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Miscellaneous Mathematics Questions
(A good collection of questions)
Topics:
1
...
Vector Algebra
3
...
Calculus
5
...
Conics
7
...
Integral Calculus
9
...
Discrete Mathematics

Dr
...
Khandelwal
(Ph
...
Mathematics, CSIR-JRF, GATE)

1

LINEAR ALGEBRA
1
...
Given an nxn matrix B and with determinant D, let B 0 be obtained by first
multiplying the ith row by k and then interchanging the ith and jth rows
...
Let a, b, c, be the real numbers
...
Rank of the matrix A=
 2
6


2 4 2 2
1 0 5 1 
 is
1 32 3 31 
6 6 12 0

(A) 4
(B) 3
(C) 2
(D) 1
5
...
The det(A), where A = 
 0 9
 0 5
0 0

(A) < n
0
0
2
0
4

0
0
0
6
3

0
0
0
7
4




 is



(A) 720
(B) 21
(C) 42
(D) 138
7
...
If (A + B)−1 exists, then
(A) A−1 and B −1 both exist
(B) A−1 + B −1 exist
(C) at least one of A−1 and B −1 exist
(D) nothing can be said
VECTOR ALGEBRA
9
...
Let →
a = ˆj − kˆ and →
b satisfying →
a × b +→
c =0
→
−
→
−
and a
...
A unit vector perpendicular to the plane passing through the points P(1,-1,0),
Q(2,1,-1) and R(-1,1,2) is
(A)

√1 ˆ
i − √12 kˆ
2

(B) − √12ˆi −
(C)

ˆ
√1 k
2

√1 ˆ
i + √12 kˆ
2

(D) − √12ˆi +

ˆ
√1 k
2

ˆ B = −2ˆi + 3kˆ
12
...
If the vectors →
a = ˆi − ˆj + 2k,
b = 2ˆi + 4ˆj + kˆ and →
mutually orthogonal then (λ, µ) is equal to
(A) (2,-3)
(B) (-2,3)
(C) (3,-2)
(D) (-3,2)
PROBABILITY DISTRIBUTIONS
14
...

x
1 2
3
4 5
6
7
2
2
2
p(x) k 2k 2k 2k k 7k + k 2k + k
The value of k
(A) 1
(B) -1
(C) -0
...
1
15
...
If a= P (X ≥
3) then P (|X| ≤ 3) equals
(A) a
(B) 1-a
(C) 2a
(D) 1-2a
16
...
An urn contains 2 white and 3 black balls
...
The ball drawn is replaced and 2 more ball of the same colour
are added to the urn
...
A discrete random variable X has pmf P (X = i) = 2 , i = 1, 2, 3,
...
The value of lim+ xsin(x) is
x→0

5

(A) -1
(B) 1
(C) 0
(D) 2


x2 + 3a,
bx + 2,
derivable at x ∈ R?

20
...
If u = 4xz4y and error in x, y, z be 0
...
004
(B) 0
...
001
(D) 0
...
If u = x1 2 x2 x3 and error in x1 , x2 , x3 be 0
...
02
(B) 0
...
04
(D) 0
...
The map f (x) = a0 cos |x| + a1 sin |x| + a2 |x|3 is differential at x = 0 if and only
if
(A) a1 = 0 and a2 = 0
(B) a0 = 0 and a1 = 0
(C) a1 = 0
(D) a0 , a1 and a2 any real number
24
...
If u = sin
, then the value of x ∂u
+ y ∂u
+ z ∂u
is
x8 +y 8 +z 8
∂x
∂y
∂z
(A) 7tan(u)
(B) −7sin(u)
(C) −7tan(u)
(D) 7sin(u)
2

2

2

∂u
+ y 2 ∂∂yu2 is
26
...
The value of a and b such that lim x(1+acosx)−bsinx
x3
x→0

(A) a = −5/2, b = −3/2
(B) a = −5/2, b = 3/2
(C) a = 5/2, b = −3/2
(D) a = 5/2, b = 3/2
28
...
The maximum area of a right angled triangle with hypotenuse h is
(A) h2 /6
(B) h2 /8
(C) h2 /2
(D) h2 /4

7

30
...
The maximum and minimum values of 2(x2 − y 2 ) − x4 + y 4 respectively are
(A) 1, -1
(B) -1, -2
(C) 2, -1
(D) 2, -2
COMPLEX ANALYSIS


 2 +3 
32
...
The sum of the series 1 + cos α + cos 2α +
...
One of the possible values of −8 − 8 3i
, where i = −1 is
√
(A) − 3 − i
√
(B) 1 − i 3
√
(C) − 3 + i
√
(D) −1 + i 3
35
...
, ω n−1 are the nth roots of unity, then 1 + 2ω + 3ω 2 +
...
The set of complex number z satisfying the equation: (3 + 7i)z + (10 − 2i)¯
z+
100 = 0 is
(A) a straight line
(B) a pair of intersecting straight line
(C) a point
(D) a pair of distinct parallel straight lines
CONIC SECTION
a
37
...
Area of the triangle formed by the lines joining the vertex of the parabola
x2 = 12y to the ends of its latus rectum is
(A) 9 sq
...
units
(C) 27 sq
...
units
39
...

A point P(x,y) is taken on the rod in such a way that AP=6 cm
...
The equation of circle passing through (1,-1) and (-2,2) and whose centre lies
on the line x + y = 3 is
(A) (x − 2)2 + (y − 1)2 = 4
(B) (x − 1)2 + (y − 2)2 = 9
(C) (x + 1)2 + (y − 1)2 = 4
(D) (x + 2)2 + (y + 1)2 = 9
41
...
The point (7
...
A plane passes through a fixed point (a, b, c)
...
The equation of the line passing through the point (−2, 3, −2) and perpendicz
x+2
y−1
z+1
y
=
=
is
ular to the two lines x = = and
2
3
−3
2
5
x+2
2
x+2
(B)
2
x+2
(C)
−2
x+2
(D)
2
(A)

y−3
7
y−3
=
3
y−3
=
7
y−3
=
−7
=

z+2
4
z+2
=
2
z+2
=
−4
z+2
=
*
4
=

44
...
The coordinates of the point where the line through the points A(3,4,1) and
B(5,1,6) crosses the XY -plane are


23
(A) 13
,
,
0
5 5


, 13 , 0
(B) 23
5 5


5 5
(C) 13
, 23 , 0


5 5
(D) 23
, 13 , 0
46
...
Area bounded by the curves y = cos(x) and y = sin(x) between the ordinates
is
x = 0 and x = 3π
2
√
(A) 4 2 + 2
√
(B) 4 2 − 1
√
(C) 4 2 + 1
√
(D) 4 2 − 2
48
...
The area of the region {(x, y) : 0 ≤ y ≤ x2 + 1, 0 ≤ y ≤ x + 1, 0 ≤ x ≤ 2} is
(A)

23
6

11

(B)
(C)
(D)

21
6
23
8
21
4

50
...
The solution of

dy
dx

=

1−x
y

represents

(A) a family of circle with centre at (0,0)
(B) a family of circle with centre at (1,0)
(C) pairs of straight lines
(D) a family of straight lines with slope -1
52
...
Every solution of dx
2 + a dx + by = 0, where, a and b are constant, approaches
to zero as x → ∞, provided

(A) a > 0, b > 0
(B) a > 0, b < 0
(C) a < 0, b < 0
(D) a < 0, b > 0
d3 y

dy
54
...
The differential equation x dx
= y, y(0) = 0, x ≥ 0 has

(A) No solution
(B) an unique solution
(C) infinitely many solution
(D) None of these
DISCRETE MATHEMATICS
56
...

(B) Take two aspirins
...
Which of the following statements is the negation of the statement ”2 is even
and -3 is negative”
(A) 2 is even and -3 is not negative
(B) 2 is odd and -3 is not negative
(C) 2 is even or -3 is not negative
(D) 2 is odd or -3 is not negative
58
...
Which is the false option ?
(A) (∼ p ∧ (p V q)) ⇒ q is a tautology
...

(C) “1+2” is a statement
...

13

60
...
No
...
No

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