Name................... Reg. No...................
INDIAN INSTITUTE OF MANAGEMENT CALCUTTA
Mathematics (Qualifying) 2023
Attempt - I
Duration: 2 Hours Total Marks: 60
Instructions:
1. Clearly write your Name and Registration Number on both the question
paper and the answer sheet. Failing to write your name and registration
number may lead to a penalty of 5 marks.
2. This paper contains 20 multiple choice questions, please select the most
appropriate option.
3. Each question carries 3 marks.
4. There is NO negative marking.
5. Use of calculators is permitted but the use of laptop computers, mobile
phones, tablet computers is not permitted.
6. At the end of the examination, submit both the question paper and the
answer sheet.
7. Please feel free to use the question paper, its reverse side, and the two
rough sheets provided at the end for rough work. However, please do
NOT use the answer sheets for rough work.
1. Consider two sets, A and B defined as below.
A = {m ∈ R| both roots of x2 − (m + 1)x + m + 4 = 0 are real.}
B = (−∞, −3)
Which of the following statements is NOT true?
(a) A ∪ B = A.
(b) A ∩ B = B.
(c) A \ B = {3}.
(d) B \ A = {}.
2. A survey regarding preference of cuisine was conducted among 50 respon-
dents. It was found that 20 people prefer Italian cuisine, 19 prefer Chinese
and 25 prefer Thai. 3 people prefer Italian and Thai but not Chinese; 5
people prefer Italian and Chinese but not Thai and 12 people prefer Thai
as well as Chinese. If 3 people prefer all the three cuisines what is the
number of respondents who did not prefer any of the three cuisines?
1
(a) 10
(b) 9
(c) 7
(d) 6
3. The minimum distance of a point on the curve y = x2 − 4 from the origin
is
√
(a) 19/2
√
(b) 15/2
p
(c) 19/2
p
(d) 15/2
4. Which of the following options correctly represents the partial derivative
of the function f (x) = x3 + y 3 + xy with respect to x at the point (1,2)?
(a) 5
(b) 17
(c) 18
(d) 9
1 −1 2
5. Let A = and B = . Which of the following options correctly
1 2 2
represents A−1 B?
2
(a)
0
6
(b)
0
2 1
(c)
−1 1
1 2 1
(d) 3
−1 1
6. Consider a right-angled triangle ABC with right angle at A. If the coordi-
nates of points A and B are (2, 3) and (0, 1), respectively, and the length
of the side BC is 5 units, then which of the following statements is true
about the coordinates of point C?
(a) The sum of the x and the y coordinates of C is either 4 or 6.
(b) The product of the x and the y coordinates of C is either 0 or 8.
(c) Either the x or the y coordinate of C is negative.
(d) The larger of the x and the y coordinates of C is 4.
7. For which of the following values of k are the points (k, 2−2k), (−k+1, 2k)
and (−4 − k, 6 − 2k) collinear?
(a) 1
2
(b) 2
(c) -1
(d) -2
8. The ratio of the sum of first n even natural numbers to that of the first n
odd natural numbers is
n+1
(a) n
n
(b) n−1
n−1
(c) n
n
(d) n+1
9. Which of the following lines bisects the angle between the lines x+y−3 = 0
and 7x − y + 5 = 0?
(a) x − 3y + 10 = 0
(b) 6x + 2y − 5 = 0
(c) 3x − y + 5 = 0
(d) 2x − 6y + 10 = 0
10. The sides of a square are given by the equations x = 2, x = 3, y = 1 and
y = 2. A circle is drawn with the diagonal of the square as its diameter.
Which of the following options correctly represents the equation of the
circle?
(a) x2 + y 2 − 2x − 4y + 7 = 0
(b) x2 + y 2 − 4x − 2y + 8 = 0
(c) x2 + y 2 − 3x − 5y + 7 = 0
(d) x2 + y 2 − 5x − 3y + 8 = 0
a b c
11. If A = b c a, where a, b, and c are real positive numbers and abc = 1
c a b
such that AT A = I, then what is the value of a3 + b3 + c3 ?
(a) 2
(b) 4
(c) 2 or 4
(d) 3
12. Which of the following correctly represents the term independent of x in
the binomial expansion of ( 32 x2 − 3x
1 9
) ?
1
(a) 9
2
(b) 18
5
(c) 18
7
(d) 18
3
∂u ∂u ∂u
13. If u = x2 y + y 2 z + z 2 x , then ∂x + ∂y + ∂z is equal to
(a) 0
(b) (x + y + z)
(c) (x2 + y 2 + z 2 )
(d) (x + y + z)2
14. Consider a function f : N → N defined as
(
x(x + 1) if x is even
f (x) =
x+3 if x is odd
For what value of x, 8f (x + 1) − f (x) = 2?
(a) 5
(b) 10
(c) 15
(d) 20
dy
15. If xy = (x + y)k and dx = xy , then the value of k is:
(a) 2
(b) 3
(c) 4
(d) 5
16. How many distinct 4-digit even numbers can be formed if repetition of
digits is allowed, and if ‘5’ is one of the digits in the number, then ‘7’ has
to be the next digit in the number?
(a) 360
(b) 365
(c) 370
(d) None of the above
17. Let the function f (x) be such that
(
x4 if x2 < 1
f (x) =
x if x2 ≥ 1
Which of the following statements is NOT true?
(a) limx→1 f (x) exists.
(b) limx→−1 f (x) exists.
(c) The function is continuous at x = 1.
(d) The function is not continuous at x = −1.
18. (2x + x2 )ex dx
R
4
(a) x2 ex + C, where C is the constant of integration.
(b) x3 ex + C, where C is the constant of integration.
(c) xex + C, where C is the constant of integration.
(d) None of the above
19. Which of the following statements is TRUE about the function f ′ (x) =
(x − 1)(x − 2)3 + 3(x − 1)2 (x − 2)2 ?
(a) f (x) has a local maxima at x = 1.
(b) f (x) has a local minima at x = 2.
(c) f (x) has a local maxima at x = 5/4.
(d) f (x) has a local minima at x = 3.
20. The area of the region included between the curves y = x2 /4b and y 2 =
4ax is:
(a) 8ab/3
(b) 16ab/3
(c) 32ab/3
(d) 64ab/3
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Name................... Reg. No...................
Mathematics (Qualifying) 2023
ROUGH SHEET
6
Name................... Reg. No...................
Mathematics (Qualifying) 2023
ROUGH SHEET
