Tiêu đề K-2618 (Mathematical Science) (Paper II).pdf ✅

Test Paper : II Test Subject : Mathematical Sciences Test Subject Code : K-2618 Roll No. OMR Sheet No. : ____________________ Name & Signature of Invigilator/s Signature : _________________________________ Name : _________________________________ Time : 2 Hours Maximum Marks : 200 Number of Pages in this Booklet : 16 Number of Questions in this Booklet : 100 K-2618 1 ±Üâ.£.®æãà./P.T.O. 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 test 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. A»Ü¦ìWÜÚWæ ÓÜãaÜ®æWÜÙÜá 1. D ±Üâo ̈Ü ÊæáàÆᤩ¿áÈÉ J ̈ÜXÔ ̈Ü ÓܧÙÜ ̈ÜÈÉ ̄ÊÜá3⁄4 ÃæãàÇ... ®ÜíŸÃÜ®Üá° ŸÃæÀáÄ. 2. D ±Ü£ÅPæ¿áá ŸÖÜá BÁáR Ë«Ü ̈Ü ®ÜãÃÜá (100) ±ÜÅÍæ°WÜÙÜ®Üá° JÙÜWæãíw ̈æ. 3. ±ÜÄàPæÒ¿á ±ÝÅÃÜí»Ü ̈ÜÈÉ, ±ÜÅÍæ° ±ÜâÔ¤Pæ¿á®Üá° ̄ÊÜáWæ ̄àvÜÇÝWÜáÊÜâ ̈Üá. Êæã ̈ÜÆ 5 ̄ËáÐÜWÜÙÜÈÉ ̄àÊÜâ ±ÜâÔ¤Pæ¿á®Üá° ñæÃæ¿áÆá ÊÜáñÜᤠPæÙÜX®Üíñæ PÜvÝx¿áÊÝX ±ÜÄàQÒÓÜÆá PæãàÃÜÇÝX ̈æ. (i) ±ÜÍæ°±ÜâÔ¤PæWæ ±ÜÅÊæàÍÝÊÜPÝÍÜ ±Üvæ¿áÆá, D Öæã©Pæ ±Üâo ̈Ü Aíb®Ü ÊæáàÈÃÜáÊÜ ±æà±ÜÃ... ÔàÆ®Üá° ÖÜÄÀáÄ. ÔrPÜRÃ... ÔàÇ... CÆÉ ̈Ü A¥ÜÊÝ ñæÃæ ̈Ü ±ÜâÔ¤Pæ¿á®Üá° ÔÌàPÜÄÓÜ ̧æàw. (ii) ±ÜâÔ¤Pæ¿áÈÉ®Ü ±ÜÅÍæ°WÜÙÜ ÓÜíTæ ÊÜáñÜᤠ±ÜâoWÜÙÜ ÓÜíTæ¿á®Üá° ÊÜááS±Üâo ̈Ü ÊæáàÇæ ÊÜáá©ÅÔ ̈Ü ÊÜÞ×£Áãí©Wæ ñÝÙæ ®æãàwÄ. ±ÜâoWÜÙÜá/±ÜÅÍæ°WÜÙÜá PÝOæ¿Þ ̈Ü A¥ÜÊÝ ©Ì±ÜÅ£ A¥ÜÊÝ A®ÜáPÜÅÊÜáÊÝXÆÉ ̈Ü A¥ÜÊÝ CñÜÃÜ ¿ÞÊÜâ ̈æà ÊÜÂñÝÂÓÜ ̈Ü ̈æãàÐܱÜäÄñÜ ±ÜâÔ¤Pæ¿á®Üá° PÜãvÜÇæ 5 ̄ËáÐÜ ̈Ü AÊÜ ̃ JÙÜWæ, ÓÜíËàPÜÒPÜÄí ̈Ü ÓÜÄ CÃÜáÊÜ ±ÜâÔ¤PæWæ Ÿ ̈ÜÇÝÀáÔPæãÙÜÛ ̧æàPÜá. B ŸÚPÜ ±ÜÅÍ氱ܣÅPæ¿á®Üá° Ÿ ̈ÜÇÝÀáÓÜÇÝWÜáÊÜâ©ÆÉ, ¿ÞÊÜâ ̈æà ÖæaÜác ÓÜÊÜá¿áÊÜ®Üã° PæãvÜÇÝWÜáÊÜâ©ÆÉ. 4. ±ÜÅ£Áãí ̈Üá ±ÜÅÍæ°WÜã (A), (B), (C) ÊÜáñÜᤠ(D) Gí ̈Üá WÜáÃÜá£Ô ̈Ü ®ÝÆáR ±Ü¿Þì¿á EñܤÃÜWÜÚÊæ. ̄àÊÜâ ±ÜÅÍæ°¿á G ̈ÜáÃÜá ÓÜÄ¿Þ ̈Ü EñܤÃÜ ̈Ü ÊæáàÇæ, PæÙÜWæ PÝ~Ô ̈Üíñæ AívÝPÜꣿá®Üá° PܱÝ3XÓÜ ̧æàPÜá. E ̈ÝÖÜÃÜOæ : A B C D (C) ÓÜÄ¿Þ ̈Ü EñܤÃÜÊÝX ̈ÝaWÜ. 5. D ±ÜÅÍæ° ±Ü£ÅPæ¿á hæãñæ¿áÈÉ PæãqrÃÜáÊÜ OMR EñܤÃÜ ÖÝÙæ¿áÈÉ ̄ÊÜá3⁄4 EñܤÃÜWÜÙÜ®Üá° ÓÜãbÓÜñÜPÜR ̈Üáa. OMR ÖÝÙæ¿áÈÉ AívÝPÜꣿáÆÉ ̈æ ̧æàÃæ ¿ÞÊÜâ ̈æà ÓܧÙÜ ̈ÜÈÉ EñܤÃÜÊÜ®Üá° WÜáÃÜá£Ô ̈ÜÃæ, A ̈ÜÃÜ ÊÜåèÆÂÊÜÞ±Ü®Ü ÊÜÞvÜÇÝWÜáÊÜâ©ÆÉ. 6. OMR EñܤÃÜ ÖÝÙæ¿áÈÉ Pæãor ÓÜãaÜ®æWÜÙÜ®Üá° hÝWÜÃÜãPÜñæÀáí ̈Ü K©Ä. 7. GÇÝÉ PÜÃÜvÜá PæÆÓÜÊÜ®Üá° ±ÜâÔ¤Pæ¿á Pæã®æ¿áÈÉ ÊÜÞvÜñÜPÜR ̈Üáa. 8. ̄ÊÜá3⁄4 WÜáÃÜáñÜ®Üá° Ÿ×ÃÜíWܱÜwÓÜŸÖÜá ̈Ý ̈Ü ̄ÊÜá3⁄4 ÖæÓÜÃÜá A¥ÜÊÝ ¿ÞÊÜâ ̈æà bÖæ°¿á®Üá°, ÓÜíWÜñÜÊÝ ̈Ü ÓܧÙÜ ÖæãÃÜñÜá ±ÜwÔ, OMR EñܤÃÜ ÖÝÙæ¿á ¿ÞÊÜâ ̈æà »ÝWÜ ̈ÜÈÉ ŸÃæ ̈ÜÃæ, ̄àÊÜâ A®ÜÖÜìñæWæ ̧Ý«ÜÂÃÝWÜᣤàÄ. 9. ±ÜÄàPæÒ¿áá ÊÜááX ̈Ü®ÜíñÜÃÜ, PÜvÝx¿áÊÝX OMR EñܤÃÜ ÖÝÙæ¿á®Üá° ÓÜíËàPÜÒPÜÄWæ ̄àÊÜâ ×í£ÃÜáXÓÜ ̧æàPÜá ÊÜáñÜᤠ±ÜÄàPÝÒ PæãsÜw¿á ÖæãÃÜWæ OMR®Üá° ̄Êæã3⁄4í©Wæ Pæãívæã¿áÂPÜãvÜ ̈Üá. 10. ±ÜÄàPæÒ¿á ®ÜíñÜÃÜ, ±ÜÄàPÝÒ ±ÜÅÍ氱ܣÅPæ¿á®Üá° ÊÜáñÜᤠ®ÜPÜÆá OMR EñܤÃÜ ÖÝÙæ¿á®Üá° ̄Êæã3⁄4í©Wæ ñæWæ ̈ÜáPæãívÜá ÖæãàWÜŸÖÜá ̈Üá. 11. ̄àÈ/PܱÜâ3 ̧ÝÇ...±ÝÀáíp... ±æ®... ÊÜÞñÜÅÊæà E±ÜÁãàXÔÄ. 12. PÝÂÆáRÇæàoÃ..., Ë ̈Üá®Ý3⁄4®Ü E±ÜPÜÃÜ| A¥ÜÊÝ ÇÝW... pæàŸÇ... CñÝ©¿á E±ÜÁãàWÜÊÜ®Üá° ̄Ðæà ̃ÓÜÇÝX ̈æ. 13. ÓÜÄ AÆÉ ̈Ü EñܤÃÜWÜÚWæ Má| AíPÜ CÃÜáÊÜâ©ÆÉ . 14. PܮܰvÜ ÊÜáñÜᤠCíXÉàÐ... BÊÜ꣤WÜÙÜ ±ÜÅÍ氱ܣÅPæWÜÙÜÈÉ ¿ÞÊÜâ ̈æà Äࣿá ÊÜÂñÝÂÓÜWÜÙÜá PÜívÜáŸí ̈ÜÈÉ, CíXÉàÐ... BÊÜ꣤WÜÙÜÈÉÃÜáÊÜâ ̈æà Aí£ÊÜáÊæí ̈Üá ±ÜÄWÜ~ÓÜ ̧æàPÜá. Test Booklet SERIAL N o. (Figures as per admission card)
*K2618* Total Number of Pages : 16 Paper II 2 K-2618 1. For all x > 0, the inequality ex > xt holds if and only if (A) t > e–1 (B) t > e (C) t < e (D) t < e–1 2. Let the sequence {xn } be defined by x1 = 1, x2 = 2 and xn+1 = xn + xn – 1, for n ≥ 2. Then limn n n x →∞ x + = 1 (A) 3 1 2 − (B) 1 2 2 + (C) 1 3 2 + (D) 1 5 2 + 3. lim cos n k k n →∞ = ∏ = π 1 2 (A) π 2 (B) 2 π (C) 0 (D) 1 4. 1 5 5 π π π ∫ x cos x dx = − (A) 1 (B) π (C) 5π (D) 0 Mathematical Sciences Paper – II Note : This paper contains hundred (100) objective type questions of two (2) marks each. Answer all questions. 5. Let f : 2 →  be some function. Then which one of the following statement is true ? (A) If f is continuous at a point, then f has partial derivatives at that point (B) If f has partial derivatives at a point, then f is continuous at that point (C) If f has partial derivatives at a point, then f is differentiable at that point (D) If f is differentiable at a point, then f has partial derivatives at that point 6. The sum of the series 15 16 15 16 21 24 15 16 21 24 27 32 + ⋅ + ⋅ ⋅ + ... is (A) 64 9 (B) 47 9 (C) 56 9 (D) 55 9 7. The sum of the series 1 1 3 1 5 1 7 1 2 1 4 1 9 1 11 1 13 1 15 1 6 1 8 +++ −−+ + + + −−+ ... is (A) 3 2 log2 (B) 5 2 log 2 (C) 1 2 log2 (D) log 23
*K2618* Total Number of Pages : 16 K-2618 3 Paper II 8. The series 1 1 1 2 1 k k k k + −( )      ∑  = ∞ is (A) Convergent (B) Oscillating (C) Divergent (D) Conditionally convergent 9. Let {an } be a sequence of positive real numbers such that lim . n n n a →∞ a + < 1 1 Then limn an →∞ = (A) 0 (B) 1 (C) ∞ (D) limit does not exist 10. If an > 0 for all n ≥ 1 and an n= ∞ ∑ 1 converges, then a an n n + = ∞ ∑ 1 1 (A) converges (B) diverges (C) oscillates (D) converges to the same sum as an n= ∞ ∑ 1 11. lim ( )! ! n n n n n →∞ + ( ) + − 1 = 1 (A) e (B) e2 (C) e–1 (D) e–2 12. Which one of the following series diverges ? (A) 1 1 2 2 3 1 4 1 5 2 6 + − ++− + ... (B) ∑ ( ) − = ∞ − 1 1 1 n n n n n (C) ( ) log − ∑ = ∞ 1 2 n n n (D) ( ) − ∑ = ∞ 2 2 1 n n n 13. Consider the subspace of 6 spanned by the columns of the following 6 × 8 matrix . What is its dimension ? 0 2 0 3 10 9 10 1 5 000 8 10 9 10 2 5 0000 1 0 4 3 5 000 0000 0 0000 0 354 5 0 0 − − − − . . . . 0 0000 0                     (A) 6 (B) 5 (C) 4 (D) 3 14. What is the shortest distance in 3 between the point (1, 2, 3) and the plane x + y + z = 0 ? (A) 6 (B) 3 2 (C) 2 3 (D) 6
*K2618* Total Number of Pages : 16 Paper II 4 K-2618 15. Which one of the following is an eigenvalue of the matrix below ? 4444 5555 7777 2222 −−−−             (A) 4 (B) 5 (C) –7 (D) 2 16. If a 3 × 3 real matrix A has eigenvalues 2, 3 2 and − 1 2,which one of the following is an eigenvalue of 8A3 + 5I – 4A–2, where I denotes the identity matrix of size 3 × 3 ? (A) – 12 (B) – 11 (C) – 10 (D) – 9 17. Suppose that M is a 4 × 7 real matrix that is row equivalent to the following matrix 0 1 23020 0000 1 3 0 000000 1 0000000             Which of the following statement about M is not necessarily true ? (A) All entries in the first column of M are zero (B) The first three rows of M are linearly independent (C) The third and fourth column of M are linearly dependent (D) The second, fifth and seventh columns of M span its column space 18. Consider the similarity equivalence relation on 6 × 6 real matrices with characteristic polynomial (t – 6)6 and minimal polynomial (t – 6)3 . How many equivalence classes are there ? (A) 3 (B) 4 (C) 5 (D) 6 19. Let A be 4 × 5 real matrix. Consider the system Ax = b of linear equations where x is a 5 × 1 column matrix of indeterminates and b is some fixed 4 × 1 column matrix with real entries. Suppose that A is row equivalent to the matrix R below and that c and d below are both solutions to Ax = b. (The entry z in d is unknown at the moment) R c =  − − −            =              1 2 1 3 0 0 000 1 0 000 0 0 000 0 1 2 3 4 5 ,    =                 , . d z 3 4 5 5 What is the value of z ? (A) 4 (B) 5 (C) 6 (D) 7 20. The rank and signature of the quadratic form –xy + z2 in three variables over the reals is (A) (3, 3) (B) (2, 1) (C) (3, 1) (D) (3, 2) 21. Which one of the following is false ? (A) A skew-symmetric matrix of odd order is singular (B) Two similar matrices have the same minimal polynomial (C) A matrix B is nilpotent if and only if its trace is zero (D) If a matrix A is similar to a diagonal matrix, then A is similar to its transpose