Indian Statistical Institute of Kolkata

Can anyone please give me the details of getting in UG courses in the Indian Statistical Institute of Kolkata?

[Quick Sam]

Indian Statistical Institute of Kolkata was established back in the year of 1931 and at that time it was having an expenditure of Rs 250 and now it is having expenditure of over Rs 15,000,000.

Courses Offered:
B.Stat. (Hons.)
B.Math. (Hons.)

Important Dates for Entrance Exam:
On-line Application Starts: 03 February 2015
On-line Application Ends: 03 March 2015
Payment of Application Fee at SBI Branches Starts: 06 February 2015
Payment of Application Fee at SBI Branches Ends: 10 March 2015
ISI Admission Test: 10 May 2015

These dates for the year 2015 have been passed so these dates can be taken as tentative for the year of 2016

Entrance Exam Centers:
SL TEST CENTRE CODE SL TEST CENTRE CODE
1 AGARTALA AG
2 AHMEDABAD AD
3 BALURGHAT BL
4 BENGALURU BG
5 BHOPAL BP
6 BHUBANESWAR BH
7 CHANDIGARH CH
8 CHENNAI CN
9 COCHIN CO
10 DELHI DH
11 DHANBAD DB
12 DIBRUGARH DG
13 DURGAPUR DP
14 GUNTUR GT
15 GUWAHATI GH
16 HYDERABAD HY
17 IMPHAL IM
18 JAIPUR JP
19 KHARAGPUR KH
20 KOLKATA CC
21 LUCKNOW LU
22 MANGALORE MN
23 MUMBAI MB
24 NAGPUR NG
25 NAINITAL NL
26 PATNA PT
27 PUNE PU
28 RAIPUR RP
29 SHILLONG SL
30 SILIGURI SG
31 SRINAGAR SR
32 SURAT ST
33 TEZPUR TZ
34 VARANASI VN
35 VISAKHAPATNAM VP

Application Fees:
Rs 600.00 for all applicants in the general category and Rs 300.00 for SC/ST/OBC/Physically Challenged candidates

Eligibility Criteria:
B.Stat (Hons):
Candidates must have passed Higher Secondary (10+2) or equivalent with Mathematics and English

B.Math (Hons):
Candidates must have passed Higher Secondary (10+2) or equivalent with Mathematics and English

Previous year question papers of entrance exam for B.Stat Hons and B.Math Hons
1. The system of inequalities
a − b2 ≥
1
4
, b − c2 ≥
1
4
, c − d2 ≥
1
4
, d − a2 ≥
1
4
has
(A) no solutions (B) exactly one solution
(C) exactly two solutions (D) infinitely many solutions.
2. Let log12 18 = a. Then log24 16 is equal to
(A)
8 − 4a
5 − a
(B)
1
3 + a
(C)
4a − 1
2 + 3a
(D)
8 − 4a
5 + a
.
3. The number of solutions of the equation tan x+sec x = 2 cos x, where 0 ≤ x ≤ _,
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?
Z 2014
1
2014
tan−1 x
x
dx
(A)
_
4
log 2014 (B)
_
2
log 2014 (C) _ log 2014 (D)
1
2
log 2014
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◦ (D) cos2 72◦
1
9. Let z1, z2 be nonzero complex numbers satisfying |z1 + z2| = |z1 − z2|. The
circumcentre of the triangle with the points z1, z2, and the origin as its vertices
is given by
(A)
2
(z1 − z2) (B)
1
3
(z1 + z2) (C)
1
2
(z1 + z2) (D)
1
3
(z1 − z2).
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?
(A) 308 (B) 364 (C) 616 (D) _8
2__17
7 _
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
(A)
3r
2
(B)
4r
3
(C)
6r
5
(D)
8r
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 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?
(A)
4
√5
(B)
8
√5
(C)
12
√5
(D)
16
√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) {∅, {∅}, {∅, {∅}}}
17. Let f(x) =
1
x − 2 • The graphs of the functions f and f−1 intersect at
(A) (1 + √2, 1 + √2) and (1 − √2, 1 − √2)
(B) (1 + √2, 1 + √2) and (√2,−1 − 1
√2 )
(C) (1 − √2, 1 − √2) and (−√2,−1 + 1
√2 )
(D) (√2,−1 − 1
√2 ) and (−√2,−1 + 1
√2 )
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 △AiAjAk, 1 ≤ i < j < k ≤ 18, are isosceles but not
equilateral?
(A) 63 (B) 70 (C) 126 (D) 144
20. The limit lim
x→0
sin_ x
x
exists only when
(A) _ ≥ 1 (B) _ = 1
(C) |_| ≤ 1 (D) _ is a positive integer.
21. Consider the region R = {(x, y) : x2 + y2 ≤ 100, sin(x + y) > 0}. What is the
area of R?
(A) 25_ (B) 50_ (C) 50 (D) 100_ − 50
22. Consider a cyclic trapezium whose circumcentre is on one of the sides. If 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
23. Consider the function f(x) = (loge 4 + √2x
x !)2
for x > 0. Then,
(A) f decreases upto some point and increases after that
(B) f increases upto some point and decreases after that
(C) f increases initially, then decreases and then again increases
(D) f decreases initially, then increases and then again decreases.
24. What is the number of ordered triplets (a, b, c), where a, b, c are positive 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′(3) > 0. Then the limit
lim
x→∞

f _3 +
3
x_
f(3)

x
(A) exists and is equal to 3 (B) exists and is equal to e
(C) exists and is always equal to f(3) (D) need not always exist.
26. Let z be a non-zero complex number such that ____
z −
1
z ____
= 2. What is the
maximum value of |z|?
(A) 1 (B) √2 (C) 2 (D) 1 + √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
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(Ac
1 ∩ Ac
2) = (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′′(x) exists, and f′′(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 = 1
2 (a + b) for any such f
(B) A(c) attains its maximum at c = 1
2 (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 △ABC, 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
B C
M
N
P
Q
X
Y
?
(A) 45◦ (B) 471
2
◦ (C) 50◦ (D) 55◦
Short-Answer Type Test
Time: 2 hours
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
Xi=1
ai =
99
Xj=0
cj .
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 PQR in R2. Let A be a point lying on △PQR 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 defined on an
interval (a, b) such that for each x ∈ (a, b), f′′(x) = g(x) and g′′(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 + v2 ≤ 1} .
Derive the length of the longest line segment that can be enclosed inside the
region A.
7. Let f : [0,∞) → R be a non-decreasing continuous function. Show then that
the inequality
(z − x)
z
Zy
f(u)du ≥ (z − y)
z
Zx
f(u)du
holds for any 0 ≤ x < y < z.
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 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.

Contact Details:
Indian Statistical Institute
No. 203,
Barrackpore Trunk Road
Kolkata,
West Bengal 700108
India

Map Location:
[MAP]Indian Statistical Institute West Bengal[/MAP]