Tiêu đề K-2616 Mathematical Science (Paper - II).pdf ✅

Total Number of Pages : 8 Paper II 2 K-2616 1. Let an = (–1)n n 1 log(n 1) 2 4 + + . Then the sequence {an} (A) is not convergent (B) converges to 0 (C) converges to a non-zero limit (D) oscillates between − ∞ and ∞ 2. If f, g are real-valued functions, then max.(f, g) = (A) 2 f + g − f − g (B) 2 f − g − f + g (C) 2 f + g + f − g (D) 2 f − g + f − g 3. Let f : [a, b] → IR take the value 1 at rational points and –1 at irrational points. Then the upper and the lower Riemann integrals are respectively (A) b – a and –(b – a) (B) b – a and 0 (C) 1 and –1 (D) 1 and 0 4. Which one of the following has positive Lebesgue measure in IR ? (A) The cantor set (B) Z (C) Q (D) (a, b) – Q, where a < b MATHEMATICS Paper – II 5. The closed unit ball of a normed linear space X is norm compact if and only if (A) X is finite dimensional (B) X is reflexive (C) X is a Hilbert space (D) X is a Banach space 6. Suppose the matrix ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ − − − + + = yi 1 xi 1 3 2i 0 1 zi 3 x 2i yi A is Hermitian. Then (x, y, z) must be (A) (0, 3, 3) (B) (3, 0, 3) (C) (3, 3, 0) (D) (0, 0, 3) 7. The 2-dimensional subspaces of IR3 can be geometrically described as (A) All planes (B) All planes passing through the origin (C) All lines passing through the origin (D) The only planes x = 0, y = 0, z = 0 8. Let ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ = 0 0 3 0 2 3 1 1 2 A . The characteristic polynomial of A is given by (A) ± (λ + 1) (λ − 2) (λ − 3) (B) ± (λ −1) (λ + 2) (λ − 3) (C) ± (λ −1) (λ − 2) (λ + 3) (D) ± (λ −1) (λ − 2) (λ − 3) Note : This paper contains fifty (50) objective type questions. Each question carries two (2) marks. All questions are compulsory.
Total Number of Pages : 8 K-2616 3 Paper II 9. Let M be the set of all 2× 2 matrices with real entries and let f : M → IR be the determinant map Then (A) f is one-one and onto (B) f is neither one-one nor onto (C) f is one-one but not onto (D) f is onto but not one-one 10. The dimension of the vector space M = { [ ] aij m×n : aij ∈ } over the field IR of real numbers is (A) m + n (B) 2mn (C) 2(m + n) (D) mn 11. The radius of convergence of the power series ∑ n n 2 z is (A) 1 (B) 0 (C) 2 (D) 2 1 12. Let f be an analytic function in . A sufficient condition for f to be constant is (A) 0 n 1 f ⎟ = ⎠ ⎞ ⎜ ⎝ ⎛ for n = 1, 2, 3 ..... (B) f(n) = 0 for n = 0, ± 1, ± 2, ..... (C) f(0) = 0 (D) f(z) = 0 for z = 0, ± 1, ± i 13. If 1, ω, 2 ω are the cube roots of unity, then the value of ( )3 2 ω+ ω is (A) 0 (B) 1 (C) –1 (D) 3 14. The modulus of the complex number ez for any complex number z is (A) z e (B) eRez (C) eImz (D) e z 15. The locus of z +1 = z −1 in the complex plane is (A) A straight line (B) A circle (C) An ellipse (D) A parabola 16. If x is a positive integer satisfying x ≡ 3 mod 7 and x ≡ –3 mod 11, then (A) x = 3 (B) No such x exists (C) Exactly one such x exists (D) There are infinitely many such x 17. The number of primitive 11th roots of unity is (A) 1 (B) 2 (C) 10 (D) 5 18. The number of subgroups of a cyclic group of order 20 is (A) 6 (B) 2 (C) 10 (D) 1 19. For which prime P in the following, both – 1 and 2 are quadratic residues modulo P ? (A) 7 (B) 17 (C) 11 (D) 13 20. Let P be prime and let F be a field with P2 number of elements. Then the number of ideals in F is (A) 1 (B) P2 (C) 2 (D) P2 – 1 21. In X = [0, 1] with the topology given by the metric d(x, y) = x − y , [ ) 2 0, 1 is (A) Closed (B) Open (C) Both open and closed (D) Neither open nor closed