INFOMATHS
HCU OLD QUESTIONS
be a 3  3 matrix. Let x and y be the values such that
matrix A is singular. What is x + y? HCU-2011
(a) 0
(b) 3
(c) 1/2
(d) 2
9. It is given that square matrix A is orthogonal and
also that det A is not equal to 1. Then, HCU - 2009
(a) |A| is zero
(b) |A| > 1
(c) |A| cannot be determined
(d) None of the
above
 i  i
 1  1
8
10. If A  
and B  

 then A equals
 i i 
 1 1 
to
HCU- 2009
(a) 64 B (b) 128 B (c) -128 B (d) -64 B
11. If A is a 3  3 matrix and A’ A = I and |A| = 1 then
the value of |(A – 1)| =
HCU- 2009
(a) 1
(b) – 1
(c) 0
(d) N.O.T
12. If a, b, c are the roots of x3 + px2 + q = 0, then
a b c
MATRICES
1. Let A be an n  n non-singular matrix over ℂ where
n  3 is an odd integer. Let a  ℝ. Then the equation
det(aA) - a det(A)
holds for
HCU-2012
(a) All values of a
(b) No value of a
(c) Only two distinct values of a
(d) Only three distinct values of a
2. How many matrices of the form
2 2

 x 3 3


2 1 y
3 3



z s t




are orthogonal, where x, y, z, s and t are real
numbers.
HCU-2012
(a) 1
(b) 2
(c) 0
(d) infinity
3. Let A be an n  n-skew symmetric matrix with a11,
a22, ….. ann as diagonal entries. Then which of the
following is correct?
HCU-2012
(a) a11a22 … ann = a11 + a22 + …. + ann
(b) a11a22 … ann = (a11 + a22 + …. + ann)2
(c) a11a22 + … + ann = (a11 + a22 + …. + ann)3
(d) all of the above
4. Consider the system of equations
8x + 7y + z = 11
x + 6y + 7z = 27
13x – 4y – 19z = - 20
How many solutions does this system have?
HCU-2012
(a) Single (b) Finite (c) Zero (d) Infinite
5. If A is a 2  2 real matrix such that A – 3I and A – 4I
are not invertible, then A2 is
HCU-2011
(a) 12A – 7I
(b) 7A – 12I
(c) 7A + 10I
(d) 12I
6. If A and B are 3  3 matrices with |A| = 4 and |B| =
3, which of the following is generally false?
HCU-2011
(a) 3|B| = 9
(b) 2|A| = 32
(c) |AB| = 12
(d) |A + B| = 7
7. The n  n matrix P is idempotent if P2 = P and
orthogonal if P'P = I. Which of the following is
false?
HCU-2011
(a) If P and Q are idempotent n  n matrices and
PQ= QP = 0, then P + Q is idempotent
(b) If P is idempotent then – P is idempotent
(c) If P and Q are orthogonal n  n matrices then PQ
is orthogonal
 1  3


2 
(d) P   2
 3
1 


 2
2 
8. Let
4 1 0 
A   2 1 2 
 x y 1
b c a
HCU- 2009
c b a
(a) p
(b) p2
(c) p3
(d) q
13. The following system of equations
6x + 5y + 4z = 0
3x + 2y + 2z = 0
12x + 9y + 8z = 0 has
HCU- 2009
(a) no solutions
(b) a unique solution
(c) more than one but finite number of solution
(d) infinite solutions
a b 
3
14. Let A  
 be a 2  2 matrix such that A = 0.
c d 
The sum of all the elements of A2 is
HCU- 2009
(a) 0
(b) a + b + c + d
(c) a2 + b2 + c2 + d2
(d) a3 + b3 + c3 + d3
15. The eigen vectors of a real symmetric matrix
corresponding to different eigen values are
HCU- 2009
(a) Singular
(b) Orthogonal
(c) Non-singular
(d) None of the
above
DEFINITE INTEGRAL
16. For a >1, the value of
(a) 2
(b)
D
17. Evaluate
1
 1 x
2
dx


2a

0
dx
is
a  2a cos x  1
HCU-2011
2
(c)

a2 1
(d) 0
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0


(c)
(d) 
2
2
18. Consider the region bounded by the graphs y = ex, y
= 0, x =1 and x = t, where t < 1.The area of this
region is atmost
HCU-2011
(a) unbounded
(b) e
(c) 0
(d) 1n
(a) - 
1
(b)
INFOMATHS/MCA/MATHS/
INFOMATHS
27. Suppose A = i – j – k, B = i – j + k and C = - i + j +
k, where i, j, k are unit vectors. Pick the odd one out
among the following:
HCU-2012
(a) A  (B  C)
(b) (A  B)  C
(c) A  C
(d) A  B
28. Consider the following equalities formed for any
three vectors A, B and C.
HCU-2012
I. (A  B)C = A(B  C)
II. (A  B)  C = A  (B  C)
III. A  (B  C) = (A  B)  C
IV. A  (B + C) = (A  B) + (A  C)
(a) Only I is true
(b) I, III and IV are true
COMPLEX NUMBERS
(c)
Only
I
and
IV
are
true
(d) All are true
21. Let A be the set of all complex numbers that lie on
29.
A
line
makes
angles
,
,
 and  with the four
the circle whose radius is 2 and centre lies at the
diagonals
of
a
cube.
Then
the
sum
origin. Then
2
2
2
2
cos

+cos

+cos

+
cos

B = {1 + 5z|z  A}
is
HCU-2011
describes
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(a)
4/3
(b)
0
(c)
1/3
(d) 1
(a) a circle of radius 5 centred at (-1, 0)
30.
Let
A
=
2i
–
3j
+
k
and
B
=
i
+
2j
+
k
be
two
(b) a straight line
vectors. The vector perpendicular to both A and B
(c) a circle of radius 5 with centre at (-1, 0).
having length 10 is
HCU-2011
(d) a circle of radius 10 centred at (-1, 0)

50
i

30
j

70
k
50
i

30
j  70k
22. Consider a set of real numbers T = {t1, t2, ….,}
(a)
(b)
83
83
defined as
j
j
(c)
Both
A
and
B
(d)
None
of
the above
 1  3   1  3 





t j  
  
 . This set is
31. If p  q  13 and p  q  1 and p  3 , then
2
2

 

HCU-2012


(a) an unbounded infinite set
the angle between p and q is
HCU-2011
(b) an infinite bounded set
4

2

(a)
(b)
(c)
(d)
(c) a finite set with |T| > 319
6
3
6
6
(d) a finite set with |T| < 10
DIFFERENTIAL EQUATIONS
19. The general solution of the differential equation
dy x  y 2

HCU-2011
dx
2 xy
(a) y2 = (ln |x| + C)x
(b) y = (ln |x| + C)x
(c) y2 = (ln |x| + C)
(d) y = (ln |x| + C)x2
20. The differential equation, whose solutions are all the
circles in a plane, is given by
HCU-2011
(a) (1 + y')2 y'" – 3y'y"2 = 0
(b) xy' + y = 0
(c) (1 + y')2y'" + 3'y"2 = 0
(d) yy" + y'2 + 1 = 0
THREE DIMENSIONAL GEOMETRY
VECTORS
23. The value of x for which the volume of 32. Find the point at which the line joining the points A
(3, 1,-2) and B(-2, 7, -4) intersects the XY-plane.
parallelepiped formed by the vectors I + xj + k, j +
HCU-2012
xk and xi + k is minimum is
HCU - 2009
(a) (5, -6, 0)
(b) (8, -5, 0)
1
1
(a) – 3
(b)
(c) 3
(d)
(c) (1, 8, 0)
(d) (4, -5, 0)
3
3
33. If x + y + z = 0 and x3 + y3 + z3 – kxyz = 0, then only



one of the following is true. Which one is it?
24. If a  i  2 j  3k , b  2i  j  k and c is a vector
HCU-2012
 
 
 
(a)
k
=
3
whatever
be
x,
y
and
z
satisfying
a  c  a  b and a . c  0 then
(b) k = 0 whatever be x, y and z.

(c) k = + 1 or -1 or 0
3 | c |2 
HCU - 2009
(d) If none of x, y, z is zero, then k = 3
3 10
34. Consider the lines given by (x = a1z + b1, y = c1z +
(a) 0
(b) 3/4
(c) 30/4 (d)
2
d1) and (x = a2z + b2, y = c2z + d2). The condition by

 
 
 
  
which these lines would be perpendicular is given by
25. If d   ( a  b )   ( b  c )  v( c  a ), v( c  a ), a
HCU-2011
   
 
1
(a) a1c1 – a2c2 + 1 = 0
(b) a1c1 + a2c2 – 1 = 0
.( b  c )  and d .( a  b  c )  3 then     v is
(c) a1a2 – c1c2 = 1
(d) a1a2 + c1c2 + 1= 0
3
HCU - 2009 35. The image P' of the point P(p, q, r) in the plane 2x +
y+z=6
HCU-2011
(a) 6
(b) 9
(c) 1
(d) 0
(a) (p, q, - r)
 

26. Let a, b and c are three non-zero vectors, no two
1
1
(b)
12  p  2q  2r  ,  6  2 p  2q  r  ,


3
3
of which are collinear. If a  2 b is collinear with
1







 6  2 p  q  2r 
c and b  3 c is collinear a , then a  2 b  6 c is
3
HCU - 2009
1
1


(c)
 6  2 p  q  2r  , 12  p  2q  r  ,
3
3
(a) 0
(b) parallel to a


1
(c) parallel to b
(d) parallel to c
 6  2 p  q  2r 
3
2
INFOMATHS/MCA/MATHS/
INFOMATHS
1
1
x 1 y  2 z  3
(c)


 6  2 p  2q  r  ,  6  2 p  2q  r  ,
3
3
1
2
2
3x  1 3 y  2 z  8
1
(d)


12  p  2q  2r 
2
3
1
3
37.
The
straight
line
through
the point (-1, 3, 3) pointing
36. The image of the line from the point P given in
in the direction of the vector (1, 2, 3) hits the xy
Question 37 and it’s reflection P' about the plane 2x
plane at the point
HCU-2011
+ y + z = 6 is given by
HCU-2011
(a)
(2,
1,
0)
(b)
(-2,
1, 0)
3x  1 3 y  5 3z  8
(a)


(c) (1, 3, 0)
(d) never
4
2
1
x 1 y  2 z  3
(b)


2
1
4
(d)
3
INFOMATHS/MCA/MATHS/