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Title: Mathematics Study Material
Description: Best Mathematics Study Material Issued by Central Board Of Secondary Education (CBSE) For Class 12th...It Also Contains Points to remember And Value Based Questions.............I Hope This Material is useful for you.........Thank you...........& All the Best for your studies..
Description: Best Mathematics Study Material Issued by Central Board Of Secondary Education (CBSE) For Class 12th...It Also Contains Points to remember And Value Based Questions.............I Hope This Material is useful for you.........Thank you...........& All the Best for your studies..
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REVIEW TEAM : 2014-15
Sl
...
Name
Designation
Dr
...
Sh
...
Ms
...
Sh
...
of
Periods
Weightage
(Marks)
(i)
Relations and Functions
30
10
(ii)
Algebra
50
13
(iii)
Calculus
80
44
(iv)
Vector and Three Dimensional Geometry
30
17
(v)
Linear Programming
20
16
(vi)
Probability
30
Total :
240
100
Unit I : RELATIONS AND FUNCTIONS
1
...
Functions
...
Binary operations
...
Inverse Trigonometric Functions
(15Periods)
Definition, range, domain, principal value branches
...
Elementary properties of inverse trigonometric
functions
...
Matrices
(25 Periods)
Concept, notation, order, equality, types of matrices, zero and identity
[Class XII : Maths]
[2]
matrix, transpose of a matrix, symmetric and skew symmetric matrices
...
Noncommutativity of multiplication of matrices and existence of non-zero
matrices whose product is the zero matrix (restrict to square matrices of
order 2)
...
Invertible
matrices and proof of the uniqueness of inverse, if it exists; (Here all
matrices will have real entries)
...
Determinants
(25 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
...
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
...
Continuity and Differentiability
(20 Periods)
Continuity and differentiability, derivative of composite functions, chain rule,
derivatives of inverse trigonometric functions, derivative of implicit function
...
Logarithmic differentiation
...
Second order derivatives
...
2
...
Sample
problems (that illustrate basic principles and understanding of the subject
as well as real-life situations)
...
Integrals
(20 Periods)
Integration as inverse process of differentiation
...
[3]
[Class XII : Maths]
dx
x 2 a2 ,
px q
dx
2
x a
ax 2 bx c dx ,
2
,
dx
2
a x
2
px q
2
ax bx c
ax 2 bx c dx and
,
dx
dx
,
2
ax bx c
ax bx c
2
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)
...
4
...
5
...
Formation of differential equation whose general solution is given
...
Solutions
of linear differential equation of the form:
dy
py q , where p and q are functions of x or constants
dx
dx
px q , where p and q are functions of y or constants
dy
Unit IV : VECTORS AND THREE-DIMENSIONAL GEOMETRY
1
...
Direction cosines
and direction ratios of a vector
...
Definition, Geometrical Interpretation, properties and applications of scalar
(dot) product of vectors, vector (cross) product of vectors, scalar triple
product of vectors, projection of a vector on a line
...
Three-Dimensional Geometry
(15 Periods)
Direction cosines and direction ratios of a line joining two points
...
Cartesian and vector equation of a plane
...
Distance of a point from
a plane
...
Linear Programming : Introduction, related terminology such as constraints,
objective function, optimization
...
P
...
P
...
Unit VI : PROBABILITY
1
...
Conditional probability, independent
events, total probability, Baye’s theorem, Random variable and its probability
distribution, mean and variance of a random variable
...
Marks per Question
Total Number of Questions in
2013-14
2014-15
1
10
6
4
12
13
6
7
7
Total
29
26
[5]
[Class XII : Maths]
MATHEMATICS (CODE NO
...
No
...
Marks 100
Typology of Questions
Learning Outcomes
& Testing
Competencies
Reasoning
Analytical Skills
Critical thinking
Derivative
Marks
%
Weightage
2
3
1
20
20%
2
...
Application – (Use abstract
information in concrete situation, to
apply knowledge to new situations;
Use given content to interpret a
situation, provide an example, or
solve a problem)
1
3
2
25
25%
4
...
Remembering – (Knowledge based
Simple recall questions, to know
specific facts, terms, concepts,
principles, or theories; Identify,
define, or recite, information)
*
*
*
*
Very
Long Long
Short Answer Answer
Answer
I
II
(1 M) (4 M) (6 M)
5
...
No
...
Relations and Functions
9
2
...
Matrices and Determinants
17
23
5
...
Applications of Derivatives
47
7
...
Applications of Integrals
85
9
...
Vectors
101
11
...
Linear Programming
122
13
...
A × B = {(a, b) : a A, b B}
...
and number of relations = 2rs
is also a relation defined on set A, called the void (empty) 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
...
i
...
x1, x2 A such that x1 x2 f(x1)
f(x2)
...
[9]
[Class XII : Maths]
Onto function (surjective) : A function f : A B is said to be onto iff
Rf = B i
...
b B, there exists a A such that f(a) = b
A function which is not one-one is called many-one function
...
Bijective Function : A function which is both injective and surjective is
called bijective function
...
Similarly fog can be
defined
...
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
...
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
...
Binary operation * defined on set A is said to be commutative iff
a * b = b * a a, b A
...
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
...
If A is the set of students of a school then write, which of following relations
are Universal, Empty or neither of the two
...
Is the relation R in the set A = {1, 2, 3, 4, 5} defined as
R = {(a, b) : b = a + 1} reflexive?
3
...
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 gof
...
Let g, f : R R be defined by
g x
6
...
write fogx
f
x
3
f : R R defined by
f x
2x 1
5
be an invertible function, write f –1(x)
...
If f x
8
...
a * b = a + b + ab, write 3 * 2
[11]
[Class XII : Maths]
(ii)
9
...
If n(A) = n(B) = 3, then how many bijective functions from A to B can be
formed?
10
...
Is f : N N given by f(x) = x2 one-one? Give reason
...
If f : R A, given by
f(x) = x2 – 2x + 2 is onto function, find set A
...
If f : A B is bijective function such that n (A) = 10, then n (B) = ?
14
...
R = {(a, b) : a, b N, a b and a divides b}
...
Is f : R R, given by f(x) = |x – 1| one-one? Give reason
17
...
18
...
If f x log
1 x
1 x2
19
...
ab
5
20
...
21
...
2x 3
7
(a)
f : R R , f (x )
(b)
f : R R, f(x) = |x + 1|
(c)
f : R – {2} R, f x
[Class XII : Maths]
3x 1
x 2
[12]
(d)
f : R [–1, 1], f(x) = sin2x
22
...
C
...
of a and b
...
23
...
Show
3x 4
4x
...
Let R be the relation on set A = {x : x Z, 0 x 10} given by
R = {(a, b) : (a – b) is divisible by 4}
...
Also, write all elements related to 4
...
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
...
Let * be a binary operation on Q such that a * b = a + b – ab
...
Prove that * is commutative and associative
...
If * is a binary operation defined on R – {0} defined by a * b
check * for commutativity and associativity
...
2a
b2
, then
If A = N × N and binary operation * is defined on A as
(a, b) * (c, d) = (ac, bd)
...
(ii)
Find the identity element for * in A (If it exists)
...
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
...
Let * be a binary operation on set Q defined by a * b ab , show that
4
(i)
4 is the identity element in Q
...
16
,
a
a Q 0
...
1
is bijective where R+ is the
2x
32
...
, 12} and R be a relation in A × A defined by (a, b)
R (c, d) if ad = bc (a, b), (c, d), A × A
...
Also obtain the equivalence class [(3, 4)]
...
If ‘*’ is a binary operation on R defined by a * b = a + b + ab
...
Find the identify element
...
34
...
Are they equal?
35
...
f : R R, g : R R given by f(x) = [x], g(x) = |x| then find
fog
1
...
3
3
R1 : is universal relation
...
R3 : is neither universal nor empty
...
No, R is not reflexive
...
(5, 7) R
4
...
(fog)(x) = x x R
[Class XII : Maths]
[14]
1
, find f 1 x
...
f 1 x
7
...
x
1
,x
2x 1
2
(i)
3 * 2 = 11
(ii)
1369
27
9
...
3
11
...
12
...
n(B) = 10
14
...
No, R is not reflexive a, a R a N
16
...
e
...
17
...
e = 5
20
...
21
...
(c)
One-one, but not onto
...
[15]
[Class XII : Maths]
22
...
Elements related to 4 are 0, 4, 8
...
f 1 x
26
...
27
...
28
...
(ii)
(1, 1) is identity in N × N
32
...
0 is the identity element
...
(fog) (x) = x2 + x
(gof) (x) = x2 – x + 1
Clearly, they are unequal
...
f 1 x
36
...
etc
...
If sin x and , then = sin–1x etc
...
sin (sin–1x) = x x [–1, 1]
cos (cos–1x) = x x [–1, 1] etc
...
x y
tan1 x tan 1 y tan 1
;
1 xy
xy 1
...
1 x 2
[Class XII : Maths]
[18]
1
...
sin
sin
3 2
3 2
...
–1
1
...
1
3
1 1
1
cos tan 1 3
2
2
What is the value of the following functions (using principal value)
...
– sec
3
3
–1
–1 3
1
...
(v)
sin
–1
–1
5
tan
...
tan–1 (1) + cot–1 (1) + sin–1 (1)
...
–1
(ii)
–1
4
sin
...
4
–1
1 cos x
1 cos x –
1 cos x
x
...
Prove that
1
tan
1 1 cos x
cos x
cot
1 sin x
1 cos x
4
5
...
–1
x 0, 2
...
sin
cos
a
a
a2 – x 2
Prove that
cot
–1
–1 8
–1
–1 8
–1 300
...
Prove that tan
8
...
2
4
2
1 x
1 x
3x
2
...
m
m n
Prove that tan1 tan1
, m, n 0
n
m n 4
10
...
x 2 1 1
2x 2
Solve for x, cos1 2 tan1
x 1 2
1 x 2
3
12
...
Solve for x ,
14
...
Evaluate
1
3
tan cos –1
11
2
16
...
Prove that
1
cot tan1 x tan1 cos 1 1 2 x 2 cos 1 2 x 2 1 , x 0
x
18
...
Solve for x, 2 tan–1(cos x) = tan–1 (2 cosec x)
20
...
If tan–1a + tan–1b + tan–1c = then
sin–1 x 1 x x 1 x 2
prove that
22
...
a + b + c = abc
If cos–1x + cos–1y + cos–1z = , prove that x2 + y2 + z2 + 2xyz = 1
[Hint : Let cos–1 x = A, cos–1 y = B, cos–1 z = c then A + B + C =
or A + B = – c
Take cos on both the sides]
...
6
2
...
3
[21]
(iv)
–
6
2
(iv)
2
–
6
(viii)
...
1
11
...
5
3
19
...
Hint:
15
...
4
Let
20
tan–1 a =
tan–1 b =
tan–1 c =
then given,
take tangent on both sides,
tan ( ) = tan
[Class XII : Maths]
[22]
tan
2 3
12
11 3
3 11
si –1
n
x – sin–1 x
...
The numbers or functions are called the elements of the matrix
...
Square Matrix : An mxn matrix is said to be a square matrix of order n
if m = n
...
e
...
Row Matrix : A matrix having only one row is called a row matrix
i
...
B bij 1xn is a row matrix of order 1xn
...
Diagonal Matrix : A square matrix is called a diagonal matrix if all its non
diagonal elements are zero
...
Identity Matrix : A scalar matrix in which each diagonal element is 1, is
called an identity matrix or a unit matrix
...
I = [eij]n × n
where,
1 if i j
eij
0 if i j
[23]
[Class XII : Maths]
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
...
Properties of the transpose of a matrix
...
Also a square matrix A is symmetric if A´ = A
...
Also a square matrix A is skew - symmetric, if
A´ = – A
...
It is denoted
by det A or |A|
...
Area of triangle 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
...
[Note : Correctness of adj A can be checked by using
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 only a non-singular matrix exists
...
adj A
A
Properties
(i)
AA–1 = A–1A = I
(ii)
(A–1)–1 = A
(iii)
(AB)–1 = B–1A–1
(iv)
(AT)–1 = (A–1)T
(v)
A
1
1
, A 0
A
Solution of system of equations using matrix :
If AX = B is a matrix equation then its solution is X = A–1B
...
(ii)
If |A| = 0 and (adj A) B 0 then system is inconsistent and has
no solution
...
[25]
[Class XII : Maths]
1
...
2
...
Find the value of a23 + a32 in the matrix A = [aij]3 × 3
0
0
and B i
i
i
, find AB
...
i 2 j 3 if i j
4
...
5
...
0 9
6
...
7 5
7
...
6
8
...
2x 3 x 1
9
...
For what value of x the matrix
3 4 x 5
10
...
[Class XII : Maths]
[26]
11
...
If
13
...
sin 30
If A
sin 60
15
...
9
2
has no inverse
...
4
cos 30
, what is |A|
...
1 3 2
4 5 6
...
Find the minor of a23 in
17
...
...
Find the value of x such that the points (0, 2), (1, x) and (3, 1) are
collinear
...
Area of a triangle with vertices (k, 0), (1, 1) and (0, 3) is 5 unit
...
20
...
21
...
What is the number of all possible matrices of order 2 × 3 with each entry
0, 1 or 2
...
Find the area of the triangle with vertices (0, 0), (6, 0) and (4, 3)
...
If
2x 4 6 3
, find x
...
x y
If A z
1
26
...
1
3
4
6
8
9x 12x
27
...
28
...
Given a square matrix A of order 3 × 3 such that |A| = 12 find the value
of |A adj A|
...
If A is a square matrix of order 3 such that |adj A| = 81 find |A|
...
Let A be a non-singular square matrix of order 3 × 3 find |adj A| if
|A| = 10
...
2 1
If A
find
3 4
33
...
0
34
...
Construct a 3 × 3 matrix A = [aij] whose elements are given by
A 1 1
...
13
36
...
2 0 1
37
...
3
38
...
39
...
40
...
41
...
Prove that the product of the matrices
cos2
cos2
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
...
12 7
[29]
[Class XII : Maths]
44
...
Hence find
A–1
...
4
If A
2
46
...
4 5 6 2 4 6
47
...
1 4
48
...
5
3x – y = 5; 6x – 2y = 3
49
...
...
3 1
By using elementary column transformation, find the inverse of A
...
cos sin
If A
and A + A´ = I, then find the general value of
...
52
...
x 2 x 3 x 2a
x 3 x 4 x 2b 0 if a, b, c are in A
...
x 4 x 5 x 2c
54
...
c
56
...
2
a ab
b
2
2
b
2
2
2
a b
2 2 2
4a b c
...
z
2
2 2 2
4a b c
...
x c
Show that :
2
x
yz
(i)
(ii)
61
...
a
c a
r p
z x
2
x a
a
a
2
a
c
ab
58
...
2
Given
y
z
2
2
z y z z x x y yz zx xy
...
2
If A
2
0
A
2
5
4
and B 2
1
1
2
3
verify that AB A B
...
Find the product AB and
1
also find (AB)–1
...
Solve the following equation for x
...
a x
[31]
[Class XII : Maths]
63
...
sin
cos
Use matrix method to solve the following system of equations :
5x – 7y = 2, 7x – 5y = 3
...
66
...
3 –2 7
1 1 0
2 2 –4
2 3 4 and B –4 2 4
If A
are two square matrices, find AB
0 1 2
2 –1 5
and hence solve the system of linear equations :
x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7
...
S ol t
ve he f l i
olow ng syst
em of equat ons by m at x m et
i
ri
hod, w here
x 0,
y 0, z 0
2 3 3
1 1 1
3 1 2
10,
10,
13
...
1
Find
where A 2
3
equations :
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
[Class XII : Maths]
[32]
69
...
If we subtract the second number from
twice the first number, we get 3
...
Represent it algebraically and find the numbers
using matrix method
...
Compute the inverse of the matrix
...
1
6
2
1
If the matrix A 0
3
1
5 and verify that A–1 A = I3
...
72
...
73
...
Also show that A 1
...
Show that
2
c
ba
ca
75
...
a
a c
Show that
a b
2
ab
c
2
ac
a
cb
b –c
b
b a
2
2 2 2
bc
2
a b
4a b c
2
c b
2
2
2
c a a b c a b c
c
cos sin 0
If A sin cos 0 , verify that A
...
A = |A| I3
...
2 1 1
For the matrix A 1 2 1 , verify that A3 – 6A2 + 9A – 4I = 0, hence
1 1 2
find A–1
...
Find the matrix X for which
3 2
7 5
...
1 1 2 1
...
xy
x z
yz
xz
80
...
x y 2
a
ab
a b c
2a 3a 2b 4a 3b 2c a 3
...
If x, y, z are different and y
z
83
...
x2
1 x 3
y 2 1 y 3 0, show that xyz = – 1
...
P
...
15 1
[34]
84
...
Using properties of determinants prove that
bc
b 2 bc c 2 bc
a 2 ac
ac
a 2 ab b 2 ab
86
...
1
...
0 1
1 0
3
...
4
...
9 6
0 29
...
3 5
3 1
...
AB = [26]
...
x = 5
9
...
0
1
11
...
12
...
k
14
...
15
...
–4
3
2
[35]
1
...
3
18
...
54
...
22
...
9 sq
...
x = ± 2
25
...
0
27
...
8 3
6 5
...
1728
30
...
100
32
...
|AB| = – 11
34
...
3 3 2 5 2
4 5
2
...
4
11
7 7
A
1 18
7
7
9
2
1
5
7
7
7
7
, B
4
4 12 5
7
7
7
7
40
...
44
43
...
12 5
17
...
k
21
...
2 2
[Class XII : Maths]
41
...
A –1
[36]
1 4 3
...
x = 9, y = 14
46
...
2 0
48
...
Inverse does not exist
...
2 1
A –1
...
2n
61
...
2
...
A
1
67
...
1
69
...
A
2
11
4
1
2
1
1
2
1
6
...
, n z
3
, y
1
...
19
10 2 3
1
2 0 1
5 1 0
0 1 3
70
...
x = 3, y = 2, z = 1
...
2
1 1 1
73
...
16 3
X
...
x = 1, y = 1, z = 1
...
A
83
...
3
CHAPTER 5
CONTINUITY AND DIFFERENTIATION
A function f(x) is said to be continuous at x = c iff lim f x f c
x c
i
...
, 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
...
Every polynomial function is continuous on R
...
f (x), f (x) + c, cf (x), | f (x)| are also continuous at x = a
...
f x
is continuous at x = a provided g(a) 0
...
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
...
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
d 1
cot x
dx
d
cosec1x
dx
1
1 x2
1
1 x2
1
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
...
[Class XII : Maths]
[40]
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
...
1
...
2
...
3
...
4
...
Write the number of points of discontinuity of f(x) = [x] in [3, 7]
...
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
...
tan3x
, x 0
2K ,
7
...
8
...
r
...
cos x
...
If f(x) = x2g(x) and g(1) = 6, g´(x) = 3 find value of f ´ (1)
...
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]
[Class XII : Maths]
(iv)
(v)
11
...
sin1 x 7 2
(vi)
Discuss the continuity of following functions at the indicated points
...
2,
x 0
(ii)
sin 2x
3x , x 0
g x
at x 0
...
(v)
12
...
x x , x 1
f x
at x 1
...
3x 2 kx 5,
For what value of k, f x
1 3x
0 x 2
is continuous
2 x 3
x 0, 3
...
For what values of a and b
f x
[Class 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
...
Prove that f(x) = |x + 1| is continuous at x = –1, but not derivable at
x = –1
...
For what value of p,
x p sin 1 x x 0
f x
is derivable at x = 0
...
tan
x
1 x2
dx
16
...
1 x
dy
?
If y sin 2 tan1
then
1 x
dx
18
...
If x 1 y 2 y 1 x 2 a then show that
20
...
If (x + y)m + n = xm
...
2x
2x
w
...
t
...
Find the derivative of tan1
1 x 2
1 x 2
23
...
r
...
loga(cos x)
...
If xy + yx + xx = mn, then find the value of
25
...
dx
dy
1 y 2
...
dx
1 x 2
dy y
...
dx
d 2y
...
x = aet (sint – cos t)
If
dy
at x is 1
...
1
2
If y sin x 1 x x 1 x then find –
28
...
x
Differentiate x
30
...
1 sin x
If y tan1
1 sin x
x
x
dy
...
dx
w
...
t
...
dy
y
x
, if cos x cos y
dx
Hint : sin
1 sin x
dy
where 2 x find dx
...
2
2
2
32
...
a
33
...
r
...
x
34
...
If y = sin–1x, find
36
...
a cos
If y e
38
...
dx
sin a
d 2y
in terms of y
...
then show that
1,
dx 2 a y
a2 b 2
1
[Class XII : Maths]
x
, 1 x 1 show that 1 x 2
,
d 2y
dy
x
a2 y 0
2
dx
dx
d 2 y 2a 2 x 2
...
Verify Rolle's theorem for the function, y = x2 + 2 in the interval [a, b]
where a = –2, b = 2
...
Verify Mean Value Theorem for the function, f(x) = x2 in [2, 4]
7
...
x
3
...
Points of discontinuity of f(x) are 4, 5, 6, 7 i
...
four points
...
R
4
...
because lim f x 3 f 3
...
4
6
...
2
7
...
–cot x
9
...
(i)
(ii)
e log2
(iii) 6 (x – 1)5
(iv)
0
7 x2 x
...
3
log3 e
3x 5
(iii) Continuous
x
1
...
x
loge 5
2
x loge x
...
k = 11
13
...
15
...
16
...
1
17
...
x
1 x 2
...
x
y 1
y x log y
dy x 1 log x yx
...
d 2y
1
cosec sec 4
...
dy
1
1
...
x log x
29
...
x x log x 1 log x
...
dy y tan x logcos y
dx x tan y logcos x
31
...
dx
2
33
...
sec2y tany
...
1 log log x
, x 1
x
x
[Class XII : Maths]
[46]
CHAPTER 6
APPLICATIONS OF DERIVATIVES
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
...
r
...
x at x = x0
...
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)
...
(ii)
A function f(x) is said to be decreasing (non-increasing) on an
interval (a, b)
...
Alternatively if f ´(x) 0 x (a, b), then f (x) is decreasing
function in (a, b)
...
dx x 0 , y 0
[47]
[Class XII : Maths]
where
dy
slope of the tangent at the point x 0, y 0
...
(i)
(ii)
If
If tangent at x = x0 is parallel to x-axis then
1
Slope of the normal to the curve at the point ( 0, y0) is given by dy
x
...
dy
0
...
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 f x f ´x x
[Class XII : Maths]
[48]
Let f be a function
...
First Derivative Test : Let f be a function defined on an open interval 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
...
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
...
Second Derivative Test : Let f be a function defined on an interval I and
let c I
...
f (c) is local maximum value of f
...
f (c) is
local minimum value of f
...
1
...
2 cm/sec
...
2
...
7 cm/sec
...
If the radius of a soap bubble is increasing at the rate of
4
...
At the instant when the radius of the circular wave is 10 cm,
how fast is the enclosed area increasing?
1
cm sec
...
[49]
[Class XII : Maths]
5
...
Find the marginal revenue when x = 7
...
Find the maximum and minimum values of function f (x) = sin 2x + 5
...
Find the maximum and minimum values (if any) of the function
f (x) = – |x – 1| + 7 x R
...
Find the value of ‘a’ for which the function f (x) = x2 – 2ax + 6, x > 0 is
strictly increasing
...
Write the interval for which the function f (x) = cos x, 0 x 2 is
decreasing
...
What is the interval on which the function f x
increasing?
log x
, x 0, is
x
4 3
x is increasing?
3
11
...
Write the interval for which the function f x
13
...
14
...
15
...
16
...
17
...
At what point on the curve y = x2 does the tangent make an angle of 45°
with positive direction of the x-axis?
19
...
[Class XII : Maths]
[50]
1
is strictly decreasing
...
What is the slope of the normal to the curve y = 5x2 – 4 sin x at x = 0
...
Find the point on the curve y = 3x2 + 4 at which the tangent is perpendicular
1
to the line with slope
...
Find the point on the curve y = x2 where the slope of the tangent is equal
to the y – co-ordinate
...
If the curves y = 2ex and y = ae–x intersect orthogonally (cut at right
angles), what is the value of a?
24
...
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
...
Find the rate of change of the area of a circle with respect to its radius
...
r
...
its radius when its radius is 3 cm?
27
...
28
...
4
2
29
...
30
...
31
...
Find the points on the curve
at which the y co-ordinate is changing 8 times as fast as the x co-ordinate
...
A ladder 5 metres long is leaning against a wall
...
How
fast is its height on the wall decreasing when the foot of the ladder is 4
metres away from the wall?
[51]
[Class XII : Maths]
33
...
Find the rate at which the radius of the
balloon increases when the radius is 15 cm
...
A man 2 metres high walks at a uniform speed of 6 metres per minute
away from a lamp post 5 metres high
...
35
...
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
...
The length x of a rectangle is decreasing at the rate of 2 cm/sec and the
width y is increasing as the rate of 2 cm/sec when x = 12 cm and y = 5 cm
...
37
...
c/sec
...
How fast is the height of the sand cone
increasing when height is 4 cm?
38
...
The length of the rectangle is always equal to the square of the breadth
...
5
cm?
39
...
40
...
x 2x 3
2
41
...
42
...
43
...
44
...
[Class XII : Maths]
[52]
45
...
4 4
16
...
Find the points on the curve 4y = x3 where slope of the tangent is
47
...
Find the equation of the tangent to the curve given by x = 1 – cos ,
x y
1 touches the curve y = be–x/a at the point where the
a b
curve crosses the y-axis
...
...
50
...
51
...
52
...
Prove that f x
54
...
x 1
2
x3
x 2 9x , x 1 2 is strictly increasing
...
is
2
increasing or decreasing
...
Find the least value of 'a' such that the function f (x) = x2 + ax + 1 is strictly
increasing on (1, 2)
...
5
Find the interval in which the function f x 5x 2 3x 2 , x 0 is strictly
decreasing
...
Show that the function f (x) = tan–1 (sin x + cos x), is strictly increasing on
the interval 0,
...
59
...
,
8 8
Show that the function f x
sin x
is strictly decreasing on 0,
...
No
...
1
60
...
0
...
64
...
009 3
...
02
...
80 4
...
0
...
65
...
001) where f(x) = x3 – 7x2 + 15
...
Find the approximate value of f (3
...
67
...
68
...
69
...
70
...
2 times the radius of the base
...
Show that the semi vertical angle of right circular cone of given surface
1
area and maximum volume is sin1
...
A point on the hypotenuse of a triangle is at a distance a and b from the
sides of the triangle
...
73
...
...
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
...
Find the intervals in which the function f (x) = (x + 1)3 (x – 3)3 is strictly
increasing or strictly decreasing
...
Find the local maximum and local minimum of f (x) = sin 2x – x,
77
...
Also find the points on which the tangents
are parallel to x-axis
...
A solid is formed by a cylinder of radius r and height h together with two
hemisphere of radius r attached at each end
...
2
2
1
metre min
...
constant but radius r is increasing at the rate of
79
...
80
...
81
...
[55]
[Class XII : Maths]
82
...
83
...
x2 y2
1
a2 b 2
84
...
Given that the perimeter is 16 metres
...
85
...
A soldier is placed
at the point (3, 2)
...
Find a point on the parabola y2 = 4x which is nearest to the point (2,
–8)
...
A square piece of tin of side 24 cm is to be made into a box without top
by cutting a square from each corner and folding up the flaps to form the
box
...
88
...
The total perimeter of the window is 30 metres
...
89
...
meter, show that the maximum value of the box is 13
...
90
...
One of the two pieces
is to be made into a square and other into a circle
...
Show that the height of the cylinder of maximum volume which can be
inscribed in a sphere of radius R is
92
...
...
Show that the altitude of the right circular cone of maximum volume that
4r
can be inscribed is a sphere of radius r is
...
Prove that the surface area of solid cuboid of a square base and given
volume is minimum, when it is a cube
...
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
...
Show that the right triangle of maximum area that can be inscribed in a
circle is an isosceles triangle
...
A given quantity of metal is to be cast half cylinder with a rectangular box
and semicircular ends
...
1
...
8 cm/sec
...
4
...
3
...
4
...
5
...
208
...
Minimum value = 4, maximum value = 6
...
Maximum value = 7, minimum value does not exist
...
a 0
...
[0, ]
10
...
x 1
12
...
0,
...
Maximum value = 4, minimum valve = 0
...
a > 1
...
R
17
...
1 1
,
...
(2, – 3)
20
...
(1, 7)
[57]
[Class XII : Maths]
1
...
(0, 0), (2, 4)
23
...
1
–
...
8r
26
...
72
28
...
a > 0
...
4, 11
33
...
2b
29
...
80
...
8
cm sec
...
34
...
4
cm sec
...
(a) 0 cm/sec
...
37
...
48
38
...
11 cm/sec
...
7 1
,
...
y
42
...
48x – 24y = 23
45
...
128
8 128 8
,
,
,
...
49
...
50
...
31
and 4,
...
2
...
3
52
...
53
...
Increasing in , Decreasing in 0,
4 2
55
...
56
...
60
...
2083
61
...
9907
62
...
06083
63
...
1925
64
...
002
65
...
995
66
...
46
68
...
5
Strictly increasing in 0, , 2
4 4
...
4 4
75
...
76
...
value
6
3
2
6
Local minima at x
6
Local minimum value
77
...
[59]
[Class XII : Maths]
Points are (2, 29) and (3, 28)
...
3
metres min
...
x + y tan – a sec = 0
...
(0, 0), (–1, –2) and (1, 2)
...
x + y = 3
82
...
xx 0
a
2
yy 0
b
2
1,
y y0
2
a y0
x x0
b2 x 0
0
...
6 3
85
...
(4, –4)
87
...
60
30
,
...
144
36
m,
m
...
4R 3
3 3
[Class XII : Maths]
[60]
CHAPTER 7
INTEGRALS
Integration is the reverse process of Differentiation
...
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
...
2
...
n 1
n –1
ax b n 1
c
n 1 a
n
ax b dx
1
log ax b c
a
3
...
tan x
...
4
...
– log cos x c log sec x c
...
cot x dx
8
...
dx
2
cosec x cot x dx
11
...
cosec x dx
13
...
15
...
1 x
17
...
19
...
2
1 x
2
2
x
2
a
2
log cosec x – cot x c
...
–1
–1
x c, x 1
...
1
x
2
1
a
2
1
x
2
dx
1
log
2a
dx
1
[Class XII : Maths]
1
a
a x
c
...
dx sec
1
a
– cosec x c
...
tan x
...
dx tan x c
...
dx tan
2
sec
9
...
10
...
log sin x c
...
x a
–1
x
c
...
21
...
23
...
25
...
1
a
2
2
– x
1
2
2
a x
1
x
2
–a
2
2
dx sin
–1
x
c
...
sin
2
x
2
2
c
...
a
2
2
log x
a x
log x
x
2
c
...
2
1
...
f x dx k f x dx
...
k f x g x dx k f x dx k g x dx
...
x
x
e f x f ' x dx e f x c
...
f x dx
2
...
n
f x f ´x dx
f x n 1
c
...
f ´ x
f x
n
f x n 1
dx
c
...
g x dx
f x
...
g x dx dx
...
a
b
f x dx
a
where
f a f a h f a 2h
lim h
...
or
h
b
1
...
c
a
b
b
b
f x dx
f t dt
...
a
a
b
4
...
a
3
...
a
(ii)
f x dx f a x dx
...
f x 0; if f x is odd function
...
a
f x dx 20 f x dx ,
if f(x) is even function
...
0
2a f x dx , if f 2a x f x
f x dx
0
0,
if f 2a x f x
Evaluate the following integrals
1
...
1
sin
x cos
1
x dx
...
1
2
...
1
x
x8
8
6
...
e
1
5
...
1
7
...
4 3 cos x
8
...
cos 2x 2sin2 x
dx
...
2
11
...
8 x
dx
...
1
a log x
e x log a dx
...
2
12
...
dx
[Class XII : Maths]
1
13
...
15
...
e
17
...
x
dx
...
x 1
21
...
sec x
...
dx
...
dx
...
2
dx
...
18
...
16
...
x cos 1 dx
...
cos x sin dx
...
cot x
...
26
...
27
...
28
...
29
...
32
...
1
1 cos x
dx
...
0
34
...
xe ex
2
x log x dx
...
x
1 sin x
cos x dx
...
[Class XII : Maths]
[66]
1
dx
35
...
a f x f a b x dx
...
1x x dx
...
If
40
...
42
...
9 4x 2
f x
b
37
...
4
a
41
...
e
log x 1 log x
sin x dx
...
dx
...
b
4
44
...
a
f x dx f a b x dx
...
1
sec x tanx dx
...
sin2 x
1 cos x dx
...
1 tan x
1 tan x dx
...
ax b x
c x dx
...
(i)
(iii)
x cosec
tan–1x 2
1 x
4
dx
...
[67]
(ii)
(iv)
b
x 1
x 1
x 1
x 1
dx
...
[Class XII : Maths]
(v)
cos x cos 2x cos 3x dx
...
sin x cos x
2
2
2
2
cos
cot
5
3
x dx
...
dx
...
[Hint : Take sec2 x as numerator]
3
cos x cos x a
6
(xi)
51
...
(xii)
sin x cos x
sin x cos x
dx
...
dx
...
(vii)
2x 1
x 2
[Class XII : Maths]
dx
...
1
3x
[Hint : put x2 = t]
dx
...
[68]
x
x
x
2
2
dx
...
(x)
3x 2
x
2
x 1 dx
...
[Hint : Multiply and divide by
52
...
(iii)
cos
(iv)
7
1
...
cos 2
x 1
x 1 x 2 x 3 dx
...
(vi)
x 2 1 x 2 2
x 3 3 x 2 4 dx
...
tan x dx
...
1
[69]
[Class XII : Maths]
53
...
3
x dx
...
sec2 x and take sec x as first function]
(iii)
e
ax
cos bx c dx
...
[Hint : put 3x = tan ]
(v)
cos
(vi)
1 sin 2x
dx
...
(xi)
e
x
(xii)
log log x log x
2
(xiii)
6x 5
(xiv)
x
1
dx
...
dx
...
x 1
dx
...
2 sin 2x
dx
...
2
x
2
4x 3 dx
...
[70]
[Hint : put log x = t x = et ]
54
...
(ii)
0
0
1
(iii)
cos 2x log sin x dx
...
x
1 x 2 3 2
0
x2
1
dx
...
(vi)
x cos x
x
5x
2
1
2
dx
...
0
x sin x
x
sin x
as
Hint : Write
1 cos x
1 cos x
1 cos x
55
...
(ii)
1
0
1
(iii)
–1
x
1 sin x dx
...
(iv)
x sin x
1 cos
0
2
dx
...
2
2
(v)
Hint :
2
1
f x dx
2
2
[71]
1
f x dx
1
2
f x dx
f x dx
1
[Class XII : Maths]
2
(vi)
x sin x cos x
sin
4
0
(vii)
a
dx
...
2
cos x b sin x
0
Hint : Use
56
...
(v)
1
59
...
sin1 x cos 1 x
sin
[Class XII : Maths]
1
x cos 1 x
a
x 2 x 4 – x 5 dx
log x log sin x
2x
dx
...
6
e
1
cos x
sec x cosec x
58
...
a
f x dx
dx , x 0, 1
[72]
a x
a x
dx
...
1
60
...
(ii)
x
2x
dx
x 1 x
3
x 1 x 3
2
dx
(iv)
[73]
x
x
4
2
4
dx
4
dx
– 16
[Class XII : Maths]
2
(v)
cot x dx
...
x
x
1
4
dx
...
Evaluate the following integrals as limit of sums :
4
(i)
2
2x 1 dx
...
2x
4
4 dx
...
0
1
5
(v)
2
0
3
(iii)
x
x
2
3 x dx
...
Evaluate
1
(i)
cot
1
1
x x 2 dx
0
(ii)
dx
sin x 2 cos x 2 sin x cos x
1
(iii)
0
63
...
[Class XII : Maths]
2
dx
(iv)
2 log sin x
– log sin 2x dx
...
[74]
3 sin 2 cos
5 cos2 4 sin d
...
0 x tan
67
...
/2
2
x dx
–1
66
...
log sin x dx
2
...
2e – 2
2
3
...
4
...
log8 9
16
5
...
log log log x
8
...
0
9
...
10
...
f (x) + c
13
...
2 32 2
32
x x 1 c
3
3
15
...
e
a
17
...
2
x 13 2 2 x 11 2 c
...
log x 1
20
...
x cos2 + c
22
...
cos
1
c
...
log sec x tan x 2 c
24
...
log sin x 2 c
26
...
4 2x
2
2
2
27
...
3
28
...
2 log |sec x/2| + c
...
1
log x e e x c
...
x log x 2 c
32
...
0
34
...
1 1 2x
sin c
3
2
36
...
–1
38
...
1
40
...
x + log x + c
...
1
log sec x tan x c
...
1 sin3x
2
sin x c or sin3 x c
3
2
3
44
...
0
46
...
x – sin x + c
48
...
2
a c x
log a c
b c x
log b c
[Class XII : Maths]
c1
...
2
a
(i)
1
1
log cosec tan1 x 2 2 c
...
1 2
1
x x x 2 1 log x x 2 1 c
...
1
12x 6 sin 2x 3 sin 4x 2 sin 6x c
...
32
2
2
6
(viii)
4
cot6 x
cot x
c
...
2
2
2
a sin x b cos x c
(x)
(xi)
tan x – cot x – 3x + c
...
–2 cosec a cos a tan x
...
sin–1 (sin x – cos x) + c
...
C
3 log x 2
[77]
[Class XII : Maths]
(iii)
1
5 – 1 2x
log
5
c
5 1 2x
(iv)
x 4
sin1
c
...
x 2
log cos x
(i)
x 1
x
log
7
[Class XII : Maths]
x
7
1
2 x 1
c
5
2x 1
2
x x 1
7 4
c
1
2
2 3
x x 1
log x
2
8
1
2
cos x cos x c
2
1
sin
7
c
1
[78]
(ii)
(iii)
1 cos x
log
c
2 3 cos x
2
log cos 2
3
(iv)
1
log 1 cos c
...
1
3
2
(ix)
tan
2x tan
2
1
a cos bx
3x
1
c b sin bx c c1
log 1 9x
2
c
3
[79]
[Class XII : Maths]
(v)
2 x sin x cos x c
(vi)
3
x 4 1
x
x
1
tan x –
c
...
(viii)
2
x a
2ax x
2
2
a
2
sin
2
(x)
x log log x –
c
...
(xii)
1
x
x x 1
e
c
...
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
1
(xiv)
(xv)
log x 1
3
1
2
1
x 1
6
2
x
2
4x 3
3
log x
1
log x 2
tan
2 x 1
c
3
1
3
3
2
x 2
2
x
2
x
2
4x 3
4x 3 c
2
(xvi)
x 2
2
[Class XII : Maths]
x
2
4 x 8 2 log x 2
[80]
x
2
4x 8 c
1
54
...
(ii)
20
–
4
1
...
2
...
5
2
/2
...
(i)
8
5
...
(i)
e
...
4
2
...
2ab
...
(v)
log 2
...
57
...
(iv)
–x cos x + sin x + c
...
(i)
(ii)
59
...
(i)
1
1
1 2
3
x
3 2
x x2 c
1
2
log 1 2 c
3
x
a
3 1
2
x 4 log x
5
log x 1
4
3
log x 1
4
log x
2
1
1
2
(ii)
(iii)
1
log x 1
5
1
1
2x
1
x
2
(v)
4 –
10
log
81
8
x 2
x 2
tan
log x 3
1
tan
x
c
...
[Class XII : Maths]
1
10
log x 1
8
(iv)
2
log x
[82]
–1
x
c
...
tan
–1
x c
...
(i)
14
...
26
...
c
(i)
(iii)
141
26
...
2
...
8
(iv)
1
log
...
1
1
2
log 1 cos x log 1 cos x log 1 2cos x c
...
3log 2 sin
65
...
e 2x
2cos 3x 3 sin3x c
...
log 2
2
4
c
...
a
k
Y
y = f( x )
A2
O A
y
1
A1
B (k , 0 ) x = b
X
x = a
1
...
2
...
Find the area enclosed by the ellipse
x
a
[85]
2
2
y
b
2
25 x
...
Find the area of region in the first quadrant enclosed by x–axis, the line
y = x and the circle x2 + y2 = 32
...
Find the area of region {(x, y) : y2 4x, 4x2 + 4y 2 9}
6
...
7
...
8
...
9
...
(–1, 0), (1, 3) and (3, 2)
(b)
(–2, 2) (0, 5) and (3, 2)
Using integration, find the area bounded by the lines
...
11
...
12
...
13
...
14
...
16
...
25 x
x
2
9
line 2x + 3y = 6
...
...
Find the area of region bounded by the curve x 2 = 4y and line
x = 4y – 2
...
Using integration find the area of region in first quadrant enclosed by
3 y and the circle x2 + y2 = 4
...
Find smaller of two areas bounded by the curve y = |x| and x2 + y2 = 8
...
Find the area lying above x-axis and included between the circle
x2 + y2 = 8x and inside the parabola y2 = 4x
...
Using integration, find the area enclosed by the curve y = cos x,
y = sin x and x-axis in the interval 0,
...
Sketch the graph y = |x – 5|
...
0
23
...
24
...
1
25 – sq
...
4
2
3
...
units
...
5 x2
ab sq
...
8
...
2
6
9
8
9
4
4
...
units
3
8
2 3 sq
...
units
7
...
units
2 ab
sq
...
(a) 4 sq
...
units
2
[Hint
...
units
2
[
Hint
: Coordinate of vertices are (– 1, 1) (0, 5) (3, 2)]
11
...
units
4
2
12
...
3 sq
...
1 sq
...
units
2
15
...
1
3
9
sq
...
3
2 sq
...
sq
...
units
3
19
...
units
...
4
8 3 sq
...
2
22
...
units
...
384 sq
...
24
...
units
4
2
2 sq
...
[Class XII : Maths]
[88]
CHAPTER 9
DIFFERENTIAL EQUATIONS
Differential Equation : Equation containing derivatives of a dependant
variable with respect to an independent variable is called 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
...
Now eliminate the arbitrary constants from these equations
...
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
dy
dx
f x, y
g x, y
[89]
where f(x, y) and g(x, y) are both
[Class XII : Maths]
homogeneous functions of the same degree in x and y i
...
, of the
form
dy
y
F is called a homogeneous differential equation
...
(iii)
dv
...
For finding solution of this
type of equations, we find integrating factor (I
...
) e
Solution is y I
...
Q
...
F
...
c
dx
Similarly, differential equations of the type dy Px Q where P
and Q are constants or functions of y only can be solved
...
Write the order and degree of the following differential equations
...
dx
5
4
(iii)
d y
dx
(v)
(vii)
4
d 2y
sin x 2
...
dy
log
0
...
2
dy
1
dx
2
k
3
sin x
...
2
...
(i)
dy
x
5
x
2
2
...
(iv)
dx
(iii)
dy
x
3
e
x
e
dx
(v)
dy
1 cos 2x
(vi)
...
1 2y
...
dy
dx
dx
3
...
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)
2
2
y
b
2
2
0
...
(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
2
2
(iii)
x d y
dy
B
x
y 0
...
dx
x
x 2 a2
Verify that y log x
equation :
a2
(vi)
2
x2
d 2y
dx
2
x
satisfies the differential
dy
0
...
(vii)
(viii)
6
...
Form the differential equation representing the family of curves
(y – b)2 = 4(x – a)
...
(i)
dy
y cot x sin 2x
...
(iii)
dy
dx
(iv)
1
...
x
dy
cos x sin x
...
ydx x y
ye dx y
y
3
3
dy
0
2 xe
y
dy
Solve each of the following differential equations :
dy
dy
2
2y
...
(iii)
x 1 y dx y 1 x dy 0
...
(xy2 + x) dx + (yx2 + y) dy = 0; y(0) = 1
...
dx
(vii)
8
...
(ii)
x
2
dy
x
2
2
xy y
...
[Class XII : Maths]
x
x
y sin
dx x sin y dy
...
y
1– y
dx
1 x
3xy
(ix)
2
y
dy
dx
y
y
tan
...
dx
2
2
...
(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
...
(iv)
10
...
Show that the differential equation
dy
dx
x 2y
x y
is homogeneous and
solve it
...
Show that the differential equation :
(x2 + 2xy – y2) dx + (y2 + 2xy – x2) dy = 0 is homogeneous and solve it
...
Solve the following differential equations :
(i)
dy
2y cos 3x
...
3e tan y dx 1 e
x
x
sec2 y dy
0
Solve the following differential equations :
(i)
(x3 + y3) dx = (x2y + xy2)dy
...
y
y
– x y sin x cos dy 0
...
y
(v)
xe
x
y x
dy
0 if y(e) = 0
dx
(vi)
(vii)
14
...
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
(iii)
x
x
1 e y dx e y 1 x dy 0
...
[95]
[Class XII : Maths]
15
...
dy
y cot x 2 x x 2 cot x given that y(0) = 0
...
(i)
order = 1,
degree = 1
(ii)
order = 2, degree = 1
(iii)
order = 4,
degree = 1
(iv)
order = 5, degree is not defined
...
(i)
x
6
x
3
...
(vi)
2 log |3x + 1| + 3log |1 – 2y| = c
...
e 1
e–1/x
[96]
e
(vi)
3
[Class XII : Maths]
2
(iv)
1
x
c
6
log x 2
(iii)
x
y loge e e
4
(v)
3
(ii)
y
6
(iii)
order = 1, degree is not defined
sec x
tan
1
x
(vii)
e
4
...
(vi)
d y
dx
2
2
d y
2
dx
2y 0
dx
2
(viii)
dy
dy
dx
2
(ii)
c
y
x
c
y sin x
= y
dy
dx
2
4 loge
x 1
16
3
(iii)
dx
2
0
2 sin x
y sin x
xy
d y
3
3
6
...
(i)
(iii)
(iv)
(v)
xy
4
y
c
4
cy x 2 1 2y
1 x
1
log
2
x 2
2
1 y
1 y
1 y
1 y
2
2
2
2
1
2 sec y c
c
1 x
2
1 y
2
c
1
1 2
[97]
[Class XII : Maths]
(vi)
1
log y
4
cos x
4
1
6
cos x xe
6
(vii)
log tan y
cos 2x
x
e
x
c
3
1 cos 2x
x
– cos 2x x 1 e c
16
3
c
4
8
...
(i)
x
y
cos x y
e
x
x
2xy
y
2
y
c
1
x
2
y
2
x
y
10
...
x
2
2
2
2xy
2xy
1
dy
dy
y
y '
xy y
3
2
[Class XII : Maths]
c
x
y
1
x c
2
(ii)
0
2y x
x
2
2
dy
dx
0
dx
(iv)
y sin
dx
(iii)
2
c
x
2
2
3
3
2
x yy '
2 3 tan
1
2
x 2y
c
3x
y
[98]
,
y 2x
dy
dx
3 sin 3x
2 cos 3x
2x
2
2
(ii)
y
y x log c x y
(ii)
cx
(iii)
y
xy cos c
x
(iv)
(v)
y x log log x , x 0
(vi)
c x y
(ii)
y
(iv)
2y = sin x
y
13
(iii)
13
...
(i)
cosec x
3
x 3
2
y
x
2
y
2
2
2
2
2
x y
2
...
(i)
y = tan x – 1 + ce tan x
sin x
cos x
c
x
x
x
(iii)
15
...
[99]
[Class XII : Maths]
CHAPTER 10
VECTORS
A quantity that has magnitude as well as direction is called a vector
...
Two or more vectors which are parallel to same line are called collinear
vectors
...
r
...
origin (0, 0, 0) is denoted by OP ,
where OP ai b j c k and OP a 2 b 2 c 2
...
If two vectors a and b are represented in magnitude and direction by the
two sides of a triangle taken in order, then their sum a b is represented
in magnitude and direction by third side of triangle taken in opposite order
...
If a is any vector and is a scalar, then a is a vector collinear with
a and a
a
...
Any vector a can be written as a a a , where a is a unit vector in
the direction of a
...
If C divides AB in ratio m : n externally,
m n
mb na
then c
...
a
b
c
Also l , m , n and l2 + m2 + n2 = 1
...
Scalar product of two vectors a and b is denoted as a
...
b a b cos , where is the angle between a and b (0 )
...
e
...
a b 0 a o, b o or
2
a a a , so i l
j j
a b
...
If a a 1i a 2 a 3 k and b b 1l b 2 b 3 k , then
j
j
a b = a1a2 + b1b2 + c1c2
...
b
Projection of a on b
and projection vector of
b
a
...
b
Cross product or vector product of two vectors a and b is denoted as
a b and is defined as a b a b sin n
...
Cross product of two vectors is not commutative i
...
, a × b b × a ,
but a × b b × a
...
i i k k o
...
a b
a b is the area of parallelogram whose adjacent sides are
a and b
...
2
If a , b and c forms a triangle, then area of the triangle
...
j
a2
b2
1
1
1
a b
b c =
c a
...
b × c and is denoted as a b c
[Class XII : Maths]
[102]
Geometrically, absolute value of scalar triple product a b c represents
volume of a parallelepiped whose coterminous edges are a , b and c
...
1
...
2
...
When is
4
...
What is the angle between a and b , If
a b 3 and a b 3 3
...
Write a unit vector which makes an angle of
7
...
If A is the point (4, 5) and vector AB has components 2 and 6 along
x-axis and y-axis respectively then write point B
...
9
...
11
...
What is the point of trisection of PQ nearer to P if positions of P and Q
are 3i 3 – 4k and 9i 8 10k respectively?
j
j
Write the vector in the direction of 2i 3 2 3 k , whose magnitude is
j
10 units
...
j
13
...
If
15
...
What is
17
...
a 2, b 2 3 and a b , what is the value of a b ?
j
a i 4k
is per pendicular to
a , if a b
...
What is the length of side BC ?
19
...
Find
21
...
23
...
x a 12
...
k
...
i ?
j
j
What is the angle between x and y if x
...
25
...
27
...
If a , b and c are unit vectors with a b c 0 , then what
is the value of a
...
c c
...
28
...
Points L, M, N divides the sides BC, CA, AB
1 : 4, 3 : 2, 3 : 7 respectively
...
j
The scalar product of vector i k with a unit vector along the sum
of the vectors 2i 4 – 5k and i 2 3k is equal to 1
...
value of
...
Show that a b + c makes equal angles with
31
...
3
j
j
If 3i and 2i 3k then express in the form of
1 2 , where 1 is parallel to and 2 is perpendicular
to
...
If a , b ,
that a
c are three vectors such that a b c 0 then prove
ve
b b c c a
...
34
...
36
...
If
i
j
j
i
Let a , b 3 – k and c 7 – k , find a vector d which
is perpendicular to a and b and c
...
j
j
If a i k , c – k are the given vectors then find a vector
b satisfying the equation a b c ,
a
...
Find a unit vector perpendicular to plane ABC, when position vectors of A,
B, C are 3i – 2k ,
j
37
...
Evaluate
39
...
41
...
43
...
a
sin
i
2 a
j
2 a
1
...
k respectively
...
if a is a unit vector
...
a
2
a
2
a b
2
b
...
^
a i j k , b i 2k and c xi x 2
j
j
lies in the plane of a and b , then find the value of x
...
If c
Prove that angle between any two diagonals of a cube is cos
1
1
...
a
6
Prove that the normal vector to the plane containing three points with
position vectors a , b and c lies in the direction of vector
b c c a a b
...
46
...
a , b , c are position vectors of the vertices A, B, C of a triangle ABC
1
then show that the area of ABC is
a b b c c a
...
I
f
i
Dot product of a vector with vectors 3k , 3 2k
i
j
j
...
Find the vectors
...
j
If a 5i 7k , b i k , find such that a b and
j
a b are orthogonal
...
Let
50
...
51
...
53
...
55
...
a and b be vectors such that
then find a b
...
Prove that a , b and c are mutually perpendicular to
,
each other and b 1 c a
...
Find volume of parallelepiped whose coterminous edges are given by
ˆ
ˆ
ˆ
ˆ
vectors a 2iˆ 3 j 4k , b iˆ 2 jˆ k , and c 3iˆ jˆ 2k
...
c iˆ j k are coplanar
...
For any three vectors a , b and
a b
b c
[107]
c , prove that
c a 2 a b c
[Class XII : Maths]
57
...
c a are coplanar
...
a
...
3
...
x and y are like parallel vectors
...
126 sq units
...
6
...
(6, 11)
8
...
12
...
,
1
3
...
4 6 4 3
...
7
13
...
14
...
–9
16
...
...
5
19
...
units
...
13
21
...
23
...
4
25
...
3
29
...
3
1
1 3
i i 3k
...
60°
34
...
5 2
2
i j k
...
38
...
x = – 2
47
...
73
50
...
3
52
...
1
1
3
i k
...
53
...
The coordinates of point R which divides line segment PQ where
P(x1, y1, z1) and Q(x2, y2, z2) in the ratio m : n internally are
mx 2 nx 1 my 2 ny 1 mz 2 nz 1
...
,
m n ,
m n
m n
Direction ratios of a line through (x1, y1, z1) and (x2, y2, z2) are x2 – x1,
y2 – y1, z2 – z1
...
Vector equation of a line through point a and parallel to vector
b is r a b
...
a
b
c
[Class XII : Maths]
[110]
(i)
Vector equation of line through two points having position vectors
a and b is r a b a
...
2
1
2
1
2
1
Angle ‘’ between lines r a 1 b 1 and r a 2 µ b 2 is given
b1 b 2
by cos
...
Two lines are perpendicular to each other if
b 1 b 2 0 or a1a2 + b1b2 + c1c2 = 0
...
c
...
(ii)
(iii)
Passing through a and normal to n is r a
...
r
...
Passing through three non collinear points is
r a b a c a 0
[111]
[Class XII : Maths]
x x1
or x 2 x 1
x3 x1
(iv)
(v)
y y1
y2 y1
y3 y1
x
y
z
1
...
Angle ‘’ between planes r n 1 d 1 and r n 2 d 2 is
n1 n 2
given by cos
...
2
2
2
2
2
b1 c1 a2 b2 c 2
Two planes are perpendicular to each other iff n 1
...
2
a1
Two planes are parallel iff
n 1 n 2 for some scaler
(i)
a1
b
c
1 1
...
n
(ii)
Distance of a point (x1, y1, z1) from plane ax + by + cz = d is
0 or
from plane
ax 1 by 1 cz 1 d
...
...
Equation of plane containing
these lines is r a 1 b 1 b 2 0
...
(i)
n d is
r
[Class XII : Maths]
2
[112]
(ii)
x x1
y y1
z z1
and
a1
b1
c1
y y2
z z2
are coplanar Iff
b2
c2
Two lines
x x2
a2
x2 x1
a1
a2
y2 y1
b1
b2
z2 z1
c1
0
c2
and equation of plane containing them is
x x1 y y1 z z1
a1
b1
c1
0
...
b n
b
–
90°
(ii)
The angle between line
x x1
y y1
z z1
and plane
a1
b1
c1
a2x + b2y + c2 z = d is given as
sin
(iii)
a 1a 2 b 1b 2 c 1c 2
2
a1
2
2
2
2
2
...
n d
b n 0 or a1a2 + b1b2 + c1c2 = 0
...
n =d
[113]
[Class XII : Maths]
1
...
What is the angle between the lines 2x = 3y = – z and 6x = – y = – 4z?
3
...
3
4
1
4
...
5
...
...
7
...
8
...
Find the angle between the planes 2x – 3y + 6z = 9 and xy – plane
...
Write equation of a line passing through (0, 1, 2) and equally inclined to
co-ordinate axes
...
What is the perpendicular distance of plane 2x – y + 3z = 10 from origin?
12
...
What is the distance between the planes 2x + 2y – z + 2 = 0 and
4x + 4y – 2z + 5 = 0
...
What is the equation of the plane which cuts off equal intercepts of unit
length on the coordinate axes
...
Are the planes x + y – 2z + 4 = 0 and 3x + 3y – 6z + 5 = 0 intersecting?
16
...
Write the vector equation of the plane which is at a distance of 8 units from
j
the origin and is normal to the vector 2i 2k
...
What is equation of the plane if the foot of perpendicular from origin to this
plane is (2, 3, 4)?
19
...
i
j
20
...
3
plane 2x + y – 2z + 4 = 0?
21
...
– i 5 – k 5 0
...
Write the line 2x = 3y = 4z in vector form
...
The line
x 4
2y 4
k z
lies exactly in the plane
1
2
2
2x – 4y + z = 7
...
25
...
Also show that (3, 9, 4) lies on that plane
...
Find the equation
r 5i 3 6k
j
of the planes r
27
...
If l1, m1, n1, and l2, m2, n2 are direction cosines of two mutually perpendicular
lines, show that the direction cosines of line perpendicular to both of them
are
m1n2 – n1m2, n1l2 – l1n2, l1m2 – m1l2
...
Find vector and Cartesian equation of a line passing through a point with
position vector 2i – k and which is parallel to the line joining the
j
j
points with position vectors –i 4 k and i 2 2k
...
Find the equation of the plane passing through the point (3, 4, 2) and
(7, 0, 6) and is perpendicular to the plane 2x – 5y = 15
...
Find equation of plane through line of intersection of planes
j
r 2i 6 12 0 and r 3i 4k 0 which is at a unit
j
distance from origin
...
Find the image of the point (3, –2, 1) in the plane 3x – y + 4z = 2
...
Find the equation of a line passing through (2, 0, 5) and which is parallel
to line 6x – 2 = 3y + 1 = 2z – 2
...
Find image (reflection) of the point (7, 4, – 3) in the line
x
1
y –1
2
z 2
...
Find equations of a plane passing through the points (2, –1, 0) and
(3, –4, 5) and parallel to the line 2x = 3y = 4z
...
Find distance of the point (– 1, – 5, – 10) from the point of intersection of
x 2
line
3
y 1
4
z 2
and the plane x – y + z = 5
...
Find equation of the plane passing through the points (2, 3, – 4) and
(1, –1, 3) and parallel to the x–axis
...
Find the distance of the point (1, –2, 3) from the plane x – y + z = 5,
measured parallel to the line
x
2
y
3
z
...
Find the equation of the plane passing through the intersection of two
plane 3x – 4y + 5z = 10, 2x + 2y – 3z = 4 and parallel to the line
x = 2y = 3z
...
Find the distance between the planes 2x + 3y – 4z + 5 = 0 and
r
...
i
j
40
...
[Class XII : Maths]
[116]
41
...
Find the point of
5
x 2
y 4
1
3
intersection
...
Find the shortest distance between the lines
i
j
r 2 3k 2 3 4k and
l
j
r 2i 4 5k 3 4 5k
...
Find the distance of the point (–2, 3, –4) from the line
x 2
3
2y 3
3z 4
measured parallel to the plane 4x + 12y – 3z + 1 = 0
...
Find the equation of plane passing through the point (–1, –1, 2) and
perpendicular to each of the plane
r 2i 3 3k 2 and r 5 4 k 6
...
x
y
z
x 2
y 1
z 1
...
j
i
j
47
...
contains the line
Check the coplanarity of lines
r –3 5k –3 5
i
j
i
j
k
r – 2 5k µ – 2 5
i
j
i
j
k
If they are coplanar, find equation of the plane containing the lines
...
Find shortest distance between the lines :
x 8
y 9
z 10
x 15
y 29
z 5
and
...
Find shortest distance between the lines :
k
r 1 2 3 2
i
j
r 1 2 1 – 2 1
...
51
...
A vari e pl
abl
ane i at a const
s
ant di ance 3p
st
from the origin and meets the
coordinate axes in A, B and C
...
A vector n of magnitude 8 units is inclined to x–axis at 45°, y axis at
60° and an acute angle with z-axis
...
m
...
Also find the length of the
i
j
perpendicular
...
A line makes angles , , , with the four diagonals of a cube
...
3
54
...
Also find the inclination of this plane with xy-plane
...
3
...
90°
x 2
y 3
z 5
...
r 2 3 k 3
i
j
i
j
k
5
...
2
7
...
0
2
1
8
...
10
...
cos–1 (6/7)
...
14
–2
13
...
x + y + z = 1
15
...
–2x + y – 3z = 8
17
...
2x + 3y + 4z = 29
19
...
0 (line is parallel to plane)
21
...
r o 6 4j 3
i
k
24
...
r –51i – 15j 50k 173
28
...
5x + 2y – 3z – 17 = 0
30
...
(0, –1, –3)
32
...
1
2
3
33
...
29x – 27y – 22z = 85
35
...
7y + 4z = 5
1
11
21
22
...
10
3 3
5x – 7y + 11z + 4 = 0
...
2
2
1
[Class XII : Maths]
37
...
2 29
38
...
40
...
1
3
1
2 , – 2 , – 2
42
...
17
units
...
2
44
...
2x – 7y + 4z + 14 = 0
47
...
14 units
...
r
49
...
1,
2, 3 ,
54
...
234
[120]
1
6
2 k 2
i
j
CHAPTER 12
LINEAR PROGRAMMING
Linear programming is the process used to obtain minimum or maximum
value of the linear objective function under known linear constraints
...
Constraints : The linear inequalities or inequations or restrictions on the
variables of a linear programming problem
...
To Find Feasible Region : Draw the graph of all the linear inequations
and shade common region determined by all the constraints
...
Optimal Feasible Solution : Feasible solution which optimizes the objective
function is called optimal feasible solution
...
Solve the following L
...
P
...
Determine graphically the minimum value of the objective function
z = – 50x + 20 y, subject to the constraints
2x – y – 5
3x + y 3
2x – 3y 12
x 0, y 0
3
...
150 and Rs
...
A can
stitch 6 shirts and 4 pants per day, while B can stitch 10 shirts and 4 pants
per day
...
4
...
A consists of 10% nitrogen and
6% phosphoric acid and B consists of 5% nitrogen and 10% phosphoric
acid
...
If A costs Rs
...
5 per kg, determine how much of each type of
fertiliser should be used so that nutrient requirements are met at minimum
cost
...
A man has Rs
...
Market price of one share of S1 is Rs 180 and S2
is Rs
...
He wishes to purchase a maximum of ten shares only
...
11 and of type S2 yields Rs
...
A company manufactures two types of lamps say A and B
...
Lamp A requires 2 hours of the
cutter’s time and 1 hours of the finisher’s time
...
The cutter has 100 hours and
finishers has 80 hours of time available each month
...
7
...
13
...
Assuming that he can sell all
that he produces, how many of each type of lamps should be manufactured
to obtain maximum profit?
7
...
He
has only Rs
...
A fan and
sewing machine cost Rs
...
240 respectively
...
22 and sewing machine at a profit of Rs
...
Assuming
that he can sell whatever he buys, how should he invest his money to
maximise his profit?
8
...
2 per
km on petrol
...
5 per km
...
100 to spend on petrol and wishes
to cover the maximum distance within one hour
...
P
...
and
then solve it graphically
...
A producer has 20 and 10 units of labour and capital respectively which
he can use to produce two kinds of goods X and Y
...
To produce one unit
of Y, 3 units of labour and one unit of capital is required
...
80 and Rs
...
A factory owner purchases two types of machines A and B for his factory
...
How many machines of each type should
he buy to maximise the daily output
...
A manufacturer makes two types of cups A and B
...
If the
profit on each cup A is 75 paise and on B is 50 paise, find how many cups
of each type should be manufactured to maximise the profit per day
...
A company produces two types of belts A and B
...
2 and Rs
...
50 per belt respectively
...
The company can produce almost 1000 belts
of type B per day
...
Almost 400
buckles for belts of type A and 700 for type B are available per day
...
An Aeroplane can carry a maximum of 200 passengers
...
400
is made on each first class ticket and a profit of Rs
...
The airline reserves at least 20 seats for first class
...
Determine, how many tickets of each type must
be sold to maximize profit for the airline
...
A diet for a sick person must contain atleast 4000 units of vitamins, 50
units of minerals and 1400 units of calories
...
5 and Rs
...
One unit of food
A contains 200 unit of vitamins, 1 unit of minerals and 40 units of calories
whereas one unit of food B contains 100 units of vitamins, 2 units of
minerals and 40 units of calories
...
What is balanced diet and what is the importance of
balanced diet in daily life?
1
...
Max z = 180 at the two corner points (0, 20) and (15, 15)
...
No minimum value
...
Minimum cost = Rs
...
4
...
of fertiliser A and 80 kg of fertilisers B; minimum cost Rs
...
Values promoted are keeping the productivity of the soil so that vegetables
and fruits are free from chemicals
...
Maximum Profit = Rs
...
6
...
[Class XII : Maths]
[124]
7
...
Profit = Rs
...
8
...
Max
...
9
...
760
...
Type A : 4;
Type B : 3
11
...
Maximum profit Rs
...
of belts of type A = 200 No
...
13
...
of first class tickets = 40, No
...
14
...
A diet containing all the nutrients in appropriate quantity is called balanced
diet
...
[125]
[Class XII : Maths]
CHAPTER 13
PROBABILITY
Conditional Probability : If A and B are two events associated with any
random experiment, then P(A/B) represents the probability of occurrence
of event-A knowing that event B has already occurred
...
P(A B) = P(A and B) = P(B) × P(A/B)
Similarly
P(A B C) = P(A) × P(B/A) × P(C/AB)
Multiplication Theorem on Probability : If the events A and B are
associated with any random experiment and the occurrence of one depends
on the other then
P(A B) = P(A) × P(B/A) where P(A) 0
When the occurrence of one does not depend on the other then these
events are said to be independent events
...
, En be a partition of sample
space and E1, E2
...
A be any event
associated with sample space S, which occurs with E1 or E2,
...
P(A/E1) + P(E2)
...
+ P(En)
...
[Class XII : Maths]
[126]
Bayes' theorem : Let S be the sample space and E1, E2
...
If A is any event which occurs with E1, or E2 or
...
P Ei P A Ei
P Ei A
n
P E P A E
i
i
i 1
Random variable : It is real valued function whose domain is the sample
space of random experiment
...
xn
P(X):
P(x1)
P(x2)
P(x3)
...
Bernoulli Trials : Trials of random experiment are called Bernoulli trials if:
(i)
Number of trials is finite
...
(iii)
Each trial has exactly two outcomes-either success or failure
...
[127]
[Class XII : Maths]
Binomial Distribution :
P(X = r) = nCr qn–r pr, where r = 0, 1, 2,
...
1
...
4, P(B) = 0
...
6
2
...
5, P(B) = 0
...
8
3
...
The probability that the enemy will
be killed by one bullet is 0
...
What is the probability that the enemy is still
alive?
4
...
20 cards are numbered 1 to 20
...
What is the
probability that the number on the card will be a multiple of 4?
6
...
Find the probability of getting at least one
head
...
The probability that a student is not a swimmer is
1
...
8
...
5 and P(A B) = 0
...
A random variable X has the following probability distribution
...
[Class XII : Maths]
[128]
3
k
4
5
15k 1
1
15
15
10
...
k
P X 2k
3k
if X 0
if X = 1 , find k
...
A problem in Mathematics is given to three students whose chances of
1 1
1
,
and
...
12
...
If the outcome is an even number, what is the probability
that it is a prime?
13
...
Find P (not A and not B)
...
In a class of 25 students with roll numbers 1 to 25, a student is picked up
at random to answer a question
...
15
...
They fire a volley
...
16
...
Find the probability of getting an even number
on the first die or a total of 8
...
A and B throw a die alternatively till one of them throws a ‘6’ and wins the
game
...
18
...
[129]
[Class XII : Maths]
19
...
4 and backward with probability
0
...
Find the probability that at the end of eleven steps he is one step away
from the starting point
...
Two cards are drawn from a pack of well shuffled 52 cards one by one with
replacement
...
Find the
probability distribution for the number of successes
...
In a game, a man wins a rupee for a six and looses a rupee for any other
number when a fair die is thrown
...
Find the expected value of the
amount he wins/looses
...
Suppose that 10% of men and 5% of women have grey hair
...
What is the probability that the selected
person is male assuming that there are 60% males and 40% females
...
A card from a pack of 52 cards is lost
...
What is the probability that they both are
diamonds?
24
...
Find the probability that there is at least one defective egg
...
Find the variance of the number obtained on a throw of an unbiased die
...
In a hurdle race, a player has to cross 8 hurdles
...
Bag A contains 4 red, 3 white and 2 black balls
...
One ball is transferred from bag A to bag B and
then a ball is drawn from bag B
...
Find
the probability that the transferred ball is black
...
If a fair coin is tossed 10 times, find the probability of getting
...
exactly six heads,
(ii) at least six heads,
at most six heads
...
From the past experience, it is known that the
probabilities that he will come by train, bus, scooter by other means of
[Class XII : Maths]
[130]
3 1 1
2
, ,
and
...
When he arrives, he is late
...
A man is known to speak truth 3 out of 4 times
...
Find the probability that it is actually a six
...
An insurance company insured 2000 scooter drivers, 4000 car drivers and
6000 truck drivers
...
01, 0
...
15
respectively
...
What is
the probability that he is a scooter driver? Which mode of transport would
you suggest to a student and why?
32
...
One card is drawn from the
remaining cards
...
33
...
One ball is drawn at random from one of the bags and is found
to be red
...
34
...
Let
be the probability that he knows the answer and
4
1
be the probability that he guesses
...
What is the probability
4
that the student knows the answer, given that he answered correctly?
35
...
If she gets 5 or 6, she tosses a coin three
times and notes the number of heads
...
If she obtained
exactly one head
...
In a bolt factory machines A, B and C manufacture 60%, 30% and 10%
of the total bolts respectively, 2%, 5% and 10% of the bolts produced by
[131]
[Class XII : Maths]
them respectively are defective
...
What is the probability that it has
been manufactured by machine A?
37
...
Two balls are drawn from one of the urns
...
38
...
Find the mean
and variance for the number of face cards obtained
...
Write the probability distribution for the number of heads obtained when
three coins are tossed together
...
40
...
The probabilities that the first and the second groups will win
are 0
...
4 respectively
...
7 and the corresponding probability is 0
...
Find the probability that the new product introduced
was by the second group
...
0
...
3
10
3
...
3)3
4
...
1
4
6
...
4
5
8
...
k
10
...
3
4
12
...
3
8
14
...
5
6
17
...
(i) p
19
...
3678 or
16
...
4)5
5
(0
...
X
0
1
2
P(X)
81/169
72/169
16/169
91
54
22
...
25
...
12 4
5 5
28
...
1
17
3
4
24
...
6
31
10
35
...
105
512
(ii)
193
512
(iii)
29
...
53
64
3
By speaking truth, integrity of character develops
...
8
31
...
[133]
[Class XII : Maths]
32
...
25
52
34
...
8
11
36
...
5
7
38
...
X
P(X)
40
...
This contains 29
questions
...
Time allowed : 3 hours
Maximum marks : 100
General Instructions
1
...
2
...
Section A comprises of 10 questions of one mark each, Section
B comprises of 12 questions of four marks each and Section C comprises
of 7 questions of six marks each
...
All questions in Section A are to be answered in one word, one sentence
or as per the exact requirement of the question
...
There is no overall choice
...
You
have to attempt only one of the alternatives in all such questions
...
Use of calculators is not permitted
...
1
...
5
Find the value of x, given that 2 * (x * 5) = 10
...
1
If sin sin1 cos1 x 1, then find the value of x
...
3
If 2
5
4 1
x 0
y
7
10
1
0
, find (x – y)
...
Solve the following matrix equation for x :
1
[x, 1] –2
0
0
0
5
...
1
Write the antiderivative of 3 x
x
3
7
...
dx
9 x2
0
8
...
If a and b are two unit vectors such that a + b is also a unit vector, then
find the angle between a and b
...
Write the vector equation of the plane, passing through the point (a, b, c)
and parallel to the plane r
...
11
...
9} and R be the relation in A×A defined by (a, b) R
(c, d) if a + d = b + c for (a, b), (c, d) in A × A prove that R is an
equivalence relation
...
Prove that cot
1
1 sin x 1 sin x
x
2 ; x 0, 4
1 sin x 1 sin x
OR
Prove that 2 tan
[Class XII : Maths]
1
1 5 2
1 1
1
5 sec 7 2 tan 8 4
[136]
13
...
y z x
2z
2x
Differentiate tan
cos
1
1
2y
3
z x y (x y z )
2x
1 x
x
2
with respect to
2
2x 1 x , when x 0
2
d y
1 dy
y dx
2
15
...
Fi
nd t
he i erval i w hi
nt
s n
ch t
he f
unct on
i
dx
2
y
0
x
f(x) = 3x4 – 4x3 – 12x2 + 5
is (a) Strictly Increasing
(b) Strictly Decreasing
OR
Find the equations of the tangent and normal to the curve
3
3
x a sin and y a cos at
4
6
17
...
(x
3) x
2
3x 18 dx
Find the particular solution of the differential equation
e
x
1 y
2
dx
y
dy 0 , given that y = 1 when x = 0
x
[137]
[Class XII : Maths]
19
...
dy
2
2xy 2
dx
x 1
Prove that, for any three vectors a, b , c
(x
2
1)
a b , b c , c a 2 a, b , c
OR
Vectors a , b and c are such that a b c 0 and a 3, b 5
and c 7 find the angle between a and b
...
Show that the lines
x 1
y 3
z 5
and
3
5
7
x 2
y 4
z 6
intersect
...
1
3
5
22
...
If a
family has two children, what is the conditional probability that both are
girls? Given that
(i)
the youngest is a girl
(ii)
at least one is a girl
...
23
...
The school P wants to award
Rs
...
y each and Rs
...
1000
...
1500 to award its 4, 1 and 3 students on the respective
values (By giving the same award money for the three values as before)
...
600, using
matrices, find the award money for each value
...
[Class XII : Maths]
[138]
24
...
3
/3
25
...
Find the area of the region in the first quadrant enclosed by the x-axis, the
line y = x and circle x2 + y2 = 32
27
...
OR
Find the distance of the point (–1, –5, –10) from the point intersection of
the line
r 2i j 2k (3i 4 j 2k ) and the plane r
...
28
...
He has only Rs
...
An electronic sewing machine costs him Rs
...
240
...
22 and a manually operated sewing machine at a profit
of Rs
...
Assuming that he can sell all the items that can buy, how should
he invest his money in order to maximise his profit? Make it as a LPP and
Solve it graphically
...
A card from a pack of 52 playing cards is lost
...
Find the probability of the lost card being a
spade
...
Find the probability distribution of
number of defective bulbs
...
[139]
[Class XII : Maths]
SECTION A
Q
...
Marks
1-10
...
x = 25, 2
...
2x
7
...
1
, 3
...
x = 2, 5
...
5,
r (ai
9
...
(i j k ) 0
or
r
...
(a, b ) A A
a + b = b + a (a, b) R (a, b) R is reflexive
e
1m
For (a, b), (c, d) A × A
If (a, b) R (c, d) i
...
a + d = b + c
c + b = d + a
then (c, d) R (a, b) R is symmetric
1m
For (a, b), (c, d), (e, f) A × A
If (a, b) R (c, d) & (c, d) R (e, f) i
...
a + d = b + c & c + f = d + e
Adding, a + d + c + f = b + c + d + e a + f = b + e
then (a, b) R (e, f) R is transitive
e
e
R is reflexive, symmetric and transitive
[Class XII : Maths]
[140]
1m
hence R is an equivalence relation
[(2, 5)] = {(1, 4), (2, 5), (3, 6), (4, 7), (5, 8), (6, 9)}
12
...
1
1
1
1
1 5 2
sec 7
8
1
1
5
8 tan 1 1
1
7
1
40
1
1
tan
3
1½+½m
1
2
...
(x + y + z) + (x + y + z)2} = (x + y + z)3
14
...
y x
x
log y = x log x,
[Class XII : Maths]
Taking log of both sides
[142]
½m
1 dy
log x 1
,
y dx
1 d y
1 dy
2
2
y dx
y dx
Diff
...
r
...
“x”
2
2
d y
dx
16
...
w
...
t
...
6
sin x cos x
=
½m
2
2
dx
sin x
...
cos x
1
3 dx
sin 2 x
...
cos 2 x
sec
2
½m
2
½m
x cos ec x 3 dx
= tan x – cot x – 3x + c
1½m
(Accept – 2 cot2x –3x + c also)
OR
x
3 x
2
3 x 18 dx
1
2x 3 x
2
1 2
...
e
2
3x 18
3/2
9
2x 3 x
8
2
x
y
x
dy xe dx
x
2
1 y dx
81
3
log x
2
2
3x 18
y
1 y
2
x
3x 18 c
1m
dy
2
Integrating both sides
xe
x
dx
xe
x
e
1
2
x
2y
dy
2
1 y
1 y
2
1+1m
c
x
For x = 0 y = 1, c = –1 solution is : e ( x 1)
19
...
x
2
2
y x
y x
1 2
1 log
2
2
1)
2x
2 dx
x 1
2
1
1
x
2
1m
2
e
log( x
2
1)
2
x
2
2
x
...
a b, b c,
= a b b
c a a b
...
b c a
...
c a b
...
b a b
...
b a a
...
b c b
...
b c 2 a, b, c
1m
OR
a b c 0
a b
a
2
2
b
a b c
½m
2
2
c c
2
2a
...
9 25 2 a b cos 49, being angle between a & b
cos
21
...
3
...
5v + 6)
1m
lines intersect if
3u – 1 = v + 2, 5u – 3 = 3v + 4, 7u – 5 = 5v + 6 for some u & v 1m
or 3u – v = 3
...
(2), 7u – 5v = 11
...
[Class XII : Maths]
[146]
1
Point of intersection of lines is : 2 ,
22
...
Here
3x + 2y + z = 1000
4x + y + 3z = 1500
x + y + z
3
4
1
2
1
1
1
3
1
= 600
1½m
x
1000
y 1500 or A
...
e
...
100 for discipline, Rs
...
300 for punctuality
One more value like sincerity, truthfulness etc
...
1m
For correct figure
½m
Let radius, height and slant height of cone be r, h & l
2
r
2
+ h
= l 2, l (constant)
2
2
h l h
3
dv
2
2
l 3h
dh
3
dv
0 h
dh
l
3
d v
dh
2
cos
l
3
I
/6
1
[Class XII : Maths]
r
1m
l
½m
3
l
3
3
cos
dx
1
3
½m
2 l
3
0
1m
, volume is maximum
h
l
/3
25
...
at h
2
2
l
1
2
r h
3
Volume of cone (V)
V
cot x
/3
/6
1
1
3
sinx
sin x
[148]
cos x
1m
dx
1m
sin
x
6
3
/3
sin
x
6
3
/6
/3
I
/6
cos x
cos x
sin x
1m
dx
1m
/3
Adding we get, 2 I
dx
cos
x
6
3
/3
dx x / 6
3
6
6
2m
/6
I
12
1m
-axis
26
...
units
27
...
e
...
2 3 1
...
2 2 5
1½m
0
1m
Point of intersection is 2i j 2k or (2, –1, 2)
Distance =
28
...
P
...
is : Maximise P = 22x + 18y
Subject to
360 x 240 y 5760
or
3x 2y 48
x y 20
x 0, y 0
For correct graph
2m
2m
vertices of feasible region are
[Class XII : Maths]
½m
[150]
A (0,20), B (8, 12), C (16, 0) & O (0, 0,)
P (A) = 360, P (B) = 392, P (C) = 352
½m
For maximum P, Electronic machines = 8,
Manual Machines = 12
29
...
4
P (E 1 / A )
1
...
4
13
1+1m
C3
51
C3
10
49
1m
OR
X = No
...
All questions are compulsory
...
The question paper consists of 26 questions divided into three sections A,
B and C
...
3
...
4
...
However, internal choice has been provided
...
5
...
Question number 1 to 6 carry one mark each
...
k 3
If A a
5
0 is skew symmetric matrix,
then write the value of ‘k’
...
Find the slope of normal to the curve,
y = x2 –3x + 7
at point
P (1, 5)
sin x
3
...
5
...
b
then
2
write the angle between a and b
...
and 2i 3k
?
x 1
y 1
1 z
and r i 2i j 3pk are
4
2
2
perpendicular, then write the value of ’p’
...
7
...
3
2
Two schools A and B decided to award prizes to their students for two
values honesty and punctuality
...
18000 for two values to 2 and 3 students respectively while School B
decided to award Rs
...
What is the amount given for honesty and for punctuality
...
Which value you prefer to be rewarded most and why?
9
...
10
...
3ax 1
x 1
2
5x 7 , 1 x 3
If f(x) =
is a continuous function, find the value of
x 3
2bx 8
a and b
...
If xy + xx =1, find
13
...
dx
f(x) = 2x3 – 9x2 + 12x + 25 is (a) increasing (b) decreasing
...
1
14
...
dx
OR
5
Find
1
x
x
x
6 x
2
tan
1
dx
x dx
15
...
Form the differential equation of the family of curves y = A e2x – B e–x
17
...
Find the perpendicular distance of the point (1, 0, 0) from the line
b 13,
a
...
x 1
y 1
z 10
2
3
8
[155]
[Class XII : Maths]
19
...
If the
coin is tossed twice, find the probability distribution of number of tails
...
20
...
Show
w
x 3
that f is a bijective function
...
21
...
Find the dimensions of
triangular poster so that the wastage of paper is minimum
...
Write steps to stop the wastage of paper
...
Find the area of the region bounded by
R =
x , y : 6x y
2
16 x
2
Find the area bounded by the parabolas y2 = –4x and x2 = 2y
23
...
x /y
x
1 y dy 0, given that y (o) = 2
Find the equation of the plane passing through the point (1, –2, 3) and
containing the line r 2i j k i 2 j 4k
...
Also
find the image of P in this line
...
A speaks truth 8 times out of 10 times
...
He reports that it
was 5
...
What can you say
about truthfulness of a person
...
If a person rides his motor cycle at 25 km/h, he has to spend Rs
...
If he rides at a faster speed of 40 km/h, the petrol cost
increases to Rs
...
He has Rs
...
Express this as L
...
P
...
What values are being promoted here?
Section – A
1
...
3
...
2
3
6
...
1
22 sq
...
A
8
...
3000
Amount given for punctuality = Rs
...
x = –31/7
11
...
dy
x (1 log x ) yx
y
dx
x log x
13
...
14
...
3
2
x
1x
1
1
tan x
log x
3
3 2
2
2
16
...
19
...
24 units
X
0
1
3/8
P(X) 1/16
2
9/16
Section - C
1
(x )
3x 3
x 2
20
...
32
4 3
3 3 sq
...
23
...
e
24
...
OR
Each side is r 3 cm
...
units
...
3
2
OR
(1, 6, 0), 2 6 unit & image is (–3, 8, –2)
25
...
26
...
(
50
40
km at 25 km/hr and
km at 40 km/hr) The values
3
3
promoted are the safety of life and saving petrol (energy)
...
All questions are compulsory
...
The question paper consists of 26 questions divided into three sections A,
B and C
...
3
...
4
...
However, internal choice has been provided
...
5
...
Question number 1 to 6 carry one mark each
...
Solve for x :
1
2
x
1
1 1
0
2 3
2
6
in
1
2
...
Find x if for unit vector a,
ˆ
3
0
5
x
5
4
7
a
...
ˆ
ˆ
[159]
[Class XII : Maths]
4
...
5
...
x
e
x
dx
2
ˆ
ˆ
ˆ
Find the value of iˆ
...
(iˆ k ) k
...
7
...
1 x 1
x 1
x 2 tan x 2 4 , then find the value of ‘x’
...
1
ab
b
2
1
cb
0
If A
tan /2
that
ac
bc
c
2
1 a
2
b
2
c
2
1
tan /2
w
and I is the identity matrix of order 2, show
0
cos
I A (I A )
sin
sin
cos
OR
3 1
2
If A
, show that A – 5A + 7 I = 0
...
[Class XII : Maths]
[160]
10
...
OR
Find the value of a and b such that the function defined by
5
f (x ) ax b
21
if
if x 2
2 x 10
if x 10
is a continuous function
...
x
If tan
x
12
...
3
is
(b) strictly decreasing
...
Find the equation of the plane passing through the point (1, 2, 1) and
perpendicular to the line joining the points (1, 4, 2) and (2, 3, 5)
...
15
...
If there are two posts for a job, find
the probability distribution for selection of his relatives
...
17
...
r
...
cos–1 x2
...
ˆ
The scalar product of the vector iˆ jˆ k with a unit vector along the
ˆ
ˆ
sum of vectors 2iˆ 4 jˆ 5k and iˆ 2 jˆ 3k is equal to one
...
19
...
x
2
3
6
3x
2x
1
...
x 2
Question number 20 to 26 carry 6 marks each
...
Of the students in a college, it is known that 60% reside in hostel and 40%
are day scholars
...
At the end of year, one student is chosen
at random from the college and he has A grade, what is the probability that
the student is a hostler?
What main values are developed in hostel life?
21
...
Check whether ‘*’ is commutative & associative?
4
Find the identity element in Q w
...
t
...
22
...
He has 30 workers and 17 units of
capital, which he uses to produce two types of goods A and B
...
If A and B are
sold at Rs
...
200 per unit respectively, how should he use his
resources to maximise the total revenue? Form the above as LPP and
solve it graphically
...
In a house, a window is in the form of a rectangle surmounted by a semi
circular opening
...
Find the
dimensions of the window such that it admits maximum light and air in the
house
...
...
Draw a rough sketch of the region enclosed between the circles x2+y2=4
and (x – 2)2 + y2 = 4
...
OR
Find the area of the region bounded by the parabola y = x2 and y x
25
...
Show that
log tan cot d log 2
...
x dx as limit of sums
...
1
7
2
...
x 4
4
...
6
...
x
1
9
...
a = 2, b = 1
12
...
1
sec(x a )
...
x – y + 3z = 2,
15
...
Such persons who are biased should be avoided in the committee
...
16
...
e
18
...
x = 1, 2, –15
e
Section - C
20
...
‘ * ’ is commutative and associative
...
22
...
1900
Yes
...
23
...
8
...
units
...
units
...
3
25
...
OR
27
2
[165]
[Class XII : Maths]
Title: Mathematics Study Material
Description: Best Mathematics Study Material Issued by Central Board Of Secondary Education (CBSE) For Class 12th...It Also Contains Points to remember And Value Based Questions.............I Hope This Material is useful for you.........Thank you...........& All the Best for your studies..
Description: Best Mathematics Study Material Issued by Central Board Of Secondary Education (CBSE) For Class 12th...It Also Contains Points to remember And Value Based Questions.............I Hope This Material is useful for you.........Thank you...........& All the Best for your studies..