TYPICAL QUESTIONS &
ANSWERS
PART
- I
OBJECTIVE TYPE QUESTIONS
Each
Question carries 2 marks.
Choose
correct or the best alternative in the following:
Q.1 The value of limit 
is
(A) 0 (B)
1
(C)
2 (D) does not exist
Ans: D
Q.2 If 
, then 
equals
(A)
0 (B) u
(C) 2u (D) 3u
Ans:
A
Q.3 Let 
. Then the value of 
is
(A)
(B) 0
(C) 
(D) 
Ans: B
Q.4 The value of 
is
(A)
1 (B)
(C) 
(D) 3
Ans:
A
Q.5 The solution of 
is
(A)
(B) 
(C) 
(D) 
Ans:
C
Q.6 The solution of 
is
(A)
sin x (B) cos x
(C) x sin x (D) x cos x
Ans:
D
Q.7 . Let
and 
be elements of 
. The set of vectors 
is
(A)
linearly independent (B) linearly dependent
(C) null (D) none of these
Ans: A
Q.8 The eigen values of the matrix 
are
(A)
and 1 (B) 0,
1 and 2
(C) –1, –2 and 4 (D) 1, 1
and –1
Ans: A
Q.9 Let 
, 
, 
be the Legendre
polynomials of order 0, 1, and 2, respectively.
Which of the following statement is correct?
(A) 
(B) 
(C) 
(D) 
Ans: B
Q.10 Let 
be the Bessel function
of order n. Then 
is equal to
(A) 
(B) 
(C) 
(D)
Ans: D
Q.11 The value of limit 
(A) 0 (B)
(C) 
(D) does not exist
Ans: D
Q.12 Let a function f(x, y) be continuous and possess first and second
order partial derivatives at a point (a, b).
If 
is a critical point and
, 
, 
then the point P is a
point of relative maximum if
(A) 
(B) 
(C) 
(D) 
Ans: B
Q.13 The triple integral 
gives
(A) volume of region T (B) surface area of region T
(C) area of region T (D) density of region T
Ans: A
Q.14 If 
then matrix A is
called
(A) Idempotent Matrix (B)
Null Matrix
(C) Transpose Matrix (D)
Identity Matrix
Ans: A
Q.15 Let 
be an eigenvalue of
matrix A then 
, the transpose of A, has an eigenvalue as
(A) 
(B) 
(C) (D) 
Ans: C
Q.16 The system of equations is said to be inconsistent, if it has
(A) unique solution (B) infinitely many solutions
(C) no solution (D)
identity solution
Ans: C
Q.17 The differential equation 
is an exact
differential equation if
(A) 
(B) 
(C) 
(D) 
Ans: B
Q.18 The integrating factor of the differential
equation 
is
(A) 
(B)
(C) xy (D)
Ans: D
Q.19 The functions 
defined on an interval
I, are always
(A) linearly dependent (B) homogeneous
(C) identically zero or
one (D) linearly
independent
Ans: D
Q.20 The value of 
, the second derivative of Bessel function in terms of 
and 
is
(A) 
(B) 
(C) 
(D)
Ans: C
Q.21 The value of limit 
is
(A) 0 (B) 1
(C) -1 (D) does not exist
Ans:
A
Q.22 If 
, the total
differential of the function at the point (1, 2) is
(A) e (dx + dy) (B) e2(dx + dy)
(C) e4(4dx + dy) (D) 4e4(dx + dy)
Ans:
D
Q.23 Let 
, x > 0, y > 0
then 
equals
(A) 0 (B) 2u
(C) u (D) 3u
Ans:
B
Q.24 The value of the integral 
over the domain E
bounded by planes x
= 0, y = 0, z = 0, x + y + z = 1 is
(A) 
(B) 
(C) 
(D) 
Ans: C
Q.25 The value of α so that 
is an integrating
factor of the differential equation 
is
(A) -1 (B) 1
(C)
(D) 
Ans: C
Q.26 The complementary function for the solution of the differential
equation 
is obtained as
(A) Ax + Bx -3/2 (B) Ax + Bx 3/2
(C) Ax2 + Bx (D) Ax -3/2 + Bx 3/2
Ans: A
Q.27 Let 
be elements of R3.
The set of vectors 
is
(A) linearly independent (B) linearly dependent
(C) null (D) none of these
Ans: A
Q.28 The value of µ for which the rank of the matrix 
is equal to 3 is
(A) 0 (B) 1
(C) 4 (D) -1
Ans: B
Q.29 Using the recurrence relation, for Legendre’s polynomial (n
+ 1) 
, the value of P2 (1.5) equals to
(A) 1.5 (B) 2.8
(C) 2.875 (D) 2.5
Ans: C
Q.30 The value of Bessel function J2(x) in terms of J1(x) and J0(x) is
(A) 2J1(x) – x J0(x) (B) 
(C)
(D) 
Ans: D
Q.31 The value of the
integral 
where C is the contour
is
(A) 
. (B)
.
(C) 0. (D) 
.
Ans: C
Because z = 1 is a pole for given function f and it lies outside the circle
|z|
= ½ .
Therefore, by Cauchy’s Theorem 
Q.32 If X has a Poisson
distribution such that 
then the variance of
the distribution is
(A) 1. (B) -1.
(C) 2. (D) 0.
Ans: A
Because P (x = 2) = 9 P (x = 4) + 90 P (x = 6)
=> 
=> 
Because m¹0, Therefore, 3m2 + m4 – 4 = 0
=> m = 1
Q.33 The vector field function 
is called solenoidal
if
(A)
curl =0. (B) div =0.
(C)
grad =0. (D) grad div =0.
Ans: B
A
vector field 
is solenoidal if div = 0
Q.34 The number of distinct real roots
of 
in the interval 
is
(A) 0. (B) 2.
(C)
3. (D) 1.
Ans: D
=> (cos x – sin x)2 (sin x + 2 cos x) = 0
Its
only root which lies in 
.
Q.35 The solution of : 
is
(A) 
. (B)
.
(C) 
. (D)
.
Ans: A
2y sin x + cos 2x = a
I.F. 
Therefore, the solution is given as
=> 2y sin x + cos 2x = a
Q.36 If 
then 
is equal to
(A) 
. (B) 
.
(C) 
. (D) 
.
Ans: A
where u and v are homogeneous functions
of order 6 and 0 respectively. Using Euler’s theorem 
= 6 u + 0 v
= 6 u.
Q.37 The value of Legendre’s Polynomial, 
is
(A) 1. (B) -1.
(C) 
. (D)
0.
Ans:
D
By Rodrigue’s formula,
Pn(0) = 0 if n is odd.
Q.38 The
value of integral 
over the region bounded by the line y = x and the curve 
is
(A) 
. (B) 
.
(C)
. (D) 
.
Ans: C
=
Q.39 The value of the integral 
where C is the
semi-circular arc above the real axis is
(A) . (B)
.
(C) 
. (D) 
.
Ans:
A
Let z = eiq then 
= i p
Q.40 Residue at z = 0 of
the function 
is
(A) 
. (B) 
.
(C) 
. (D) 
.
Ans:
B
Let
Residue
= coefficient of 
Q.41 In solving any problem, odds against A are 4 to 3 and odds in favour of B in solving the same problem are 7 to 5. The probability that the problem will be solved is
(A)
. (B) 
.
(C)
. (D) 
.
Ans:
B
P(A) = 
, P(B) = 
. Probability problem will be solved i.e. P(AÈB)
P(AÈB) = P(A) + P(B) – P(AB)
Because A & B are independent, So P(AB) = P(A) P(B)
P(AÈB) = 
Q.42 The
value of the integral 
over the area in the
first quadrant by the curve 
is
(A) 
. (B) 
.
(C)
. (D) 
.
Ans: D
over x2 –
2ax + y2 = 0
=
Q.43 The
surface 
will be orthogonal to
the surface 
at the point 
for values of a and b
given by
(A)
a = 0.25, b = 1. (B) a = 1, b = 2.5.
(C) a = 1.5, b = 2. (D) 
.
Ans:
A
a= 0.25, b = 1
Let F = ax2 – byz – (a + 2) = 0
G = 4x2y + z3 – 4 = 0
These
surfaces will be orthogonal if 
Also since (1, -1, 2) lies on F
\ a + 2b – a – 2 = 0 
b = 1 , thus a = 
Q.44 If
and 
and if z = u + v then 
equals
(A)
4 v. (B) 4 u.
(C) 2 u. (D) 4 u + v.
Ans: C
z = u + v
i.e.
u is homogeneous function of degree 2 and v is homogeneous function of degree
0. By Euler’s Theorem,
Q.45 The
series 
equals
(A) 
. (B) 
.
(C) 
. (D)
.
Ans: C
Q.46 The
value of integral 
, where 
is a Legendre
polynomial of degree 3, equals
(A) 
. (B) 0.
(C)
. (D) 
.
Ans: D
As 
Q.47 For what values of x,
the matrix 
is singular?
(A) 0, 3 (B)
3, 1
(C) 1, 0 (D) 1, 4
Ans: A
The matrix is singular if its determinant is zero. Solving determinant, we get equations
x(x-3)2=0.
Q.48 If 
then 
(A) 3 ab (B) 2 abz
(C) abz (D) 3 abz
Ans: B
Because
Q.49 The value of the integral 
is
(A)
. (B) 2.
(C)
-2. (D) 0.
Ans: D
Since it is an odd function.
Q.50 If
and 
then div 
(A) 5 (B) 5u
(C) 
(D) 0
Ans: B
Q.51 The solution of the differential
equation 
is given as
(A)
(B)
(C) 
(D)
Ans: A
Dividing by z, we get
,
Let 1/(log z)=u, then above differential equation becomes
Q.52 The value of the integral 
where C is the circle 
is given as
(A) (B)
(C) 0 (D) 
Ans: C
The given function has a pole at z=1, which lies outside the circle C. So by Cauchy’s theorem integral is zero.
Q.53 The value of the Legendre’s
polynomial 
if
(A) 
(B) 
(C) 
(D)
Ans: D
By orthogonal property of Legendre’s polynomial.
Q.54 Two
persons A and B toss an unbiased coin alternately on the understanding that the
first who gets the head wins. If A
starts the game, then his chances of winning is
(A) 
(B) 
(C) 
(D) 
Ans: C
Probability of getting head=1/2= probability of getting tail.
If A starts the game, then in first chance either A wins the game, in second case A fails, B fails and A won the match and so on, we get an infinite series. Let HA, HB, TA, TB, denotes the getting of head and tails by A and B respectively.
P(wining of A)=P(HA)+P(TATBHA)+ P(TATBTATBHA)+…….
=
This is an infinite G.P. series with common ratio 1/4. Thus
P(winning of A) = 
.
Q.55 The value of limit 
(A) equals 0. (B)
equals 
.
(C) equals 1. (D) does not exist.
Ans: D
Let y= mx2 be equation of curve. As x→0, y also tends to zero.
=
, which depends on m.
Thus it does not exist.
Q.56 If 
then 
equals
(A) 
. (B) 
.
(C) 
. (D) 
.
Ans: A
Eliminating 
we get
Q.57 The function 
has
(A) a minimum at (0, 0).
(B)
neither minimum nor maximum at (0, 0).
(C)
a minimum at (1, 1).
(D)
a maximum at (1, 1).
Ans: B
f (x, y) = y2 – x3
fx = - 3x2 = 0 , fy = 2y = 0
gives (0,0) is a critical point.
∆f (x, y) = f(∆x,∆y)= (∆y)2 –(∆ x)3
> 0 , if (∆y)2 >(∆ x)3
< 0 , if (∆y)2 < (∆ x)3
This means in the neighborhood of (0,0) f changes sign. Thus (0,0) is neither a point of maximum nor minimum.
Q.58 The
family of orthogonal trajectories to the family 
, where k is an arbitrary constant, is
(A) 
. (B) 
.
(C) 
. (D) 
.
Ans: A
y = (x – k)2 Diff. w.r.t. x
y1 = 2(x
– k) => y1 = 2
For orthogonal trajectories y1 is replaced by -1/y1.
Therefore, -1/y1 = 2
=> 2dy + dx = 0
Integrating, we get y3/2 = ¾ (c-x)
Q.59 Let
be two linearly
independent solutions of the differential equation 
. Then 
, where 
are constants is a solution of this differential equation
for
(A)
. (B)
.
(C) no value of 
. (D)
all real .
Ans: B
yy” – (y’)2 = 0
Because, y1, y2 are solutions
Therefore, y1y1” – (y1’)2 = 0
y2y2” – (y2’)2 = 0
Now (c1y1 + c2y2) (c1y1 + c2y2)” – ((c1y1 + c2y2)’)2
= (c1y1 + c2y2) (c1y1” + c2y2”) – (c1y1’2 + c2y2’2) - 2 c1y1’c2y2’
= c12(y1y1” – (y1’)2) + c22 (y2y2” – (y2’)2) + c1 c2 (y1 y2”+y2y1” - 2y1’y2’)
= 0, if c1c2 = 0.
Q.60 If A, B are two square matrices of
order n such that AB=0, then rank of
(A) at least one of A, B is less than n.
(B)
both A and B is less than n.
(C)
none of A, B is less than n.
(D)
at least one of A, B is zero.
Ans: B
Since A, B are square matrix of order n such that AB = 0, then rank of both A and B is less than n.
Q.61 A 
real matrix has an
eigen value i, then its other two eigen values can be
(A) 0, 1. (B) -1, i.
(C) 2i, -2i. (D)
0, -i.
Ans: D
Because i is
one eigen value so another eigen value must be – i.
Q.62 The
integral
, n>1, where 
is the Legendre’s
polynomial of degree n, equals
(A) 1. (B)
.
(C) 0. (D) 2.
Ans: C
Let I = 
Let cosq = t. –sinqdq = dt
= 0 
Q.63 The value of limit 
is
(A) 0 (B) 1
(C) limit does not exist (D) -1
Ans.: A
Language of the question is not up to the mark in the sense that its statement does not go with all the alternatives consequently, change is in order.
The suggested change is either satisfies the statement given in the alternative (C) or
assumes the value given in one of three remaining alternatives A, B and D.
Q.64 If 
then the value
of 
is equal to
(A) 0 (B) 
(C) 
(D) 
Ans.: B
Since 
taking log on both sides we get log(u)=y log(x)
Q.65 If 
, then the value of 
is
(A) z (B) 2z
(C) tan(z) (D) sin(z)
Ans.: C
If
u(x,y) = 
, is a homogeneous function of degree n, then from
Euler’s theorem
= nu.
Here 
= 
is a homogeneous function of degree 1.
Therefore 
u = sin z
From u = sin z; 
, 
Q.66 The value of integral 
is equal to
(A) 
(B) 
(C) 
(D)
Ans.: B
Q.67 The differential equation of a family of circles having the radius r and the centre on the x-axis is given by
(A) 
(B) 
(C) 
(D) 
Ans.: A
Let (h,0) be
centre on x-axis. Thus eq. of circle is 
Differentiating, w.r.
to x, we get 
Eliminating h
between 
and 
We get 
.
Q.68 The solution of the
differential equation 
satisfying the initial conditions y(0)=1, y(π/2) =
2 is
(A) y = 2cos(x) + Sin(x) (B) y = cos(x) + 2 sin(x)
(C) y = cos(x) + Sin(x) (D) y = 2cos(x) + 2 sin(x)
Ans.: B
On
solving the differential equation , we get y = Acosx + Bsin x, Since
y(0)=1, 
Thus, y = cos(x)
+ 2 sin x
Q.69 If the matrix 
then
(A) C=Acos(θ) – Bsin(θ) (B) C=Asin(θ) + Bcos(θ)
(C) C=Asin(θ) – Bcos(θ) (D) C=Acos(θ) + Bsin(θ)
Ans.: D
Q.70 The three vectors (1,1,-1,1), (1,-1,2,-1) and (3,1,0,1) are
(A) linearly independent (B) linearly dependent
(C) null vectors (D) none of these.
Ans.: B
Let a,b,c be three constants such that a(1,1,-1,1)+b (1,-1,2,-1) +c(3,1,0,1)=(0,0,0,0).
This yields a + b + 3c = 0, a – b + c = 0, -a + 2b = 0, a – b + c = 0.
On solving, we get a = 2b = -2c → b = - c. Since a, b, c are non-zero, therefore three vectors are linearly dependent.
Q.71 The value of 
is equal to
(A) 1 (B) 0
(C) 
(D) 
Ans.: B
Q.72 The value of the integral 
is
(A) 
(B) 
(C) 
(D) 
Ans.: C
. Here v=1.
Q.73 The value of limit 
is
(A) limit does not exist (B) 0
(C) 1 (D) -1
Ans.: A
Consider the path y = mx2 As (x,y)→(0,0), we get x →0. Therefore
which depends on m.
Thus limit does not exist.
Q.74 If then the value of 
is equal to
(A) 0 (B)
(C) (D)
Ans.: B
Since
taking log on both sides we get log(u)=y log(x)
Q.75 If 
, then the value of 
is
(A) u (B) 2u
(C) 3u (D) 0
Ans.: D
Let 
Here 
and 
are homogenous
functions of degree zero.
Consequently 
Or 
; similarly 
.
; 
= 0 + 0 = 0.
Q.76 The value of integral 
is equal to
(A) 22 (B) 26
(C) 5 (D) 25
Ans.: B 
Q.77 The solution of the differential equation 
is given by
(A) 
(B) 
(C) 
(D) 
Ans.: A
Let x + y = t, Differentiating w r to x we get
Or 
,
; integrating we get
→ 
Or 
or 
.
Q.78 The solution of the
differential equation 
is
(A) 
(B) 
(C) 
(D) 
Ans.: A
The solution of
differential equation is given as C.F.
P.I. =
In writing the C.F. we have used the roots of the auxiliary equation
i.e. m = 1, 2. For writing
the P.I we have used 
; 
Q.79 If 3x+2y+z= 0, x+4y+z=0, 2x+y+4z=0, be a system of equations then
(A) system is inconsistent
(B) it has only trivial solution
(C) it can be reduced to a single equation thus solution does not exist
(D) Determinant of the coefficient matrix is zero.
Ans. B
, then system has only trivial solution.
Q.80 If λ is an eigen value of a non-singular matrix A then the eigen value of A-1 is
(A) 1/ λ (B) λ
(C) -λ (D) -1/ λ
Ans. A By definition of A-1.
Q.81 The product of eigen value
of the matrix 
is
(A) 3 (B) 8
(C) 1 (D) -1
Ans.: B
Eigen values are 1,2,4.
Thus product = 8.
Q.82 The value of the integral 
is
(A) 
(B) 
(C) 
(D) 
Ans.: C 
. Here v=2.
Q.83 If 
then
(A) 
(B) 
(C) 
(D) 
Ans.: B
Since is a homogeneous
function of degree 0. Thus by Euler’s theorem .
Q.84 If 
, then the value of 
is
(A) 1 (B) r
(C) 1/r (D) 0
Ans.: B
Q.85 The value of integral 
is equal to
(A) -4 (B) 3
(C) 4 (D) -3
Ans.: C
Q.86 The solution of differential equation 
under condition y(1)=1
is given by
(A) 
(B) 
(C) 
(D) 
Ans.: B
The given differential is a particular case of linear differential equation of first order
. Here 
. Multiplying throughout by x, it can be written as
; Integrating w.r. to x we get
; Given y(1) = 1; 
or 
which is alternative
B.
Q.87 The particular integral of the differential equation 
is
(A) 
(B) 
(C) 
(D) 
Ans.: A
P.I.
is a case of failure
of 
; 
In
such cases 
.
Q.88 The product of the eigen
values of 
is equal to
(A) 6 (B) -8
(C) 8 (D) -6
Ans.: C
.
The eigenvalues are 1,2,4. Thus product of eigen values = 8.
Q.89 If 
then matrix A is equal to
(A) 
(B) 
(C) 
(D) 
Ans.: D
Q.90 The value of 
(m being an integer
< n) is equal to
(A) 1 (B) -1
(C) 2 (D) 0
Ans.: D
Using
Rodrigue formula 
can be expressed as
Q.91 The value of the 
is
(A) 
(B) 
(C) 
(D) 
Ans.: A
.
.
.
.