Title K-2620 (Paper-II) (Mathematical Sciences).pdf ✅

Instructions for the Candidates 1. Write your roll number in the space provided on the top of this page. 2. This paper consists of Hundred multiple-choice type of questions. 3. At the commencement of examination, the question booklet will be given to you. In the first 5 minutes, you are requested to open the booklet and compulsorily examine it as below : (i) To have access to the Question Booklet, tear off the paper seal on the edge of the cover page. Do not accept a booklet without sticker seal or open booklet. (ii) Tally the number of pages and number of questions in the booklet with the information printed on the cover page. Faulty booklets due to pages/questions missing or duplicate or not in serial order or any other discrepancy should be got replaced immediately by a correct booklet from the invigilator within the period of 5 minutes. Afterwards, neither the Question Booklet will be replaced nor any extra time will be given. 4. Each item has four alternative responses marked (A), (B), (C) and (D). You have to darken the circle as indicated below on the correct response against each item. Example : A B C D where (C) is the correct response. 5. Your responses to the questions are to be indicated in the OMR Sheet kept inside this Booklet. If you mark at any place other than in the circles in the OMR Sheet, it will not be evaluated. 6. Read the instructions given in OMR carefully. 7. Rough Work is to be done in the end of this booklet. 8. If you write your name or put any mark on any part of the OMR Answer Sheet, except for the space allotted for the relevant entries, which may disclose your identity, you will render yourself liable to disqualification. 9. You have to return the OMR Answer Sheet to the invigilators at the end of the examination compulsorily and must not carry it with you outside the Examination Hall. 10. You can take away question booklet and carbon copy of OMR Answer Sheet after the examination. 11. Use only Blue/Black Ball point pen. 12. Use of any calculator, electronic gadgets or log table etc., is prohibited. 13. There is no negative marks for incorrect answers. 14. In case of any discrepancy found in the Kannada translation of a question booklet the question in English version shall be taken as final. Name & Signature of Invigilator/s Signature : _________________________________ Name : _________________________________ Time : 2 Hours Maximum Marks : 200 Number of Pages in this Booklet : 24 Number of Questions in this Booklet : 100 Book let SERIA L N o. Paper : II Subject : Mathematical Sciences Subject Code : 26 Roll No. OMR Sheet No. : ____________________ K – 2620 1 ±Üâ.£.®æãà./P.T.O. A»Ü¦ìWÜÚWæ ÓÜãaÜ®æWÜÙÜá 1. D ±Üâo ̈Ü ÊæáàÆᤩ¿áÈÉ J ̈ÜXÔ ̈Ü ÓܧÙÜ ̈ÜÈÉ ̄ÊÜá3⁄4 ÃæãàÇ... ®ÜíŸÃÜ®Üá° ŸÃæÀáÄ. 2. 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Total Number of Pages : 24 Paper II 2 K-2620 Mathematical Sciences Paper – II Note : This paper contains hundred (100) objective type questions of two (2) marks each. All questions are compulsory. 1. The number of pairs of integers a and b satisfying 0 < a < b and ab = ba is (A) 0 (B) 1 (C) 2 (D) infinite 2. Let ( )!( )! !( )! ( ) 2 3 0 4 n n n n I n= ∞ ∑ −− and 1 3 2 2 1 n II n n n + = ∞ ∑ −− ( ) then (A) Series (I) converges and series (II) diverges (B) Series (I) diverges and series (II) converges (C) Both the series (I) and (II) converge (D) Both the series (I) and (II) diverge 3. The series ( ) − + = + ∞ ∑ 1 2 1 1 3 5 n n n n is (A) Divergent (B) Convergent (C) Conditionally convergent (D) Absolutely convergent 4. The radius of convergence of the power series 1 1 1 2 1 1 +       +       +            =  ∞ ∑ n n n n z n n ... is (A) e 4 (B) 4 e (C) 4e (D) e4
Total Number of Pages : 24 K-2620 3 Paper II 5. The set [e, π] ∩ Q is (A) compact (B) connected (C) compact but not connected (D) neither compact nor connected 6. Suppose that g : → is continuous on  and g(x) = 0, for every rational x. Then (A) g 7 ( ) < 0 (B) g 7 ( ) > 0 (C) g 7 ( ) = 0 (D) g 7 ( ) ≠ 0 7. Let fn : →be differentiable for each n = 1, 2, ..., with | ( f x n ′ ) | ≤ 1, for all n and x. Assume lim ( ) ( ) n nf x g x →∞ = . Then (A) g is continuous for all x (B) g is continuous only for x > 0 (C) g is continuous only for x < 0 (D) g is not continuous 8. The sum of the series 1 1 1 2 2 1 2 3 3 2 2 2 2 2 2 ! ! ! + .... + + + + + is (A) 6 e (B) 6 17e (C) 17 6 e (D) 6 17 e 9. The value of lim n n n →∞ ( + ) 3 1 2 is (A) 1 (B) + ∞ (C) 0 (D) 2
Total Number of Pages : 24 Paper II 4 K-2620 10. Which one of the following statements is false ? (A) If f is bounded and has finitely many discontinuities on [a, b], then f is Riemann integrable on [a, b] (B) If f is monotonic on [a, b], then f is Riemann integrable on [a, b] (C) If f is continuous on [a, b], then f is Riemann integrable on [a, b] (D) If f and g are not Riemann integrable on [a, b], then fg is not Riemann integrable on [a, b] 11. Let f : 2 →1 be defined by f x y xy x y ( , ) = +2 2 , if (x, y) ≠ (0, 0) and f(0, 0) = 0. Which one of the following statement is true ? (A) D1 f(0, 0) = 0 (B) f is not continuous at (0, 0) (C) D2 f(0, 0) = 0 (D) f is continuous at (0, 0) 12. lim ... n n n →∞ n + + + +       1 1 2 2 3 1 = (A) 0 (B) ∞ (C) 1 (D) limit does not exist 13. Let R denote an arbitrary 3 × 4 matrix of rank 2 and O denote the 3 × 4 matrix all of whose entries are 0. What is the rank of the following 12 × 16 matrix ? RORO R R R R OROR OOO R − −             (A) 2 (B) 4 (C) 6 (D) 8 14. Let A be a real 2 × 2 matrix such that A8 = I but A4 ≠ I, where I denotes the identity matrix of size 2 × 2. Then the trace of A equals (A) ± 2 (B) 0 (C) ±1 (D) ± 1 2