1. LIST OF MEMBERS WHO PREPARED
QUESTION BANK FOR MATHEMATICS
FOR CLASS XII
TEAM MEMBERS
Sl. No.
Name
Designation
1.
Sh. S.B. Tripathi
(Group Leader)
(M. 9810233862)
R.S.B.V., Jheel Khuranja
Delhi – 31.
2.
Sh. Sanjeev Kumar
(Lecturer Maths)
(M. 9811458610)
R.S.V.V., Raj Niwas Marg,
Delhi.
3.
Dr. R.P. Singh
(Lecturer Maths)
(M. 9818415348)
R.P.V.V., Gandhi Nagar
Delhi – 31
4.
Sh. Joginder Arora
(Lecturer Maths)
(M. 9953015325)
R.P.V.V., Hari Nagar
Delhi.
5.
Sh. Manoj Kumar
(Lecturer Maths)
(M. 9818419499)
R.P.V.V., Kishan Kunj
Delhi.
6.
Miss Saroj
(Lecturer Maths)
(M. 9899240678)
G.G.S.S.S., No. 1, Roop Nagar
Delhi-110007
2. REVIEW OF SUPPORT MATERIAL : 2012
GROUP LEADER
1.
Dr. Vandita Kalra
(Vice Principal)
GGSSS, Kirti Nagar,
Delhi.
TEAM MEMBERS
1.
Sh. Sanjeev Kumar
(PGT Maths)
R.P.V.V., Raj Niwas Marg,
Delhi.
2.
Ms. Saroj
(PGT Maths)
G.G.S.S.S., No. 1, Roop Nagar
Delhi.
3.
Ms. Anita Saluja
(PGT Maths)
G.G.S.S.S., Kirti Nagar
Delhi.
XII – Maths
2
3. CLASS XII
MATHEMATICS
Units
Weightage (Marks)
(i)
Relations and Functions
10
(ii)
Algebra
13
(iii)
Calculus
44
(iv)
Vector and Three Dimensional Geometry
17
(v)
Linear Programming
06
(vi)
Probability
10
Total : 100
Unit I : RELATIONS AND FUNCTIONS
1.
Relations and Functions
(10 Periods)
Types of Relations : Reflexive, symmetric, transitive and equivalence
relations. One to one and onto functions, composite functions, inverse of
a function. Binary operations.
2.
Inverse Trigonometric Functions
(12 Periods)
Definition, range, domain, principal value branches. Graphs of inverse
trigonometric functions. Elementary properties of inverse trigonometric
functions.
Unit II : ALGEBRA
1.
Matrices
(18 Periods)
Concept, notation, order, equality, types of matrices, zero matrix, transpose
of a matrix, symmetric and skew symmetric matrices. Addition,
multiplication and scalar multiplication of matrices, simple properties of
addition, multiplication and scalar multiplication. Non-commutativity of
3
XII – Maths
4. multiplication of matrices and existence of non-zero matrices whose
product is the zero matrix (restrict to square matrices of order 2). Concept
of elementary row and column operations. Invertible matrices and proof
of the uniqueness of inverse, if it exists; (Here all matrices will have real
entries).
2.
Determinants
(20 Periods)
Determinant of a square matrix (up to 3 × 3 matrices), properties of
determinants, minors, cofactors and applications of determinants in finding
the area of a triangle. adjoint and inverse of a square matrix. Consistency,
inconsistency and number of solutions of system of linear equations by
examples, solving system of linear equations in two or three variables
(having unique solution) using inverse of a matrix.
Unit III : CALCULUS
1.
Continuity and Differentiability
(18 Periods)
Continuity and differentiability, derivative of composite functions, chain
rule, derivatives of inverse trigonometric functions, derivative of implicit
function. Concept of exponential and logarithmic functions and their
derivatives. Logarithmic differentiation. Derivative of functions expressed
in parametric forms. Second order derivatives. Rolle’s and Lagrange’s
mean Value Theorems (without proof) and their geometric interpretations.
2.
Applications of Derivatives
(10 Periods)
Applications of Derivatives : Rate of change, increasing/decreasing
functions, tangents and normals, approximation, maxima and minima (first
derivative test motivated geometrically and second derivative test given
as a provable tool). Sample problems (that illustrate basic principles and
understanding of the subject as well as real-life situations).
3.
Integrals
(20 Periods)
Integration as inverse process of differentiation. Integration of a variety of
functions by substitution, by partial fractions and by parts, only simple
integrals of the type to be evaluated.
dx
x 2 a2 ,
XII – Maths
dx
2
x a
2
,
dx
2
a x
2
,
4
dx
dx
,
2
ax bx c
ax bx c
2
5. px q
ax 2 bx c dx ,
px q
2
ax bx c
ax 2 bx c dx and
dx , a 2 x 2 dx ,
px q
x 2 a 2 dx ,
ax 2 bx c dx
Definite integrals as a limit of a sum, Fundamental Theorem of Calculus
(without proof). Basic properties of definite integrals and evaluation of
definite integrals.
4.
Applications of the Integrals
(10 Periods)
Application in finding the area under simple curves, especially lines, area
of circles/parabolas/ellipses (in standard form only), area between the
two above said curves (the region should be clearly identifiable).
5.
Differential Equations
(10 Periods)
Definition, order and degree, general and particular solutions of a
differential equation. Formation of differential equation whose general
solution is given. Solution of differential equations by method of separation
of variables, homogeneous differential equations of first order and first
degree. Solutions of linear differential equation of the type :
dy
p x y q x , where p x and q x are function of x.
dx
Unit IV : VECTORS AND THREE-DIMENSIONAL GEOMETRY
1.
Vectors
(12 Periods)
Vectors and scalars, magnitude and direction of a vector. Direction cosines/
ratios of vectors. Types of vectors (equal, unit, zero, parallel and collinear
vectors), position vector of a point, negative of a vector, components of
a vector, addition of vectors, multiplication of a vector by a scalar, position
vector of a point dividing a line segment in a given ratio. Scalar (dot)
product of vectors, projection of a vector on a line. Vector (cross) product
of vectors. Scalar triple product of vectors.
2.
Three-Dimensional Geometry
(12 Periods)
Direction cosines/ratios of a line joining two points. Cartesian and vector
equation of a line, coplanar and skew lines, shortest distance between
two lines. Cartesian and vector equation of a plane. Angle between
5
XII – Maths
6. (i) two lines, (ii) two planes, (iii) a line and a plane. Distance of a point
from a plane.
Unit V : LINEAR PROGRAMMING
(12 Periods)
1.
Linear Programming : Introduction, definition of related terminology such
as constraints, objective function, optimization, different types of linear
programming (L.P.) problems, mathematical formulation of L.P. problems,
graphical method of solution for problems in two variables, feasible and
infeasible regions, feasible and infeasible solutions, optimal feasible
solutions (up to three non-trivial constraints).
Unit VI : PROBABILITY
1.
Probability
(18 Periods)
Multiplication theorem on probability. Conditional probability, independent
events, total probability, Baye’s theorem, Random variable and its
probability distribution, mean and variance of haphazard variable. Repeated
independent (Bernoulli) trials and Binomial distribution.
XII – Maths
6
7. CONTENTS
S.No.
Chapter
Page
1.
Relations and Functions
9
2.
Inverse Trigonometric Functions
3 & 4. Matrices and Determinants
17
23
5.
Continuity and Differentiation
39
6.
Applications of Derivatives
47
7.
Integrals
61
8.
Applications of Integrals
85
9.
Differential Equations
90
10.
Vectors
101
11.
Three-Dimensional Geometry
111
12.
Linear Programming
122
13.
Probability
127
Model Papers
141
7
XII – Maths
9. CHAPTER 1
RELATIONS AND FUNCTIONS
IMPORTANT POINTS TO REMEMBER
Relation R from a set A to a set B is subset of A × B.
A × B = {(a, b) : a A, b B}.
If n(A) = r, n (B) = s from set A to set B then n (A × B) = rs.
and no. of relations = 2rs
is also a relation defined on set A, called the void (empty) relation.
R = A × A is called universal relation.
Reflexive Relation : Relation R defined on set A is said to be reflexive
iff (a, a) R a A
Symmetric Relation : Relation R defined on set A is said to be symmetric
iff (a, b) R (b, a) R a, b, A
Transitive Relation : Relation R defined on set A is said to be transitive
if (a, b) R, (b, c) R (a, c) R a, b, c R
Equivalence Relation : A relation defined on set A is said to be
equivalence relation iff it is reflexive, symmetric and transitive.
One-One Function : f : A B is said to be one-one if distinct elements
in A has distinct images in B. i.e. x1, x2 A s.t. x1 x2 f(x1) f(x2).
OR
x1, x2 A s.t. f(x1) = f(x2)
x1 = x2
One-one function is also called injective function.
9
XII – Maths
10.
Onto function (surjective) : A function f : A B is said to be onto iff
Rf = B i.e. b B, there exist a A s.t. f(a) = b
A function which is not one-one is called many-one function.
A function which is not onto is called into.
Bijective Function : A function which is both injective and surjective is
called bijective.
Composition of Two Function : If f : A B, g : B C are two
functions, then composition of f and g denoted by gof is a function from
A to C given by, (gof) (x) = g (f (x)) x A
Clearly gof is defined if Range of f domain of g. Similarly fog can be
defined.
Invertible Function : A function f : X Y is invertible iff it is bijective.
If f : X Y is bijective function, then function g : Y X is said to be
inverse of f iff fog = Iy and gof = Ix
when Ix, Iy are identity functions.
–1.
g is inverse of f and is denoted by f
Binary Operation : A binary operation ‘*’ defined on set A is a function
from A × A A. * (a, b) is denoted by a * b.
Binary operation * defined on set A is said to be commutative iff
a * b = b * a a, b A.
Binary operation * defined on set A is called associative iff a * (b * c)
= (a * b) * c a, b, c A
If * is Binary operation on A, then an element e A is said to be the
identity element iff a * e = e * a a A
Identity element is unique.
If * is Binary operation on set A, then an element b is said to be inverse
of a A iff a * b = b * a = e
Inverse of an element, if it exists, is unique.
XII – Maths
10
11. VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)
1.
If A is the set of students of a school then write, which of following
relations are. (Universal, Empty or neither of the two).
R1 = {(a, b) : a, b are ages of students and |a – b| 0}
R2 = {(a, b) : a, b are weights of students, and |a – b| < 0}
R3 = {(a, b) : a, b are students studying in same class}
2.
Is the relation R in the set A = {1, 2, 3, 4, 5} defined as R = {(a, b) : b
= a + 1} reflexive?
3.
If R, is a relation in set N given by
R = {(a, b) : a = b – 3, b > 5},
then does elements (5, 7) R?
4.
If f : {1, 3} {1, 2, 5} and g : {1, 2, 5} {1, 2, 3, 4} be given by
f = {(1, 2), (3, 5)}, g = {(1, 3), (2, 3), (5, 1)}
Write down gof.
5.
Let g, f : R R be defined by
g x
6.
If
x 2
, f x 3 x 2. Write fog.
3
f : R R defined by
f x
2x 1
5
be an invertible function, write f–1(x).
x
x 1 Write
,
x 1
7.
If f x
8.
Let * is a Binary operation defined on R, then if
(i)
fo f x .
a * b = a + b + ab, write 3 * 2
11
XII – Maths
12. (ii)
9.
a *b
a b 2
3
, Write 2 * 3 * 4.
If n(A) = n(B) = 3, Then how many bijective functions from A to B can be
formed?
10.
If f (x) = x + 1, g(x) = x – 1, Then (gof) (3) = ?
11.
Is f : N N given by f(x) = x2 is one-one? Give reason.
12.
If f : R A, given by
f(x) = x2 – 2x + 2 is onto function, find set A.
13.
If f : A B is bijective function such that n (A) = 10, then n (B) = ?
14.
If n(A) = 5, then write the number of one-one functions from A to A.
15.
R = {(a, b) : a, b N, a b and a divides b}. Is R reflexive? Give reason?
16.
Is f : R R, given by f(x) = |x – 1| is one-one? Give reason?
17.
f : R B given by f(x) = sin x is onto function, then write set B.
18.
1 x
2x
, show that f
2f x .
If f x log
1 x
1 x2
19.
If ‘*’ is a binary operation on set Q of rational numbers given by a * b
then write the identity element in Q.
20.
If * is Binary operation on N defined by a * b = a + ab a, b N. Write
the identity element in N if it exists.
SHORT ANSWER TYPE QUESTIONS (4 Marks)
21.
ab
5
Check the following functions for one-one and onto.
2x 3
7
(a)
f : R R , f (x )
(b)
f : R R, f(x) = |x + 1|
(c)
f : R – {2} R, f x
XII – Maths
3x 1
x 2
12
13. (d)
f : R [–1, 1], f(x) = sin2x
22.
Consider the binary operation * on the set {1, 2, 3, 4, 5} defined by
a* b = H.C.F. of a and b. Write the operation table for the operation *.
23.
Let f : R
4
4
R
3
3
be a function given by f x
1
that f is invertible with f x
4x
. Show
3x 4
4x
.
4 3x
24.
Let R be the relation on set A = {x : x Z, 0 x 10} given by
R = {(a, b) : (a – b) is multiple of 4}, is an equivalence relation. Also, write
all elements related to 4.
25.
Show that function f : A B defined as f x
3x 4
where
5x 7
7
3
A R , B R is invertible and hence find f –1.
5
5
26.
Let * be a binary operation on Q. Such that a * b = a + b – ab.
(i)
(ii)
27.
Prove that * is commutative and associative.
Find identify element of * in Q (if it exists).
If * is a binary operation defined on R – {0} defined by a * b
check * for commutativity and associativity.
28.
2a
b2
, then
If A = N × N and binary operation * is defined on A as
(a, b) * (c, d) = (ac, bd).
(i)
Check * for commutativity and associativity.
(ii)
Find the identity element for * in A (If it exists).
29.
Show that the relation R defined by (a, b) R(c, d) a + d = b + c on
the set N × N is an equivalence relation.
30.
Let * be a binary operation on set Q defined by a * b ab , show that
4
(i)
4 is the identity element of * on Q.
13
XII – Maths
14. (ii)
Every non zero element of Q is invertible with
a 1
16
,
a
a Q 0 .
1
is bijective where R+ is the
2x
set of all non-zero positive real numbers.
31.
Show that f : R+ R+ defined by f x
32.
Consider f : R+ [–5, ) given by f(x) = 9x2 + 6x – 5 show that f is
1
invertible with f
x 6 1
.
3
33.
If ‘*’ is a binary operation on R defined by a * b = a + b + ab. Prove that
* is commutative and associative. Find the identify element. Also show
that every element of R is invertible except –1.
34.
If f, g : R R defined by f(x) = x2 – x and g(x) = x + 1 find (fog) (x) and
(gof) (x). Are they equal?
35.
f : [1 ) [2, ) is given by f x x
,
36.
f : R R, g : R R given by f(x) = [x], g(x) = |x| then find
fog
1
, find f 1 x .
x
2
2
and gof .
3
3
ANSWERS
1.
R1 : is universal relation.
R2 : is empty relation.
R3 : is neither universal nor empty.
2.
No, R is not reflexive.
3.
(5, 7) R
4.
gof = {(1, 3), (3, 1)}
5.
(fog)(x) = x x R
XII – Maths
14
15. 5x 1
2
6.
f 1 x
7.
fof x
8.
x
1
,x
2x 1
2
(i)
3 * 2 = 11
(ii)
1369
27
9.
6
10.
3
11.
2
2
Yes, f is one-one x 1, x 2 N x 1 x 2 .
12.
A = [1, ) because Rf = [1, )
13.
n(B) = 10
14.
120.
15.
No, R is not reflexive a, a R a N
16.
f is not one-one functions
f(3) = f (–1) = 2
3 – 1 i.e. distinct element has same images.
17.
B = [–1, 1]
19.
e = 5
20.
Identity element does not exist.
21.
(a)
Bijective
(b)
Neither one-one nor onto.
(c)
One-one, but not onto.
(d)
Neither one-one nor onto.
15
XII – Maths
16. 22.
*
1
2
3
4
5
1
1
1
1
1
1
2
1
2
1
2
1
3
1
1
3
1
1
4
1
2
1
4
1
5
1
1
1
1
5
24.
Elements related to 4 are 0, 4, 8.
25.
f 1 x
26.
0 is the identity element.
27.
Neither commutative nor associative.
28.
7x 4
5x 3
(i)
Commutative and associative.
(ii)
(1, 1) is identity in N × N
33.
0 is the identity element.
34.
(fog) (x) = x2 + x
(gof) (x) = x2 – x + 1
Clearly, they are unequal.
x x2 4
2
35.
f 1 x
36.
fog
2
0
3
gof
2
1
3
XII – Maths
16
17. CHAPTER 2
INVERSE TRIGONOMETRIC FUNCTIONS
IMPORTANT POINTS
sin–1 x, cos–1 x, ... etc., are angles.
If sin x and , then = sin–1x etc.
2 2
Function
Domain
Range
(Principal Value Branch)
sin–1x
2 , 2
cos–1x
[–1, 1]
[0, ]
tan–1x
R
,
2 2
cot–1x
R
(0, )
sec–1x
R – (–1, 1)
0,
cosec –1x
[–1, 1]
R – (–1, 1)
2 , 2 0
2
sin–1 (sin x) = x x ,
2 2
cos–1 (cos x) = x x [0, ] etc.
sin (sin–1x) = x x [–1, 1]
cos (cos–1x) = x x [–1, 1] etc.
17
XII – Maths
18.
1
sin1x cosec –1 x 1, 1
x
tan–1x = cot–1 (1/x) x > 0
sec–1x = cos–1 (1/x), |x| 1
sin–1(–x) = – sin–1x x [–1, 1]
tan–1(–x) = – tan–1x x R
cosec–1(–x) = – cosec–1x |x| 1
cos–1(–x) = – cos–1x × [–1, 1]
cot–1(–x) = – cot–1x x – R
sec–1(–x) = – sec–1x |x| 1
sin1 x cos 1 x
tan–1 x cot –1 x
, x 1, 1
2
x R
2
sec –1 x cosec –1x
x 1
2
x y
tan1 x tan 1 y tan 1
;
1 xy
xy 1.
x y
tan1 x tan1 y tan 1
;
1 xy
xy 1.
2x
2 tan–1 x tan –1
, x 1
1 x 2
2x
2 tan1 x sin1
, x 1
,
1 x 2
1 x 2
2 tan1 x cos 1
, x 0.
1 x 2
XII – Maths
18
19. VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)
1.
Write the principal value of
–1
–
–1
(i)
sin
(iii)
tan
(v)
cot
(vii)
sin
3 2
–1
3 2.
(ii)
cos
1
–
3
(iv)
cosec–1 (– 2).
–1
1
.
3
(vi)
sec–1 (– 2).
1
3
1 1
1
cos tan 1 3
2
2
2.
What is value of the following functions (using principal value).
(i)
tan
–1
–1 2
1
.
– sec
3
3
–1
–1 3
1
.
– – cos
2
2
(ii)
sin
(iv)
cosec
(iii)
tan–1 (1) – cot–1 (–1).
(v)
sin
–1
5
tan
.
6
2 sec
–1
2 .
tan–1 (1) + cot–1 (1) + sin–1 (1).
(vi)
–1
(viii)
–1
4
sin
.
5
cosec
–1
(vii)
tan
3
cosec
.
4
SHORT ANSWER TYPE QUESTIONS (4 MARKS)
3.
Show that tan
–1
1 cos x
1 cos x –
19
1 cos x
x
.
1 – cos x
4
2
x [0, ]
XII – Maths
20. 4.
Prove
1
tan
5.
tan
Prove
6.
1 1 cos x
cos x
cot
1 sin x
1 cos x
4
–1
x 0, 2 .
2
2
x
a x
–1 x
–1
.
sin
cos
a
a
a2 – x 2
Prove
cot
–1
–1 8
–1
–1 8
–1 300
.
2 tan cos
tan 2 tan sin
tan
161
17
17
1
7.
Prove
tan
8.
Solve
cot
–1
2
1 x
1 x2
2x cot
–1
1
1 2
cos x .
2
4
2
1 x
1 x
3x
2
.
4
9.
m
m n
Prove that tan1 tan 1
, m, n 0
n
m n 4
10.
1
1 y 2 x y
2x 1
cos 1
Prove that tan sin1
1 x 2 2
1 y 2 1 xy
2
11.
x 2 1 1
2x 2
Solve for x, cos1 2 tan 1
x 1 2
1 x 2
3
12.
Prove that tan1
13.
Solve for
14.
1
1
32
Prove that 2 tan–1 tan –1 tan –1
4
5
43
XII – Maths
x,
1
1
1
1
tan1 tan1 tan1
3
5
7
8 4
tan cos 1 x sin tan1 2 ; x 0
20
21. 1
3
tan cos –1
11
2
15.
Evaluate
16.
a cos x b sin x
a
tan –1 x
Prove that tan–1
b cos x a sin x
b
17.
Prove that
1
cot tan1 x tan1 cos 11 2 x 2 cos 1 2 x 2 1 , x 0
x
18.
a b
b c
c a
tan –1
tan –1
0 where a, b,
Prove that tan–1
1 ab
1 bc
1 ca
c > 0
19.
Solve for x, 2 tan–1(cos x) = tan–1 (2 cosec x)
20.
Express
21.
If tan–1a + tan–1b + tan–1c = then
sin–1 x 1 x x 1 x 2
in simplest form.
prove that a + b + c = abc
22.
If sin–1x > cos–1x, then x belongs to which interval?
ANSWERS
(ii)
6
(iii)
–
6
3
(vi)
2
3
(vii)
.
6
2.
(i)
0
(ii)
3
(iii)
(v)
(vi)
5
(vii)
(i)
–
(v)
1.
3
21
(iv)
–
6
2
(iv)
2
–
6
(viii)
.
4
XII – Maths
22. 8.
1
11.
13.
5
3
19.
x
22.
1
, 1
2
21.
Hint:
15.
.
4
Let
20
tan–1 a =
tan–1 b =
tan–1 c =
then given,
take tangent on both sides,
tan ( ) = tan
XII – Maths
22
tan
2 3
12
11 3
3 11
sin–1 x – sin–1 x.
23. CHAPTER 3 & 4
MATRICES AND DETERMINANTS
POINTS TO REMEMBER
Matrix : A matrix is an ordered rectangular array of numbers or functions.
The numbers or functions are called the elements of the matrix.
Order of Matrix : A matrix having ‘m’ rows and ‘n’ columns is called the
matrix of order mxn.
Square Matrix : An mxn matrix is said to be a square matrix of order n
if m = n.
Column Matrix : A matrix having only one column is called a column
matrix i.e. A = [aij]mx1 is a column matrix of order mx1.
Row Matrix : A matrix having only one row is called a row matrix
i.e. B bij 1xn is a row matrix of order 1xn.
Zero Matrix : A matrix having all the elements zero is called zero matrix
or null matrix.
Diagonal Matrix : A square matrix is called a diagonal matrix if all its non
diagonal elements are zero.
Scalar Matrix : A diagonal matrix in which all diagonal elements are
equal is called a scalar matrix.
Identity Matrix : A scalar matrix in which each diagonal element is 1, is
called an identity matrix or a unit matrix. It is denoted by I.
I = [eij]n ×
where,
1 if i j
eij
0 if i j
n
23
XII – Maths
24.
Transpose of a Matrix : If A = [ai j ]m × n be an m × n matrix then the
matrix obtained by interchanging the rows and columns of A is called the
transpose of the matrix. Transpose of A is denoted by A´ or AT.
Properties of the transpose of a matrix.
(i)
(iii)
(A´)´ = A
(ii)
(A + B)´ = A´ + B´
(kA)´ = kA´, k is a scalar
(iv)
(AB)´ = B´A´
Symmetrix Matrix : A square matrix A = [aij ] is symmetrix if aij = aji
i, j. Also a square matrix A is symmetrix if A´ = A.
Skew Symmetrix Matrix : A square matrix A = [aij] is skew-symmetrix,
if aij = – aji i, j. Also a square matrix A is skew - symmetrix, if A´ = –
A.
Determinant : To every square matrix A = [aij] of order n × n, we can
associate a number (real or complex) called determinant of A. It is denoted
by det A or |A|.
Properties
(i)
|AB| = |A| |B|
(ii)
|kA|n × n = kn |A|n × n where k is a scalar.
Area of triangles with vertices (x1, y1), (x2, y2) and (x3, y3) is given
by
x1
1
x2
2
x3
y1 1
y2 1
y3 1
x1
The points (x1, y1), (x2, y2), (x3, y3) are collinear x 2
x3
y1 1
y2 1 0
y3 1
Adjoint of a Square Matrix A is the transpose of the matrix whose
elements have been replaced by their cofactors and is denoted as adj A.
Let
n
adj A = [Aji]n ×
XII – Maths
A = [aij]n ×
n
24
25. Properties
(i)
A(adj A) = (adj A) A = |A| I
(ii)
If A is a square matrix of order n then |adj A| = |A|n–1
(iii)
adj (AB) = (adj B) (adj A).
[Note : Correctness of adj A can be checked by using
A.(adj A) = (adj A) . A = |A| I ]
Singular Matrix : A square matrix is called singular if |A| = 0, otherwise
it will be called a non-singular matrix.
Inverse of a Matrix : A square matrix whose inverse exists, is called
invertible matrix. Inverse of only a non-singular matrix exists. Inverse of
a matrix A is denoted by A–1 and is given by
A
1
1
. adj . A
A
Properties
(i)
(ii)
(A–1)–1 = A
(iii)
(AB)–1 = B–1A–1
(iv)
AA–1 = A–1A = I
(AT)–1 = (A–1)T
Solution of system of equations using matrix :
If AX = B is a matrix equation then its solution is X = A–1B.
(i)
If |A| 0, system is consistent and has a unique solution.
(ii)
If |A| = 0 and (adj A) B 0 then system is inconsistent and has
no solution.
(iii)
If |A| = 0 and (adj A) B = 0 then system is either consistent and
has infinitely many solutions or system is inconsistent and has no
solution.
25
XII – Maths
26. VERY SHORT ANSWER TYPE QUESTIONS (1 Mark)
1.
4
x 3
If
y 4 x y
5 4
3 9 , find x and y.
2.
i
If A
0
3.
Find the value of a23 + a32 in the matrix A = [aij]3
0
0
and B i
i
i
, find AB .
0
× 3
if i j
2i – j
where aij
.
i 2 j 3 if i j
4.
If B be a 4 × 5 type matrix, then what is the number of elements in the
third column.
5.
5 2
3 6
If A
and B 0 1 find 3 A 2B.
0 9
6.
2 3
1 0
If A
and B 2 6 find A B ´.
7 5
7.
2
If A = [1 0 4] and B 5 find AB .
6
8.
x 2
4
If A
is symmetric matrix, then find x.
2x 3 x 1
9.
3
0 2
2 0
4 is skew symmetrix matrix.
For what value of x the matrix
3 4 x 5
10.
2 3
If A
P Q where P is symmetric and Q is skew-symmetric
1 0
matrix, then find the matrix Q.
XII – Maths
26
27. 11.
a ib
Find the value of c id
12.
If
13.
k
For what value of k, the matrix
3
14.
sin 30
If A
sin 60
15.
2x 5
5x 2
c id
a ib
3
0, find x .
9
2
has no inverse.
4
cos 30
, what is |A|.
cos 60
Find the cofactor of a12 in
2
6
1
3
0
5
5
4
7
.
1 3 2
4 5 6 .
3 5 2
16.
Find the minor of a23 in
17.
1 2
Find the value of P, such that the matrix
is singular.
4 P
18.
Find the value of x such that the points (0, 2), (1, x) and (3, 1) are
collinear.
19.
Area of a triangle with vertices (k, 0), (1, 1) and (0, 3) is 5 unit. Find the
value (s) of k.
20.
If A is a square matrix of order 3 and |A| = – 2, find the value of |–3A|.
21.
If A = 2B where A and B are square matrices of order 3 × 3 and |B| =
5, what is |A|?
22.
What is the number of all possible matrices of order 2 × 3 with each entry
0, 1 or 2.
23.
Find the area of the triangle with vertices (0, 0), (6, 0) and (4, 3).
24.
If
2x 4 6 3
, find x .
1 x
2
1
27
XII – Maths
28. 25.
x y
If A z
1
y z
x
1
z x
y , write the value of det A.
1
26.
a11 a12
If A
such that |A| = – 15, find a11 C21 + a12C22 where Cij is
a21 a22
cofactors of aij in A = [aij].
27.
If A is a non-singular matrix of order 3 and |A| = – 3 find |adj A|.
28.
5 3
If A
6 8 find adj A
29.
Given a square matrix A of order 3 × 3 such that |A| = 12 find the value
of |A adj A|.
30.
If A is a square matrix of order 3 such that |adj A| = 8 find |A|.
31.
Let A be a non-singular square matrix of order 3 × 3 find |adj A| if |A| =
10.
32.
2 1
If A
find
3 4
33.
3
If A 1 2 3 and B 4 find |AB|.
0
A 1 1 .
SHORT ANSWER TYPE QUESTIONS (4 MARKS)
34.
x y
Find x, y, z and w if
2x y
2x z
–1
0
3x w
35.
Construct a 3 × 3 matrix A = [a ij] whose elements are given by
1 i j if i j
aij = i 2 j if i j
2
XII – Maths
28
5
.
13
29. 36.
1 2 3
3 0 1
Find A and B if 2A + 3B =
and A 2B 1 6 2 .
2 0 1
37.
1
If A 2 and B 2 1 4 , verify that (AB)´ = B´A´.
3
38.
3 1
3
2 2 1 P Q where P is a symmetric and Q
Express the matrix
4 5 2
is a skew-symmetric matrix.
39.
cos
If A =
sin
sin
cos n sin n
n
, then prove that A sin n cos n
cos
where n is a natural number.
40.
2 1
5 2
2 5
Let A
, B 7 4 , C 3 8 , find a matrix D such that
3 4
CD – AB = O.
41.
1 3 2 1
1 x 1 2 5 1 2 0
Find the value of x such that
15 3 2 x
42.
Prove that the product of the matrices
cos 2
cos 2
cos sin
cos sin
and
2
2
cos sin sin
cos sin sin
is the null matrix, when and differ by an odd multiple of
43.
.
2
5 3
2
–1
If A
show that A – 12A – I = 0. Hence find A .
12 7
29
XII – Maths
30. 44.
2 3
2
If A
find f(A) where f(x) = x – 5x – 2.
4 7
45.
4
If A
2
46.
1 2 3
Find the matrix X so that X
4 5 6
47.
2 3
1 2
–1
–1 –1
If A
and B 1 3 then show that (AB) = B A .
1 4
48.
3
, find x and y such that A2 – xA + yI = 0.
5
7 8 9
2 4 6 .
Test the consistency of the following system of equations by matrix
method :
3x – y = 5; 6x – 2y = 3
49.
Using elementary row transformations, find the inverse of the matrix
6 3
, if possible.
A
1
2
50.
3 1
By using elementary column transformation, find the inverse of A
.
5 2
51.
cos
If A
sin
sin
and A + A´ = I, then find the general value of .
cos
Using properties of determinants, prove the following : Q 52 to Q 59.
52.
a b c
2a
2a
3
2b
b c a
2b
a b c
2c
2c
c a b
53.
x 2 x 3 x 2a
x 3 x 4 x 2b 0 if a, b, c are in A.P .
x 4 x 5 x 2c
54.
sin cos sin
sin cos sin 0
sin cos sin
XII – Maths
30
31. b
2
c
b
55.
c
56.
a
2
c
59.
2
a ab
b
2
2
b
2
2
2
a b
2 2 2
4a b c .
2
a b
a
p q 2 p
x y
x
b
ac c
2
ac
bc
b
x b
b
c
b
q
y
c
r .
z
2
2 2 2
4a b c .
2
c
2
c
x x a b c .
x c
Show that :
2
x
yz
(i)
(ii)
61.
2
bc
x
60.
a
c a
r p
z x
2
x a
a
a
2
a
c
ab
58.
2
2
b c
q r
y z
a
57.
2
y
z
2
2
z y z z x x y yz zx xy .
xy
y
zx
If the points (a, b) (a´, b´) and (a – a´, b – b´) are collinear. Show
that ab´ = a´b.
2
If A
2
0
Given A
2
5
4
and B 2
1
1
2
3
verity that AB A B .
5
0
2
and B 1
0
1
1
0 . Find the product AB and
1
also find (AB)–1.
62.
Solve the following equation for x.
a x
a x
a x
a x
a x
a x
a x
a x 0.
a x
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XII – Maths
32. 63.
0
If A
tan
2
that,
tan
2
and I is the identity matrix of order 2, show
0
cos
I A I A
sin
64.
sin
cos
Use matrix method to solve the following system of equations : 5x – 7y
= 2, 7x – 5y = 3.
LONG ANSWER TYPE QUESTIONS (6 MARKS)
65.
Obtain the inverse of the following matrix using elementary row operations
0 1 2
A 1 2 3 .
3 1 1
66.
1
1 1 2 2 0
0
9 2 3
2 3
Use product
to solve the system of equations
3 2 4 6 1 2
x – y + 2z = 1, 2y – 3z = 1, 3x – 2y + 4z = 2.
67.
Solve the following system of equations by matrix method, where x 0,
y 0, z 0
2 3 3
1 1 1
3 1 2
10,
10,
13.
x y z
x y z
x y z
68.
1
where A 2
3
equations :
Fi
nd A–1,
2
3
3
3
2 , hence solve the system of linear
–4
x + 2y – 3z = – 4
2x + 3y + 2z = 2
3x – 3y – 4z = 11
XII – Maths
32
33. 69.
The sum of three numbers is 2. If we subtract the second number from
twice the first number, we get 3. By adding double the second number
and the third number we get 0. Represent it algebraically and find the
numbers using matrix method.
70.
Compute the inverse of the matrix.
3
A 15
5
71.
If the matrix
1
6
2
1
A 0
3
1
5 and verify that A–1 A = I3.
5
1
2
2
2
1
and B –1 0
3
1
4
2
3
0
0
–1 ,
2
then
compute (AB)–1.
72.
Using matrix method, solve the following system of linear equations :
2x – y = 4, 2y + z = 5, z + 2x = 7.
1
0 1 1
A 2 3I
.
if A 1 0 1 . Also show that A 1
2
1 1 0
73.
Find A
74.
1
Find the inverse of the matrix A 1
0
column transformations.
75.
2 3
Let A
and f(x) = x2 – 4x + 7. Show that f (A) = 0. Use this result
1 2
2
3
2
2
0 by using elementary
1
to find A5.
76.
cos sin 0
If A sin cos 0 , verify that A . (adj A) = (adj A) . A = |A| I3.
0
0
1
33
XII – Maths
34. 77.
2 1 1
For the matrix A 1 2 1 , verify that A3 – 6A2 + 9A – 4I = 0, hence
1 1 2
find A–1.
78.
Find the matrix X for which
3 2
7 5 . X
79.
1 1 2 1
.
2 1 0 4
By using properties of determinants prove the following :
1 a2 b 2
2ab
2b
2ab
1 a2 b 2
2a
2b
2a
1 a2 b 2
y z 2
xy
zx
2
81.
xy
x z
yz
xz
80.
yz
3
2xyz x y z .
x y 2
a
ab
a b c
2a 3a 2b 4a 3b 2c a 3 .
3a 6a 3b 10a 6b 3c
x
82.
If x, y, z are different and y
z
83.
3
1 a 2 b 2 .
x 2 1 x 3
y 2 1 y 3 0. Show that xyz = – 1.
z2
1 z 3
If x, y, z are the 10th, 13th and 15th terms of a G.P. find the value of
log x
log y
log z
XII – Maths
10 1
13 1 .
15 1
34
35. 84.
Using the properties of determinants, show that :
1 a 1
1
1 1 1
1
1 b 1
abc 1 abc bc ca ab
a b c
1
1
1 c
85.
Using properties of determinants prove that
bc
b 2 bc
c 2 bc
a 2 ac
ac
c 2 ac ab bc ca
a 2 ab b 2 ab
86.
I
f
3
ab
3 2 1
A 4 1 2 , find A–1 and hence solve the system of equations
7 3 3
3x + 4y + 7z = 14, 2x – y + 3z = 4, x + 2y – 3z = 0.
ANSWERS
1.
x = 2, y = 7
2.
0 1
1 0
3.
11.
4.
4
5.
9 6
0 29 .
6.
3 5
3 1 .
7.
AB = [26].
8.
x = 5
9.
x = – 5
10.
0
1
11.
a2 + b2 + c2 + d2.
12.
x = – 13
13.
k
14.
|A| = 1.
15.
46
16.
–4
3
2
35
1
.
0
XII – Maths
39. CHAPTER 5
CONTINUITY AND DIFFERENTIATION
POINTS TO REMEMBER
A function f(x) is said to be continuous at x = c iff lim f x f c
x c
i.e., lim – f x lim f x f c
x c
x c
f(x) is continuous in (a, b) iff it is continuous at x c c a, b .
f(x) is continuous in [a, b] iff
(i)
(ii)
(iii)
f(x) is continuous in (a, b)
lim f x f a ,
x a
lim f x f b
x b –
Trigonometric functions are continuous in their respective domains.
Every polynomial function is continuous on R.
If f (x) and g (x) are two continuous functions and c R then at x = a
(i)
(ii)
g (x) . f (x), f (x) + c, cf (x), | f (x)| are also continuous at x = a.
(iii)
f (x) ± g (x) are also continuous functions at x = a.
f x
is continuous at x = a provided g(a) 0.
g x
f (x) is derivable at x = c in its domain iff
39
XII – Maths
40. lim
x c
f x f c
f x f c
lim
, and is finite
x c
x c
x c
The value of above limit is denoted by f´(c) and is called the derivative
of f(x) at x = c.
d
dv
du
u · v u · v ·
dx
dx
dx
d u
dx v
If y = f(u) and u = g(t) then
v·
du
dv
u·
dx
dx
v2
dy
dt
If y = f(u),
dy
du
du
f ´ u .g´ t
Chain Rule
dt
x = g(u) then,
dy
dy
du
f ´u
.
dx
du
dx
g ´u
d 1
sin x
dx
1
1 x
2
,
d 1
1
tan x
,
dx
1 x2
d
1
sec 1 x
,
dx
x x2 1
d x
e
dx
ex,
d 1
cos x
dx
1
1 x
2
d 1
1
cot x
dx
1 x 2
d
1
cosec 1x
dx
x x2 1
d
1
log x
dx
x
f (x) = [x] is discontinuous at all integral points and continuous for all x
R – Z.
Rolle’s theorem : If f (x) is continuous in [ a, b ], derivable in (a, b) and
f (a) = f (b) then there exists atleast one real number c (a, b) such that
f´ (c) = 0.
XII – Maths
40
41.
Mean Value Theorem : If f (x) is continuous in [a, b] and derivable in
(a, b) then there exists atleast one real number c (a, b) such that
f ´ c
f b – f a
.
b a
f (x) = logex, (x > 0) is continuous function.
VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)
1.
For what value of x, f(x) = |2x – 7| is not derivable.
2.
Write the set of points of continuity of g(x) = |x – 1| + |x + 1|.
3.
What is derivative of |x – 3| at x = – 1.
4.
What are the points of discontinuity of f x
5.
Write the number of points of discontinuity of f(x) = [x] in [3, 7].
6.
x 3 if x 2
f x
4 if x 2 is a continuous function for all
The function,
2x if x 2
x 1 x 1
.
x 7 x 6
x R, find .
tan3x
, x 0
2K ,
7.
For what value of K, f x sin2x
x 0
is continuous x R .
8.
Write derivative of sin x w.r.t. cos x.
9.
If f(x) = x2g(x) and g(1) = 6, g´(1) = 3 find value of f´(1).
10.
Write the derivative of the following functions :
(i)
(iii)
log3 (3x + 5)
e 6 loge
x 1
(ii)
e
log 2 x
,x 1
41
XII – Maths
42. (iv)
sec 1 x cosec 1 x , x 1.
(v)
sin1 x 7 2
(vi)
logx 5, x > 0.
SHORT ANSWER TYPE QUESTIONS (4 MARKS)
11.
Discuss the continuity of following functions at the indicated points.
(i)
(ii)
sin 2x
3x , x 0
g x
at x 0.
3
x 0
2
(iii)
x 2 cos 1 x x 0
f x
0
x 0
(iv)
f(x) = |x| + |x – 1| at x = 1.
(v)
12.
x x
, x 0
f x x
at x 0.
2,
x 0
x x , x 1
f x
at x 1.
0
x 1
at x 0.
3x 2 kx 5,
For what value of k, f x
1 3x
0 x 2
is continuous
2 x 3
x 0, 3.
13.
For what values of a and b
f x
XII – Maths
x 2
a
x 2
a b
x 2
2b
x 2
if x –2
if x –2
is continuous at x = 2.
if x –2
42
43. 14.
Prove that f(x) = |x + 1| is continuous at x = –1, but not derivable at
x = –1.
15.
For what value of p,
x p sin 1 x x 0
f x
is derivable at x = 0.
0
x 0
1
2
2x
dy
1
1
2 tan1 , 0 x 1 find
,
.
tan
x
1 x2
dx
16.
If y
17.
1 x
dy
?
If y sin 2 tan1
then
1 x
dx
18.
If 5x + 5y = 5x+y then prove that
19.
If x 1 y 2 y 1 x 2 a then show that
20.
If
21.
If (x + y)m +
22.
2x
2x
w.r.t. sin1
.
Find the derivative of tan1
1 x 2
1 x 2
23.
Find the derivative of loge(sin x) w.r.t. loga(cos x).
24.
If xy + yx + xx = mn, then find the value of
25.
If x = a cos3, y = a sin3 then find
dy
5y x 0 .
dx
dy
1 y 2
.
dx
1 x 2
1 x 2 1 y 2 a x y then show that
n
= xm . yn then prove that
43
dy
1 y 2
.
dx
1 x 2
dy y
.
dx x
dy
.
dx
d 2y
.
dx 2
XII – Maths
44. 26.
x = aet (sint – cos t)
If
y = aet (sint + cost) then show that
dy
at x is 1.
dx
4
dy
.
dx
27.
1
2
If y sin x 1 x x 1 x then find –
28.
If y x loge x loge x
29.
x
Differentiate x
30.
Find
31.
1 sin x
If y tan1
1 sin x
32.
1
If x sin loge
a
33.
Differentiate log x log x , x 1 w .r .t . x
34.
If sin y = x sin (a + y) then show that
35.
If y = sin–1x, find
36.
If
37.
a cos
If y e
38.
If y3 = 3ax2 – x3 then prove that
39.
Verify Rolle's theorem for the function, y = x2 + 2 in the interval [a, b]
where a = –2, b = 2.
40.
Verify Mean Value Theorem for the function, f(x) = x2 in [2, 4]
x
x
then find
dy
.
dx
w.r.t. x.
dy
y
x
, if cos x cos y
dx
1 sin x
dy
where 2 x find dx .
1 sin x
y then show that (1 – x2) y´´ – xy´ – a2y = 0.
2
dy sin a y
.
dx
sin a
d 2y
in terms of y.
dx 2
d 2y
b 4
x2 y2
2 3.
2 1, then show that
dx 2 a y
a2 b
XII – Maths
1
x
, 1 x 1 show that 1 x 2
,
44
d 2y
dy
x
a2 y 0
2
dx
dx
d 2 y 2a 2 x 2
.
dx 2
y5
45. ANSWERS
1.
x = –7/2
2.
R
3.
–1
4.
x = 6, 7
5.
Points of discontinuity of f(x) are 4, 5, 6, 7 i.e. four points.
Note : At x = 3, f(x) = [x] is continuous. because lim f x 3 f 3 .
x 3
3
.
4
6.
7
.
2
7.
k
8.
–cot x
9.
15
10.
(i)
(ii)
e log2
(iii) 6 (x – 1)5
(iv)
0
7 x2 x
.
2 1 x 7
(vi)
(i) Discontinuous
(ii)
Discontinuous
(iv)
continuous
(v)
11.
3
log3 e
3x 5
(iii) Continuous
x
1
.log2 e.
x
loge 5
2
x loge x
.
(v) Discontinuous
12.
k = 11
13.
a = 0, b = –1.
15.
p > 1.
16.
0
22.
1
17.
x
1 x 2
.
23.
–cot2x logea
24.
x
y 1
y x log y
dy x 1 log x yx
.
dx
x y log x xy x 1
45
XII – Maths
46. 25.
d 2y
1
cosec sec 4 .
dx 2 3a
28.
x log x
29.
x
dy
1
x x . x x log x 1 log x
.
dx
x logx
30.
dy y tan x logcos y
dx x tan y logcos x
31.
dy
1
.
dx
2
27.
2log x
x 1
logx
log logx .
x
logx
Hint. : sin
x
x
cos for x , .
2
2
2
1 log log x
, x 1
x
x
33.
log x log x
35.
sec2y tany.
XII – Maths
dy
1
1
.
dx
1 x 2 2 x 1 x
46
47. CHAPTER 6
APPLICATIONS OF DERIVATIVES
POINTS TO REMEMBER
Rate of Change : Let y = f(x) be a function then the rate of change of
y with respect to x is given by
dy
f ´ x where a quantity y varies with
dx
another quantity x.
dy
or f ´ x 0 represents the rate of change of y w.r.t. x at x = x0.
dx x x 0
If x = f(t) and y = g (t)
By chain rule
dy dy
dx dt
(i)
dx
dx
if
0.
dt
dt
A function f(x) is said to be increasing (non-decreasing) on an
interval (a, b) if x1 x2 in (a, b) f(x1) f (x2) x1, x2 (a, b).
Alternatively if f ´ x 0 0 x a, b , then f(x) is increasing
function in (a, b).
(ii)
A function f(x) is said to be decreasing (non-increasing) on an
interval (a, b). If x1 x2 in (a, b) f(x1) f(x2) x1, x2 (a, b).
Alternatively if f´(x) 0 x (a, b), then f(x) is decreasing
function in (a, b).
The equation of tangent at the point (x0, y0) to a curve y = f(x) is given
by
dy
y y0
x x 0 .
dx x 0 , y 0
47
XII – Maths
48. where
dy
slope of the tangent at the point x 0, y 0 .
dx x 0 ,y 0
dy
does not exist then tangent is parallel to y-axis at
dx x 0 , y 0
(x0, y0) and its equation is x = x0.
(i)
(ii)
If
If tangent at x = x0 is parallel to x-axis then
dy
0
dx x x 0
1
Slope of the normal to the curve at the point (x0, y0) is given by dy
dx x x 0
Equation of the normal to the curve y = f(x) at a point (x0, y0) is given by
y y0
1
x x 0 .
dy
dx x 0 ,y 0
dy
0. then equation of the normal is x = x0.
dx x 0 ,y 0
If
dy
If dx
x 0 ,y 0 does not exit, then the normal is parallel to x-axis and the
equation of the normal is y = y0.
Let
y = f(x)
x = the small increment in x and
y be the increment in y corresponding to the increment in x
Then approximate change in y is given by
dy
dy
x
dx
or
dy = f ´(x) x
The approximate change in the value of f is given by
f x x
XII – Maths
x ´ f x x
f
48
49.
Let f be a function. Let point c be in the domain of the function f at which
either f´ (x) = 0 or f is not derivable is called a critical point of f.
First Derivative Test : Let f be a function defined on an open interval I.
Let f be continuous at a critical point c I. Then if,
(i)
(ii)
f´(x) changes sign from negative to positive as x increases through
c, then c is a point of local minima.
(iii)
f´(x) changes sign from positive to negative as x increases through
c, then c is called the point of the local maxima.
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 inflexion.
Second Derivative Test : Let f be a function defined on an interval I and
let c I. Then
(i)
x = c is a point of local maxima if f´(c) = 0 and f´´(c) < 0.
f (c) is local maximum value of f.
(ii)
x = c is a point of local minima if f´(c) = 0 and f"(c) > 0. f(c) is
local minimum value of f.
(iii)
The test fails if f´(c) = 0 and f´´(c) = 0.
VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)
1.
The side of a square is increasing at the rate of 0.2 cm/sec. Find the rate
of increase of perimeter of the square.
2.
The radius of the circle is increasing at the rate of 0.7 cm/sec. What is
the rate of increase of its circumference?
3.
If the radius of a soap bubble is increasing at the rate of
4.
A stone is dropped into a quiet lake and waves move in circles at a speed
of 4 cm/sec. At the instant when the radius of the circular wave is 10 cm,
how fast is the enclosed area increasing?
1
cm sec. At
2
what rate its volume is increasing when the radius is 1 cm.
49
XII – Maths
50. 5.
The total revenue in rupees received from the sale of x units of a product
is given by
R(x) = 13x2 + 26x + 15. Find the marginal revenue when x = 7.
6.
Find the maximum and minimum values of function f(x) = sin 2x + 5.
7.
Find the maximum and minimum values (if any) of the function
f(x) = – |x – 1| + 7 x R .
8.
Find the value of a for which the function f(x) = x2 – 2ax + 6, x > 0 is
strictly increasing.
9.
Write the interval for which the function f(x) = cos x, 0 x 2 is
decreasing.
10.
What is the interval on which the function f x
increasing?
log x
, x 0, is
x
4 3
x is increasing?
3
11.
For which values of x, the functions y x 4
12.
Write the interval for which the function f x
13.
Find the sub-interval of the interval (0, /2) in which the function
f(x) = sin 3x is increasing.
14.
Without using derivatives, find the maximum and minimum value of
y = |3 sin x + 1|.
15.
If f (x) = ax + cos x is strictly increasing on R, find a.
16.
Write the interval in which the function f(x) = x9 + 3x7 + 64 is increasing.
17.
What is the slope of the tangent to the curve f = x3 – 5x + 3 at the point
whose x co-ordinate is 2?
18.
At what point on the curve y = x2 does the tangent make an angle of 45°
with positive direction of the x-axis?
19.
Find the point on the curve y = 3x2 – 12x + 9 at which the tangent is
parallel to x-axis.
XII – Maths
50
1
is strictly decreasing.
x
51. 20.
What is the slope of the normal to the curve y = 5x2 – 4 sin x at x = 0.
21.
Find the point on the curve y = 3x2 + 4 at which the tangent is perpendicular
1
to the line with slope .
6
22.
Find the point on the curve y = x2 where the slope of the tangent is equal
to the y – co-ordinate.
23.
If the curves y = 2ex and y = ae–x intersect orthogonally (cut at right
angles), what is the value of a?
24.
Find the slope of the normal to the curve y = 8x2 – 3 at x
25.
Find the rate of change of the total surface area of a cylinder of radius
r and height h with respect to radius when height is equal to the radius
of the base of cylinder.
26.
Find the rate of change of the area of a circle with respect to its radius.
How fast is the area changing w.r.t. its radius when its radius is 3 cm?
27.
For the curve y = (2x + 1)3 find the rate of change of slope at x = 1.
28.
Find the slope of the normal to the curve
x = 1 – a sin ; y = b cos2 at
1
.
4
2
29.
If a manufacturer’s total cost function is C(x) = 1000 + 40x + x2, where
x is the out put, find the marginal cost for producing 20 units.
30.
Find ‘a’ for which f (x) = a (x + sin x) is strictly increasing on R.
SHORT ANSWER TYPE QUESTIONS (4 MARKS)
31.
A particle moves along the curve 6y = x3 + 2. Find the points on the curve
at which the y co-ordinate is changing 8 times as fast as the x coordinate.
32.
A ladder 5 metres long is leaning against a wall. The bottom of the ladder
is pulled along the ground away from the wall at the rate of 2 cm/sec.
How fast is its height on the wall decreasing when the foot of the ladder
is 4 metres away from the wall?
51
XII – Maths
52. 33.
A balloon which always remain spherical is being inflated by pumping in
900 cubic cm of a gas per second. Find the rate at which the radius of
the balloon increases when the radius is 15 cm.
34.
A man 2 meters high walks at a uniform speed of 5 km/hr away from a
lamp post 6 metres high. Find the rate at which the length of his shadow
increases.
35.
Water is running out of a conical funnel at the rate of 5 cm3/sec. If the
radius of the base of the funnel is 10 cm and altitude is 20 cm, find the
rate at which the water level is dropping when it is 5 cm from the top.
36.
The length x of a rectangle is decreasing at the rate of 5 cm/sec and the
width y is increasing as the rate of 4 cm/sec when x = 8 cm and y = 6 cm.
Find the rate of change of
(a)
Perimeter
(b) Area of the rectangle.
37.
Sand is pouring from a pipe at the rate of 12c.c/sec. The falling sand
forms a cone on the ground in such a way that the height of the cone is
always one-sixth of the radius of the base. How fast is the height of the
sand cone increasing when height is 4 cm?
38.
The area of an expanding rectangle is increasing at the rate of 48 cm2/
sec. The length of the rectangle is always equal to the square of the
breadth. At what rate is the length increasing at the instant when the
breadth is 4.5 cm?
39.
Find a point on the curve y = (x – 3)2 where the tangent is parallel to the
line joining the points (4, 1) and (3, 0).
40.
Find the equation of all lines having slope zero which are tangents to the
curve y
1
.
x 2x 3
2
41.
Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1.
42.
Find the equation of the normal at the point (am2, am3) for the curve
ay2 = x3.
43.
Show that the curves 4x = y2 and 4xy = k cut as right angles if k2 = 512.
44.
Find the equation of the tangent to the curve y 3x 2 which is parallel
to the line 4x – y + 5 = 0.
XII – Maths
52
53. 45.
Find the equation of the tangent to the curve
x y a at the point
a2 a2
, .
4 4
16
.
3
46.
Find the points on the curve 4y = x3 where slope of the tangent is
47.
Show that
48.
Find the equation of the tangent to the curve given by x = a sin3t,
x y
1 touches the curve y = be–x/a at the point where the
a b
curve crosses the y-axis.
y = b cos3 t at a point where t
49.
.
2
Find the intervals in which the function f(x) = log (1 + x) –
x
, x 1
1 x
is increasing or decreasing.
50.
Find the intervals in which the function f(x) = x3 – 12x2 + 36x + 17 is
(a)
Increasing
(b)
Decreasing.
51.
Prove that the function f(x) = x2 – x + 1 is neither increasing nor decreasing
in [0, 1].
52.
Find the intervals on which the function f x
53.
Prove that f x
54.
Find the intervals in which the function f(x) = sin4x + cos4x, 0 x
x
is decreasing.
x 1
2
x3
x 2 9 x, x 1, 2 is strictly increasing. Hence find
3
the minimum value of f (x).
is
2
increasing or decreasing.
55.
Find the least value of 'a' such that the function f(x) = x2 + ax + 1 is strictly
increasing on (1, 2).
53
XII – Maths
54. 3
56.
5
Find the interval in which the function f x 5x 2 3x 2 , x 0 is strictly
decreasing.
57.
Show that the function f(x) = tan–1 (sin x + cos x), is strictly increasing on
the interval 0, .
4
58.
59.
Show that the function f x cos 2x is strictly increasing on
4
3 7
.
,
8 8
Show that the function f x
sin x
is strictly decreasing on 0, .
2
x
Using differentials, find the approximate value of (Q. No. 60 to 64).
1
1
60.
0.009 3 .
62.
0.0037 2 .
61.
255 4 .
63.
0.037.
1
64.
25.3 .
65.
Find the approximate value of f (5.001) where f(x) = x3 – 7x2 + 15.
66.
Find the approximate value of f (3.02) where f (x) = 3x2 + 5x + 3.
LONG ANSWER TYPE QUESTIONS (6 MARKS)
67.
Show that of all rectangles inscribed in a given fixed circle, the square
has the maximum area.
68.
Find two positive numbers x and y such that their sum is 35 and the
product x2y5 is maximum.
69.
Show that of all the rectangles of given area, the square has the smallest
perimeter.
70.
Show that the right circular cone of least curved surface area and given
volume has an altitude equal to
XII – Maths
54
2 times the radium of the base.
55. 71.
Show that the semi vertical angle of right circular cone of given surface
1
area and maximum volume is sin1 .
3
72.
A point on the hypotenuse of a triangle is at a distance a and b from the
sides of the triangle. Show that the minimum length of the hypotenuse is
2
a3
3
2 2
b3
.
73.
Prove that the volume of the largest cone that can be inscribed in a
8
sphere of radius R is
of the volume of the sphere.
27
74.
Find the interval in which the function f given by f(x) = sin x + cos x,
0 x 2 is strictly increasing or strictly decreasing.
75.
Find the intervals in which the function f(x) = (x + 1)3 (x – 3)3 is strictly
increasing or strictly decreasing.
76.
Find the local maximum and local minimum of f(x) = sin 2x – x,
x .
2
2
77.
Find the intervals in which the function f(x) = 2x3 – 15x2 + 36x + 1 is
strictly increasing or decreasing. Also find the points on which the tangents
are parallel to x-axis.
78.
A solid is formed by a cylinder of radius r and height h together with two
hemisphere of radius r attached at each end. It the volume of the solid
1
metre min. How
2
fast must h (height) be changing when r and h are 10 metres.
is constant but radius r is increasing at the rate of
79.
Find the equation of the normal to the curve
x = a (cos + sin ) ; y = a (sin – cos ) at the point and show
that its distance from the origin is a.
80.
For the curve y = 4x3 – 2x5, find all the points at which the tangent passes
through the origin.
81.
Find the equation of the normal to the curve x2 = 4y which passes
through the point (1, 2).
55
XII – Maths
56. 82.
Find the equation of the tangents at the points where the curve 2y = 3x2
– 2x – 8 cuts the x-axis and show that they make supplementary angles
with the x-axis.
83.
Find the equations of the tangent and normal to the hyperbola
at the point (x0, y0).
x2 y2
1
a2 b 2
84.
A window is in the form of a rectangle surmounted by an equilateral
triangle. Given that the perimeter is 16 metres. Find the width of the
window in order that the maximum amount of light may be admitted.
85.
A jet of an enemy is flying along the curve y = x2 + 2. A soldier is placed
at the point (3, 2). What is the nearest distance between the soldier and
the jet?
86.
Find a point on the parabola y2 = 4x which is nearest to the point (2,
–8).
87.
A square piece of tin of side 18 cm is to be made into a box without top
by cutting a square from each cover and folding up the flaps to form the
box. What should be the side of the square to be cut off so that the
volume of the box is the maximum.
88.
A window in the form of a rectangle is surmounted by a semi circular
opening. The total perimeter of the window is 30 metres. Find the
dimensions of the rectangular part of the window to admit maximum light
through the whole opening.
89.
An open box with square base is to be made out of a given iron sheet of
area 27 sq. meter, show that the maximum value of the box is 13.5 cubic
metres.
90.
A wire of length 28 cm is to be cut into two pieces. One of the two pieces
is to be made into a square and other in to a circle. What should be the
length of two pieces so that the combined area of the square and the
circle is minimum?
91.
Show that the height of the cylinder of maximum volume which can be
inscribed in a sphere of radius R is
92.
2R
3
. Also find the maximum volume.
Show that the altitude of the right circular cone of maximum volume that
4r
can be inscribed is a sphere of radius r is
.
3
XII – Maths
56
57. 93.
Prove that the surface area of solid cuboid of a square base and given
volume is minimum, when it is a cube.
94.
Show that the volume of the greatest cylinder which can be inscribed in
a right circular cone of height h and semi-vertical angle is
4
h 3 tan2 .
27
95.
Show that the right triangle of maximum area that can be inscribed in a
circle is an isosceles triangle.
96.
A given quantity of metal is to be cast half cylinder with a rectangular box
and semicircular ends. Show that the total surface area is minimum when
the ratio of the length of cylinder to the diameter of its semicircular ends
is : ( + 2).
ANSWERS
1.
0.8 cm/sec.
2. 4.4 cm/sec.
3.
2 cm3/sec.
4. 80 cm2/sec.
5.
Rs. 208.
6.
Minimum value = 4, maximum value = 6.
7.
Maximum value = 7, minimum value does not exist.
8.
a 0.
9.
[0, ]
10.
(0, e]
11.
x 1
12.
(– , 0) U (0, )
13.
0, .
6
14.
Maximum value = 4, minimum valve = 0. 15. a > 1.
16.
R
17. 7
18.
1 1
, .
2 4
19. (2, – 3)
20.
1
4
21. (1, 7)
57
XII – Maths
58. 1
.
2
22.
(0, 0), (2, 4)
23.
24.
1
– .
4
25. 6r
26.
2 cm2/cm
27. 72
28.
30.
a > 0.
31.
4, 11
33.
a
.
2b
29. Rs. 80.
32.
8
cm sec.
3
1
cm sec.
34.
2.5 km/hr.
35.
4
cm sec.
45
36.
(a) –2 cm/min, (b) 2 cm2/min
37.
1
cm sec.
48
38.
7.11 cm/sec.
39.
7 1
, .
2 4
40.
y
42.
2x + 3my = am2 (2 + 3m2)
44.
48x – 24y = 23
45.
2x + 2y = a2
46.
128
8 128 8
,
,
,
.
3 27 3
27
48.
y = 0
49.
Increasing in (0, ), decreasing in (–1, 0).
50.
Increasing in (– , 2) (6, ), Decreasing in (2, 6).
XII – Maths
31
and 4, .
3
58
1
.
2
59. 25
.
3
52.
(– , –1) and (1, ).
53.
54.
Increasing in , Decreasing in 0,
4 2
55.
a = – 2.
56.
Strictly decreasing in (1, ).
60.
0.2083
61.
3.9961
62.
0.06083
63.
0.1925
64.
5.03
65.
–34.995
66.
45.46
68.
25, 10
74.
5
Strictly increasing in 0, , 2
4 4
.
4
5
Strictly decreasing in ,
.
4 4
75.
Strictly increasing in (1, 3) (3, )
Strictly decreasing in (–, –1) (–1, 1).
76.
Local maxima at x
Local max. value
6
3
2
6
Local minima at x
6
Local minimum value
77.
3
2
6
Strictly increasing in (–, 2] [3, )
Strictly decreasing in (2, 3).
59
XII – Maths
60. Points are (2, 29) and (3, 28).
78.
3
metres min.
79.
x + y tan – a sec = 0.
80.
(0, 0), (–1, –2) and (1, 2).
81.
x + y = 3
82.
5x – y – 10 = 0 and 15x + 3y + 20 = 0
83.
xx 0
a
2
yy 0
b
2
1,
y y0
2
a y0
x x0
b 2x 0
0.
16
84.
6 3
85.
5
86.
(4, –4)
87.
3cm
88.
60
30
,
.
4 4
90.
112
28
cm,
cm.
4
4
XII – Maths
60
61. CHAPTER 7
INTEGRALS
POINTS TO REMEMBER
Integration is the reverse process of Differentiation.
Let
These integrals are called indefinite integrals and c is called constant of
integration.
From geometrical point of view an indefinite integral is collection of family
of curves each of which is obtained by translating one of the curves
parallel to itself upwards or downwards along y-axis.
d
F x f x then we write
dx
f x dx F x c .
STANDARD FORMULAE
1.
2.
x n 1
c
x dx n 1
log x c
n
n 1
n –1
ax b n 1
c
n 1 a
n
ax b dx
1
log ax b c
a
3.
sin x dx
5.
tan x . dx
4.
– cos x c.
n 1
n 1
cos x dx
sin x c.
– log cos x c log sec x c .
61
XII – Maths
62. 6.
cot x dx
8.
cosec x . dx
2
cosec x cot x dx
11.
sec x dx
12.
cosec x dx
13.
x
x
e dx e c .
15.
1
1
16.
1 x
17.
x
18.
19.
20.
2
1 x
2
2
x
2
a
2
14.
–1
–1
x
2
a
2
1
x
2
dx
1
x c.
–1
log
2a
dx
1
x c , x 1.
1
a
a x
c.
a x
log
2a
dx
x
a dx
x c, x 1 .
1
1
XII – Maths
log cosec x – cot x c .
dx sec
1
a
– cosec x c .
dx sin
1
x
sec x . tan x . dx
x . dx tan x c.
log sec x tan x c .
dx tan
2
sec
9.
– cot x c.
10.
2
7.
log sin x c .
tan
x a
c.
x a
–1
x
c.
a
62
ax
c
log a
sec x c .
63. 21.
22.
23.
1
a
2
2
– x
1
2
2
a x
1
x
2
– a
2
2
dx sin
c.
2
dx log x
dx log x
x
2
25.
a x dx
x
2
a x
2
2
a
2
x
2
2
x
2
a dx
x
sin
a
2
2
2
2
a x
c.
2
a
2
2
2
a x
a x dx
x
a
24.
26.
–1
c.
–1 x
c.
a
2
2
log x
a x
log x
x
2
c.
2
x
2
a
2
2
a
2
2
a
2
c.
2
RULES OF INTEGRATION
1.
k .f x dx k f x dx .
2.
k f x g x dx k f x dx k g x dx .
INTEGRATION BY SUBSTITUTION
1.
2.
f ´ x
f x dx
log f x c .
n
f x f ´x dx
f x n 1
c.
n 1
63
XII – Maths
64. 3.
f ´ x
f x
n
f x n 1
dx
c.
–n 1
INTEGRATION BY PARTS
f x . g x dx
f x .
g x dx – f ´ x . g x dx dx .
DEFINITE INTEGRALS
b
f x dx
F b F a , where F x
f x dx .
a
DEFINITE INTEGRAL AS A LIMIT OF SUMS.
b
f x dx
a
where
f a f a h f a 2h
lim h
..... f a n 1 h
h0
b a
h
b
.
or
h
a
n
f x dx lim h
f a rh
h0
r 1
PROPERTIES OF DEFINITE INTEGRAL
b
1.
a
c
b
f x dx
a
f t dt .
a
b
f x dx f x dx f x dx.
a
a
b
4.
2.
b
b
3.
b
a
f x dx – f x dx .
(i)
a
XII – Maths
c
b
f x dx
a
f a b x dx .
a
(ii)
0
64
a
f x dx
f a x dx.
0
65. a
5.
f x 0; if f x is odd function.
–a
a
6.
a
f x dx 20 f x dx ,
if f(x) is even function.
a
2a
7.
0
2a f x dx , if f 2a x f x
f x dx
0
0,
if f 2a x f x
VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)
Evaluate the following integrals
1
1.
3.
1
sin
x cos
1
x dx .
2.
e
x
dx .
1
1
x
x8
8
6.
x log x log log x dx .
e
1
5.
x
99
cos4 x dx .
1
7.
0
4 3 sin x
log
dx .
4 3 cos x
8.
9.
cos 2x 2sin2 x
dx .
cos 2 x
10.
2
11.
x
c
c x dx .
8 x
dx .
x 8
4.
1 sin2 x dx .
1
a log x
e x log a dx .
2
7
sin
x dx .
2
12.
65
d
f x dx .
dx
XII – Maths
66. 1
sin2 x cos2 x dx .
15.
log e
dx .
e
17.
2
x
x
e x dx .
x
dx .
2
19.
x 1
21.
cos
23.
sec x .log sec x tan x dx .
2
1
dx .
14.
16.
dx .
ex
a x dx .
18.
13.
x x 1
x
x 1
e
dx .
x
dx .
20.
22.
x cos 1 dx .
24.
cos x sin dx .
26.
1
x 2 dx .
28.
x cos x dx .
30.
32.
1
ax ax dx .
x
1
1
3
25.
cot x .logsin x dx .
27.
x 2 3 log x dx .
29.
1
1 cos x
dx .
sin x
x 1
33.
0
34.
0 x dx
x e 1 e x 1
dx .
x e ex
2
x log x dx .
31.
x
1 sin x
cos x dx .
2
XII – Maths
where [ ] is greatest integer function.
66
67. 35.
2
2
0 x dx
where [ ] is greatest integer function.
f x
b
36.
a f x f a b x dx .
38.
If
40.
x
1
1x x dx .
39.
37.
a f x dx b f x dx .
2 x dx .
1
a
1
0 1 x 2
, then what is value of a.
4
b
a
SHORT ANSWER TYPE QUESTIONS (4 MARKS)
41.
(i)
x cosec
tan–1x 2
1 x
4
dx .
(ii)
1
x 1
x 1
x 1
x 1
dx .
cos x a
(iii)
sin x a sin x b dx .
(iv)
cos x a dx .
(v)
cos x cos 2x cos 3x dx .
(vi)
cos
(viii)
cot
2
(vii)
sin
(ix)
4
x cos x dx .
sin x cos x
2
2
2
dx .
(x)
a sin x b cos x
6
(xi)
2
6
sin x cos x
2
2
dx .
(xii)
sin x cos x
67
5
3
x dx.
4
x cosec x dx .
1
dx .
3
cos x cos x a
sin x cos x
dx .
sin 2x
XII – Maths
68. 42.
Evaluate :
(i)
x
x
4
x
2
dx .
1
*(ii)
x 6 log x
(iii)
1 x x
(v)
(vii)
(ix)
(xi)
43.
dx
7 log x 2
(iv)
x a x b
5x 2
2
dx .
.
2
1
3x
1
2
dx .
dx .
(viii)
2x 1
x 2
4x x
(x)
dx .
2
3x 2
x
2
x 1 dx .
Evaluate :
dx
(i)
x x
(ii)
1 cos x 2 3 cos x dx .
(iii)
cos
XII – Maths
(vi)
7
1
.
sin x
sin cos
2
d .
cos 2
68
(xii)
x
x
1
9 8x x
2
sin x
dx .
sin x
x
2
dx .
2
dx .
6x 12
2
1 x – x dx .
sec x 1 dx .
69. (iv)
(v)
(vii)
(ix)
(xi)
44.
x 1
x 1 x 2 x 3 dx .
x
2
x 2
x 2 x 1
dx .
(vi)
.
(viii)
dx
2x 1 x
sin x
sin 4x
2
4
(x)
dx .
(xii)
tan x dx .
x 2 1 x 2 2
x 3 3 x 2 4 dx .
dx
sin x 1 2 cos x .
x
x
4
x
x
2
1
x
2
2
9
dx .
1
dx .
81
Evaluate :
(i)
x
5
(iii)
e
ax
(v)
3
(ii)
cos bx c dx .
cos
sin
(vi)
1 sin 2x
dx .
1 cos 2x
e
2x
(ix)
2ax x dx .
(xi)
e
x
(xii)
log log x log x
2
x
3
(viii)
x dx.
(vii)
sec
(iv)
sin x dx .
e
x
x
(x)
e
3
x dx .
6x
–1
1 9x
–1
tan
2
dx .
x dx .
x 1
dx .
2x 2
x 2
1
x 12
dx .
2 sin 2x
dx .
1 cos 2x
1
dx .
2
69
XII – Maths
70. (xiii)
(xiv)
x 2
(xv)
2x 5
(xvi)
45.
6x 5
x
2
2
6 x x dx .
x 3
dx .
x 3
x
2
4x 3 dx .
4x 8 dx .
Evaluate the following definite integrals :
4
(i)
sin x cos x
9 16 sin 2x
2
dx .
(ii)
0
0
1
(iii)
cos 2x log sin x dx .
x
0
1 x
1 x
1 2
2
2
(iv)
dx .
sin
1
x
1 x 2 3 2
0
dx .
2
(v)
sin
0
2
sin 2x
4
4
dx .
(vi)
x cos x
x
5x
2
1
2
dx .
4x 3
2
(vii)
x sin x
1 cos x dx .
0
46.
Evaluate :
3
(i)
x 1 x 2 x 3 dx . (ii)
1
XII – Maths
0
70
x
1 sin x dx .
71.
4
(iii)
2
log 1 tan x dx .
log sin x dx .
(iv)
0
0
(v)
x sin x
1 cos
2
0
dx .
x
3
2x x
3
f x dx where f x x 3 x 2
3x 2
2
when 2 x 1
2
(vi)
when 1 x 1
when 1 x 2.
2
(vii)
x sin x cos x
sin
0
4
4
(viii)
a
x
2
0
47.
dx .
x cos x
2
2
dx .
2
cos x b sin x
Evaluate the following integrals
(i)
3
6
1
(iii)
1
(v)
1
dx
1
(ii)
tan x
1 sin x
log
1 sin x
x tan x
sin
0
dx .
sec x cosec x
(iv)
e
0
a
dx .
(iv)
0
a
71
1
2x
dx .
1 x2
e
cos x
cos x
e
a x
cos x
dx .
dx .
a x
XII – Maths
72. 1
48.
2x dx
where [ ] is greatest integer function.
0
log x log sin x
49.
e
51.
dx .
sin2x dx .
52.
sin x
e
sin x sin2x dx .
b
4
53.
log x 1 log x
50.
sin x dx .
54.
dx .
a
f x dx f a b x dx .
4
a
55.
1
sec x tanx dx .
56.
sin2 x
1 cos x dx .
57.
1 tan x
1 tan x dx .
58.
ax b x
c x dx .
59.
Evaluate
sin1 x cos 1 x
b
dx , x 0, 1
(i)
sin
(ii)
1
x
1
x
x 2 1 log x 2 1 2 log x
dx
4
x
(iii)
(iv)
(v)
XII – Maths
1
x cos 1 x
dx
x2
x sin x cos x
2
1
sin
dx
3
x
dx
a x
(vi)
6
72
sin x cos x
sin 2x
dx
73.
2
(vii)
sin
–
x cos x dx
2
2
(viii)
x
2
dx ,
where [x] is greatest integer function
1
3
2
(ix)
x sin x dx .
1
LONG ANSWER TYPE QUESTIONS (6 MARKS)
60.
Evaluate the following integrals :
(i)
(iii)
x
x
5
5
4
dx .
(ii)
x
2x
dx
x 1 x
3
x 1 x 3
2
dx
(iv)
x
x
4
x
4
4
2
dx
4
dx
– 16
2
(v)
cot x dx .
tan x
(vi)
0
(vii)
–1
1 x
0
61.
x tan
x
2 2
1
dx .
1
dx .
Evaluate the following integrals as limit of sums :
4
(i)
2
2x 1 dx .
(ii)
2
x
2
3 dx .
0
73
XII – Maths
74. 3
2
3x
(iii)
4
2x 4 dx .
3x
(iv)
e
2x
dx .
0
1
5
x
(v)
2
2
3 x dx .
2
62.
Evaluate
1
cot
(i)
1
1
x x 2 dx
0
dx
sin x 2 cos x 2 sin x cos x
(ii)
1
(iii)
log 1 x
0
1 x2
63.
1
sin x sin2x dx .
65.
sec
3
x dx .
2
dx
2 log sin x
(iv)
– log sin 2x dx .
0
3 sin 2 cos
64.
5 cos2 4 sin d .
66.
e
2x
cos 3x dx .
ANSWERS
1.
2.
x c.
2e – 2
2
3.
tan x + c.
4.
8x
x9
x2
8log x
c.
log8 9
16
5.
0
6.
log | log (log x) | + c
XII – Maths
74
75. 8.
x a 1 a x
c
a 1 loga
7.
0
9.
tan x + c
10.
0
11.
x c 1 c x
c
c 1 log c
12.
f(x) + c
13.
tan x – cot x + c
14.
2 32 2
32
x x 1 c
3
3
15.
log |x| + c
16.
e
a
17.
2x e x
c
log 2e
18.
2
x 13 2 2 x 11 2 c .
3
19.
log x 1
20.
2e
21.
x cos2 + c
22.
log x cos 1
c.
cos
23.
log sec x tan x 2 c
24.
log cos x sin
c
sin
25.
logsin x 2 c
26.
x4
1
3x 2
2
3 log x c.
4 2x
2
1
c.
x 1
2
2
x
x
log e a c
c
27.
1
log 2 3log x c .
3
28.
log |x + cos x| c
29.
2 log |sec x/2| + c.
30.
1
log x e e x c .
e
75
XII – Maths
76. 31.
x log x 2 c
32.
a
33.
0
34.
1
35.
36.
b a
2
37.
–1
38.
0
39.
1
40.
0
2
2 1
x 2 log ax
2x c .
2
a
(i)
1
1
log cosec tan1 x 2 2 c .
2
x
(ii)
41.
1 2
1
x x x 2 1 log x x 2 1 c.
2
2
1
(iii)
(iv)
sin a b
sin x a
sin x b
c
x cos 2a – sin 2a log |sec (x – a)| + c.
1
(v)
log
12x 6 sin 2x 3 sin 4x 2 sin 6x c .
48
2
3
sin x
1
5
(vi)
sin x
(vii)
1
1
1
1
2x sin 2x sin 4x sin 6x c .
32
2
2
6
(viii)
4
cot6 x
cot x
c.
6
4
3
sin x c .
5
1
(ix)
a2
b
2
2
2
2
2
c.
a sin x b cos x
[Hint : put a2 sin2 x + b2 cos2 x = t]
XII – Maths
76
77. (x)
–2 cosec a cos a tan x . sin a c .
[Hint. : Take sec2 x as numerator]
(xi)
(xii)
42.
tan x – cot x – 3x + c.
sin–1 (sin x – cos x) + c.
(i)
1
tan
3
(ii)
(iii)
log
–1
2x 2
3
2 log x 1
1
c.
[Hint : put x2 = t]
C
[Hint : put log x = t]
3 log x 2
1
log
5
5 – 1 2x
c
5 1 2x
(iv)
x 4
sin1
c.
5
(v)
2 log
x a
x b c
(vi)
cos sin
1
cos x
sin . log sin x
cos
Hint :
5
(vii)
(viii)
(ix)
2
log 3 x 2 x 1
6
4x x
sin x
sin x
11
tan
1
3 2
x 3 log x
2
2
2
1
2
2
sin x sin
sin x
3 x 1
c
2
6x 12 2 3 tan
4 sin
2
sin x sin c
1
x 3
c
3
x 2
c
2
77
XII – Maths
78. (x)
1
1 x x
3
2 2
1
3
2x 1 1 x x 2
8
5
sin
1
16
1
2
x x x 1
7
2
c
1
2
2 3
x x 1
log x
2
8
3
2
(xi)
x 2
(xii)
log cos x
x 1
2 x 1
c
5
1
2
cos x cos x c
2
[Hint : Multiply and divide by
43.
(i)
1
log
7
(ii)
(iii)
x
x
7
7
c
1
1 cos x
log
c
2 3 cos x
2
1
log cos 2
3
(iv)
(v)
9
log 1 cos c .
3
log x 3
10
4
log x 2
15
x 4 log
x 2 2
1
x 1 c
6
c
x 1
(vi)
x
2
3
tan
x
1 x
3 tan c
2
3
1
[Hint : put x2 = t]
(vii)
XII – Maths
2
17
log 2x 1
1
log x
17
2
4
1
34
78
tan
1
x
2
c
sec x 1 ]
79. (viii)
1
log 1 cos x
2
1
log 1 cos x
6
2
log 1 2 cos x c
3
[Hint : Multiply Nr and Dr by sin x and put cos x = t]
(ix)
1
1 sin x
log
8
(x)
1
1 sin x
log
2
(xi)
x
x
1
2
1
tan
44.
(i)
2 sin x
c
c
1
tan x
tan x 1
log
2 tan x
2 2
tan x
1
3 2
2 sin x
x 1
1
tan
1
1
log
4 2
x 1
2
2
(xii)
1
2 tan x 1
c
2 tan x 1
x2 9
c
3 2
1 3
3
3
x cos x sin x c
3
1
sec x tan x log sec x tan x c
2
[Hint : Write sec3x = sec x . sec2 x and take sec x as first function]
(ii)
(iii)
e
ax
2
a b
(iv)
2x tan
2
1
a cos bx
3x
1
c b sin bx c c1
log 1 9x
2
c
[Hint : put 3x = tan ]
3
(v)
(vi)
2 x sin x cos x c
3
x 4 1
x
x
1
tan x –
c.
4
12
4
79
XII – Maths
80. 1
(vii)
(ix)
(x)
e
2x
tan x c .
2
x a
2ax x
e
(viii)
2
2
a
2
1
sin
2
x
c.
2x
x a
c
a
x x 1
e
c.
x 1
(xi)
ex tan x + c.
(xii)
x log log x –
x
c . [Hint : put log x = t x = et ]
log x
(xiii)
2 6 x x
2 32
25
2x 1
2
1 2x 1
8
6 x x
sin
c
5
8
4
(xiv)
(xv)
1
x 2 x 2 9 3 log x
2
x
2
9 c
2
2
x
2
4x 3
3
1
log x 2
3
2
x 2
2
x
2
x
2
4x 3
4x 3 c
2
(xvi)
x 2
2
1
45.
(i)
XII – Maths
x
2
4 x 8 2 log x 2
log 3.
(ii)
20
80
–
4
x
2
4x 8 c
81. (iii)
(v)
–
4
1
.
2
[Hint : put x2 = t]
(vii)
4
15
(vi)
(viii)
47.
(i)
(iii)
log 2.
2
6
log .
5
2
x
sin x
Hint :
dx .
1 cos x
1 cos x
8.
(ii)
log 2.
(iv)
8
–
log 2.
2
2
.
4
95/12.
Hint :
(vii)
8
1
(v)
1
25
/2.
(iii)
–
.
5 – 10 log
(i)
2
(vi)
46.
(iv)
2
1
f x dx
2
1
f x dx
2
2
f x dx
f x dx
1
1
2
.
16
Hint : Use
2
.
2ab
.
(ii)
12
0.
(iv)
81
a
a
f x
0
f a x
0
log 2.
2
/2.
XII – Maths
82. (v)
2
(vi)
4
48.
1
2
49.
–x cos x + sin x + c.
50.
x + log x + c.
51.
1
log sec x tan x c .
2
52.
1 sin3x
sin x
3
2
53.
2 2
54.
0
55.
log |1 + sin x| + c
56.
x – sin x + c
57.
log |cos x + sin x| + c
58.
a.
a c x b c x C.
log a c log b c
(i)
2 2x 1
2 x x2
sin1 x
x c
(ii)
–2 1 x cos 1 x
(iii)
59.
XII – Maths
1
1
1 2
3
x
3 2
x x2 c
1
2
log 1 2 c
x 3
82
83. (iv)
sin x x cos x
c
x sin x cos x
(v)
x a tan1 x ax c
(vi)
2 sin1
(vii)
0
(viii)
2
(ix)
60.
(i)
a
3 1
2
3 5
3
1
2 .
x 4 log x
5
log x 1
4
3
log x 1
4
log x
2
1
1
tan
–1
x c.
2
1
x
log
2
(ii)
(iii)
1
x 1
–
x 1
log x 1
5
1
1
1
2x
1
x
(vi)
x
x log
x
2
log x
1
4 –
10
81
log x 3
8
log
x 2
1
tan
x 2
2
4
tan
10
log x 1
2
(v)
–1
2
8
(iv)
tan
1
c.
1
–1
x
c.
2
27
2 x 3
c.
x
c.
2
2.
1
2 2
tan
1
x2
1
2x
1
4 2
83
log
x
x
2
2
2x 1
2x 1
c
XII – Maths
84. (vii)
/8.
(i)
14.
(iii)
26.
(iv)
1
26
61.
(ii)
127 e
.
2
(i)
log 2
2
(ii)
(iii)
log 2.
8
(iv)
62.
.
2
141
(v)
8
.
3
1
log .
2
2
1
tan x x
log
c
5
2 tan x 1
63.
1
1
2
log 1 cos x log 1 cos x log 1 2cos x c.
6
2
3
64.
3log 2 sin
65.
1
1
sec x tan x log sec x tan x c.
2
2
66.
e 2x
2cos 3x 3 sin3x c .
13
XII – Maths
4
c.
2 sin
84
85. CHAPTER 8
APPLICATIONS OF INTEGRALS
POINTS TO REMEMBER
AREA OF BOUNDED REGION
Area bounded by the curve y = f(x), the x axis and between the ordinates,
x = a and x = b is given by
b
Area =
f x dx
a
y
y
y = f(x)
O
a
a
x
b
b
O
x
y = f(x)
Area bounded by the curve x = f(y) the y-axis and between abscissas,
y = c and y = d is given by
d
Area =
f y
dy
c
y
y
d
d
x = f(y)
x = f(y)
c
c
x
O
85
O
x
XII – Maths
86.
Area bounded by two curves y = f(x) and y = g(x) such that 0 g(x)
f(x) for all x [a, b] and between the ordinate at x = a and x = b is given
by
Y
y = f(x)
B
A
y = g(x)
a
O
X
b
b
Area =
f x – g x dx
a
Required Area
k
b
f x dx f x dx.
a
k
Y
y = f(x)
A2
O A
y
1
A1
X
B(k, 0) x = b
x=a
LONG ANSWER TYPE QUESTIONS (6 MARKS)
1.
Find the area enclosed by circle x2 + y2 = a2.
2.
Find the area of region bounded by x , y : x 1 y
3.
Find the area enclosed by the ellipse
x
a
XII – Maths
86
2
2
y
b
2
2
1
2
25 x .
87. 4.
Find the area of region in the first quadrant enclosed by x–axis, the line
y = x and the circle x2 + y2 = 32.
5.
Find the area of region {(x, y) : y2 4x, 4x2 + 4y 2 9}
6.
Prove that the curve y = x2 and, x = y2 divide the square bounded by
x = 0, y = 0, x = 1, y = 1 into three equal parts.
7.
Find smaller of the two areas enclosed between the ellipse
x
a
and the line
2
2
y
b
2
2
1
bx + ay = ab.
8.
Find the common area bounded by the circles x 2 + y 2 = 4 and
(x – 2)2 + y2 = 4.
9.
Using integration, find the area of the region bounded by the triangle
whose vertices are
(a)
10.
(–1, 0), (1, 3) and (3, 2)
(b)
(–2, 2) (0, 5) and (3, 2)
Using integration, find the area bounded by the lines.
(i)
x + 2y = 2, y – x = 1 and 2x + y – 7 = 0
(ii)
y = 4x + 5,
y = 5 – x
and 4y – x = 5.
11.
Find the area of the region {(x, y) : x2 + y2 1 x + y}.
12.
Find the area of the region bounded by
y = |x – 1| and y = 1.
13.
Find the area enclosed by the curve y = sin x between x = 0 and x
3
2
and x-axis.
14.
Find the area bounded by semi circle y
15.
16.
Find area of smaller region bounded by ellipse
2
Find area of region given by {(x, y) : x2 y |x|}.
25 x
x
2
9
y
and x-axis.
2
1 and straight
4
line 2x + 3y = 6.
87
XII – Maths
88. 17.
Find the area of region bounded by the curve x 2 = 4y and line
x = 4y – 2.
18.
Using integration find the area of region in first quadrant enclosed by
3 y and the circle x2 + y2 = 4.
x-axis, the line x
19.
Find smaller of two areas bounded by the curve y = |x| and x2 + y2 = 8.
20.
Find the area lying above x-axis and included between the circle
x2 + y2 = 8x and the parabola y2 = 4x.
21.
Using integration, find the area enclosed by the curve y = cos x,
y = sin x and x-axis in the interval 0, .
2
22.
Sketch the graph y = |x – 5|. Evaluate
6
x 5 dx .
0
23.
Find area enclosed between the curves, y = 4x and x2 = 6y.
24.
Using integration, find the area of the following region :
x , y :
x 1 y
5 x2
ANSWERS
1.
a2 sq. units.
2.
1
25 – sq. units.
4
2
3.
ab sq. units
5.
8.
10.
2
6
9
8
9
8
4.
1
sin
1
sq. units
3
8
2 3 sq. units
3
(a) 6 sq. unit
XII – Maths
7.
(4 – 8) sq. units
2 ab
sq. units
4
9.
(a) 4 sq. units (b) 2 sq. units
[Hint. Coordinate of vertices are (0, 1) (2, 3) (4, – 1)]
88
89. (b)
15
sq.
[
Hint
: Coordinate of vertices are (– 1, 1) (0, 5) (3, 2)]
2
11.
1
sq. units
4
2
12.
13.
3 sq. units
14.
1 sq. units
25
sq. units
2
1
15.
17.
3
9
sq. units
16.
3
2 sq. units
2
sq. units
18.
8
sq. unit
3
19.
2 sq. unit.
20.
4
8 3 sq. units
3
21.
2
22.
13 sq. units.
23.
8 sq. units.
24.
1
5
sq. units
4
2
2 sq. units.
89
XII – Maths
90. CHAPTER 9
DIFFERENTIAL EQUATIONS
POINTS TO REMEMBER
Differential Equation : Equation containing derivatives of a dependant
variable with respect to an independent variable is called differential
equation.
Order of a Differential Equation : The order of a differential equation is
defined to be the order of the highest order derivative occurring in the
differential equation.
Degree of a Differential Equation : Highest power of highest order
derivative involved in the equation is called degree of differential equation
where equation is a polynomial equation in differential coefficients.
Formation of a Differential Equation : We differentiate the family of
curves as many times as the number of arbitrary constant in the given
family of curves. Now eliminate the arbitrary constants from these
equations. After elimination the equation obtained is differential equation.
Solution of Differential Equation
(i)
Variable Separable Method
dy
f x, y
dx
We separate the variables and get
f(x)dx = g(y)dy
Then
(ii)
f x dx
g y dy
c is the required solutions.
Homogenous Differential Equation : A differential equation of
the form
XII – Maths
dy
dx
f x, y
g x , y
90
where f(x, y) and g(x, y) are both
91. homogeneous functions of the same degree in x and y i.e., of the
form
dy
y
F is called a homogeneous differential equation.
x
dx
For solving this type of equations we substitute y = vx and then
dy
dv
. The equation can be solved by variable
dx
dx
separable method.
(iii)
v x
Linear Differential Equation : An equation of the from
dy
Py Q where P and Q are constant or functions of x only
dx
is called a linear differential equation. For finding solution of this
type of equations, we find integrating factor (I.F .) e
Solution is y I .F .
Q. I.F . dx
P dx .
c
VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)
1.
Write the order and degree of the following differential equations.
(i)
dy
(ii)
cos y 0.
dx
5
4
(iii)
d y
dx
(v)
(vii)
4
d 2y
sin x 2 .
dx
d 2y
1
2
dx
dx
d 3y
3
dx
2
d 2y
2
dx
5
(iv)
d y
dx
13
dy
dy
dx
5
2
2
3
d y
dx
2
4.
dy
log
0.
dx
32
.
(vi)
2
dy
1
dx
2
k
d y
dx
2
.
3
sin x .
91
(viii)
dy
dy
tan
0
dx
dx
XII – Maths
92. 2.
Write the general solution of following differential equations.
(i)
dy
x
5
x
2
2
.
(ii)
x .
(iv)
dx
(iii)
dy
x
3
e
x
e
dx
(v)
dy
1 cos 2x
(vi)
.
dy
dx
1 cos 2y
5
x y
.
1 2y
.
3x 1
Write integrating factor of the following differential equations
(i)
dy
y cos x sin x
dx
(ii)
dy
2
y sec x sec x tan x
dx
(iii)
x
(v)
x
(vii)
4.
dy
dx
dx
3.
(ex + e–x) dy = (ex – e–x)dx
x
2
dy
4
y x .
dx
dy
3y x
dx
dy
1
dx
1 x
2
dy
y log x x y
dx
(iv)
(vi)
3
x
dy
y tan x sec x
dx
y sin x
Write order of the differential equation of the family of following curves
(i)
(iii)
(v)
y = Aex + Bex + c
(ii)
Ay = Bx2
(x – a)2 + (y – b)2 = 9
(iv)
Ax + By2 = Bx2 – Ay
(vi)
y = a cos (x + b)
x
a
(vii)
XII – Maths
2
2
y
b
2
2
0.
y = a + bex+c
92
93. SHORT ANSWER TYPE QUESTIONS (4 MARKS)
5.
(i)
Show that y e
1 x 2 d
2
dx
(ii)
y
2
m sin
x
1
x
is a solution of
dy
2
m y 0.
dx
Show that y = sin(sin x) is a solution of differential equation
2
d y
dx
2
tan x
dy
2
y cos x 0.
dx
2
2
(iii)
x d y
dy
B
x
y 0.
Show that y Ax
is a solution of
2
dx
x
dx
(iv)
Show that y = a cos (log x) + b sin (log x) is a solution of
x
2
2
d y
dx
(v)
dy
y 0.
dx
x
Verify that y log x
equation :
a2
(vi)
2
x2
x 2 a2
d 2y
dx
2
x
satisfies the differential
dy
0.
dx
Find the differential equation of the family of curves
y = ex (A cos x + B sin x), where A and B are arbitrary constants.
(vii)
(viii)
6.
Find the differential equation of an ellipse with major and minor
axes 2a and 2b respectively.
Form the differential equation representing the family of curves
(y – b)2 = 4(x – a).
Solve the following differential equations.
(i)
dy
y cot x sin 2x .
(ii)
x
dy
2
2y x log x .
dx
dx
93
XII – Maths
94. (iii)
dx
dy
(iv)
1
sin x
. y cos x
x
3
cos x
,
x 0.
x
dy
cos x sin x .
dx
(v)
(vi)
7.
ydx x y
ye dx
y
3
y3
dy
0
2 xe
y
dy
Solve each of the following differential equations :
dy
dy
2
2y
.
dx
dx
(i)
y x
(ii)
cos y dx + (1 + 2e–x) sin y dy = 0.
(iii)
x 1 y dy y 1 x dx 0.
2
1 x 2 1 –
(iv)
(v)
(vi)
2
y
2
dy
xy dx 0.
(xy2 + x) dx + (yx2 + y) dy = 0; y(0) = 1.
dy
3
3
x
y sin x cos x xy e .
dx
(vii)
8.
tan x tan y dx + sec2 x sec2 y dy = 0
Solve the following differential equations :
(i)
x2 y dx – (x3 + y3) dy = 0.
(ii)
x
2
dy
x
2
2
xy y .
dx
(iii)
XII – Maths
x 2 y 2 dx
2 xy dy 0,
94
y 1 1.
95. x
x
y sin
dx x sin y dy .
y
y
(iv)
dy
(vi)
x
dy
(viii)
2xy
dx
9.
2
y
1– y
dx
1 x
3xy
(ix)
(v)
y
2
(vii)
2
dy
dx
dy
y
y
tan .
x
x
e
x y
2 y
x e .
dx
2
2
.
dx x 2
xy dy 0
Form the differential equation of the family of circles touching
y-axis at (0, 0).
(ii)
Form the differential equation of family of parabolas having vertex
at (0, 0) and axis along the (i) positive y-axis (ii) positive x-axis.
(iii)
Form differential equation of family of circles passing through
origin and whose centre lie on x-axis.
(iv)
10.
(i)
Form the differential equation of the family of circles in the first
quadrant and touching the coordinate axes.
Show that the differential equation
dy
dx
x 2y
x 2y
is homogeneous and
solve it.
11.
Show that the differential equation :
(x2 + 2xy – y2) dx + (y2 + 2xy – x2) dy = 0 is homogeneous and solve it.
12.
Solve the following differential equations :
(i)
dy
2y cos 3x .
dx
dy
2
y cos x 2 sin x cos x if y 1
2
dx
(ii)
sin x
(iii)
3e tan y dx 1 e
x
x
sec2 y dy
95
0
XII – Maths
96. 13.
Solve the following differential equations :
(i)
(x3 + y3) dx = (x2y + xy2)dy.
(ii)
x dy – y dx
(iii)
y
y
y x cos y sin dx
x
x
x
2
2
y dx .
y
y
– x y sin x cos dy 0.
x
x
(iv)
x2dy + y(x + y) dx = 0 given that y = 1 when x = 1.
y
(v)
xe
x
y x
dy
0 if y(e) = 0
dx
(vi)
(vii)
16.
(x3 – 3xy2) dx = (y3 – 3x2y)dy.
dy
dx
y
y
cosec 0 given that y 0 when x 1
x
x
Solve the following differential equations :
2
dy
tan x y .
dx
(i)
cos x
(ii)
x cos x
dy
y x sin x cos x 1.
dx
(iii)
x
x
1 e y dx e y 1 x dy 0.
y
(iv)
(y – sin x) dx + tan x dy = 0, y(0) = 0.
XII – Maths
96
