ISI Admission Test 2016 Syllabus and Sample Paper B.Stat B.Math UGA and UGB

FOR ISI EXAM PREPARATION

ISI 2016
Syllabus and Sample
Paper · B.Stat B.Math
UGA and UGB
EXAM YEAR TYPE SUBJECT

ISI 2016 Syllabus and Sample Paper B.Stat B.Math UGA and UGB

Notes · Sample Papers · Previous Year Papers · Mock Tests

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Test Codes: UGA (Multiple-choice Type) and
UGB (Short Answer Type), 2016

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Questions will be set on the following and related topics.

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Algebra: Sets, operations on sets. Prime numbers, factorization of inte-
gers and divisibility. Rational and irrational numbers. Permutations and
combinations, Binomial Theorem. Logarithms. Polynomials: Remainder
Theorem, Theory of quadratic equations and expressions, relations between
roots and coefficients. Arithmetic and geometric progressions. Inequalities
involving arithmetic, geometric & harmonic means. Complex numbers.

Geometry: Plane geometry. Geometry of 2 dimensions with Cartesian and
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polar coordinates. Equation of a line, angle between two lines, distance from
a point to a line. Concept of a Locus. Area of a triangle. Equations of circle,

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parabola, ellipse and hyperbola and equations of their tangents and normals.
Mensuration.
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Trigonometry: Measures of angles. Trigonometric and inverse trigonomet-
ric functions. Trigonometric identities including addition formulae, solutions
of trigonometric equations. Properties of triangles. Heights and distances.

Calculus: Sequences - bounded sequences, monotone sequences, limit of a

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sequence. Functions, one-one functions, onto functions. Limits and continu-

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ity. Derivatives and methods of differentiation. Slope of a curve. Tangents
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and normals. Maxima and minima. Using calculus to sketch graphs of func-
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tions. Methods of integration, definite and indefinite integrals, evaluation of

e m area using integrals.
la
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ag Reference (For more sample questions)
Test of Mathematics at the 10 + 2 level, Indian Statistical Institute. Pub-
lished by Affiliated East-West Press Pvt. Ltd., 105, Nirmal Tower, 26
Barakhamba Road, New Delhi 110001.

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Sample Questions for UGA

Instructions. UGA is a multiple choice examination. In each of the fol-
lowing questions, exactly one of the choices is correct. You get four marks
for each correct answer, one mark for each unanswered question, and zero
marks for each incorrect answer.

1 Define an = (12 + 22 + . . . + n2 )n and bn = nn (n!)2 . Recall n! is the
product of the first n natural numbers. Then,
(A) an < bn for all n > 1 (B) an > bn for all n > 1
(C) an = bn for infinitely many n (D) None of the above

2 The sum of all distinct four digit numbers that can be formed using
the digits 1, 2, 3, 4, and 5, each digit appearing at most once, is
(A) 399900 (B) 399960 (C) 390000 (D) 360000

3 The last digit of (2004)5 is
(A) 4 (B) 8 (C) 6 (D) 2

4 The coefficient of a3 b4 c5 in the expansion of (bc + ca + ab)6 is
   
12! 6 6
(A) (B) 3! (C) 33 (D) 3
3!4!5! 3 3

5 Let ABCD be a unit square. Four points E, F , G and H are chosen
on the sides AB, BC, CD and DA respectively. The lengths of the
sides of the quadrilateral EF GH are α, β, γ and δ. Which of the
following is always true? √
(A) 1 ≤ α2 + β 2 + γ 2 + δ 2 ≤ 2 2
√ √
(B) 2 2 ≤ α2 + β 2 + γ 2 + δ 2 ≤ 4 2
(C) 2 ≤ α2 + β 2 + γ 2 + δ 2 ≤ 4
√ √
(D) 2 ≤ α2 + β 2 + γ 2 + δ 2 ≤ 2 + 2

6 If log10 x = 10log100 4 then x equals
(A) 410 (B) 100 (C) log10 4 (D) none of the above

7 z1 , z2 are two complex numbers with z2 6= 0 and z1 6= z2 and satisfying
z 1 + z2 z1
= 1. Then is
z 1 − z2 z2
2

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(A) real and negative
(B) real and positive
(C) purely imaginary
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(D) none of the above need to be true always

o mnumbers x satisfying the inequality x (x+1)(x−2) ≥
8 The set of allcreal m .
m . 3

s e
0 is
(A)ethe interval [2, ∞)
s (B) the interval [0, ∞)
g l a
g la(C) the interval [−1, ∞) (D) none of the above a
a
9 The minimum area of the triangle formed by any tangent to the ellipse
2
x2
a2
+ yb2 = 1 and the coordinate axes is
2
(C) (a+b)
2 2 2 2
(A) ab (B) a +b
2 2 (D) a +ab+b
3

10 Let A be the fixed point (0, 4) and B be a moving point (2t, 0).
Let M be the mid-point of AB and let the perpendicular bisector
of AB meet the y-axis at R. The locus of the mid-point P of M R is
(A) y + x2 = 2 (B) x2 + (y − 2)2 = 1/4
(C) (y − 2)2 − x2 = 1/4
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(D) none of the above
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11 The sides of a triangle are given to be x . + x + 1, 2x + 1 and x − 1.
e mthe triangle is
2 2

a
Then the largest of the three angles
l s of

g
 
x
x+1 a
(A) 75◦ (B) π radians (C) 120 ◦
(D) 135 ◦

12 Two poles, AB of length two metres and CD of length twenty me-
tres are erected vertically with bases at B and D. The two poles
are at a distance not less than twenty metres. It is observed that
tan ∠ACB = 2/77. The distance between the two poles is
(A) 72m (B) 68m (C) 24m (D) 24.27m

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13 If A, B, C are the angles of a triangle and sin2 A + sin2 B = sin2 C,
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then C is equal to
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(A) 30◦ (B) 90◦ (C) 45◦
las(D) none of the above

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1
ag
ag 14 In the interval (−2π, 0), the function f (x) = sin
x 3

(A) never changes sign
(B) changes sign only once
(C) changes sign more than once, but finitely many times
(D) changes sign infinitely many times
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15 The limit
(ex − 1) tan2 x
lim
x→0 x3
(A) does not exist (B) exists and equals 0
(C) exists and equals 2/3 (D) exists and equals 1

16 Let f1 (x) = ex , f2 (x) = ef1 (x) and generally fn+1 (x) = efn (x) for all
d
n ≥ 1. For any fixed n, the value of fn (x) is equal to
dx
(A) fn (x) (B) fn (x)fn−1 (x)
(C) fn (x)fn−1 (x) · · · f1 (x) (D) fn+1 (x)fn (x) · · · f1 (x)ex

17 If the function
( 2
x −2x+A
sin x if x 6= 0
f (x) =
B if x = 0
is continuous at x = 0, then
(A) A = 0, B = 0 (B) A = 0, B = −2
(C) A = 1, B = 1 (D) A = 1, B = 0

18 A truck is to be driven 300 kilometres (kms.) on a highway at a con-
stant speed of x kms. per hour. Speed rules of the highway require
that 30 ≤ x ≤ 60. The fuel costs ten rupees per litre and is consumed
at the rate 2 + (x2 /600) litres per hour. The wages of the driver are
200 rupees per hour. The most economical speed (in kms. per hour)
to drive the truck is
√ √
(A) 30 (B) 60 (C) 30 3.3 (D) 20 33

Z 1 Z a
et e−t
19 If b = dt then dt is
0 t+1 a−1 t − a − 1
(A) bea (B) be−a (C) −be−a (D) −bea

20 In the triangle ABC, the angle ∠BAC is a root of the equation
√
3 cos x + sin x = 1/2.
Then the triangle ABC is
(A) obtuse angled (B) right angled
(C) acute angled but not equilateral (D) equilateral

21 Let n be a positive integer. Consider a square S of side 2n units with
sides parallel to the coordinate axes. Divide S into 4n2 unit squares
by drawing 2n − 1 horizontal and 2n − 1 vertical lines one unit apart.
A circle of diameter 2n − 1 is drawn with its centre at the intersection
4

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of the two diagonals of the square S. How many of these unit squares
contain a portion of the circumference of the circle?
(A) 4n − 2 (B) 4n (C) 8n − 4 (D) 8n − 2
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six feet tall. is walking at a speed of 10 feet/second from the lantern
22 A lantern is placed
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m point on the wall. When he is midway between the
to the enearest
l as
l a s
lantern and the wall, the rate of change (in ft./sec.) in the length of
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(A) 2.4
is
(B) 3 (C) 3.6 (D) 12

23 An isosceles triangle with base 6 cms. and base angles 30◦ each is
inscribed in a circle. A second circle touches the first circle and also
touches the base of the triangle at its midpoint. If the second circle is
situated outside the triangle, then its radius (in cms.) is
√ √ √ √
(A) 3 3/2 (B) 3/2 (C) 3 (D) 4/ 3

24 Let n be a positive integer. Define
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f (x) = min{|x − 1|, |x − 2|, . . . , |x − n|}.
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Z n+1
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Then f (x)dx equals
l a
g (C) (n + 2)
(n +a3)
0
(n + 4) (n + 2)
(A) (B) (D)
4 4 2 4
25 Let S = {1, 2, . . . , n}. The number of possible pairs of the form (A, B)
with A ⊆ B for subsets A and B of S is
n   
n n
X n n
(A) 2 (B) 3 (C) (D) n!
k n−k
k=0

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26 The number of maps f from the set {1, 2, 3} into the set {1, 2, 3, 4, 5}

m
such that f (i) ≤ f (j) whenever i < j is
(D) 30 .c
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s em 27 Consider three boxes, each containing 10 balls labelled 1, 2,l.a. . , 10.
gby n , the
g la Suppose one ball is drawn from each of the boxes. Denote a
a
i
label of the ball drawn from the i-th box, i = 1, 2, 3. Then the number
of ways in which the balls can be chosen such that n < n < n is 1 2 3
(A) 120 (B) 130 (C) 150 (D) 160

28 Let a be a real number. The number of distinct solutions (x, y) of
the system of equations (x − a)2 + y 2 = 1 and x2 = y 2 , can only be
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(A) 0, 1, 2, 3, 4 or 5 (B) 0, 1 or 3
(C) 0, 1, 2 or 4 (D) 0, 2, 3, or 4

29 The maximum of the areas of the isosceles triangles with base on the
positive x-axis and which lie below the curve y = e−x is:
(A) 1/e (B) 1 (C) 1/2 (D) e

30 Suppose a, b and n are positive integers, all greater than one. If an +bn
is prime, what can you say about n?
(A) The integer n must be 2
(B) The integer n need not be 2, but must be a power of 2
(C) The integer n need not be a power of 2, but must be even
(D) None of the above is necessarily true
31 Water falls from a tap of circular cross section at the rate of 2 me-
tres/sec and fills up a hemispherical bowl of inner diameter 0.9 metres.
If the inner diameter of the tap is 0.01 metres, then the time needed
to fill the bowl is
(A) 40.5 minutes (B) 81 minutes

(C) 60.75 minutes (D) 20.25 minutes
32 The value of the integral
Z 5π/2 −1 (sin x)
etan
dx
etan
−1 (sin x)
+ etan
−1 (cos x)
π/2

equals (A) 1 (B) π (C) e (D) none of these
33 The set of all solutions of the equation cos 2θ = sin θ + cos θ is given
by
(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
34 The number
 210 11
11
is
10 2 10 2 10 2 10 2 10






(A) strictly larger than 1 2 3 4 5
10 2 10 2 10 2 10 2





(B) strictly larger than 1 2 3 4 but strictly smaller than
10 2 10 2 10 2 10 2 10






1 2 3 4 5

2 10
2 10
2 10
2
(C) less than or equal to 10 1




2
3
10 2 10 2 10 2 10 2 10
4
(D) equal to 1 2 3 4 5 .
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35 The value of
" ( r !) r r #
1 5 2 8
sin−1 cot sin−1 1− + cos−1 + sec−1
2 6 3 3
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is
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(A) 0
e m (B) π/6 (C) π/4 (D) π/2
l as
l as ag
g
36 Which of the following graphs represents the function
a f (x) = Z √
x
2
e−u /x du, for x>0 and f (0) = 0?
0

(A) (B)

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c. (D)o
(C)
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as
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2 ag
     2 2 2 3 2 n
1 3 n
37 If a = 1 +
n 1+ 1+ 2
··· 1 + , then
2 2 2
n n n n
2
lim a−1/n
n→∞ n

is √
(A) 0 (B) 1 (C) e (D) e/2
38 The function x(α − x) is strictly increasing on the interval 0 < x < 1
if and only if
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(A) α ≥ 2 (B) α < 2

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(C) α < −1
e
(D) α > 2

e m las
las intersect at a point P outside the circle. If ∠AOC = 43 andg∠BP D =
39 Consider a circle with centre O. Two chords AB and CD extended
a ◦

ag ◦
18 , then the value of ∠BOD is
(A) 36◦ (B) 29◦ (C) 7◦ (D) 25◦
40 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?
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10 7 4 10 7
(A) 2 (B)
3 4 3 4
    
10 7 10 14
(C) 2 (D)
3 3 4
41 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 intersects the major and minor axes
at C and D respectively, then the ratio P C : P D equals
(A) 2 (B) 1/2 (C) 4 (D) 1/4
42 The set of complex numbers z satisfying the equation

(3 + 7i)z + (10 − 2i)z̄ + 100 = 0

represents, in the complex plane,
(A) a straight line
(B) a pair of intersecting straight lines
(C) a pair of distinct parallel straight lines
(D) a point
43 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.

44 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
d e f
, and are in
a b c
(A) A.P. (B) G.P. (C) H.P. (D) none of the above.

45 The number of solutions of the equation sin−1 x = 2 tan−1 x is
(A) 1 (B) 2 (C) 3 (D) 5.

46 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◦ .

47 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.
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48 Let L be the point (t, 2) and M be a point on the y-axis such that LM
has slope −t. Then the locus of the midpoint of LM , as t varies over
all real values, is
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(A) y = 2 + 2x2 (B) y = 1 + x2

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(C) y = 2 − 2x2 (D) y = 1 − x2 .
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49 Let f : (0, 2) ∪ (4, 6) → R be a differentiable function. Suppose also
l as
as
that f ”(x) = 1 for all x ∈ (0, 2) ∪ (4, 6). Which of the following is
l ag
agALWAYS true?
(A) f is increasing
(B) f is one-to-one
(C) f (x) = x for all x ∈ (0, 2) ∪ (4, 6)
(D) f (5.5) − f (4.5) = f (1.5) − f (0.5)
50 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
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(C) a straight line
(D) none of the above.
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51 Let P be a variable point on a circle C and Q be a fixed point outside

ag
C. If R is the mid-point of the line segment P Q, then the locus of R is
(A) a circle (B) an ellipse
(C) a line segment (D) segment of a parabola

52 N is a 50 digit number. All the digits except the 26th from the right
are 1. If N is divisible by 13, then the unknown digit is
(A) 1 (B) 3 (C) 7 (D) 9.

53 Suppose a < b. The maximum value of the integral

m
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Z b 
3
m
2
−x−x dx

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a 4

e m over all possible values of a and b is s
la(D) 2 .
las (A)
3
(B)
4
(C)
3
ag
ag 4 3 2 3
1 1 1
54 For any n ≥ 5, the value of 1 + + + ··· + n lies between
2 3 2 −1
n n
(A) 0 and (B) and n
2 2
(C) n and 2n (D) none of the above.
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55 Let ω denote a cube root of unity which is not equal to 1. Then the
number of distinct elements in the set
(1 + ω + ω 2 + · · · + ω n )m : m, n = 1, 2, 3, · · ·


is
(A) 4 (B) 5 (C) 7 (D) infinite.

56 The value of the integral
Z 3
dx
2 loge x
(A) is less than 2 (B) is equal to 2
(C) lies in the interval (2, 3) (D) is greater than 3.

57 The area of the region bounded by the straight lines x = 12 and x = 2,
and the curves √given by the equations y = loge x and x
1 5 3 1
√ y =5 2 is
(A) log 2 (4 + 2) − 2 loge 2 + 2 (B) log 2 (4 − 2) − 2 loge 2
1
e √ 5 3
e
(C) log 2 (4 − 2) − 2 loge 2 + 2 (D) none of the above
e

58 In a win-or-lose game, the winner gets 2 points whereas the loser gets
0. Six players A, B, C, D, E and F play each other in a preliminary
round from which the top three players move to the final round. After
each player has played four games, A has 6 points, B has 8 points and
C has 4 points. It is also known that E won against F. In the next set
of games D, E and F win their games against A, B and C respectively.
If A, B and D move to the final round, the final scores of E and F are,
respectively,
(A) 4 and 2 (B) 2 and 4 (C) 2 and 2 (D) 4 and 4.

59 The number of ways in which one can select six distinct integers from
the set {1, 2, 3, · · · , 49}, such that no two consecutive integers are se-
lected, is
     
49 48 43
(A) −5 (B)
6 5 6
   
25 44
(C) (D) .
6 6
60 Let n ≥ 3 be an integer. Assume that inside a big circle, exactly n
small circles of radius r can be drawn so that each small circle touches
the big circle and also touches both its adjacent small circles. Then,
the radius of the big circle is
(A) r cosec πn (B) r(1 + cosec 2π
n )
π
(C) r(1 + cosec 2n ) (D) r(1 + cosec πn )
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61 If n is a positive integer such that 8n + 1 is a perfect square, then
(A) n must be odd
(B) n cannot be a perfect square
m
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(C) 2n cannot be a perfect square

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(D) none of the above
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62 Let C denote the set of all complex numbers. Define
l as
l as A = {(z, w) |z, w ∈ C and |z| = |w|}
ag
ag B = {(z, w) |z, w ∈ C, and z 2 = w2 }.
Then,
(A) A = B (B) A ⊂ B and A 6= B
(C) B ⊂ A and B 6= A (D) none of the above

63 Let f (x) = a0 +a1 |x|+a2 |x|2 +a3 |x|3 , where a0 , a1 , a2 , a3 are constants.
(A) f (x) is differentiable at x = 0 whatever be a0 , a1 , a2 , a3
(B) f (x) is not differentiable at x = 0 whatever be a0 , a1 , a2 , a3
m
Then

.co
(C) f (x) is differentiable at x = 0 only if a1 = 0
(D) f (x) is differentiable at x = 0 only if a1 = 0, a3 = 0

e m
2
64 If f (x) = cos(x) − 1 + x2 , then
l as
ag
(A) f (x) is an increasing function on the real line
(B) f (x) is a decreasing function on the real line
(C) f (x) is increasing on −∞ < x ≤ 0 and decreasing on 0 ≤ x < ∞
(D) f (x) is decreasing on −∞ < x ≤ 0 and increasing on 0 ≤ x < ∞

65 The number of roots of the equation x2 + sin2 x = 1 in the closed
interval [0, π2 ] is
(A) 0 (B) 1 (C) 2 (D) 3

m
m .co
66 The set of values of m for which mx2 −6mx+5m+1 > 0 for all real x is

.co em
(A) m < 14 (B) m ≥ 0

m s
1
(C) 0 ≤ m ≤ 4 (D) 0 ≤ m < 41

s e 67 The digit in the unit’s place of the number 1! + 2! + 3! + . . g la99! is
g la a (D) 7
.+

a (A) 3 (B) 0 (C) 1

3 3 3
68 The value of lim 1 +2 n+...+n
4 is:
n→∞
3
(A) 4 (B) 41 (C) 1 (D) 4

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n
69 For any integer n ≥ 1, define an = 1000
n! . Then the sequence {an }
(A) does not have a maximum
(B) attains maximum at exactly one value of n
(C) attains maximum at exactly two values of n
(D) attains maximum for infinitely many values of n

70 The equation x3 y + xy 3 + xy = 0 represents
(A) a circle (B) a circle and a pair of straight lines
(C) a rectangular hyperbola (D) a pair of straight lines

71 For each positive integer n, define a function fn on [0, 1] as follows:

 0 if x=0
π 1


sin if 0<x≤


2n n




2π 1 2




 sin if < x ≤
2n n n



fn (x) = 3π 2 3
sin if <x≤
2n n n




. . .





 .. .. ..



 sin nπ if n − 1 < x ≤ 1.



2n n
Z 1
Then, the value of lim fn (x) dx is
n→∞ 0
1 2
(A) π (B) 1 (C) (D) .
π π
72 Let d1 , d2 , . . . , dk be all the factors of a positive integer n including 1
and n. If d1 + d2 + . . . + dk = 72, then d11 + d12 + · · · + d1k is:
k2 72 72
(A) 72 (B) k (C) n (D) none of the above

73 A subset W of the set of real numbers is called a ring if it contains 1
and if for all a, b ∈ W , the numbers na − b and ab are also inoW . Let
S = 2mn | m, n integers and T = pq | p, q integers, q odd . Then


(A) neither S nor T is a ring (B) S is a ring T is not a ring
(C) T is a ring S is not a ring (D) both S and T are rings
74 A rod AB of length 3 rests on a wall. P is a point on AB such that
AP : P B = 1 : 2. If the rod slides along the wall, then the locus of P
lies on
(A) 2x + y + xy = 2
(B) 4x2 + y 2 = 4
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(C) 4x2 + xy + y 2 = 4
(D) x2 + y 2 − x − 2y = 0.
75 Consider the equation x2 + y 2 = 2007. How many solutions (x, y) exist
m
co
such that x and y are positive integers?

om .
(A) None

. c
(B) Exactly two
em
m
(C) More than two but finitely many
e l as
l as
(D) Infinitely many.
76 Consider the functions f1 (x) = x, f2 (x) = 2 + loge x, x > 0 (where e is
ag
agthe base of natural logarithm). The graphs of the functions intersect
(A) once in (0, 1) and never in (1, ∞)
(B) once in (0, 1) and once in (e2 , ∞)
(C) once in (0, 1) and once in (e, e2 )
(D) more than twice in (0, ∞).
77 Consider the sequence
n
X r
un = , n ≥ 1.
2r
r=1

m
.co
Then the limit of un as n → ∞ is
(A) 1 (B) 2 (C) e (D) 1/2.

e m
78 Suppose that z is any complex number which is not equal to any of

as
{3, 3ω, 3ω 2 } where ω is a complex cube root of unity. Then
l
+ g +
z − 3 za− 3ω z − 3ω
1 1 1
2

equals
2 +3z 2 2 2
(A) 3z
(z−3)3
(B) 3zz 3+3ωz
−27
(C) z 3 −3z3z
2 +9z−27 (D) z 33z−27 .
79 Consider all functions f : {1, 2, 3, 4} → {1, 2, 3, 4} which are one-one,
onto and satisfy the following property:
if f (k) is odd then f (k + 1) is even, k = 1, 2, 3.
The number of such functions is
m
.co
(A) 4 (B) 8 (C) 12 (D) 16.
m
.co
80 A function f : R → R is defined by

e m
s
− x1

m

f (x) = e , x > 0
s e 0 x ≤ 0.
g la
g la Then a
a (A) f is not continuous
(B) f is differentiable but f 0 is not continuous
(C) f is continuous but f 0 (0) does not exist
(D) f is differentiable and f 0 is continuous.
81 The last digit of 9! + 39966 is
(A) 3 (B) 9 (C) 7 (D) 1.
13

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82 Consider the function
2x2 + 3x + 1
f (x) = , 2 ≤ x ≤ 3.
2x − 1
Then
(A) maximum of f is attained inside the interval (2, 3)
(B) minimum of f is 28/5
(C) maximum of f is 28/5
(D) f is a decreasing function in (2, 3).
83 A particle P moves in the plane in such a way that the angle between
the two tangents drawn from P to the curve y 2 = 4ax is always 90◦ .
The locus of P is
(A) a parabola (B) a circle (C) an ellipse (D) a straight line.
84 Let f : R → R be given by
f (x) = |x2 − 1|, x ∈ R.
Then
(A) f has a local minima at x = ±1 but no local maximum
(B) f has a local maximum at x = 0 but no local minima
(C) f has a local minima at x = ±1 and a local maximum at x = 0
(D) none of the above is true.
85 The number of triples (a, b, c) of positive integers satisfying
2a − 5b 7 c = 1
is
(A) infinite (B) 2 (C) 1 (D) 0.
86 Let a be a fixed real number greater than −1. The locus of z ∈ C
satisfying |z − ia| = Im(z) + 1 is
(A) parabola (B) ellipse (C) hyperbola (D) not a conic.
87 Consider the function f : R \ {1} → R \ {2} given by
2x
f (x) = .
x−1
Then
(A) f is one-one but not onto
(B) f is onto but not one-one
(C) f is neither one-one nor onto
(D) f is both one-one and onto.
88 Consider a real valued continuous function f satisfying f (x+1) = f (x)
for all x ∈ R. Let
Z t
g(t) = f (x) dx, t ∈ R.
0

Define h(t) = limn→∞ g(t+n)
n , provided the limit exists. Then
(A) h(t) is defined only for t = 0
(B) h(t) is defined only when t is an integer
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(C) h(t) is defined for all t ∈ R and is independent of t
(D) none of the above is true.
89 Consider the sequence a1 = 241/3 , an+1 = (an + 24)1/3 , n ≥ 1. Then
m
co
the integer part of a100 equals

om .
(A) 2 (B) 10 (C) 100 (D) 24.

. c
90 Let x, y ∈ (−2, 2) and xy = −1. Then the minimum value of
em
e m 4 9
l as
s
+

l a 4−x 2 9 − y2
ag
agis
(A) 8/5 (B) 12/5 (C) 12/7 (D) 15/7.
91 What is the limit of
√
 1 n 2 + n
1+ 2
n +n
as n → ∞?
(A) e (B) 1 (C) 0 (D) ∞.
4 2
92 Consider the function f (x) = x + x + x − 1, x ∈ (−∞, ∞). The
function
(A) is zero at x = −1, but is increasing near x = −1
m
.co
(B) has a zero in (−∞, −1)
(C) has two zeros in (−1, 0)
e m
l as
(D) has exactly one local minimum in (−1, 0).
93 Consider a sequence of 10 A’s and 8 B’s placed in a row. By a run we

ag
mean one or more letters of the same type placed side by side. Here
is an arrangement of 10 A’s and 8 B’s which contains 4 runs of A and
4 runs of B:
AAAB B AB B B AAB AAAAB B
In how many ways can 10 A’s and 8 B’s be arranged in a row so that
there are 4
runs

of A and 49
runs

of B?
(A) 2 93 73 (B) 3 73 (C) 10

8

(D) 10

8

4 4 5 5 .
94 Suppose n ≥ 2 is a fixed positive integer and
f (x) = xn |x|, x ∈ R.
m
m .co
.co
Then
(A) f is differentiable everywhere only when n is even
e m
e m (B) f is differentiable everywhere except at 0 if n is odd
las
las (C) f is differentiable everywhere
ag
g
(D) none of the above is true.
a 95 The line 2x + 3y − k = 0 with k > 0 cuts the x axis and y axis at
points A and B respectively. Then the equation of the circle having
AB as diameter is
(A) x2 + y 2 − k2 x − k3 y = k 2
(B) x2 + y 2 − k3 x − k2 y = k 2
(C) x2 + y 2 − k2 x − k3 y = 0
15

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(D) x2 + y 2 − k3 x − k2 y = 0.
96 Let α > 0 and consider the sequence
(α + 1)n + (α − 1)n
xn = , n = 1, 2, . . . .
(2α)n
Then limn→∞ xn is
(A) 0 for any α > 0
(B) 1 for any α > 0
(C) 0 or 1 depending on what α > 0 is
(D) 0, 1 or ∞ depending on what α > 0 is.
97 If 0 < θ < π/2 then
(A) θ < sin θ
(B) cos(sin θ) < cos θ
(C) sin(cos θ) < cos(sin θ)
(D) cos θ < sin(cos θ).
98 Assume the following inequalities for positive integer k:
1 √ √ 1
√ < k+1− k < √ .
2 k+1 2 k
The integer part of
9999
X 1
√
k=2
k
equals
(A) 198 (B) 197 (C) 196 (D) 195.
99 Consider the sets defined by the inequalities
A = {(x, y) ∈ R2 : x4 + y 2 ≤ 1}, B = {(x, y) ∈ R2 : x6 + y 4 ≤ 1}.
Then
(A) B ⊆ A
(B) A ⊆ B
(C) each of the sets A − B, B − A and A ∩ B is non-empty
(D) none of the above is true.
100 The number of one-to-one functions from a set with 3 elements to a
set with 6 elements is
(A) 20
(B) 120
(C) 216
(D) 720
101 The minimum value of the function f (x) = x2 + 4x + x4 + x12 where
x > 0, is
(A) 9.5 (B) 10 (C) 15 (D) 20
102 The angle between the hyperbolas xy = 1 and x2 − y 2 = 1 (at their
point of intersection) is
(A) π2 (B) π3 (C) π4 (D) π6
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103 Given two complex numbers z, w with unit modulus (i.e., |z| = |w| =
1), which of the following
√ statements
√ will ALWAYS be correct?
(A) |z + w| < √ 2 and |z − w| < √ 2
m
co
(B) |z + w| ≤ √2 and |z − w| ≥√ 2

om .
(C) |z + w| ≥ √2 or |z − w| ≥ √2

. c
(D) |z + w| < 2 or |z − w| < 2
em
e m l as
a s
l Answers to selected problems. ag
g
Hints and
a
There are also other ways to solve the problems apart from the ones sketched
in the hints. Indeed, a student should feel encouraged upon finding a different
way to solve some of these problems.
Hints and Answers to selected UGA Sample Questions.

1 (B). Take the nth root of an and bn and use A.M.≥ G.M.
3 (A). As 2004 = 2000 +4, the last digits of (2004)5 and 45 are equal.
4 (D) Use binomial expansion of (bc + a (b + c))6 .
m
.co
6 (B) Let y = log10 x. Then log10 y = log100 4. Hence y = 2.
8 (D) Check for ‘test points’.

e m −1
14 (D) sin x13 changes sign at the points (nπ) 3 for all n ≥ 1.
s


x
15 (D) Observe that (e −1) tan2 x
=
l a
(ex −1) sin2 x
· x2 · cos12 x .
ag ofisdifferentiation.
x 3 x
16 (C) Use induction and chain rule
60
22 (B) Show that the height function . t
26 (C) Compute the number of maps such that f (3) = 5, f (3) = 4 etc..
Alternatively, define g : {1, 2, 3} → {1, 2, . . . , 7} by g (i) = f (i) + (i − 1).
Then, g is a strictly increasing function and its image is a subset of size 3 of
{1, 2, . . . 7}.
28 (D) Draw graphs of (x + y)(x − y) = 0 and (x − a)2 + y 2 = 1.
38 (A) Differentiate.
51 (A) Compute for C = x2 + y 2 = 1 and Q = (a, 0) for some a > 1.


R2 R2
m
.co
57 (C) Compute the integral 2x dx − log xdx.
m (D) Let s be distance between the centre of the big circle and the centrem
c. o 60of (any)
1/2 1/2

m s e
s e l a
one of the small circles. Then there exists a right angle triangle
g
la
π
with hypoteneuse s, side r and angle .
61(C) If 8n + 1 = m , then 2n is a product of two consecutive a
n

ag
2 integers.
62 (C) z 2 = w2 ⇒ z = ±w ⇒ B ⊆ A. But |i| = 1 and i2 6= 1.
63 (C) Amongst 1, |x|, |x|2 , |x|3 , only |x| is not differentiable at 0.
64 (D) Look at the derivative of f .
65 (B) Draw graphs of y = cos x and y = ±x and find the number of points
of intersections.
66 (D) Calculate the discriminant (b2 − 4ac) of the given quadratic.
17

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67 (A) The unit digit of all numbers n! with n ≥ 5 is 0.
n
i3 .
P
68 (B) Use the formula for
i=1
69 (C) Find out the first values of n for which an+1an becomes < 1.
2 2
70 (D) The equation is xy(x + y + 1) = 0.
72 (C) Multiply the given sum by n.
73 (D) Verify using the given definition of a ring.
75 (A) Observe that one of x, y is odd and the other one is even. Square
of an even number is divisible by 4 whereas square of an odd number leaves
remainder 1 when divided by 4. Compare this with the right hand side.
76 (C) Check that f1 (1) < f2 (1), f1 (e) < f2 (e) and f1 (e2 ) > f2 (e2 ).
83 (D) Note that a tangent to the parabola y 2 = 4ax has equation of the
a a
form y = mx + m . Coordinates of P satisfy two equations: y = mx + m and
x
y = − m − ma. Eliminate m.
84 (C) The function f is non-negative and it vanishes only at 1 and −1. The
derivative vanishes at x = 0 and it does not exist at x = √1, x = −1.
 n2 +√n  n2 +n  n22+ n
1 1 n +n
91 (A) Write 1 + n2 +n = 1 + n2 +n .
00 0 0
92 (D) As f = 12x2 + 2 > 0, the function f is increasing. Now f (−1) < 0
0
whereas f (0) > 0.

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Sample Questions for UGB
Instructions UGB consists of questions that will require you to provide
answers with appropriate justification.
m
co
1 Find the sum of all distinct four digit numbers that can be formed

. c om
using the digits 1, 2, 3, 4, 5, each digit appearing at most once.
em .
e m
2 How many natural numbers less than 108 are there, with sum of digits
l as
l as
equal to 7?
ag
ag
3 Consider the squares of an 8 × 8 chessboard filled with the numbers 1
to 64 as in the figure below. If we choose 8 squares with the property
that there is exactly one from each row and exactly one from each
column, and add up the numbers in the chosen squares, show that the
sum obtained is always 260.
1 2 3 4 5 6 7 8
9 10 11 12 13 14 15 16
17 18 19 20 21 22 23 24
25 26 27 28 29 30 31 32
33 34 35 36 37 38 39 40
m
.co
41 42 43 44 45 46 47 48
49 50 51 52 53 54 55 56
e m
s
57 58 59 60 61 62 63 64

l a
logg(2 + x) − x sin x
4 Consider the function

f (x) = lim a
2n
e
n→∞ 1+x 2n

defined for x > 0. Is f (x) continuous at x = 1? Justify your answer.
Show that f (x) does not vanish anywhere in the interval 0 ≤ x ≤ π2 .
Indicate the points where f (x) changes sign.
5 An isosceles triangle with base 6 cms. and base angles 30o each is
inscribed in a circle. A second circle, which is situated outside the
triangle, touches the first circle and also touches the base of the triangle
m
.co
at its midpoint. Find its radius.

m
.co m
6 Suppose a is a complex number such that

m
1 1
s e
e la
a2 + a + + 2 + 1 = 0.

las a a
If m is a positive integer, find the value of
ag
ag 1
a2m + am + m + 2m .
1
a a
7 Let an = 1 . . . 1 with 3n digits. Prove that an is divisible by 3an−1 .
8 Let f (u) be a continuous function and, for any real number u, let [u]
denote the greatest integer less than or equal to u. Show that for any
19

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x > 1,
Z x [x] Z x
X
[u]([u] + 1)f (u)du = 2 i f (u)du.
1 i=1 i

9 If a circle intersects the hyperbola y = 1/x at four distinct points
(xi , yi ), i = 1, 2, 3, 4, then prove that x1 x2 = y3 y4 .
10 Two intersecting circles are said to be orthogonal to each other if the
tangents to the two circles at any point of intersection are perpendicu-
lar to each other. Show that every circle through the points (2, 0) and
(−2, 0) is orthogonal to the circle x2 + y 2 − 5x + 4 = 0.
11 Show that the function f (x) defined below attains a unique minimum
for x > 0. What is the minimum value of the function? What is the
value of x at which the minimum is attained?
1 1
f (x) = x2 + x + + 2 for x 6= 0.
x x
Sketch on plain paper the graph of this function.
12 Show that there is exactly one value of x which satisfies the equation
2 cos2 (x3 + x) = 2x + 2−x .

13 Let S = {1, 2, . . . , n}. Find the number of unordered pairs {A, B} of
subsets of S such that A and B are disjoint, where A or B or both
may be empty.
14 An oil-pipe has to connect the oil-well O and the factory F , between
which there is a river whose banks are parallel. The pipe must cross
the river perpendicular to the banks. Find the position and nature of
the shortest such pipe and justify your answer.
15 Find the maximum value of x2 + y 2 in the bounded region, including
the boundary, enclosed by y = x2 , y = − x2 and x = y 2 + 1.
16 Let x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) where x1 , · · · , xn , y1 , · · · , yn
are real numbers. We write x > y if either x1 > y1 or for some k,
with 1 ≤ k ≤ n − 1, we have x1 = y1 , . . . , xk = yk , but xk+1 > yk+1 .
Show that for u = (u1 , . . . , un ), v = (v1 , . . . , vn ), w = (w1 , . . . , wn ) and
z = (z1 , . . . , zn ), if u > v and w > z, then u + w > v + z.
17 How many real roots does x4 + 12x − 5 have?
18 For any positive integer n, let f (n) be the remainder obtained on
dividing n by 9. For example, f (263) = 2.
(a) Let n be a three-digit number and m be the sum of its digits.
Show that f (m) = f (n).
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(b) Show that f (n1 n2 ) = f (f (n1 ) · f (n2 )) where n1 , n2 are any two
positive three-digit integers.
19 Find the maximum among 1, 21/2 , 31/3 , 41/4 , . . ..
m
om
20 Show that it is not possible to have a triangle with sides a, b and c
. co
. c
whose medians have lengths 32 a, 23 b and 45 c.
em
21 For real m
e numbers x, y and z, show that
l as
l as|x| + |y| + |z| ≤ |x + y − z| + |y + z − x| + |z + x − y|.
ag
g
22a Let X, Y , Z be the angles of a triangle.
(i) Prove that
X Y X Z Z Y
tan tan + tan tan + tan tan = 1.
2 2 2 2 2 2
(ii) Using (i) or otherwise prove that
X Y Z 1
tan tan tan ≤ √ .
2 2 2 3 3

m
m
23 Let α be a real number. Consider the function .co
e
swhich g is continuous at all x.
2
g(x) = (α + | x |)2 e(5−|x|) , −∞ < x < ∞.
(i) Determine the values of α l a
gfor which g is differentiable at all x.
for
(ii) Determine the values ofaα

24 Write the set of all positive integers in a triangular array as
1 3 6 10 15 . .
2 5 9 14 . . .
4 8 13 . . . .
7 12 . . . . .
11 . . . . . .
m
.co
Find the row number and column number where 20096 occurs. For
m
.co m
example 8 appears in the third row and second column.

m s e
s e g la
la
8 7 2

a
25 Show that the polynomial x − x + x − x + 15 has no real root.

ag
26 Let m be a natural number with digits consisting entirely of 6’s and
0’s. Prove that m is not the square of a natural number.

P.T.O.
21

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27 Let 0 < a < b.
(i) Show that amongst the triangles with base a and perimeter a + b,
the maximum area is obtained when the other two sides have
equal length 2b .
(ii) Using the result of (i) or otherwise show that amongst the quadri-
lateral of given perimeter the square has maximum area.
28 Let n ≥ 1, S = {1, 2, . . . , n}. For a function f : S → S, a subset D ⊂ S
is said to be invariant under f , if f (x) ∈ D for all x ∈ D. Note that
the empty set and S are invariant for all f . Let deg(f ) be the number
of subsets of S invariant under f .
(i) Show that there is a function f : S → S such that deg(f ) = 2.
(ii) Further show that for any k such that 1 ≤ k ≤ n there is a
function f : S → S such that deg(f ) = 2k .

29 Let
P (x) = xn + an−1 xn−1 + an−2 xn−2 + · · · + a1 x + a0
be a polynomial with integer coefficients, such that P (0) and P (1) are
odd integers. Show that:
(a) P (x) does not have any even integer as root.
(b) P (x) does not have any odd integer as root.
30 Let N = {1, 2, . . . , n} be a set of elements called voters. Let C =
{S : S ⊆ N } be the set of all subsets of N . Members of C are called
coalitions. Let f be a function from C to {0, 1}. A coalition S ⊆ N
is said to be winning if f (S) = 1; it is said to be a losing coalition
if f (S) = 0. A pair hN, f i as above is called a voting game if the
following conditions hold.
(a) N is a winning coalition.
(b) The empty set ∅ is a losing coalition.
(c) If S is a winning coalition and S ⊆ S 0 , then S 0 is also winning.
(d) If both S and S 0 are winning coalitions, then S ∩ S 0 6= ∅, i.e., S
and S 0 have a common voter.
Show that the maximum number of winning coalitions of a voting
game is 2n−1 . Find a voting game for which the number of winning
coalitions is 2n−1 .
31 Suppose f is a real-valued differentiable function defined on [1, ∞) with
f (1) = 1. Suppose, moreover, that f satisfies f 0 (x) = 1/(x2 + f 2 (x)).
Show that f (x) ≤ 1 + π/4 for every x ≥ 1.
32 If the normal to the curve x2/3 + y 2/3 = a2/3 at some point makes an
angle θ with the X-axis, show that the equation of the normal is
y cos θ − x sin θ = a cos 2θ.

33 Suppose that a is an irrational number.
22

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(a) If there is a real number b such that both (a+b) and ab are rational
numbers, show that a is√a quadratic √ surd. (a is a quadratic surd
if it is of the form r + s or r − s for some rationals r and s,
m
co
where s is not the square of a rational number).

om .
(b) Show that there are two real numbers b1 and b2 such that

. c
(i) a + b1 is rational but ab1 is irrational.
em
em
(ii) a + b2 is irrational but ab2 is rational.
l as
l as (Hint: Consider the two cases, where a is a quadratic surd
and a is not a quadratic surd, separately).
ag
34ag
Let A, B, and C be three points on a circle of radius 1.
(a) Show that the area of the triangle ABC equals
1
(sin(2∠ABC) + sin(2∠BCA) + sin(2∠CAB)) .
2
(b) Suppose that the magnitude of ∠ABC is fixed. Then show that
the area of the triangle ABC is maximized when ∠BCA = ∠CAB.
(c) Hence or otherwise show that the area of the triangle ABC is
maximum when the triangle is equilateral.

m
.co
35 In the given figure, E is the midpoint of the arc E
ABEC and ED is perpendicular to the chord BC

em
C
at D. If the length of the chord AB is l1 , and that of
s
B
D

a
BD is l2 , determine the length of DC in terms of l1
l
a, xg> 0. Show that f (x) is an increasing
A
and l2
36 (a) Let f (x) = x − xe−1/x
function on (0, ∞), and limx→∞ f (x) = 1.
(b) Using part (a) and calculus, sketch the graphs of y = x − 1, y = x,
y = x + 1, and y = xe−1/|x| for −∞ < x < ∞ using the same X
and Y axes.
37 For any integer n greater than 1, show that
2n
 
n 2n
2 < < n−1 .
n
m
.co
Y i
m
(1 − )

.co
n
i=0

e m
e m 38 Show that there exists a positive real number x 6= 2 such that log2 x =
las
las x
2 . Hence obtain the set of real numbers c such that
ag
ag log2 x
x
=c
has only one real solution.
39 Find a four digit number M such that the number N = 4 × M has the
following properties.
(a) N is also a four digit number.
23

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(b) N has the same digits as in M but in the reverse order.
40 Consider a function f on nonnegative integers such that f (0) = 1,
f (1) = 0 and f (n) + f (n − 1) = nf (n − 1) + (n − 1)f (n − 2) for n ≥ 2.
Show that
n
f (n) X (−1)k
= .
n! k!
k=0

41 Of all triangles with a given perimeter, find the triangle with the max-
imum area. Justify your answer.
42 A 40 feet high screen is put on a vertical wall 10 feet above your eye-
level. How far should you stand to maximize the angle subtended by
the screen (from top to bottom) at your eye?
43 Study the derivatives of the function
p
y = x3 − 4x
and sketch its graph on the real line.
44 Suppose P and Q are the centres of two disjoint circles C1 and C2
respectively, such that P lies outside C2 and Q lies outside C1 . Two
tangents are drawn from the point P to the circle C2 , which intersect
the circle C1 at points A and B. Similarly, two tangents are drawn
from the point Q to the circle C1 , which intersect the circle C2 at
points M and N . Show that AB = M N .
 
1 2n
45 Evaluate: lim log .
n→∞ 2n n
46 Consider the equation x5 + x = 10. Show that
(a) the equation has only one real root;
(b) this root lies between 1 and 2;
(c) this root must be irrational.
47 In how many ways can you divide the set of eight numbers {2, 3, . . . , 9}
into 4 pairs such that no pair of numbers has g.c.d. equal to 2?
48 Suppose S is the set of all positive integers. For a, b ∈ S, define
l.c.m(a, b)
a∗b=
g.c.d(a, b)
For example, 8 ∗ 12 = 6.
Show that exactly two of the following three properties are satis-
fied :
(a) If a, b ∈ S then a ∗ b ∈ S.
(b) (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c ∈ S.
(c) There exists an element i ∈ S such that a ∗ i = a for all a ∈ S.

24

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Hints and Answers to selected UGB Sample Questions.

1. The answer is 399960. For each x ∈ {1, 2, 3, 4, 5}, there are 4! such
m
co
numbers whose last digit is x. Thus the digits in the unit place of all the

om .
120 numbers add up to 4! (1 + 2 + 3 + 4 + 5). Similarly the numbers at ten’s

. c
place add up to 360 and so on. Thus the sum is 360 (1 + 10 + 100 + 1000).
e m
e m
3. Let the chosen entries be in the positions (i, ai ), 1 ≤ i ≤ 8. Thus a1 , . . . , a8
l as
as
is a permutation of {1, . . . , 8}. The entry in the square corresponding to
l ag
ag
(i, j)th place is i + 8 (j − 1). Hence the required sum is
8
P
(i + 8 (aj − 1)).
i=1
√
5. Radius is 3 2 3 . Use trigonometry.
n
7. Observe that an = an−1 1 + t + t2 where t = 103



9. Substitute y = x1 in the equation of a circle and clear denominator to
get a degree 4 equation in x. The product of its roots is the constant term,
which is 1.

m
.co
11. The function f (x) − 4 is a sum of squares and hence non-negative. So
the minimum is 4 which is attained at x = 1.
n

e m
l as
13. The number is 3 2+1 . An ordered pair (A, B) of disjoint subsets of S is
determined by 3 choices for every element of S (either it is in A, or in B or

ag
in neither of them). Hence such pairs are 3n in number. An unordered pair
will be counted twice in this way, except for the case A and B both empty.
n
Hence the number is 1 + 3 2−1 .
15. Answer is 5. The maximum is attained at points (2, 1) and (2, −1).
17. Answer is 2. Let f be the given polynomial. Then f (0) is negative and
f is positive as x tends to ±∞. √ Hence it has at least 2 real roots. Since the
derivative of f is zero only at 3 −3, it cannot have more than two real roots.
√3 1
19. Maximum is √ 3. Either check the maximum of the function x x , or
√
compare 3 3 with n n.
o m
m . c
. co β = y + z − x, γ = x + z − y, and use the triangle inequality.
21. Rewrite the given inequality in terms of the new variables α = x + y
s
−
e m
z,

s em 22. (i) Using the additive formula for tan(A/2 + B/2) and observing g l a that
g la ◦
tan(A/2 + B/2) = tan(90 − C/2) = cot(C/2), it is easy to prove a (i).
a (ii) Note that tan(A/2), tan(B/2), tan(C/2) are positive numbers. So the
arithmetic mean of tan(A/2) tan(B/2), tan(C/2) tan(B/2), tan(A/2) tan(C/2)
is greater than or equal to its geometric mean. Use this together with (i).
2
23. For any α, , h(x) = (α + x)2 e(5−x) , −∞ < x < ∞ is continuous and
differentiable at all x and note that f (x) = h(| x |), −∞ < x < ∞.
25

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(i) As | x | is a continuous function, f is a continuous function at all x for
any real number α.
(ii) As | x | is a differentiable function at all x 6= 0 f is differentiable at all
x 6= 0 for any real number α. At x = 0, find the right hand and the left
hand derivatives of f . Check that f 0 (0) exists if and only if α = 0 or α = 15 .
24. The top row has the n-th triangular number n(n+1)/2 at the n-th place.
Now (200 × 201)/2 = 20100 > 20096 > 19900 = (199 × 200)/2. So, 20100
occurs on the first row and 200-th column and is on the first row of this 200-
th slanted line. Therefore 20096 = 20100 − 4 occurs on the 200 − 4 = 196-th
column and on the 5-th row.
25. The polynomial can be re-written as x7 (x − 1) + x(x − 1) + 15. It it easily
seen that the polynomial is strictly positive when x ≤ 0 and x ≥ 1. Further
if 0 < x < 1, then | x7 (x − 1) |< 1 and | x(x − 1) |< 1. This implies that the
polynomial is strictly positive for all real numbers x.
26. Suppose n is a perfect square. If 10|n, then 102 |n. Hence n ends in
an even (possibly zero) number of 0’s. So, for some k (possibly zero), the
number 10n2k ends in 6 and has digits consisting of 0’s and 6’s. Therefore 10n2k
ends in 66 or in 06. In both cases the number is even and leaves a remainder
2 on division by 4. This is a contradiction as an even perfect square is a
multiple of 4.
28. (i) If f (i) = i + 1 for all i = 1, · · · , n − 1, with f (n) = 1 then deg(f ) = 2.
(ii) Consider disjoint subsets A1 , A2 , · · · , Ak of S such that each Ai has more
than one element. Let f be a cyclic function in each Ai . Then deg(f ) = 2k .

26

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A Model Question Paper for B.Math/B.Stat

Booklet No. Test Code : UGA
m
om
Forenoon
. co
. c em
e m l as
l as ag
ag Questions : 30 Time : 2 hours

Write your Name, Registration Number, Test Centre, Test Code and the
Number of this Booklet in the appropriate places on the Answersheet.

m
m .co
s e
l a
g of the suggested answers is correct.
This test contains 30 questions in all. For each of the 30 questions, there
are four suggested answers. Onlyaone
You will have to identify the correct answer in order to get full credit for
that question. Indicate your choice of the correct answer by darkening the
appropriate oval completely on the answersheet.
You will get
4 marks for each correctly answered question,
0 marks for each incorrectly answered question and
1 mark for each unattempted question.

m
m .co
m .co s e m
s e g la
la You are not allowed to use calculator. a
All rough work must be done on this booklet only.

ag

WAIT FOR THE SIGNAL TO START.
27

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1 The system of inequalities
1 1 1 1
a − b2 ≥ , b − c 2 ≥ , c − d 2 ≥ , d − a 2 ≥
4 4 4 4
m
co
(where a, b, c, d are real numbers) has
(A) no solutions
. c om (B) exactly one solution
em .
(C) exactly two solutions
e m
(D) infinitely many solutions.

l as
l s
2 Let log12 18 = a. Then log24 16 is equal to
a 8 − 4a 1 4a − 1 8 − 4a
ag
ag (A)
5−a
(B)
3+a
(C)
2 + 3a
(D)
5+a
.

3 The number of solutions of the equation tan x + sec x = 2 cos x, where
0 ≤ x ≤ π, is
(A) 0 (B) 1 (C) 2 (D) 3.
4 Using only the digits 2, 3 and 9, how many six digit numbers can be
formed which are divisible by 6?
(A) 41 (B) 80 (C) 81 (D) 161
5 What is the value of the following integral?
m
.co
Z 2014
tan−1 x
dx
m
1 x
e
2014

π π
l as
(A) log 2014 (B) log 2014 (C) π log 2014 (D) log 2014
1

ag
4 2 2
6 A light ray travelling along the line y = 1, is reflected by a mirror
placed along the line x = 2y. The reflected ray travels along the line
(A) 4x − 3y = 5 (B) 3x − 4y = 2 (C) x − y = 1 (D) 2x − 3y = 1.
7 For a real number x, let [x] denote the greatest integer less than or
equal to x. Then the number of real solutions of 2x − [x] = 4 is
(A) 1 (B) 2 (C) 3 (D) 4.
8 What is the ratio of the areas of the regular pentagons inscribed inside
and circumscribed around a given circle?
(A) cos 36◦ (B) cos2 36◦ (C) cos2 54◦
o
(D) cos2 72◦ m
m 9 Let z , z be nonzero complex numbers satisfying |z + z | = |z − z |. .c
. co 1 2

s e m
The circumcentre of the triangle with the points z , z , and the origin
1 2 1 2

em 1 la
1 2

s
as its vertices is given by

la a3g(z − z ).
1 1 1
(A) (z − z ) (B) (z + z ) (C) (z + z ) (D)

ag
1 2 1 2 1 2 1 2
2 3 2
10 In how many ways can 20 identical chocolates be distributed among 8
students so that each student gets at least one chocolate and exactly
two students get at least two chocolates each?   
8 17
(A) 308 (B) 364 (C) 616 (D)
2 7
1

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Page 29 of 35

11 Two vertices of a square lie on a circle of radius r, and the other two
vertices lie on a tangent to this circle. Then, each side of the square is
3r 4r 6r 8r
(A) (B) (C) (D) ·
2 3 5 5
12 Let P be the set of all numbers obtained by multiplying five distinct
integers between 1 and 100. What is the largest integer n such that
2n divides at least one element of P ?
(A) 8 (B) 20 (C) 24 (D) 25
13 Consider the function f (x) = ax3 + bx2 + cx + d, where a, b, c and
d are real numbers with a > 0. If f is strictly increasing, then the
0 00 000
function g(x) = f (x) − f (x) + f (x) is
(A) zero for some x ∈ R (B) positive for all x ∈ R
(C) negative for all x ∈ R (D) strictly increasing.

14 Let A be the set of all points (h, k) such that the area of the triangle
formed by (h, k), (5, 6) and (3, 2) is 12 square units. What is the least
possible length of a line segment joining (0, 0) to a point in A?
4 8 12 16
(A) √ (B) √ (C) √ (D) √
5 5 5 5
15 Let P = {ab c : a, b, c positive integers, a2 + b2 = c2 , and 3 divides c}.
What is the largest integer n such that 3n divides every element of P ?
(A) 1 (B) 2 (C) 3 (D) 4
16 Let A0 = ∅ (the empty set). For each i = 1, 2, 3, . . . , define the set
Ai = Ai−1 ∪ {Ai−1 }. The set A3 is
(A) ∅ (B) {∅} (C) {∅, {∅}} (D) {∅, {∅}, {∅, {∅}}}
1
17 Let f (x) = · The graphs of the functions f and f −1 intersect at
x−2
√ √ √ √
(A) (1 + √ 2, 1 + √ 2) and (1
√ − 2, 1 − 2)
(B) (1 + 2, 1 + 2) and ( 2, −1 − √12 )
√ √ √
(C) (1 − 2, 1 − 2) and (− 2, −1 + √12 )
√ √
(D) ( 2, −1 − √12 ) and (− 2, −1 + √12 )
18 Let N be a number such that whenever you take N consecutive positive
integers, at least one of them is coprime to 374. What is the smallest
possible value of N ?
(A) 4 (B) 5 (C) 6 (D) 7

19 Let A1 , A2 , . . . , A18 be the vertices of a regular polygon with 18 sides.
How many of the triangles 4Ai Aj Ak , 1 ≤ i < j < k ≤ 18, are isosceles
but not equilateral?
(A) 63 (B) 70 (C) 126 (D) 144
2

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sinα x
20 The limit lim exists only when
x→0 x
(A) α ≥ 1 (B) α = 1
(C) |α| ≤ 1
m
co
(D) α is a positive integer.

. c
is the area of R?om
21 Consider the region R = {(x, y) : x2 + y 2 ≤ 100, sin(x + y) > 0}. What

em .
e m
(A) 25π (B) 50π (C) 50 (D) 100π − 50
l as
l as
22 Consider a cyclic trapezium whose circumcentre is on one of the sides.
ag
agIf the ratio of the two parallel sides is 1 : 4, what is the ratio of the
sum of √the two
√ oblique sides to the longer parallel
√ side? √ √
(A) 3 : 2 (B) 3 : 2 (C) 2 : 1 (D) 5 : 3
( √ !)2
4 + 2x
23 Consider the function f (x) = loge for x > 0. Then,
x

(A) f decreases upto some point and increases after that
(B) f increases upto some point and decreases after that
m
.co
(C) f increases initially, then decreases and then again increases
(D) f decreases initially, then increases and then again decreases.

e m
as
24 What is the number of ordered triplets (a, b, c), where a, b, c are positive
l
ag
integers (not necessarily distinct), such that abc = 1000?
(A) 64 (B) 100 (C) 200 (D) 560
25 Let f : (0, ∞) → (0, ∞) be a function differentiable at 3, and satisfying
f (3) = 3f 0(3) > 0. Then the limit
   x
3
f 3 + x 
lim  
x→∞  f (3) 

(A) exists and is equal to 3 (B) exists and is equal to e
m
m (C) exists and is always equal to f (3) (D) need not always exist.
.co
m .co 1
s em
e la √
26 Let z be a non-zero complex number such that z − = 2. What is
s
z

la
the maximum value of |z|?
√ g
a(D) 1 + 2.
ag (A) 1 (B) 2 (C) 2
27 The minimum value of
sin x + cos x + tan x + cosec x + sec x + cot x is
√ √
(A) 0 (B) 2 2 − 1 (C) 2 2 + 1 (D) 6

3

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28 For any function f : X → Y and any subset A of Y , define
f −1 (A) = {x ∈ X : f (x) ∈ A}.
Let Ac denote the complement of A in Y . For subsets A1 , A2 of Y ,
consider the following statements:
(i) f −1 (Ac1 ∩ Ac2 ) = (f −1 (A1 ))c ∪ (f −1 (A2 ))c
(ii) If f −1 (A1 ) = f −1 (A2 ) then A1 = A2 .
Then,
(A) both (i) and (ii) are always true
(B) (i) is always true, but (ii) may not always be true
(C) (ii) is always true, but (i) may not always be true
(D) neither (i) nor (ii) is always true.
29 Let f be a function such that f 00 (x) exists, and f 00 (x) > 0 for all
x ∈ [a, b]. For any point c ∈ [a, b], let A(c) denote the area of the
region bounded by y = f (x), the tangent to the graph of f at x = c
and the lines x = a and x = b. Then
(A) A(c) attains its minimum at c = 12 (a + b) for any such f
(B) A(c) attains its maximum at c = 21 (a + b) for any such f
(C) A(c) attains its minimum at both c = a and c = b for any such f
(D) the points c where A(c) attains its minimum depend on f .

30 In 4ABC, the lines BP , BQ trisect ∠ABC and the lines CM , CN
trisect ∠ACB. Let BP and CM intersect at X and BQ and CN
intersect at Y . If ∠ABC = 45◦ and ∠ACB = 75◦ , then ∠BXY is
A

M
P
X
N ?
Q

Y

B C
◦
(A) 45◦ (B) 47 12 (C) 50◦ (D) 55◦

4

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A Model Question Paper for B.Math/B.Stat

BOOKLET No. TEST CODE : UGB
Afternoon Session
m
om . co
. c em
e
There m are 3 pages in this booklet. l as
l as
The exam has 8 questions.
ag
ag Answer as many as you can.

Time : 2 hours

o m
c
. Centre, Test Code and the
Write your Name, Registration number, Test m
e on the answer-booklet.
Number of this booklet in the appropriatesplaces

g l a
a
ALL ROUGH WORK IS TO BE DONE ON THIS BOOKLET
AND/OR THE ANSWER-BOOKLET.
CALCULATORS ARE NOT ALLOWED.

STOP! WAIT FOR THE SIGNAL TO START.
m
m .co
m .co s e m
s e g la
g la a
a

P.T.O.
5

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Page 33 of 35

1 A class has 100 students. Let ai , 1 ≤ i ≤ 100, denote the number of
friends the i-th student has in the class. For each 0 ≤ j ≤ 99, let cj
denote the number of students having at least j friends. Show that
100
X 99
X
ai = cj .
i=1 j= 1

2 It is given that the graph of y = x4 + ax3 + bx2 + cx + d (where a, b, c, d
are real) has at least 3 points of intersection with the x-axis. Prove
that either there are exactly 4 distinct points of intersection, or one of
those 3 points of intersection is a local minimum or maximum.

3 Consider a triangle P QR in R2 . Let A be a point lying on 4P QR
or in the region enclosed by it. Prove that, for any function f (x, y) =
ax + by + c on R2 ,
f (A) ≤ max {f (P ), f (Q), f (R)} .
4 Let f and g be two non-decreasing twice differentiable functions de-
fined on an interval (a, b) such that for each x ∈ (a, b), f 00 (x) = g(x)
and g 00 (x) = f (x). Suppose also that f (x)g(x) is linear in x on (a, b).
Show that we must have f (x) = g(x) = 0 for all x ∈ (a, b).

5 Show that the sum of 12 consecutive integers can never be a perfect
square. Give an example of 11 consecutive integers whose sum is a
perfect square.

6 Let A be the region in the xy-plane given by
A = {(x, y) : x = u + v, y = v, u2 + v 2 ≤ 1} .
Derive the length of the longest line segment that can be enclosed in-
side the region A.

7 Let f : [0, ∞) → R be a non-decreasing continuous function. Show
then that the inequality
Zz Zz
(z − x) f (u)du ≥ (z − y) f (u)du
y x

holds for any 0 ≤ x < y < z. [P. T. O]
8 Consider n (> 1) lotus leaves placed around a circle. A frog jumps
from one leaf to another in the following manner. It starts from some
selected leaf. From there, it skips exactly one leaf in the clockwise
direction and jumps to the next one. Then it skips exactly two leaves
in the clockwise direction and jumps to the next one. Then it skips
three leaves again in the clockwise direction and jumps to the next
one, and so on. Notice that the frog may visit the same leaf more than
6

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once. Suppose it turns out that if the frog continues this way, then all
the leaves are visited by the frog sometime or the other. Show that n
cannot be odd.
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om . co
. c em
em l as
l as ag
ag

m
m .co
s e
l a
ag

m
m .co
m .co s e m
s e g la
g la a
a

7

m . c
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