2025

Booklet Number: 10671 TEST CODE: UGA

Forenoon Time: 2 hours

. This test contains thirty (30) multiple-choice questions (MCQs).
. The questions are to be answered on a separate Optical Mark
Recognition (OMR) Answer Sheet.
" Please write your Name, Registration Number, Test Centre, Test
Code and the Number of this Question Booklet in the appropriate
places on the OMR Answer Sheet. Please do not forget to put
your signature in the designated place.
" For each of the questions there are four suggested answers, of
which only one is correct. For each question, indicate your choice
of the correct answer by darkening the appropriate circle
completelyon the OMR Answer Sheet, using ball-point pen with
BLACK ink only.
" You will score

4marks for each correctly answered question,

0mark for each incorrectly answered question, and
1mark for each unattempted question.
" ALL ROUGH WORK MUST BE DONE ONLY IN THE SPACE
AVAILABLE ON THIS QUESTION BOOKLET.
" USE OF CALCULATORS, MOBILE PHONES AND ALL
TYPES OF ELECTRONIC DEVICES IS STRICTLY
PROHIBITED.
STOP! WAIT FOR THE SIGNAL TO START.

UGA,
 UGA,

1. Let A be an m x n matrix with the (i,j)th entry given by the

real number a;, 1 <i<m, 1<jSn. Let

a= max min
ISjSn 1siSm
and B min max
1Sjsn1SISm }
Then

(A) asB but not necessarily a = ß

(B) <a but not necessarily a =B
(C) a = B
(D) nothing can be said in general

2. The coefficient of in (1 3z)° (1+ 9z?)°(1 + 3z)°is

(A) -39x5 (B) 3 x 5 (C) -38 x5 (D) 38 x 5

3. Consider the cyclic quadrilateral ABCD given below.

Assume that AB = BC, AD = CD, and A=

AD
Let
0= ZADC. Then cos 8is equal to
1 2 3
(A) (B) 5 (C);
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 4. In the ry-plane, the curve 3ry + 6ry + 2ry'0represents
(A) a pair of straight lines
(B) an ellipse
(C) a pair of straight lines and an ellipse
(D) a hyperbola

1
5. Let I- T de The
1
(A) I<6 (B) 1> 13
1 1
63 14

6. Consider two events A and B with probabilities P(A) and PB)

respectively such that 0 < P(A), P(B) <1. Define

P(A|B) = P(An B)
P(B)
Consider the following statements.
(1) P(A|B°) + P(A|B) = 1.
(II) P(A|B) + P(A|B) = 1.
Then, in general,
(A) (I) is true and (II) is false (B) (I)
is false and (II) is true
(C) both (I) and (II) are true (D)
both (I) and (II) are false
7. Let f() =7z"+ 4z -3.
Then f has
(A) exactly 1 real root
(B) exactly 3 real roots
(C) exactly 5 real roots
(D) 11 real roots

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8. Twelve boxes are placed along a circle. In each box, 1, 2,3

or 4 balls are put such that the total number of balls in any
4consecutive boxes is samne. The number of ways this can be
done is

(A) 4! (B) 44 (C) (4!)3 (D) (4)4

9. Let A = {(*,y) : ,y E [0, 1]} and B = {(z, y) : ,yE 0, 2]}.

Define f:A’ Bby f(z,y) = (a² +y, z +y). Then f is
(A) one-to-one but not onto
(B) onto but not one-to-one
(C) both one-to-one and onto
(D) neither one-to-one nor onto

10. The number of ordered pair (a, b) of positive integers with a <b
satisfying a' + b² = 2025 is
(A) 0 (B) 1 (C) 2 (D) 6

11. For each n1, let an and b, be real numbers such that an t0
and b, 0. Let

(an t ib,)" = n(a, +ib,) for alln > 6.

Then

(A) no such an, b, exist

(B) a, =nnl cos , , =nnt sin

(C) an = nicos 2,
n bn = ni sin 27r

(D) an = nai cos, bn =n sin 2

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 an-2 for n >
2, Then the
= 5a,-1 t
a, =l and a,
12. Let do = 0, determinant
value of the
ajot j0o
aj00 g9

is
(B) -5101 (C) 1 (D) 5101
(A) -1

The lengths of the three sides of a right angled triangle are in

13.
smallest angle of the triangle is
geometric progression. The
(A) tan-() (B) cos()
(C) sin"() (D) sin"()

14. The lengths of the two adjacent sides of a parallelogram are

2 cm and 3cm. The length of one diagonal is v19 cm. Then
the length of the other diagonal is
(A) V5 cm (B) V7cm (C) V15 cm (D) V21 cm

15. Consider the following statements about two similar triangles

Aj and A.
S: Lengths of the sides of A are in arithmetic progression.
S,: Lengths of the sides of A, are in geometric progresion.
S: Lengths of the sides of A, are in arithmetic progression.
S4: Lengths of the sides of A2 are in geometric progression.
Then

(A) S, implies S, but S, does not imply S

(B) S, does not imply S,, but S, implies S
(C) S, implies S3, and S, implies S
(D) S, does not imply S3, and S, does not imply S
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16. Let (a] denote the greatest integer less than or equal to zeR
and |z| has its usual meaning, that is, z| =rif z 2 0, and
|a=-, if z<0. Then the value of the integral

dr

1 1
(A) 1+ (B) 1+lge 7
log, 2
(C) 2- log, 2 (D) none of the above

1 1
17. Let f(z) = Then the functionf has
1+ < 1| 1+<z+1|
(A) neither alocal maximum nor a local minimum
(B) a local minimum at r =0, but no local maximum
(C) local maxima at r = t1, but no local minimum
(D) a local minimum at r = 0 and local maxima at r = t1

18. Let

-1, if r<0,
0, if z=0,
1, if z >0.

Then the function F defined by F(z) = f() dt is

(A) not continuous
(B) continuous, but nowhere differentiable
(C) differentiable everywhere
(D) differentiable everywhere except at 0

: Page 5 of 16:
 20
g(t)g(r - t) dt, where
19. For areal number z, let f(r) = | -20

s) - 1,0, otherwise.
ifze[0, 1),

Then f(r) is equal to

if z e [0. 1].
(A) 2-, ifr e[1,2].
0. otherwise

1+, if z¬ (0, 1),

(B) 1-z, ifr e (1, 2),
0, otherwise

1, if z e (-20,20),
0, otherwise

(D) none of the above

20. Let

L= lim (n + 100)
n’o0
-
Then
(A) 2< L< 16
(B) 16 <L<32
(C) 32 <L<243 (D) L > 243
21. Let a,b, c, d be positive
integers such that the product
abcd = 999. Then the number
of different ordered 4-tuples
(a, b, c, d) is

(A) 20 (B) 48 (C) 80 (D) 84

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22. In a certain test there are n questions. At least i questions were

wrongly answered by 2"- students, where i 1, 2,... ,n. If the
total number of wrong answers given by all students is 2047,
then n is equal to
(A) 10 (B) 11 (C) 12 (D) 13

23. Letn>3. There are n straight lines in a plane, no two of which

are parallel and no three pass through a common point. Their
points of intersection are joined. Then the number of fresh line
segments thus created is

(A)
n(n- 1)(n 2) n(n-1)(n - 2)(n -3)
(B) 6

(C) n(n-1)(n - 2)(n - 3) (D) none of the above

24. Let d be the side length of the largest possible equilateral

triangle that can be put inside a square of side length 1. Then
(A) d< 1 (B) d= 1
2 2
(C) 1<d<;31/4 (D) d2 3/A

25. Let

n² n?

Vn +n- n 1.
Then n-’00
lim a,

(A) does not exist (B) is equal to 1

(C) is equal to e (D) is equal to

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 1
tan!
26. Let n be a positive integer The value of

(A) tan"(n +1) (B) tanG)

(C) tan-n (D) tan )

27. Let f(æ) (+ 18)(r-4)z(r +4) -2. Then

(A) f has exactly one real root
(B) fhas exactly 3 distinct real roots
(C) fhas 5 distinct real roots
(D) f has a repeated root

28. Let k be a positive integer and f(r) =e-1. Then

lim f(r) +fG)+ f()++f()

0

is equal to
1
(A) 2-; (B) 2 2k+1
(C) k (D) 2t+1 1

29. Let

a,
23-1 3-1 n1
n n 2
Then lim an
n’0

(A) does not exist (B) is equal to

(C) is equal to 1 (D) is equal to

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30. A subset u, u2, U3, l4, us} of the first 90 positive

integers can
be selected in ( )ways. Let Umax max{u1, Wz, y, u4, Us}
and
Umin = min{uj, uz, U3, ú4, U,}. Then the aríthnetic mean of
Umax + Umin OVer all such subsets is

(A) 45 (B) 46 (C) 89 (D) 91

2 (13-"8lx)
34

3 7 3 «9
3×3 x 3x 34 -4!
3 S 3 .3 3
2!
"
33+ 3
3 3 34 3

32+2y2 = -G

(nt)

ja)s

a-l

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