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class 12 revision of cbse math part-I 2022-2023

Class 12 revision of cbse math part-II 2022-2023

Integration: Chapter 7

Exercise 7.1

(1) $\int 1dx = x+C$

(2) $\int x dx=\frac{x^2}{2}+C$

(3) $\int x^n dx = \dfrac{x^{n+1}}{n+1}+C$

(4) $\int e^x dx=e^x+C$

(5) $\int a^x dx = \dfrac{a^x}{\log_e a}+C$

(6) $\int \frac{1}{x}dx=\log_e x+C$

(7) $\int \sin x dx = -\cos x+C$

(8) $\int \cos x dx =\sin x +C$

(9) $\int \sec^2 xdx=\tan x+C$

(10) $\int \operatorname{cosec}^2x dx = -\cot x+C$

(11) $\int \sec x\tan x dx=\sec x+C$

(12) $\int \operatorname{cosec}x\cot x dx = -\operatorname{cosec}x+C$

(13) $\int \dfrac{1}{\sqrt{1-x^2}}dx=\sin^{-1}x+C$

(14) $\int \dfrac{1}{\sqrt{1-x^2}}dx=-\cos^{-1}x+C$

(15) $\int\dfrac{1}{1+x^2}dx=\tan^{-1}x+C$

(16) $\int\dfrac{1}{1+x^2}dx=-\cot^{-1}x+C$

(17) $\int\dfrac{1}{x\sqrt{x^2-1}}dx=\sec^{-1}x+C$

(18) $\int\dfrac{1}{x\sqrt{x^2-1}}dx=-\operatorname{cosec}^{-1}x+C$

Methods of Integration:-
1. Integration by Substitution
2. Integration using Partial Fractions
3. Integration by Parts

Exercise 7.2

Integration by substitution

⇒ Some other formulae:

(1) $\int\tan x dx =\log_e|\sec x|+C$

(2) $\int\cot xdx =\log_e|\sin x|+C$

(3) $\int \sec x dx=\log_e|\sec x+\tan x|+C$

(4) $\int \operatorname{cosec}x dx=\log_e|\operatorname{cosec}x-\cot x|+C$

Exercise 7.3

Integration using trigonometric identities

(1) $\sin 2x = 2\sin x\cos x$

(2) $\sin 2x = \dfrac{2\tan x}{1+\tan^2x}$

(3) $\cos 2x = \cos^2x-\sin^2x$

(4) $\cos 2x =1-2\sin^2x$

(5) $\cos 2x = 2\cos^2x-1$

(6) $1-\cos 2x =2\sin^2 x$

(7) $1+\cos 2x = 2\cos^2x$

(8) $\sin 3x = 3\sin x-4\sin^3x$

(9) $\cos 3x = 4\cos^3x-3\cos x$

(10)(a) $2\sin x\cos y =\sin(x+y)+\sin(x-y)$

(b) $2\cos x\sin y = \sin(x+y)-\sin(x-y)$

(c) $2\cos x\cos y=\cos(x+y)+\cos(x-y)$

(d) $2\sin x\sin y=\cos(x-y)-\cos(x+y)$

(11) (a) $\sin x+\sin y=2\sin\dfrac{x+y}{2}\cos\dfrac{x-y}{2}$

(b) $\sin x-\sin y=2\cos\dfrac{x+y}{2}\sin\dfrac{x-y}{2}$

(c) $\cos x+\cos y = 2\cos\dfrac{x+y}{2}\cos\dfrac{x-y}{2}$

(d) $\cos x-\cos y=-2\sin\dfrac{x+y}{2}\sin\dfrac{x-y}{2}$

Exercise 7.4

Integrals of Some Particular Functions

(1) $\displaystyle \int\dfrac{1}{x^2-a^2}dx=\dfrac{1}{2a}\log\left|\dfrac{x-a}{x+a}\right|+C$

(2) $\displaystyle\int\dfrac{1}{a^2-x^2}dx=\dfrac{1}{2a}\log\left|\dfrac{a+x}{a-x}\right|+C$

(3) $\displaystyle \int\dfrac{1}{x^2+a^2}dx=\dfrac{1}{a}\tan^{-1}\frac{x}{a}+C$

(4) $\displaystyle \int\dfrac{1}{\sqrt{x^2-a^2}}dx=\log\left|x+\sqrt{x^2-a^2}\right|+C$

(5) $\displaystyle\int \dfrac{1}{\sqrt{a^2-x^2}}dx=\sin^{-1}\frac{x}{a}+C$

(6) $\displaystyle \int\dfrac{1}{\sqrt{x^2+a^2}}dx=\log\left|x+\sqrt{x^2+a^2}\right|+C$

(7) $\displaystyle \int\dfrac{dx}{ax^2+bx+c}$

(8) $\displaystyle \int\dfrac{dx}{\sqrt{ax^2+bx+c}}$

(9) $\displaystyle \int\dfrac{px+q}{ax^2+bx+c}dx$

$px+q=A\frac{d}{dx}(ax^2+bx+c)+B$

$\Rightarrow px+q=A(2ax+b)+B$

(10) $\displaystyle \int\dfrac{px+q}{\sqrt{ax^2+bx+c}}dx$

From (7) to (10) types form using completing the square

$x^2+bx+c$ for completing the square add and subtract by $\left(\dfrac{\text{cofficient of x}}{2}\right)^2$

Exercise 7.5

Integration by Partial Fractions:

⇒ Partial fraction expression:

(1) $\dfrac{px+q}{(x+a)(x+b)}=\dfrac{A}{(x+a)}+\dfrac{B}{(x+b)}$

(2) $\dfrac{px+q}{(x-a)^2}=\dfrac{A}{(x-a)}+\dfrac{B}{(x-a)^2}$

(3) $\dfrac{px^2+qx+r}{(x-a)(x-b)(x-c)}=\dfrac{A}{(x-a)}+\dfrac{B}{(x-b)}+\dfrac{C}{(x-c)}$

(4) $\dfrac{px^2+qx+r}{(x-a)^2(x-b)}=\dfrac{A}{(x-a)}+\dfrac{B}{(x-a)^2}+\dfrac{C}{(x-b)}$

(5) $\dfrac{px^2+qx+r}{(x-a)(x^2+b^2)}=\dfrac{A}{(x-a)}+\dfrac{Bx+C}{(x^2+b^2)}$

Exercise 7.6

Integration by Parts

Formula

(1) $\displaystyle\int u.vdx=u\int vdx-\int\left[\frac{d}{dx}u\int vdx\right]dx+C$

ILATE

I= inverse tigonometry( $\sin^{-1}x,\cos^{-1}x...$ )

L= logrithmic function( $\log x..$ )

A= Algebraic function

T = trigonometric function

E= Exponential function

(2) $\int\left[f(x)+f'(x)\right]dx=e^xf(x)+C$

Exercise 7.7

Integrals of some more types

(1) $\displaystyle\int \sqrt{x^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\log\left|x+\sqrt{x^2-a^2}\right|+C$

(2) $\displaystyle\int \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}+\frac{a^2}{2}\log\left|x+\sqrt{x^2+a^2}\right|+C$

(3) $\displaystyle\int \sqrt{a^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}+C$

Exercise 7.9 & 7.10

$\displaystyle \int_a^bf(x) dx=\left[F(x)\right]_a^b= F(b)-F(a)$

Evaluation of Definite Integrals by Substitution

Exercise 7.11

Some Properties of Definite Integrals

(1) $P_0:\int_a^bf(x)dx = \int_a^bf(t)dt$

(2) $P_1:\int_a^bf(x)dx =-\int_b^af(x)dx\text{ and }\int_a^af(x)dx =0$

(3) $P_2:\int_a^bf(x)dx =\int_a^cf(x)dx +\int_c^bf(x)dx$

(4) $P_3:\int_a^bf(x)dx =\int_a^bf(a+b-x)dx$

(5) $P_4:\int_0^af(x)dx =\int_0^af(a-x)dx$

(6) $P_5:\int_0^{2a}f(x)dx =\int_0^af(x)dx+\int_a^bf(2a-x)dx$

(7) $P_6:(i)\int_0^{2a}f(x)dx =2\int_0^af(x)dx \text{ if } f(2a-x)=f(x)$

$(ii) \int_0^{2a}f(x)dx =0 \text{ if } f(2a-x)=-f(x)$

(8) $P_7:(i) \int_{-a}^{a}f(x) dx=2\int_0^af(x)dx,\text{If f is an even function,i.e. if} f(-x)=f(x)$

$(ii) \int_{-a}^{a}f(x) dx=0,\text{If f is an odd function,i.e. if} f(-x)=-f(x)$

Chapter 8

Area under Simple Curves

Area of region when strip is perpendicular to x-axis $=\int ydx$

Area of region when strip is perpendicular to x-axis $=\int x dy$

Exercise 8.2

Area between Two Curves $= \int_a^b(f(x)-g(x))dx$ when $f(x)\geq g(x)$

Chapter 9(Differential equation): Exercise 9.1:

Order of a differential equation:-Order of a differential equation is defined as the order of the highest order derivative of the dependent variable with respect to the independent variable involved in the given differential equation.

Ex:- $\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}+x =0$

Order of d.e. = 2

Degree of a differential equation:-By the degree of a differential equation, when it is a polynomial equation in
derivatives, we mean the highest power (positive integral index) of the highest order derivative involved in the given differential equation.

EX:- $\left(\dfrac{d^2y}{dx^2}\right)^3+\dfrac{dy}{dx}+x =0$

Degree of d.e. = 3

NOTE:- Order and degree (if defined) of a differential equation are always positive integers.

Exercise 9.2

General solution:The solution which contains arbitrary constants is called the general solution(primitive) of the differential equation.

particular solution:-The solution free from arbitrary constants i.e., the solution obtained from the general solution by giving particular values to the arbitrary constants is called a particular solution of the differential equation.

Methods of Solving First Order, First Degree Differential Equations

Exercise 9.4

Differential equations with variables separable

Exercise: 9.5

Homogeneous differential equations:- In homogeneous differential equation substitute $y = vx$ or $x=vy$

Exercise 9.6

Linear differential equations:-

(1) (i) Write the given differential equation in the form $\dfrac{dy}{dx}+Py=Q$ where P, Q are constants or functions of x only.
(ii) Find the Integrating Factor $(I.F) = e^{\int Pdx}$

(iii) Solution of differential equation of this form is
$y.(I.F) = \int Q(I.F)dx+C$

(2) (i) Write the given differential equation in the form $\dfrac{dx}{dy}+Px=Q$ where P, Q are constants or functions of x only.
(ii) Find the Integrating Factor $(I.F) = e^{\int Pdy}$

(iii) Solution of differential equation of this form is
$x.(I.F) = \int Q(I.F)dy+C$

Vector algebra:

Exercise 10.1

Let The point P(x, y, z)

Position vector of P

$\overrightarrow{OP}=x\hat{i}+y\hat{j}+z\hat{k}$

dr’s of line OP = (x, y, z)

dc’s of the line OP

$l=\dfrac{x}{\sqrt{x^2+y^2+z^2}},m=\dfrac{y}{\sqrt{x^2+y^2+z^2}},n=\dfrac{z}{\sqrt{x^2+y^2+z^2}}$

Relation between dc’s

$l^2+m^2+n^2=1$

Types of Vectors:

(1) Zero Vector: A vector whose initial and terminal points coincide, is called a zero vector (or null vector)

Ex:- $\overrightarrow{AA}=0$

(2) Unit Vector: A vector whose magnitude is unity (i.e., 1 unit) is called a unit vector. The unit vector in the direction of $\vec{a}$ given vector is denoted by $\hat{a}$ .

(3) Coinitial Vectors: Two or more vectors having the same initial point are called coinitial vectors.

(4) Collinear Vectors :Two or more vectors are said to be collinear if they are parallel to the same line, irrespective of their magnitudes and directions.

(5) Equal Vectors: Two vectors $\vec{a}$ and $\vec{b}$ are said to be equal, if they have the same magnitude and direction regardless of the positions of their initial points, and written
as $\vec{a}=\vec{b}$ .

(6) Negative of a Vector :-A vector whose magnitude is the same as that of a given vector(say, $\overrightarrow{AB}$ ), but direction is opposite to that of it, is called negative of the given vector.
For example, $\overrightarrow{BA}$ vector is negative of the vector $\overrightarrow{AB}$ , and written as $\overrightarrow{BA}= -\overrightarrow{AB}$ .

Exercise 10.2

Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$

(i) Modulus of $\vec{a}=|\vec{a}|$

$|\vec{a}|=\sqrt{a_1^2+a_2^2+a_3^2}$

(ii) Unit vector:- $\hat{a}=\dfrac{\vec{a}}{|\vec{a}|}$

$\hat{a}=\dfrac{a_1\hat{i}+a_2\hat{j}+a_3\hat{k}}{\sqrt{a_1^2+a_2^2+a_3^2}}$

Components of a vector:

Let $\overrightarrow{OP}=x\hat{i}+y\hat{j}+\hat{k}$

Scalar component of $\overrightarrow{OP}=(x, y, z)$

Vector component of $\overrightarrow{OP}=(x\hat{i}, y\hat{j}, z\hat{k})$

Remarks:
$\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$

(i) Let two vectors $\vec{a}$ and $\vec{b}$ are collinear vector

$\dfrac{a_1}{b_1}=\dfrac{a_2}{b_2}=\dfrac{a_3}{b_3}$

(ii) If $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ , then $a_1, a_2, a_3$ are also called direction ratios of .

(iii) In case if it is given that l, m, n are direction cosines of a vector, then $l\hat{i} + m\hat{j} + n\hat{k} = (cos\alpha)\hat{i} + (cos\beta) \hat{j} + (cos \gamma)\hat{k}$ is the unit vector in the direction of that vector, where $\alpha, \beta$ and $\gamma$ are the angles which the vector makes with x, y and z axes respectively.

Vector joining two points:

If $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ are any two points, then the vector joining $P_1$ and $P_2$ is the
vector $\overrightarrow {P_1P_2}$

$\overrightarrow{OP_1}=x_1\hat{i}+y_1\hat{j}+z_1\hat{k}$

$\overrightarrow{OP_2}=x_2\hat{i}+y_2\hat{j}+z_2\hat{k}$

Then, $\overrightarrow {P_1P_2}=\overrightarrow{OP_2}-\overrightarrow{OP_1}$

$= (x_2\hat{i}+y_2\hat{j}+z_2\hat{k})-(x_1\hat{i}+y_1\hat{j}+z_1\hat{k})$

$=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}$

Magnitude of $\overrightarrow {P_1P_2}$ is $|\overrightarrow {P_1P_2}|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$

Section formula: P and q are two points on line and point R divide PQ in m:n Then

$\overrightarrow{OR} =\dfrac{m\overrightarrow{OQ}+n\overrightarrow{OP}}{m+n}$

And the mid point

$\overrightarrow{OR} =\dfrac{\overrightarrow{OQ}+\overrightarrow{OP}}{2}$

Product of Two Vectors

Scalar (or dot) product of two vectors: The scalar product of two nonzero vectors $\vec{a}$ and $\vec{b}$ , denoted by $\vec{a}.\vec{b}$

$\vec{a}.\vec{b}=|\vec{a}||\vec{b}|\cos\theta$

Observation: (1) Let $\vec[a]$ and $\vec{b}$ be two nonzero vectors, then $\vec{a}.\vec{b}=0$ if and only if $\vec[a]$ and $\vec{b}$ are perpendicular to each other. i.e.

$\vec{a}.\vec{b}=0\Leftrightarrow \vec{a}\perp\vec{b}$

(2) If $\theta=0$ then, $\vec{a}.\vec{b}=|\vec{a}||\vec{b}|$

In particular, $\vec{a}.\vec{a}=|\vec{a}|^2$ in this case $\theta =0$

(3) If $\theta=\pi$ then, $\vec{a}.\vec{b}=-|\vec{a}||\vec{b}|$

(4) $\hat{i}.\hat{i}=\hat{j}.\hat{j}=\hat{k}.\hat{k}=1$

$\hat{i}.\hat{j}=\hat{j}.\hat{k}=\hat{k}.\hat{i}=0$

(5) $\cos\theta =\left(\dfrac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}\right)$

(6) The scalar product is commutative, i.e.

$\vec{a}.\vec{b}=\vec{b}.\vec{a}$

(7) If $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$

Then $\vec{a}.\vec{b}=a_1.b_1+a_2.b_2+a_3.b_3$

Projection of a vector on a line

Projection of a vector $\vec{a}$ on other vector $\vec{b}$ , is given by

$\vec{a}.\hat{b}$ or $\vec{a}\left(\dfrac{\vec{b}}{|\vec{b}}\right)$ or $\dfrac{(\vec{a},\vec{b})}{|\vec{b}|}$

Remarks: $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$

$\cos\alpha =\dfrac{a_1}{|\vec{a}|},\cos\beta =\dfrac{a_2}{|\vec{a}|},\cos\gamma=\dfrac{a_3}{|\vec{a}|}$

Vector (or cross) product of two vectors

(1) $\vec{a}\times \vec{b}$ is vector.

(2) Let $\vec{a}$ and $\vec{b}$ be two non zero vectors, then $\vec{a}\times \vec{b}=0$ if and only if $\vec{a}$ and $\vec{b}$ are parallel(or collinear)to each other i.e.

$\vec{a}\times \vec{b}=0\Leftrightarrow \vec{a}||\vec{b}$

In Particular $\vec{a}\times \vec{a}=0$

(3). If $\theta =\dfrac{\pi}{2}$ and $\vec{a}\times \vec{b}=|\vec{a}||\vec{b}|$ .

(4) $\hat{i}\times \hat{i}=\hat{j}\times \hat{j}=\hat{k}\times \hat{k}=0$

(5) $\hat{i}\times \hat{j}=\hat{k}, \hat{j}\times \hat{k}=\hat{i}, \hat{k}\times \hat{i}=\hat{j}$

(6) $\sin\theta =\dfrac{|\vec{a}\times \vec{b}|}{|\vec{a}||\vec{b}|}$

(7) $\vec{a}\times \vec{b}=-\vec{b}\times \vec{a}$

(8) $\hat(j)\times \hat{i}=-\hat{k}, \hat{k}\times \hat{j}=-\hat{i},\hat{i}\times \hat{k}=-\hat{j}$

(9) If $\vec{a}$ and $\vec{b|}$ are the adjacent sides of a triangle then its area is given as $=\dfrac{1}{2}|\vec{a}\times \vec{b}|$

(10) If $\vec{a}$ and $\vec{b|}$ are the adjacent sides of a parallelogram then its area is given as $=|\vec{a}\times \vec{b}|$

(11) If $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$

$\vec{a}\times \vec{b}=\begin{vmatrix} \hat{i} &\hat{j} &\hat{k}\\ a_1 & a_2 & a_3 \\b_1 & b_2 & b_3 \end{vmatrix}$

Three dimension geometry

Direction cosine: If $\vec{a} = a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ makes angles $\alpha,\beta,\gamma$ with x, y, and z-axis respectively

$l=\cos\alpha=\dfrac{a_1}{\sqrt{a_1^2+a_2^2+a_3^2}}, m=\cos\beta=\dfrac{a_2}{\sqrt{a_1^2+a_2^2+a_3^2}}$

$n=\cos\gamma=\dfrac{a_3}{\sqrt{a_1^2+a_2^2+a_3^2}}$

Relation between direction cosine of a line

$l^2+m^2+n^2=1$

Direction cosines of a line passing through two points:

Let $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$

Then $\vec{PQ}=(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}$

$PQ= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$

$l=\dfrac{(x_2-x_1)}{PQ},m=\dfrac{(y_2-y_1)}{PQ},n=\dfrac{(z_2-z_1)}{PQ}$

Equation of a Line in Space:

(i) it passes through a given point and has given direction,

(ii) it passes through two given points.

Equation of a line through a given point and parallel to a given vector $\vec{b}$

vector form:

$\vec{r}=\vec{a}+\lambda\vec{b}$

Cartesian form:

If point $P(x_1,y_1,z_1)$ and direction ratio = $(a,b,c)$

$\dfrac{x-x_1}{a}=\dfrac{y-y_1}{b}=\dfrac{z-z_1}{c}$

Note: –If l, m, n are the direction cosines of the line, the equation of the line is

$\dfrac{x-x_1}{l}=\dfrac{y-y_1}{m}=\dfrac{z-z_1}{n}$

Equation of a line passing through two given points:

Two points are $\vec{a}$ and $\vec{b}$ then equation of line

$\vec{r}=\vec{a}+\lambda(\vec{b}-\vec{a})$

In cartesian form:

Two points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$

Equation of line

$\dfrac{x-x_1}{x_2-x_1}=\dfrac{y-y_1}{y_2-y_1}=\dfrac{z-z_1}{z_2-z_1}$

Angle between Two Lines:

$\vec{a} = a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b} = b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ angle between them is $\theta$

$\cos\theta=\left|\dfrac{\vec{a}.\vec{b}}{|\vec{a}||\vec{b}|}\right|$

$\cos \theta =\left|\dfrac{a_1b_1+a_2b_2+a_3b_3}{\sqrt{a_1^2+a_2^2+a_3^2}\sqrt{b_1^2+b_2^2+b_3^2}}\right|$

(i) If the lines are perpendicular, $\theta =0$

$\vec{a}.\vec{b}=0$ or $a_1b_1+a_2b_2+a_3b_3=0$

(ii) If two lines are parallel $\theta =0$ or $\pi$

$\dfrac{a_1}{b_1}=\dfrac{a_2}{b_2}=\dfrac{a_3}{b_3}$

Shortest Distance between Two Lines

(i) Distance between two skew lines

$\vec{r}=\vec{a_1}+\lambda\vec{b_1}$

$\vec{r}=\vec{a_2}+\mu\vec{b_2}$

Shortest distance= $\left|\dfrac{(\vec{b_1}\times \vec{b_2})(a_2-a_1)}{|\vec{b_1}\times \vec{b_2}|}\right|$

In cartesian form:

$\dfrac{x-x_1}{a_1}=\dfrac{y-y_1}{b_1}=\dfrac{z-z_1}{c_1}$
$\dfrac{x-x_1}{a_2}=\dfrac{y-y_1}{b_2}=\dfrac{z-z_1}{c_2}$

$S.D. =\left|\dfrac{\begin{vmatrix}x_2-x_1 & y_2-y_1 & z_2-z_1\\a-1 & b_1 &c_1 \\a_2 & b_2 & c_2\end{vmatrix}}{\sqrt{(b_1c_2-b_2c_1)^2+(c_1a_2-c_2a_1)^2+(a_1b_2-a_2b_1)^2}}\right|$