑤𝑜 𝑐𝑢𝑟𝑣𝑒𝑠 𝑐 1 𝑎𝑛𝑑 𝑐 2 𝑖𝑠 𝑑𝑦 𝑑𝑥]𝑐 1 ×^

𝑑𝑦 𝑑𝑥]𝑐 2 = −

III. Increasing/Decreasing Functions  Definition of an increasing function: A function f(x) is "increasing" at a point x 0 if and only if there exists some interval I containing x 0 such that f(x 0 ) > f(x) for all x in I to the left of x 0 and f(x 0 ) < f(x) for all x in I to the right of x 0.  Definition of a decreasing function: A function f(x) is "decreasing" at a point x 0 if and only if there exists some interval I containing x 0 such that f(x 0 ) < f(x) for all x in I to the left of x 0 and f(x 0 ) > f(x) for all x in I to the right of x 0.  To find the intervals in which a given function is increasing or decreasing

Differentiate the given function y = f(x), to get f’(x)
Solve f’(x) = 0 to find the critical points.
Consider all the subintervals of R formed by the critical points.( no. of subintervals will be one more than the no. of critical points. )
Find the value of f’(x) in each subinterval.
f’(x) > 0 implies f(x) is increasing and f’(x) < 0 implies f(x) is decreasing.

VII. CONCAVITY  Definition of a concave up curve: f(x) is "concave up" at x 0 if and only if f '(x) is increasing at x 0 which means f”(x)> 0 at x 0 i.e. it is a minima.  Definition of a concave down curve: f(x) is "concave down" at x 0 if and only if f '(x) is decreasing at x 0 which means f”(x) < 0 at x 0 i.e. it is a maxima.  The first derivative test: If f '(x 0 ) exists and is positive, then f(x) is increasing at x 0. If f '(x) exists and is negative, then f(x) is decreasing at x 0. If f '(x 0 ) does not exist or is zero, then the test fails.

 Alternate method of finding extrema: If f(x) is continuous in a closed interval I, then the absolute extrema of f(x) in I occur at the critical points and/or at the endpoints of I.

VII. INDEFINITE INTEGRALS

Definition - if the derivative of F(x) is f(x) then ANTIDERIVATIVE or INTEGRAL of f(x) is F(x) , it is denoted by∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥) + 𝐶 where C is any constant of integration. The process of finding the antiderivative or integral is called INTEGRATION.

 Theorem 1. If two functions differ by a constant, they have the same derivative.  Theorem 2. If two functions have the same derivative, their difference is a constant I. FORMULA OF INTEGRATION.

∫[𝑓(𝑥) ± 𝑔(𝑥)]𝑑𝑥 = ∫ 𝑓(𝑥) 𝑑𝑥 ± ∫ 𝑔(𝑥)𝑑𝑥
(^) ∫ 𝑘𝑓(𝑥)𝑑𝑥 = 𝑘 ∫ 𝑓(𝑥)𝑑𝑥 + 𝐶
1. ∫ 𝑓(𝑔(𝑥)). 𝑔′(𝑥)𝑑𝑥 = ∫ 𝑓(𝑡)𝑑𝑡 , 𝑤ℎ𝑒𝑟𝑒 𝑔(𝑥) = 𝑡
2. ∫ 𝑓(𝑥). 𝑔(𝑥)𝑑𝑥 = 𝐹(𝑥). 𝑔(𝑥) − ∫ 𝐹(𝑥)𝑔′(𝑥)𝑑𝑥

where u is a variable, a is any constant, and e is a defined constant.

II. INTEGRAL OF TRIGONOMETRIC FUNCTIONS:

III. INTEGRAL OF POWERS OF TRIGONOMETRIC FUNCTIONS : The integrals of powers of trigonometric functions will be limited to those which may, by substitution, be written in the form (^) ∫ 𝑢𝑛𝑑𝑢

Techniques of Integration: Integrating Powers and Product of Sines and Cosines∫ 𝑠𝑖𝑛𝑚𝑥𝑐𝑜𝑠𝑛𝑥𝑑𝑥

We have two cases: both m and n are even or at least one of them is odd.

2. Case I: m or n odd Suppose n is odd - then substitute sinx = t. Indeed, we have cosxdx = dt and hence ∫ 𝒔𝒊𝒏𝒎𝒙𝒄𝒐𝒔𝒏^ 𝒙𝒅𝒙 = ∫ 𝒕𝒎(𝟏 − 𝒕𝟐)

𝒏/𝟐 𝒅𝒕.

Case II: m and n are even : Use the trigonometric identitiessin^2 u 

 cos u ,

cos^2 u ^1 2

 cos u

IV. INTEGRALS OF MULTIPLES OF SIN AND COS : for integrals ∫ 𝒔𝒊𝒏(𝒎𝒙) 𝒄𝒐𝒔(𝒏𝒙)𝒅𝒙, ∫ 𝒔𝒊𝒏(𝒎𝒙) 𝒔𝒊𝒏(𝒏𝒙)𝒅𝒙,

∫ 𝒄𝒐𝒔(𝒎𝒙) 𝒄𝒐𝒔(𝒏𝒙)𝒅𝒙^ use the transformation formula

Sin(mx).sin(nx) = 1 2 [𝑐𝑜𝑠(𝑚 − 𝑛)𝑥 − 𝑐𝑜𝑠(𝑚 + 𝑛)𝑥]
Sin(mx).cos (nx) = 12 [𝑠𝑖𝑛(𝑚 − 𝑛)𝑥 + 𝑠𝑖𝑛(𝑚 + 𝑛)𝑥]
cos(mx).cos(nx) = 1 2 [𝑐𝑜𝑠(𝑚 − 𝑛)𝑥 + 𝑐𝑜𝑠(𝑚 + 𝑛)𝑥]

V. REDUCTION FORMULA : In integrals of the form∫ 𝒕𝒂𝒏𝒏^ 𝒙𝒅𝒙 , (^) ∫ 𝒄𝒐𝒕𝒏^ 𝒙𝒅𝒙 , (^) ∫ 𝒔𝒆𝒄𝒏^ 𝒙𝒅𝒙 , (^) ∫ 𝒄𝒐𝒔𝒆𝒄𝒏^ 𝒙𝒅𝒙 Use

For (^) ∫ 𝒕𝒂𝒏𝒏^ 𝒙𝒅𝒙 , substitute tannx = tann-2x tan^2 x = tann - 2x(sec^2 x - 1) , then put tanx = t
For ∫ 𝒄𝒐𝒕𝒏^ 𝒙𝒅𝒙 , substitute cotnx = cotn-2x cot^2 x = cot n - 2x(cosec^2 x - 1) , then put cotx = t
For (^) ∫ 𝒔𝒆𝒄𝒏^ 𝒙𝒅𝒙 , substitute secnx = secn-2x sec^2 x = secn - 2x(tan^2 x + 1) , then put secx = t
For ∫ 𝒄𝒐𝒔𝒆𝒄𝒏^ 𝒙𝒅𝒙 , substitute cosecnx = cosecn-2x cosec^2 x = cosecn - 2x(cot^2 x + 1) , then put cosecx = t

VI. INTEGRALS INVOLVING √𝒂𝟐^ ± 𝒙𝟐𝑨𝑵𝑫 √𝒙𝟐^ ± 𝒂𝟐^ ---- Trigonometric substitutions may be used to eliminate radicals from integrals

for √𝑎^2 − 𝑥^2 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑥 = 𝑎 𝑠𝑖𝑛𝑡 then dx = a cost dt
for √𝑎^2 + 𝑥^2 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑥 = 𝑎 𝑡𝑎𝑛𝑡 then dx = a sec^2 t dt
for √𝑥^2 − 𝑎^2 𝑠𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒 𝑥 = 𝑎 𝑠𝑒𝑐𝑡 then dx = a sect tant dt

VII. Standard formula

∫ 1 𝑎^2 +𝑥^2 𝑑𝑥 =^

1 𝑎 tan

−1 𝑥 𝑎 + C

(^) ∫ (^) 𝑎 (^2) − 𝑥^12 𝑑𝑥 = 1 2𝑎 𝑙𝑜𝑔 |

𝑎+𝑥 𝑎−𝑥|^ + C

∫ 1 𝑥^2 − 𝑎^2 𝑑𝑥 = 1 2𝑎 𝑙𝑜𝑔 |

𝑥−𝑎 𝑥+𝑎|^ + C

∫ 1 √𝑎^2 −𝑥^2 dx =^ 𝑠𝑖𝑛

−1 𝑥 𝑎 + C

(^) ∫ 1 √𝑎^2 +𝑥^2 dx =^ 𝑙𝑜𝑔|𝑥 + √𝑎

(^2) + 𝑥 (^2) | + C

(^) ∫ (^) √𝑥 (^21) −𝑎 2 dx = 𝑙𝑜𝑔|𝑥 + √𝑥^2 − 𝑎^2 | + C
(^) ∫ √𝑎^2 − 𝑥^2 dx = 𝑥 2 √𝑎^2 − 𝑥^2 + 𝑎

2 2 𝑠𝑖𝑛

−1 𝑥 𝑎 + C

∫ √𝑎^2 + 𝑥^2 dx = 𝑥 2 √𝑎

(^2) + 𝑥 (^2) + 𝑎 2

2 𝑙𝑜𝑔|𝑥 + √𝑎^2 + 𝑥^2 | + C

(^) ∫ √𝑥^2 − 𝑎^2 dx = 𝑥 2 √𝑥^2 − 𝑎^2 − 𝑎 2

2 𝑙𝑜𝑔|𝑥 + √𝑥^2 − 𝑎^2 | + C

VIII. Integrals of the form (^) ∫ (^) 𝒂𝒙𝟐+𝒃𝒙+𝒄𝟏 𝒅𝒙 or (^) ∫ 𝟏 √𝒂𝒙𝟐+𝒃𝒙+𝒄 𝒅𝒙 : Apply completion of square method to convert

ax^2 + bx + c = a [(𝑥 + 𝑏 2𝑎)

( √4𝑎𝑐−𝑏^2 2𝑎 )

2 ] and use suitable standard formula.

IX. Integrals of the form ∫ 𝒙𝟐+𝟏 𝒙𝟒+𝝀𝒙𝟐+𝟏 𝒅𝒙 , ∫^

𝒙𝟐−𝟏 𝒙𝟒+𝝀𝒙𝟐+𝟏 𝒅𝒙 , ∫^

𝟏 𝒙𝟒+𝝀𝒙𝟐+𝟏 𝒅𝒙 𝒘𝒉𝒆𝒓𝒆 𝝀 ∈ 𝑹 ,  Divide numerator and denominator by x^2  Express denominator as (𝑥 ± (^1) 𝑥)

2 ± 𝑘^2 , ( choose the sign between x and (^1) 𝑥 as opposite of that in numerator.  Substitute x + 1 𝑥 = t or x -^

1 𝑥 = t as the case may be.  Reduce the integral to standard form and apply suitable formula.

𝑝𝑥^2 + 𝑞𝑥 + 𝑟 (𝑎𝑥 + 𝑏)(𝑐𝑥 + 𝑑)(𝑒𝑥 + 𝑓)

𝐴 𝑎𝑥 + 𝑏

𝐵 𝑐𝑥 + 𝑑

𝐶 𝑒𝑥 + 𝑓 𝑝𝑥 + 𝑞 (𝑎𝑥 + 𝑏)^2

𝐴 𝑎𝑥 + 𝑏

𝐵 (𝑎𝑥 + 𝑏)^2 𝑝𝑥^2 + 𝑞𝑥 + 𝑟 (𝑎𝑥 + 𝑏)^2 (𝑐𝑥 + 𝑑)

𝐴 𝑎𝑥 + 𝑏

𝐵 (𝑎𝑥 + 𝑏)^2

𝐶 𝑐𝑥 + 𝑑 𝑝𝑥^2 + 𝑞𝑥 + 𝑟 (𝑎𝑥 + 𝑏)^3

𝐴 𝑎𝑥 + 𝑏

𝐵 (𝑎𝑥 + 𝑏)^2

𝐶 (𝑎𝑥 + 𝑏)^3 𝑝𝑥^2 + 𝑞𝑥 + 𝑟 (𝑎𝑥 + 𝑏)(𝑐𝑥^2 + 𝑑𝑥 + 𝑒)

𝐴 𝑎𝑥+𝑏 +^

𝐵𝑥+𝐶 𝑐𝑥^2 +𝑑𝑥+𝑒, where cx

(^2) +dx+e can not be further factorised A ,B , C are real numbers to be determined by taking LCM and comparing the coefficients of like terms from the numerator.

Integrate the result of step 3. XVI. To evaluate ∫ 𝒅𝒙 𝒙(𝒙𝒏+𝒌) , 𝑛 ∈ 𝑁, 𝑛 ≥ 2  Multiply numerator and denominator by xn-  Then substitute xn^ = t , so that n x n-1^ dx = dt  Then apply partial fraction. XVII. If a rational function contains only even powers of x in both numerator and denominator  Put x^2 = y t in the given rational function  Resolve the rational function obtained in step 1 into partial fraction  Replace back y = x^2. Then integrate.

XVIII. Integration by Parts – If u and g are two functions of x then the integral of product of two functions = 1 st^ function × 𝒕𝒉𝒆 𝒊𝒏𝒕𝒆𝒈𝒓𝒂𝒍 𝒐𝒇 𝒕𝒉𝒆 𝟐𝒏𝒅𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏 - integral of the product of the derivative of 1st function and the integral of the 2nd^ function  Write the given integral∫ 𝑢(𝑥). 𝑣(𝑥) 𝑑𝑥 where you identify the two functions u(x) and v(x) as the 1st^ and 2nd function by the order I – inverse trigonometric function L – Logarithmic function A – Algebraic function T – Trigonometric function E – Exponential function  Note that if you are given only one function, then set the second one to be the constant function g(x)=1.  integrate the given function by using the formula ∫ 𝑢(𝑥). 𝑣(𝑥)𝑑𝑥 = 𝑢(𝑥) ∫ 𝑣(𝑥)𝑑𝑥 − ∫ [(^ 𝑑 𝑑𝑥 𝑢(𝑥)) (∫ 𝑣(𝑥)𝑑𝑥)] 𝑑𝑥 XIX. Integrals of the form ∫ 𝒆𝒙[𝒇(𝒙) + 𝒇′(𝒙)]^ dx  Express the integral as sum of two integrals , one containing f(x) and other containing f’(x) i.e., ∫ 𝒆𝒙[𝒇(𝒙) + 𝒇′(𝒙)]^ dx = ∫ 𝒆𝒙𝒇(𝒙)𝐝𝐱 + ∫ 𝒆𝒙𝒇′(𝒙)𝐝𝐱  Evaluate the first integral by integration by parts by taking ex^ as 2nd^ function  2 nd^ integral on R.H.S. will get cancelled by the 2nd^ term obtained by evaluating the 1st^ integral.  We get (^) ∫ 𝒆𝒙[𝒇(𝒙) + 𝒇′(𝒙)] dx = ex^ f(x) + C XX. Integrals of the type ∫ 𝒆𝒂𝒙^ 𝒔𝒊𝒏𝒃𝒙𝒅𝒙 or ∫ 𝒆𝒂𝒙^ 𝒄𝒐𝒔𝒃𝒙𝒅𝒙  Apply integration by parts twice by taking eax^ as the first function.

XXI. INTEGRATION OF SOME SPECIAL IRRATIONAL ALGEBRAIC FUNCTIONS integrals of the

form∫ 𝜑(𝑥)𝑃√𝑄 𝑑𝑥

 ∫ 1 (𝑎𝑥+𝑏)√𝑐𝑥+𝑑 𝑑𝑥:^ P and Q are both linear functions of x, put Q = t

(^2) .i.e. cx + d = t (^2).

 ∫ 1 (𝑎𝑥^2 +𝑏𝑥+𝑐)√𝑝𝑥+𝑞 𝑑𝑥:^ P is a quadratic expression and Q is linear expression of x, put Q = t

(^2).

i.e. put px + q = t^2  ∫ 1 (𝑎𝑥+𝑏)√𝑝𝑥^2 +𝑞𝑥+𝑟 𝑑𝑥^ : P is a linear expression and Q is quadratic expression of x, put P =^

1 𝑡, i.e. ax+ b = 1 𝑡.  (^) ∫ 1 (𝑎𝑥^2 +𝑏)√𝑐𝑥^2 +𝑑

dx : P and Q are pure quadratic expressions, put x= (^1) 𝑡,to obtain (^) ∫ −𝑡dt (𝑎+𝑏𝑡^2 )√𝑐+𝑑𝑡^2

, then put c+dt^2 = u^2  (^) ∫ 𝑝𝑥+𝑞 (𝑎𝑥^2 +𝑏)√𝑐𝑥^2 +𝑑

dx : P and Q are pure quadratic expressions and 𝜑(𝑥) 𝑖𝑠 𝑙𝑖𝑛𝑒𝑎𝑟, put x = t^2.

VIII. DEFINITE INTEGRAL :

The Fundamental Theorem of Calculus Let f ( x ) be continuous on [ a , b ]. If F ( x ) is any antiderivative of f ( x ),

then ∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎) 𝑏 𝑎 where b, the upper limit, and a, the lower limit, are given values.Notice that the constant of integration does not appear in the final expression of equation.

Areas above and below a curve:If the graph of y = f(x), between x = a and x = b, has portions above and

portions below the X axis, then (^) ∫𝑎 𝑏 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑏) − 𝐹(𝑎)is the sum of the absolute values of the positive areas above the X axis and the negative areas below the X axis. the value of b is the upper limit and the value of a is the lower limit.

3. Mean Value Theorem(for definite integrals) If f is continuous on  a b , , then at some

point c in  a b , ,    

1 b f c  (^) b  a  af x dx

Definite integral as the limit of a sum of all the strips between a and b, having areas of 𝑓(𝑎 + 𝑘 − 1̅̅̅̅̅̅̅ℎ). ℎ that is, ∫ 𝑓(𝑥)𝑑𝑥 = lim ℎ→0 ∑^ 𝑘=𝑛𝑘=1[𝑓(𝑥 + (𝑘 − 1)ℎ)] × ℎ 𝑏 𝑎

= lim ℎ→ ℎ[𝑓(𝑎) + 𝑓(𝑎 + ℎ) + 𝑓(𝑎 + 2ℎ) + ⋯ + 𝑓(𝑎 + (𝑛 − 1)ℎ)]

Steps :- 1. Find nh = b – a

IX. AREA UNDER THE BOUNDED REGION

 Area of the region bounded by the curve y = f(x) , the x axis and ordinates x = a and x = b is ∫ 𝑦𝑑𝑥 𝑏 𝑎 = ∫ 𝑓(𝑥)𝑑𝑥 𝑏 𝑎  Area of the region bounded by the curve x = f(y) , the y axis and ordinates y = a and y= b is (^) ∫𝑎 𝑏 𝑥𝑑𝑦=

∫ 𝑓(𝑦)𝑑𝑦 𝑏 𝑎  If y = f 1 (x) and y = f 2 (x) are two curves intersecting at the points (a, b) and (c, d) then the area enclosed between the curves is given by (^) ∫ (𝑦𝑎^ 𝑐 𝑢𝑝𝑝𝑒𝑟 𝑐𝑢𝑟𝑣𝑒 − 𝑦𝑙𝑜𝑤𝑒𝑟 𝑐𝑢𝑟𝑣𝑒)𝑑𝑥.

 If x = f 1 (y) and x = f 2 (y) are two curves intersecting at the points (a, b) and (c, d) then the area enclosed between the curves is given by ∫ (𝑥𝑢𝑝𝑝𝑒𝑟 𝑐𝑢𝑟𝑣𝑒 − 𝑥𝑙𝑜𝑤𝑒𝑟 𝑐𝑢𝑟𝑣𝑒) 𝑐 𝑎 𝑑𝑦.  WORKING RULE- I. Trace the graph of the curves and write about them in brief. II. Find the points of intersection of the curves. III. Express y in term of x befrom the equation of the curve if you are integrating w.r.t. x ( or x in term of y if you wish to integrate w.r.t. y ) as the case may be. IV. Consider the area under the bounded region as definite integral by using the concept discussed above. V. Evaluate the definite integral. VI. Write the answer in sq. units.

MATRICES AND DETERMINANTS

 DEFINITION: A matrix A = [𝒂𝒊𝒋]𝒎×𝒏 is defined as an ordered rectangular array of numbers in

m rows and n columns. 𝑨 = [

]

1. ROW MATRIX A matrix can have a single row A = [𝒂𝒊𝒋]𝟏×𝒏 = **[ a 11 a 12 a 13 … a1n]

COLUMN MATRIX - A matrix can have a single column A** = [𝒂𝒊𝒋]𝒎×𝟏 = [

]

**3. ZERO or NULL MATRIX – A matrix is called the zero or null matrix if all the entries are 0.

SQUARE MATRIX - A matrix for which horizontal and vertical dimensions are the same (i.e., an** **matrix).
DIAGONAL MATRIX - A square matrix A** = [𝒂𝒊𝒋]𝒏×𝒏 is called diagonal matrix if aij = 0 for 𝒊 ≠ 𝒋**.
SCALAR MATRIX - A diagonal matrix A** = [𝒂𝒊𝒋]𝒏×𝒏 is called the scalar matrix if all its diagonal **elements are equal.
IDENTITY MATRIX – A diagonal matrix A** = [𝒂𝒊𝒋]𝒏×𝒏 is called the identity matrix if aij = 1 for i **= j , it is denoted by In.
UPPER TRIANGULAR MATRIX - A square matrix A** = [𝒂𝒊𝒋]𝒏×𝒏 is called upper triangular matrix if aij = 0 for 𝒊 > 𝒋

9. LOWER TRIANGULAR MATRIX - A square matrix A = [𝒂𝒊𝒋]𝒏×𝒏 is called lower triangular matrix if aij = 0 for 𝒊 < 𝒋

 MATRIX OPERATIONS

1. DEFINITION: Two matrices A and B can be added or subtracted if and only if their dimensions are the same (i.e. both matrices have the identical amount of rows and **columns.

Addition** If A = [𝒂𝒊𝒋]𝒎×𝒏 and B = [𝒃𝒊𝒋]𝒎×𝒏 are matrices of the same type then the sum is a matrixC = [𝑪𝒊𝒋]𝒎×𝒏 obtained by adding the corresponding elements aij + bij i.e****. A+B = C if aij + bij =cij 3. Matrix addition is commutative , associative and distributive over multiplication -

 A + B = B + A  A + (B + C) = (A+ B) + C

 A (B + C) = AB + AC  (A+B)C= AC + BC

4. Subtraction If A = [𝒂𝒊𝒋]𝒎×𝒏 and B = [𝒃𝒊𝒋]𝒎×𝒏 are matrices of the same type then the difference is a matrix D = [𝒅𝒊𝒋]𝒎×𝒏 obtained by subtracting the corresponding elements aij - bij i.e****. A - B = C if aij - bij =dij

Equal matrices – Two matrices are said to be equal if they have the same order and their corresponding elements are also equal i.e. A = [𝒂𝒊𝒋]𝒎×𝒏 = B = [𝒃𝒊𝒋]𝒎×𝒏 if aij = bij for all I, j.
Scalar multiplication- If A = [𝒂𝒊𝒋]𝒎×𝒏 and B = [𝒃𝒊𝒋]𝒎×𝒏 are matrices of the same order and k, m are scalars then, scalar multiplication is defined as kA=[kaij].

 K(A+B) = Ka + Kb  (m+n) A = mA+ nA ^ (mk)A = m(kA) =k(mA)

Matrix Multiplication

DEFINITION: When the number of columns of the first matrix is the same as the number of rows in the second matrix then matrix multiplication can be performed.

Let A = [𝒂𝒊𝒋]𝒎×𝒏 and B = [𝒃𝒊𝒋]𝒏×𝒑. Then the product of A and B is the matrix C, which has dimensions mxp. The ij th^ element of matrix C is found by multiplying the entries of the i th^ row of A with the corresponding entries in the j th^ column of B and summing the n terms. The elements of C are:

Mathematics Study Notes: Relations, Functions, Calculus, and Geometry, Schemes and Mind Maps of Mathematics

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POINTS TO

REMEMBER IN CLASS

XII MATHEMATICS

By

Balraj Khurana

INDEX

1. Relations and functions - Pg 2

2. Inverse trigonometric functions - Pg 5

3. Calculus identities - Pg 6

4. Continuity - Pg 7

5. Differentiation - Pg 8

6. Application of derivative - Pg 9

7. Indefinite integral - Pg 11

8. Definite integral - Pg 14

9. Matrices - Pg 16

10. Determinants - Pg 19

11. Solution of system of linear equations - Pg 21

RELATIONS AND FUNCTIONS

I. RELATION

INVERSE TRIGONOMETRIC FUNCTIONS

V. SUM FORMULA

 𝑠𝑖𝑛−1𝑥 ± 𝑠𝑖𝑛−1𝑦 = 𝑠𝑖𝑛−1[𝑥√1 − 𝑦^2 ± 𝑦√1 − 𝑥^2 ].

 𝑐𝑜𝑠−1𝑥 ± 𝑐𝑜𝑠−1𝑦 = 𝑐𝑜𝑠−1[𝑥𝑦 ∓ √1 − 𝑥^2 √1 − 𝑦^2 ]

 𝑡𝑎𝑛−1𝑥 ± 𝑡𝑎𝑛−1𝑦 = 𝑡𝑎𝑛−1^ (

 2𝑠𝑖𝑛−1𝑥 = 𝑠𝑖𝑛−1[2𝑥√1 − 𝑥^2 ]

 2𝑐𝑜𝑠−1𝑥 = 𝑐𝑜𝑠−1[2𝑥^2 − 1]

 2 𝑡𝑎𝑛−1𝑥 = 𝑡𝑎𝑛−^

CALCULUS

I. ALGEBRAIC AND TRIGONOMETRICIDENTITIES

[𝑐𝑜𝑠(𝑢 − 𝑣) − 𝑐𝑜𝑠(𝑢 + 𝑣)]

[𝑐𝑜𝑠(𝑢 + 𝑣) + 𝑐𝑜𝑠(𝑢 − 𝑣)]

[𝑠𝑖𝑛(𝑢 + 𝑣) + 𝑠𝑖𝑛(𝑢 − 𝑣)]

II. CONIC SECTION FORMULA

1. Circle formula: ^ ^ ^ 

2. Parabola formula: ^ ^ ^ 

  1  2 ^2

  1  2 ^2

III. FORMULAS OF LIMITS

a. Change of base rule for logs: log^

ln

ln

a x^

x

a

IV. CONTINUITY

III. RULES OF DIFFERENTIATION

 Intermediate Value Theorem : If a function is continuous between a and b , then it takes on every value

 Mean Value Theorem(for derivatives) : If f ( ) x is a continuous function over a b , , and f(x) is

differentiable in ( a,b )then at some point c between a and b :

 Rolle’s Theorem If (i) f ( ) x is a continuous function over a b , , (ii) f(x) is differentiable in ( a,b ) (iii) f( a )

= f(b)then there exists some point c between a and b such that f’(c) = 0 ( the tangent at x = c is parallel

VI. APPLICATION OF DERIVATIVE

I. APPROXIMATIONS, DIFFERENTIALS AND ERRORS

3. Mean Value Theorem(for definite integrals) If f is continuous on  a b , , then at some

point c in  a b , ,    

MATRICES AND DETERMINANTS

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 MATRIX OPERATIONS