NEST Exam 2023 Question Paper Mathematics Shift II

1. Let f be the function on R defined by f (x) = x3 − 3x2 + ax − 1, where a ∈ R.
Then the set of all possible values of a for which f is strictly increasing is

(a) [3, ∞)
(b) (−∞, 3]
(c) [−3, 0]
(d) [0, 3]
1
2. If = teiθ , where t ∈ R and 0 ≤ θ < 2π, then the value of θ is
(1 + i)2023
π
(a)
4
3π
(b)
4
5π
(c)
4
7π
(d)
4
3. Let S be the set of all 4-digit natural numbers with the following properties:

(i) every digit of any element of S belongs to the set {0, 1, 3, 5, 7, 9},
(ii) every element of S is divisible by 5, and
(iii) no element of S is divisible by 2.

Then the number of elements in S is

(a) 180
(b) 216
(c) 360
(d) 250

1

4. If a teacher assigns homework on the nth day, the probability that she will assign
homework on the (n + 1)th day is 13 . If she does not assign homework on the
nth day, the probability that she will assign homework on the (n + 1)th day is
2
3
. If she assigned homework on a Monday then the probability that she will
assign homework on the Thursday of the week is
1
(a)
3
7
(b)
27
13
(c)
27
2
(d)
3
2π 4π 42π 44π
5. The value of sin + sin + · · · + sin + sin is
23 23 23 23
(a) −1
(b) 0
(c) 1
(d) 2

6. Let P = (a, b) be a point in the Euclidean plane, with a and b nonzero. For any
point S on the x-axis, let T be the point of intersection of the line P S with the
y-axis. Let M be the midpoint of the segment ST . Then the locus of M , as S
varies on the x-axis, is given by

(a) xy = ab
ab
(b) xy =
4
(c) xy = ay + bx
(d) 2xy = ay + bx

2

7. Let f be a differentiable function on R satisfying the conditions
Rx 1
(i) f (x) = (f (t)) 3 dt for all x ∈ R, and
0

(ii) f (x) > 0 for all x > 0.

Then the value of f (3) is
√
(a) 2 2
√
(b) 3 3
1
(c)
2
1
(d)
3
8. Let f (x) = ln x − 2023x + 2023 for all x ∈ (0, ∞). Then the number of points
at which the graph of f cuts the x axis is

(a) 0
(b) 2
(c) 3
(d) 1

9. Let N be the number of integers n such that

(i) n = 2a 3b 5c where a, b, c are non-negative integers ≤ 10, and
(ii) n is neither a square nor a cube of a natural number.

Then N is equal to

(a) 848
(b) 849
(c) 1051
(d) 1059

3

10. Let ABC be a triangle and let a, b and c denote the lengths of the sides BC,
CA and AB respectively. Let α and β be positive real numbers such that

α(∠A) + β(∠B) = (α + β)(∠C).

Then

(a) αa + βb = (α + β)c
(b) αa + βb = (α + β)c implies a = b
(c) αa + βb > (α + β)c
(d) αa + βb = (α + β)c implies αa = βb

11. For a, b ∈ R, with a > 0, let N (a, b) denote the number of elements in the set
{x ∈ R | x + a sin x = b}. Then

(a) N (a, b) = 1 for all a, b.
(b) there does not exist any a such that N (a, b) = 1 for all b.
(c) N (a, b) is finite for all a, b.
(d) there exist a, b such that N (a, b) is infinite.

12. Let N be the number of solutions of the equation

x0 + 2x1 + 2x2 + 2x3 + 2x4 + x5 = 6,

with x0 , x1 , x2 , x3 , x4 and x5 taking non-negative integer values. Then

(a) N < 50
(b) 50 ≤ N < 100
(c) 100 ≤ N < 1000
(d) 1000 ≤ N
Rb
13. For a, b > 0 let F (a, b) = | sin 2πx|dx. Then
a

(a) F (10, 11) = 2F (0, 12 )
(b) F ( 41
4 4
, 43 ) = 12 F ( 21 , 1)
(c) F ( 18 , 14 ) = F (1, 2)
(d) F ( 41
4 4
, 43 ) = 23 F (0, 34 )

4

14. Let S = {x, y, z} and f : S → N be a function. Let A be a subset of N such
that the following conditions are satisfied:

(i) if f (x) ∈ A then f (y) ∈ A, and
(ii) if f (z) ∈
/ A then f (y) ∈
/ A.

Then it follows that

(a) whenever f (x) ∈ A, f (z) ∈ A.
(b) whenever f (x) ∈
/ A, f (z) ∈
/ A.
(c) whenever f (z) ∈ A, f (x) ∈ A.
(d) whenever f (z) ∈
/ A, f (x) ∈
/ A.

15. Let a and b be non-zero vectors. Let S be the set of vectors v such that
a × v = b. Then

(a) there exists a positive real number r such that ||v|| < r for all v ∈ S.
(b) S is non-empty if and only if a · b = 0.
(c) S is contained in a plane.
(d) if v1 and v2 are in S, then there exists λ ∈ R such that v1 − v2 = λa.

16. Let C1 , C2 and C3 , be three circles having the same radius r, which touch each
other externally. Then

(a) for any circle C which is touched internally by C1 and C2 , C3 lies within C.
(b) there is no circle C touched internally by C1 , C2 and C3 .
 
(c) a circle C touched internally by C1 , C2 and C3 has radius 1 + √23 r.
(d) the radius of any circle C touched internally by C1 and C2 is at least 2r.

17. Let S = {(a, b) | a, b ∈ Z}. Let R be the equivalence relation on S defined by
(a, b)R(c, d) if a2 + b2 = c2 + d2 . For (a, b) ∈ S let F (a, b) denote the equivalence
class {(c, d) ∈ S | (a, b)R(c, d)} of (a, b). Then

(a) there exists (a, b) ∈ S such that F (a, b) has only one element.
(b) there exists (a, b) ∈ S such that F (a, b) has exactly 4 elements.
(c) there exists (a, b) ∈ S such that F (a, b) has exactly 6 elements.
(d) there exists (a, b) ∈ S such that F (a, b) has infinitely many elements.

5