NEST Exam 2023 Question Paper Mathematics Shift I

1. Let g : R 7→ R be a differentiable function such that g(x)g 0 (x) > 0 for all x ∈ R.
Then

(a) g is increasing.
(b) g is decreasing.
(c) |g| is increasing.
(d) |g| is decreasing.

2. The number of real roots of f (x) = x6 + x3 − 1 is

(a) 0
(b) 2
(c) 4
(d) 6

3. In a throw of a (biased single) dice, the probability of the outcome being a
1
number n is 41 if n is even, and 12 if n is odd. If the dice is thrown twice, then
the probability that the sum of the two outcomes is an even number is
3
(a)
8
1
(b)
2
5
(c)
8
3
(d)
4

1,
 if x > 0,
4. Define sgn(x) = −1, if x < 0,

0, if x = 0.

√
Let f : R → R be the function defined by f (x) = (x − 5) sgn(x2 − 5). Then
the number of discontinuities of f is

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

1

5. Let S be the set of all natural numbers x such that

(i) 100 ≤ x ≤ 999,
(ii) 0 appears at least once as a digit in the decimal expansion of x, and
(iii) the sum of the digits of x is 10.

Then the number of elements in S is

(a) 18
(b) 20
(c) 27
(d) 30

6. The horizontal line y = k intersects the parabola y = 2(x − 4)(x − 6) at points
A and B. If the length of AB is 8, then the value of k is

(a) 30
(b) 10
(c) 20
(d) 8
n
1 X
7. Let S(n) = 4 (l + 2)(l + 4)(l + 6). The value of lim S(n) is
n l=1 n→∞

1
(a)
6
1
(b)
2
1
(c)
4
(d) 1

2

 
0 0 α
8. Let α be a complex number such that α 6= 1 and α5 = 1. Let A =  0 α 0 
α 0 0
and I denote the identity matrix. Then the value of I + A + A + A + A4 is
2 3

 
1 0 −1
(a) (1 + α2 + α4 )  0 0 0 
−1 0 1
 
1 0 −1
(b) α(1 + α2 )  0 0 0 
−1 0 1
 
−1 0 1
(c) (1 + α2 + α4 )  0 0 0 
1 0 −1
 
−1 0 1
(d) (1 + α2 + α4 )  0 1 0 
1 0 −1

9. Let P and Q be the vertices of the parabolae y = x2 +bx+c and y = −x2 +dx+e,
respectively.

If P and Q are the points of intersection of the parabolae then the slope of the
line through P and Q is
c+e
(a)
2
c+d
(b)
2
b+d
(c)
2
b+e
(d)
2

3

10. Let ABC be a triangle with AC = 2048, AB = 512 and BC = 2000. Let P be
a point on the segment AB such that AP = 1 and Q be a point on the segment
AC such that AQ = 1024. Let R be the midpoint of P Q. Let Z be the point
of intersection of AR and BC.

Then the length of ZC is
2000
(a)
256
2000
(b)
257
1000
(c)
256
1000
(d)
257
11. Let f : R → R be a continuous function such that f (0) = 1 and

|f (x) − f (y)| ≤ | sin{(x − y)2 }| for all x, y ∈ R,

and let g be the function defined by g(x) = x2 f (x2 ) for all x ∈ R. Then the
value of g 0 (2) is

(a) 2
(b) 4
(c) 6
(d) 0

4

12. Let n ≥ 3 be an integer. Let P1 , P2 , . . . , P2n be points in the plane, which are
the vertices of a regular 2n-gon. The number of obtuse-angled triangles with
vertices contained in the set {P1 , P2 , . . . , P2n } is

(a) n(n − 1)(n − 2)
n2 (n − 1)(n − 2)
(b)
3
n(n − 1)2
(c)
2
(d) 2n(2n − 1)(2n − 2)

13. If A, B, C are 3×3 matrices with entries in R, satisfying the condition AB = AC,
then

(a) the determinant of AB is 0.
(b) either A is the zero matrix or B = C.
(c) either B = C or A is not an invertible matrix.
(d) either A is the zero matrix or the determinant of B − C is zero.

14. Let X, Y, Z be sets and f : X → Y and g : Y → Z be functions. Then

(a) g ◦ f being injective implies f injective.
(b) g ◦ f being surjective implies g surjective.
(c) g ◦ f being injective implies g injective.
(d) g being surjective implies g ◦ f surjective.

15. Let f : (0, 3) ∪ (6, 9) → R be a differentiable function such that f 0 (x) = 21 for
all x ∈ (0, 3) ∪ (6, 9). Then

(a) f is an increasing function.
(b) f is a one to one function.
(c) f (8) − f (7) = f (2) − f (1).
(d) there exists a number c in R such that f (x + 6) = f (x) + c for all x ∈ (0, 3).

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16. Let A and B be two points on the parabola y − 2x2 = 0 and O be the origin
(0, 0). If

(a) OAB is an isosceles triangle then the y coordinates of A and B are equal.
√
(b) OAB is an equilateral triangle then the length of each side is 3.
√
(c) OAB is an isosceles triangle and the two equal sides are of length 3 then
OAB is an equilateral triangle.
√
(d) OAB is an equilateral triangle then its altitude is 3.

17. Let f : [0, 1] → R be a continuous function and P be a polynomial of degree 4
with coefficients in R. If P (f (x)) = 0 for all x ∈ R, then

(a) f (x) = 0 for all x ∈ R.
(b) f is a constant function.
(c) for all continuous functions g, there exists x ∈ [0, 1] such that P (g(x)) = 0.
(d) P has at most two roots which do not belong to R.

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