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- The number of subsets of { 1 , 2 , 3 ,... , 10 } having an odd number of ele- ments is
(A) 1024 (B) 512 (C) 256 (D) 50.
- For the function on the real line R given by f (x) = |x| + |x + 1| + ex, which of the following is true?
(A) It is differentiable everywhere. (B) It is differentiable everywhere except at x = 0 and x = −1. (C) It is differentiable everywhere except at x = 1/2. (D) It is differentiable everywhere except at x = − 1 /2.
- If f, g are real-valued differentiable functions on the real line R such that f (g(x)) = x and f ′(x) = 1 + (f (x))^2 , then g′(x) equals
(A)
1 + x^2 (B) 1 + x^2 (C)
1 + x^4 (D) 1 + x^4.
- The number of real solutions of ex^ = sin(x) is
(A) 0 (B) 1 (C) 2 (D) infinite.
- What is the limit of
∑^ n
k=
e−k/n n
as n tends to ∞?
(A) The limit does not exist. (B) ∞ (C) 1 − e−^1 (D) e−^0.^5
- A group of 64 players in a chess tournament needs to be divided into 32 groups of 2 players each. In how many ways can this be done?
(A)
32!2^32
(B)
(C)
(D)
- The integral part of
n=
n equals
(A) 196 (B) 197 (C) 198 (D) 199.
- Let an be the number of subsets of { 1 , 2 ,... , n} that do not contain any two consecutive numbers. Then
(A) an = an− 1 + an− 2 (B) an = 2an− 1 (C) an = an− 1 − an− 2 (D) an = an− 1 + 2an− 2.
- There are 128 numbers 1, 2 ,... , 128 which are arranged in a circular pattern in clockwise order. We start deleting numbers from this set in a clockwise fashion as follows. First delete the number 2, then skip the next available number (which is 3) and delete 4. Continue in this manner, that is, after deleting a number, skip the next available number clockwise and delete the number available after that, till only one number remains. What is the last number left?
(A) 1 (B) 63 (C) 127 (D) None of the above.
- Let z and w be complex numbers lying on the circles of radii 2 and 3 respectively, with centre (0, 0). If the angle between the corresponding vectors is 60 degrees, then the value of |z + w|/|z − w| is:
(A)
(B)
(C)
(D)
- The number of complex roots of the polynomial z^5 − z^4 − 1 which have modulus 1 is
(A) 0 (B) 1 (C) 2 (D) more than 2.
- The number of real roots of the polynomial
p(x) = (x^2020 + 2020x^2 + 2020)(x^3 − 2020)(x^2 − 2020)
is
(A) 2 (B) 3 (C) 2023 (D) 2025.
- Which of the following is the sum of an infinite geometric sequence whose terms come from the set { 1 , 12 , 14 ,... , (^21) n ,.. .}?
(A)
(B)
(C)
(D)
- If a, b, c are distinct odd natural numbers, then the number of rational roots of the polynomial ax^2 + bx + c
(A) must be 0.
(B) must be 1.
(C) must be 2.
(D) cannot be determined from the given data.
- Let A, B, C be finite subsets of the plane such that A ∩ B, B ∩ C and C ∩ A are all empty. Let S = A ∪ B ∪ C. Assume that no three points of S are collinear and also assume that each of A, B and C has at least 3 points. Which of the following statements is always true?
(A) There exists a triangle having a vertex from each of A, B, C that does not contain any point of S in its interior.
(B) Any triangle having a vertex from each of A, B, C must contain a point of S in its interior.
(C) There exists a triangle having a vertex from each of A, B, C that contains all the remaining points of S in its interior.
(D) There exist 2 triangles, both having a vertex from each of A, B, C such that the two triangles do not intersect.
- Shubhaangi thinks she may be allergic to Bengal gram and takes a test that is known to give the following results:
- For people who really do have the allergy, the test says “Yes” 90% of the time.
- For people who do not have the allergy, the test says “Yes” 15% of the time. If 2% of the population has the allergy and Shubhaangi’s test says “Yes”, then the chances that Shubhaangi does really have the allergy are
(A) 1/ 9
(B) 6/ 55
(C) 1/ 11
(D) cannot be determined from the given data.
- If sin(tan−^1 (x)) = cot(sin−^1 (
13 17 )) then^ x^ is
(A)
(B)
(C)
172 − 132 172 +13^2 (D)
172 − 132 17 × 13.
- Let S = { 1 , 2 ,... , n}. For any non-empty subset A of S, let l(A) denote the largest number in A. If f (n) =
A⊆S l(A), that is,^ f^ (n) is the sum of the numbers l(A) while A ranges over all the nonempty subsets of S, then f (n) is
(A) 2n(n + 1) (B) 2n(n + 1) − 1 (C) 2n(n − 1) (D) 2n(n − 1) + 1.
- The area of the region in the plane R^2 given by points (x, y) satisfying |y| ≤ 1 and x^2 + y^2 ≤ 2 is
(A) π + 1 (B) 2π − 2 (C) π + 2 (D) 2π − 1.
- Let n be a positive integer and t ∈ (0, 1). Then
∑^ n
r=
r
n r
tr(1 − t)n−r
equals
(A) nt (B) (n − 1)(1 − t) (C) nt + (n − 1)(1 − t) (D) (n^2 − 2 n + 2)t.
- For any real number x, let [x] be the greatest integer m such that m ≤ x. Then the number of points of discontinuity of the function g(x) = [x^2 −2] on the interval (− 3 , 3) is
(A) 5 (B) 9 (C) 13 (D) 16.
