Firstranker's choice act subset of R?
(A) {1/n : n ? N} ? {0} ? (0, p] (Correct Answer)
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(B) {1/n : n ? N} ? {0, 1}
(C) {1, 2, 3} ? [4,5] ? {6 + 1/n : n ? N}
(D) {1 - 1/n : n ? N} ? (1,2)
Question No.2 (Question Id - 10)
Let G be an abelian group of order 16. Which of the following is true ?
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(A) There exists g ? G such that order of g is 8.
(B) If there exists a subgroup H of G of order 8, then there exists g ? G with order 8.
(C) If there exists g ? G with order 8, then G is cyclic.
(D) There is a one-to-one group homomorphism f : G ? Sm for some m = 1. (Correct Answer)
Question No.3 (Question Id - 8)
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Consider the system of linear equations :
3x + y - z = a
- x + 2y + 5z = ß
4x + z = 7
For which a and ß does this system have a unique solution ?
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(A) For no a, ß ? R there is a unique solution.
(B) a is unique but ß can be arbitrary.
(C) a and ß are both unique.
(D) For all a, ß ? R there is a unique solution. (Correct Answer)
Question No.4 (Question Id - 6)
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What are the maximum and minimum of the function f (x) = ex - x on the interval [-1/2, 2] ?
(A) 1 + 1/e and ve - 1/2
(B) 1 + 1/e and 1. (Correct Answer)
(C) ve - 1/2 and 1.
(D) ve - 1/2 and 1 + 1/e
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Question No.5 (Question Id - 1)
Let X, Y and Z be finite sets and let f : X ? Y and g : Y ? Z be maps.
Which of the following assertions is always true?
(A) If g o f is a bijection, then both g and f are bijections.
(B) If g is one to one, then g o f is also one to one.
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(C) If f is onto, then g o f is also onto.
(D) If g o f is onto, then |Z| = |Y|, where |A| denotes the number of elements in any finite set A. (Correct Answer)
Question No.6 (Question Id - 9)
Question No.7 (Question Id - 7)
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What is the value of the integral ?0log3 dx / (e-x + ex) ?
(A) p/12 (Correct Answer)
(B) log (v3 + 1 / v3 - 1) + log2
(C) v3 + 1 / v3 - 1 log2
(D) p/6
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Question No.8 (Question Id - 3)
Let an = (-1)n+1 (1 + 1/(2n+1)) for n = 1. Which of the following is correct ?
(A) lim sup an = 1 and lim inf an = -1. (Correct Answer)
(B) lim sup an = 1 and lim inf an = 1.
(C) lim sup an = -1 and lim inf an = -1.
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(D) lim sup an = -1 and lim inf an = 1.
Question No.9 (Question Id - 2)
Consider a series ? an of real numbers. Which of the following assertions is necessarily true ?
(A) If |an| = (n+1)/n3 for all n = 1, then ? an converges conditionally but it does not necessarily converge absolutely.
(B) If |an| = (n+1)/n3 for all n = 1, then ? an converges conditionally.
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(C) If 1/2 * (n+1)/n3 = an = (n+1)/n3 for all n = 1, then ? an converges absolutely. (Correct Answer)
(D) If 0 = an = (n+1)/n3 for all n = 1, then ? an converges absolutely.
Question No.10 (Question Id - 5)
The sum 1/1001 + 1/1002 + ... + 1/2000 is:
A. less than 1.
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B. more than 1/2.
C. more than log2.
D. less than log2.
Which of the above assertions are correct?
(A) A, B and C only
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(B) A, B and D only (Correct Answer)
If {xn} is bounded, then it contains a Cauchy subsequence.
If {xn} is bounded, then it cannot contain a convergent subsequence.
D. If {xn} is bounded and a subsequence of {xn} converges to a real number L, then {xn} also converges to L.
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Which of the above assertions is/are true?
(A) A only
(B) A and B only (Correct Answer)
(C) B and C only
(D) D only
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Question No.2 (Question Id - 23)
Consider the subset
N = { (1 b) : b ? R }
of the group of 2 × 2 matrices
G = { (a b) : a, b, d ? R, ad=1 }
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under matrix multiplication. Which of the following statements is correct ?
(A) N is not a subgroup of G.
(B) N is a subgroup of G, but N is not normal.
(C) N is a subgroup of G and the number of cosets of N in G is finite.
(D) N is a subgroup of G and there are infinitely many cosets of N in G. (Correct Answer)
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Question No.3 (Question Id - 22)
Let G be a group in which every element other than identity has order 2. Then, which of the following statements is necessarily true?
(A) G must be finite and abelian.
(B) G can be infinite, but G must be abelian. (Correct Answer)
(C) G is not necessarily abelian, but it must be finite.
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(D) G may be non-abelian as well as infinite.
Question No.4 (Question Id - 18)
What is the area of the portion of the sphere x² + y² + z² = R² lying between the planes z = R and z = v3R/2 ?
(A) R²(2 - v3) (Correct Answer)
(B) pR²/v3
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(C) 1/2 R²
(D) 1/4 R²
Question No.5 (Question Id - 21)
Let A ? M4x3(R), B ? M3x4(R) and C ? M4x5(R). Consider the following assertions :
A. The matrix ABC cannot have rank equal to 4.
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B. AB can have rank 3 but BC cannot have rank 4.
C. ABC and BA can have ranks at most 3.
D. Rank of AB must be less than or equal
Question No.7 (Question Id - 17)
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A function f: (a, b) ? R is said to be uniformly continuous if for every ? > 0 there exists a d > 0 such that |f (x) - f (y)| < ? whenever |x - y| < d (and d is independent of x and y).
Let f: (0, 1) ? R be the map given by f(x) = vx. Consider the following assertions:
A. f is differentiable on (0, 1).
B. f is differentiable and f ' is bounded on (0, 1).
C. f is uniformly continuous on (0, 1).
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D. f is differentiable and f ' is uniformly continuous on (0, 1).
Which of the above assertions is/are correct?
(A) A only
(B) A and C only (Correct Answer)
(C) A, B and C only
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(D) A, B and D only
Question No.8 (Question Id - 13)
Let ? anxn be a power series with real coefficients and radius of convergence R such that 0 < R < 8. Consider the following assertions:
(A) If ? anxn converges for some x with |x| = R, then ? anxn converges for every x with |x| = R.
(B) If |x| > R, then sup |? anxn| = 8.
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(C) If ? anxn diverges for some x with |x| = R, then ? anxn diverges for every x with |x| = R.
(D) Let Sk(x) = ? anxn for all k = 2. Then, {Sk(x)} is a Cauchy sequence for every x ? (-R, R).
Which of the above assertions is/are correct?
(A) A and C only
(B) B and D only (Correct Answer)
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(C) D only
(D) B only
Question No.9 (Question Id - 19)
Let S be the sphere x² + y² + z² = R², F be the vector field on R³ given by F = x³i + y³j + z³k and n denotes the unit normal vector to the surface S. What is the value of the surface integral ? F · n ds?
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A. sup{|x| sin x : x ? R} = 8 and inf{|x| sin
B. Let f: R ? R be the map given by f (x) = 2x + 3. Then, sup{f (sin(x) + 5): x ? R} = 15 and inf{ f (sin(x) + 5) : x ? R} = 11
C. Let f: R ? R be the map given by f (x) = 5x + 5. Then, inf{f (1/n) sin (1/n) : n = 1} = 0
D. Let f: R ? R be the map given by f (x) = 3x + 4. Then, sup{f (f (x)) : x ? (0, 2)} = 34
Which of the above assertions is/are correct?
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(A) B and D only (Correct Answer)
(B) B, C and D only
(C) A, C and D only
(D) A and D only
Question No.11 (Question Id - 11)
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Let X = {(x, y) ? R² : x² + y² = 1}, Y = {(x, y) ? R² : x = y} and Z = {(x, y) ? R² : y = - x}. Consider the following assertions :
A. X ? Y ? Z is an equivalence relation on R.
B. X ? Y is a reflexive relation on R but not symmetric.
C. X ? Y is an equivalence relation on R.
D. Y ? Z is an equivalence relation on R.
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Which of the above assertions are correct?
(A) A and B only
(B) A, B and D only
(C) A and D only (Correct Answer)
(D) A, C and D only
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Question No.12 (Question Id - 20)
Let V be a finite dimensional vector space over R with dim V = 2. Fix a non-zero vector v0 ? V. Consider the following assertions :
A. There is a unique basis of V containing v0.
B. There exist infinitely many bases of V containing v0.
C. There is a unique injective linear map T : V ? V such that T(v0) = v0.
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D. There exist infinitely many linear isomorphisms T : V ? V such that T(v0) = v0.
Which of the above assertions is/are correct?
(A) A only
(B) C only
(C) A and C only
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(D) B and D only (Correct Answer)
Question No.14 (Question id-24)
For which of the following does n! have 2020 as a factor?
(A) n = 8097 (Correct Answer)
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(B) n = 8085
(C) n = 8080
(D) n = 10100
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