ISI Admission Test 2026 Sample Paper B.Stat B.Math UGA

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ISI 2026
Sample Paper · B.Stat
B.Math UGA
EXAM YEAR TYPE SUBJECT

ISI 2026 Sample Paper B.Stat B.Math UGA

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Sample Question Paper-UGA

Group A
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Each of the following questions have exactly one correct option and you have to identify it.

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m . 5π/2
e tan−1 (sin x)

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1. The value of the integral
e tan−1 (sin x)
+e
dx equals
tan−1 (cos x)
l a
ofg
π/2

(a) 1. la (D) none a
g (B) π. (C) e.
a of all solutions of the equation cos 2θ = sin θ + cos θ is given by
2. The set
these.

(a) θ = 0.
(b) θ = nπ + π2 , where n is any integer.
(c) θ = 2nπ or θ = 2nπ − π2 or θ = nπ − π4 , where n is any integer.
(d) θ = 2nπ or θ = nπ + π4 , where n is any integer.
       
n n+1 n+2 n+k
3. For k ≥ 1, the value of + + + ··· + equals

m
0 1 2 k

(a) n+k+1 (B) (n + k + 1) n+k
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(C) n+k+1 n+k+1





n+k . n+1 . n+1 . (D) .
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n
" ( r !)
s
r
e r #

a
1 5 2 8
l
−1 −1 −1 −1
4. The value of sin cot sin 1− + cos + sec is

g
2 6 3 3

(a) 0. a
(B) π/6. (C) π/4. (D) π/2.

 2
2  2
3  2
n
5. If an = 1 + n12 1 + n2 2 1 + n3 2 · · · 1 + nn2 , then

2
lim a−1/n
n
n→∞

is
√
(a) 0. (B) 1. (C) e. (D) e/2.
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d10

c. o (a) 1.
6. If f (x) = ex sin x, then f (x) equals
dx10 x=0
e m
e m (B) −1.
a s (D) 32.
l at a point P outside the
(C) 10.

as g
l circle. If ∠AOC = 43 and ∠BP D = 18 , then the value of ∠BOD is a
7. Consider a circle with centre O. Two chords AB and CD extended intersect

ag
◦ ◦

(a) 36 . ◦
(B) 29 . (C) 7 .
◦
(D) 25 . ◦ ◦

8. A box contains 10 red cards numbered 1, . . . , 10 and 10 black cards numbered 1, . . . , 10. In how many
ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a
red card and a black card with the same number?

(a) 10

7
4
(B) 10

7

(C) 10 (D) 10

7
14

3 4 2 . 3 4 . 3 2 . 3 4 .

9. Let P be a point on the ellipse x2 + 4y 2 = 4 which does not lie on the axes. If the normal at the point P

m
intersects the major and minor axes at C and D respectively, then the ratio P C : P D equals
.
(a) 2. c. o(B) 1/2. (C) 4. (D) 1/4.
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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 1 of 3

10. The set of complex numbers z satisfying the equation
(3 + 7i)z + (10 − 2i)z̄ + 100 = 0
represents, in the Argand plane,
(a) a straight line. (B) a pair of intersecting straight lines. (C) a pair of distinct parallel straight lines.
(D) a point.
11. The number of triplets (a, b, c) of integers such that a < b < c and a, b, c are sides of a triangle with
perimeter 21 is
(a) 7. (B) 8. (C) 11. (D) 12.
12. Suppose a, b and c are three numbers in G.P. If the equations ax2 + 2bx + c = 0 and dx2 + 2ex + f = 0
have a common root, then ad , eb and fc are in
(a) A.P. (B) G.P. (C) H.P. (D) none of the above.
13. The number of solutions of the equation sin−1 x = 2 tan−1 x is
(a) 1. (B) 2. (C) 3. (D) 5.
14. Suppose ABCD is a quadrilateral such that ∠BAC = 50 , ∠CAD = 60 , ∠CBD = 30 and ∠BDC = 25◦ .
◦ ◦ ◦

If E is the point of intersection of AC and BD, then the value of ∠AEB is
(a) 75◦ . (B) 85◦ . (C) 95◦ . (D) 110◦ .
15. Let R be the set of all real numbers. The function f : R → R defined by f (x) = x3 − 3x2 + 6x − 5 is
(a) one-to-one, but not onto. (B) one-to-one and onto. (C) onto, but not one-to-one. (D) neither
one-to-one nor onto.
16. Suppose x, y ∈ (0, π/2) and x ̸= y. Which of the following statements is true?
(a) 2 sin(x + y) < sin 2x + sin 2y for all x, y.
(b) 2 sin(x + y) > sin 2x + sin 2y for all x, y.
(c) There exist x, y such that 2 sin(x + y) = sin 2x + sin 2y.
(d) None of the above.
17. A triangle ABC has a fixed base BC. If AB : AC = 1 : 2, then the locus of the vertex A is
(a) a circle whose centre is the midpoint of BC.
(b) a circle whose centre is on the line BC but not the midpoint of BC.
(c) a straight line.
(d) none of the above.
18. Let N be a 50 digit number. All the digits except the 26th one from the right are 1. If N is divisible by
13, then the unknown digit is
(a) 1. (B) 3. (C) 7. (D) 9.
19. Suppose a < b. The maximum value of the integral
Z b 
3 2
− x − x dx
a 4
over all possible values of a and b is
(a) 3/4. (B) 4/3. (C) 3/2. (D) 2/3.
20. For any n ≥ 5, the value of 1 + 21 + 13 + · · · + 2n1−1 lies between
(a) 0 and n/2. (B) n/2 and n. (C) n and 2n. (D) none of the above.

Group B
Each of the following questions may have multiple correct options and you have to identify all the correct options.
1. Let x be an irrational number. If a, b, c and d are rational numbers such that ax+b
cx+d is a rational number,
which of the following must be true?

2
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m
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m .co s e m
se (C) ab = cd. l a
(D) a = da=g0
(A) ad = bc
(B) ac = bd.

2. Let z = x + iy be a complex number, which satisfies the equation (z + z)z = 2 + 4i. Then

(A) y = ±2. (C) x = ±3.
(B) x = ±2. (D) y = ±1.

3. Define Sn = 21 · 34 · · · · · 2n−1
2n where n is a positive integer. Then

m
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1 1
(A) Sn < √4n+2 for some n > 2. (C) Sn < √2n+5 for all n ≥ 2.
m
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1 1

m
(B) Sn < 2n+1 for all n ≥ 2.
√ (D) Sn > 4n+2 for all n ≥ 2.
√

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2

s e
4. If n + 19n + 92 is a perfect square, then the possible values of n may be
l a
(A) −19 l a ag
ag
(C) −4
(B) −8 (D) −11

1 1 1
5. Let a, b and c be three real numbers. Then the equation x−a + x−b + x−c =0

(A) always have real roots. (C) always have real and equal roots.
(B) can have real or complex roots depending on (D) always have real roots, which are not necessar-
the values of a, b and c. ily equal.

6. Let X be the set {1, 2, 3, . . . , 10} and P the subset {1, 2, 3, 4, 5}. The number of subsets Q of X such that
m
c.(C)o 2
P ∩ Q = {3} is

em (D) 2
5
(A) 1
(B) 24
s
9

l a
7. Suppose that the equations x + bx + cag= 0 and x + cx + ab = 0 have exactly one common non-zero
root. Then
2
a 2

(A) a + b + c = 0. (C) the two roots which are not common may not
(B) the two roots which are not common must nec- be real.
essarily be real. (D) the two roots which are not common are either
both real or both not real.

tan(α−β+γ)
8. Let tan(α+β−γ) = tan β
tan γ . Then

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(A) sin(β − γ) = sin(α − β). (C) sin(β − γ) = 0.

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(B) sin(α − γ) = sin(β − γ). (D) sin 2α + sin 2β + sin 2γ = 0

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9. Let K be the set of all points (x, y) such that |x| + |y| ≤ 1. Given a point A in the plane, let FA be the

la
point in K which is closest to A. Then the points A for which FA = (1, 0) are
ag
ag (A) all points A = (x, y) with x ≥ 1. (C) all points A = (x, y) with x ≥ 1 and y = 0.
(B) all points A = (x, y) with x ≥ y + 1 and (D) all points A = (x, y) with x ≥ 0 and y = 0.
x≥1−y

10. Let tan 3θ
tan θ = k. Then

(A) k ∈ (1/3, 3) (C) sin 3θ 2k
sin θ = k−1 .
(B) k ∈
/ (1/3, 3) (D) sin 3θ 2k
sin θ > k−1

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a For more Question Papers, Sample Papers, Notes & Syllabus visit Page 3 of 3