Page 1 2012 MATHEMATICS – MA MA 1/20 MA : MATHEMATICS Duration: Three Hours Maximum Marks: 100 Read the following instructions carefully. 1. Do not open the seal of the Question Booklet until you are asked to do so by the invigilator. 2. Take out the Optical Response Sheet (ORS) from this Question Booklet without breaking the seal and read the instructions printed on the ORS carefully. 3. On the right half of the ORS, using ONLY a black ink ball point pen, (i) darken the bubble corresponding to your test paper code and the appropriate bubble under each digit of your registration number and (ii) write your registration number, your name and name of the examination centre and put your signature at the specified location. 4. This Question Booklet contains 20 pages including blank pages for rough work. After you are permitted to open the seal, please check all pages and report discrepancies, if any, to the invigilator. 5. There are a total of 65 questions carrying 100 marks. All these questions are of objective type. Each question has only one correct answer. Questions must be answered on the left hand side of the ORS by darkening the appropriate bubble (marked A, B, C, D) using ONLY a black ink ball point pen against the question number. For each question darken the bubble of the correct answer. More than one answer bubbled against a question will be treated as an incorrect response. 6. Since bubbles darkened by the black ink ball point pen cannot be erased, candidates should darken the bubbles in the ORS very carefully. 7. Questions Q.1 – Q.25 carry 1 mark each. Questions Q.26 – Q.55 carry 2 marks each. The 2 marks questions include two pairs of common data questions and two pairs of linked answer questions. The answer to the second question of the linked answer questions depends on the answer to the first question of the pair. If the first question in the linked pair is wrongly answered or is unattempted, then the answer to the second question in the pair will not be evaluated. 8. Questions Q.56 – Q.65 belong to General Aptitude (GA) section and carry a total of 15 marks. Questions Q.56 – Q.60 carry 1 mark each, and questions Q.61 – Q.65 carry 2 marks each. 9. Unattempted questions will result in zero mark and wrong answers will result in NEGATIVE marks. For all 1 mark questions, ? mark will be deducted for each wrong answer. For all 2 marks questions, ? mark will be deducted for each wrong answer. However, in the case of the linked answer question pair, there will be negative marks only for wrong answer to the first question and no negative marks for wrong answer to the second question. 10. Calculator is allowed whereas charts, graph sheets or tables are NOT allowed in the examination hall. 11. Rough work can be done on the question paper itself. Blank pages are provided at the end of the question paper for rough work. 12. Before the start of the examination, write your name and registration number in the space provided below using a black ink ball point pen. Name Registration Number MA Page 2 2012 MATHEMATICS – MA MA 1/20 MA : MATHEMATICS Duration: Three Hours Maximum Marks: 100 Read the following instructions carefully. 1. Do not open the seal of the Question Booklet until you are asked to do so by the invigilator. 2. Take out the Optical Response Sheet (ORS) from this Question Booklet without breaking the seal and read the instructions printed on the ORS carefully. 3. On the right half of the ORS, using ONLY a black ink ball point pen, (i) darken the bubble corresponding to your test paper code and the appropriate bubble under each digit of your registration number and (ii) write your registration number, your name and name of the examination centre and put your signature at the specified location. 4. This Question Booklet contains 20 pages including blank pages for rough work. After you are permitted to open the seal, please check all pages and report discrepancies, if any, to the invigilator. 5. There are a total of 65 questions carrying 100 marks. All these questions are of objective type. Each question has only one correct answer. Questions must be answered on the left hand side of the ORS by darkening the appropriate bubble (marked A, B, C, D) using ONLY a black ink ball point pen against the question number. For each question darken the bubble of the correct answer. More than one answer bubbled against a question will be treated as an incorrect response. 6. Since bubbles darkened by the black ink ball point pen cannot be erased, candidates should darken the bubbles in the ORS very carefully. 7. Questions Q.1 – Q.25 carry 1 mark each. Questions Q.26 – Q.55 carry 2 marks each. The 2 marks questions include two pairs of common data questions and two pairs of linked answer questions. The answer to the second question of the linked answer questions depends on the answer to the first question of the pair. If the first question in the linked pair is wrongly answered or is unattempted, then the answer to the second question in the pair will not be evaluated. 8. Questions Q.56 – Q.65 belong to General Aptitude (GA) section and carry a total of 15 marks. Questions Q.56 – Q.60 carry 1 mark each, and questions Q.61 – Q.65 carry 2 marks each. 9. Unattempted questions will result in zero mark and wrong answers will result in NEGATIVE marks. For all 1 mark questions, ? mark will be deducted for each wrong answer. For all 2 marks questions, ? mark will be deducted for each wrong answer. However, in the case of the linked answer question pair, there will be negative marks only for wrong answer to the first question and no negative marks for wrong answer to the second question. 10. Calculator is allowed whereas charts, graph sheets or tables are NOT allowed in the examination hall. 11. Rough work can be done on the question paper itself. Blank pages are provided at the end of the question paper for rough work. 12. Before the start of the examination, write your name and registration number in the space provided below using a black ink ball point pen. Name Registration Number MA 2012 MATHEMATICS – MA MA 2/20 Notations and Symbols used R : Set of all real numbers C : Set of all complex numbers Z : Set of all integers F : A field C n : The set of all n-tuples of complex numbers n F : The set of all n-tuples over F 12 ... n R R R ? ? ? : Cartesian product of rings 12 , ,..., n R R R ( , ) x D f x y : Partial derivative with respect to x. 2 ( , ) N ?? : Normal distribution with mean ? and variance 2 ? () EX : Expectation of X ( , ) Cov X Y : Covariance between X and Y S n : The group of all permutations on n symbols n P : The set of all polynomials of degree at most n C n : Cyclic Group of Order n Z(G) : Centre of the Group G 1 i?? Page 3 2012 MATHEMATICS – MA MA 1/20 MA : MATHEMATICS Duration: Three Hours Maximum Marks: 100 Read the following instructions carefully. 1. Do not open the seal of the Question Booklet until you are asked to do so by the invigilator. 2. Take out the Optical Response Sheet (ORS) from this Question Booklet without breaking the seal and read the instructions printed on the ORS carefully. 3. On the right half of the ORS, using ONLY a black ink ball point pen, (i) darken the bubble corresponding to your test paper code and the appropriate bubble under each digit of your registration number and (ii) write your registration number, your name and name of the examination centre and put your signature at the specified location. 4. This Question Booklet contains 20 pages including blank pages for rough work. After you are permitted to open the seal, please check all pages and report discrepancies, if any, to the invigilator. 5. There are a total of 65 questions carrying 100 marks. All these questions are of objective type. Each question has only one correct answer. Questions must be answered on the left hand side of the ORS by darkening the appropriate bubble (marked A, B, C, D) using ONLY a black ink ball point pen against the question number. For each question darken the bubble of the correct answer. More than one answer bubbled against a question will be treated as an incorrect response. 6. Since bubbles darkened by the black ink ball point pen cannot be erased, candidates should darken the bubbles in the ORS very carefully. 7. Questions Q.1 – Q.25 carry 1 mark each. Questions Q.26 – Q.55 carry 2 marks each. The 2 marks questions include two pairs of common data questions and two pairs of linked answer questions. The answer to the second question of the linked answer questions depends on the answer to the first question of the pair. If the first question in the linked pair is wrongly answered or is unattempted, then the answer to the second question in the pair will not be evaluated. 8. Questions Q.56 – Q.65 belong to General Aptitude (GA) section and carry a total of 15 marks. Questions Q.56 – Q.60 carry 1 mark each, and questions Q.61 – Q.65 carry 2 marks each. 9. Unattempted questions will result in zero mark and wrong answers will result in NEGATIVE marks. For all 1 mark questions, ? mark will be deducted for each wrong answer. For all 2 marks questions, ? mark will be deducted for each wrong answer. However, in the case of the linked answer question pair, there will be negative marks only for wrong answer to the first question and no negative marks for wrong answer to the second question. 10. Calculator is allowed whereas charts, graph sheets or tables are NOT allowed in the examination hall. 11. Rough work can be done on the question paper itself. Blank pages are provided at the end of the question paper for rough work. 12. Before the start of the examination, write your name and registration number in the space provided below using a black ink ball point pen. Name Registration Number MA 2012 MATHEMATICS – MA MA 2/20 Notations and Symbols used R : Set of all real numbers C : Set of all complex numbers Z : Set of all integers F : A field C n : The set of all n-tuples of complex numbers n F : The set of all n-tuples over F 12 ... n R R R ? ? ? : Cartesian product of rings 12 , ,..., n R R R ( , ) x D f x y : Partial derivative with respect to x. 2 ( , ) N ?? : Normal distribution with mean ? and variance 2 ? () EX : Expectation of X ( , ) Cov X Y : Covariance between X and Y S n : The group of all permutations on n symbols n P : The set of all polynomials of degree at most n C n : Cyclic Group of Order n Z(G) : Centre of the Group G 1 i?? 2012 MATHEMATICS – MA MA 3/20 Q. 1 – Q. 25 carry one mark each. Q.1 The straight lines 12 : 0, : 0 L x L y ?? and 3 :1 L x y ?? are mapped by the transformation 2 wz ? into the curves 12 , CC and 3 C respectively. The angle of intersection between the curves at 0 w ? is (A) 0 (B) /4 ? (C) /2 ? (D) ? Q.2 In a topological space, which of the following statements is NOT always true : (A) Union of any finite family of compact sets is compact. (B) Union of any family of closed sets is closed. (C) Union of any family of connected sets having a non empty intersection is connected. (D) Union of any family of dense subsets is dense. Q.3 Consider the following statements: P: The family of subsets 11 , , 1,2,... n An nn ?? ?? ? ? ? ?? ?? ?? ?? satisfies the finite intersection property. Q: On an infinite set X , a metric : d X X R ?? is defined as 0, ( , ) 1, xy d x y xy ? ? ? ? ? ? . The metric space ( , ) Xd is compact. R: In a Frechet ( 1 T ) topological space, every finite set is closed. S: If : f R X ? is continuous, where R is given the usual topology and ( , ) X ? is a Hausdorff ( 2 T ) space, then f is a one-one function. Which of the above statements are correct? (A) P and R (B) P and S (C) R and S (D) Q and S Q.4 Let H be a Hilbert space and S ? denote the orthogonal complement of a set SH ? . Which of the following is INCORRECT? (A) For 1 2 1 2 1 2 ,; S S H S S S S ?? ? ? ? ? (B) () SS ?? ? (C) {0} H ? ? (D) S ? is always closed. Q.5 Let H be a complex Hilbert space, : T H H ? be a bounded linear operator and let * T denote the adjoint of T . Which of the following statements are always TRUE? P: , , , , * x y H Tx y x T y ? ? ? Q: , , , * , x y H x Ty T x y ? ? ? R: , , , , * x y H x Ty x T y ? ? ? S: , , , * , * x y H Tx Ty T x T y ? ? ? (A) P and Q (B) P and R (C) Q and S (D) P and S Q.6 Let { , , } X a b c ? and let ? ? ,{ },{ },{ , }, a b a b X ? ?? be a topology defined on X . Then which of the following statements are TRUE? P: ( , ) X ? is a Hausdorff space. Q: ( , ) X ? is a regular space. R: ( , ) X ? is a normal space. S: ( , ) X ? is a connected space. (A) P and Q (B) Q and R (C) R and S (D) P and S Page 4 2012 MATHEMATICS – MA MA 1/20 MA : MATHEMATICS Duration: Three Hours Maximum Marks: 100 Read the following instructions carefully. 1. Do not open the seal of the Question Booklet until you are asked to do so by the invigilator. 2. Take out the Optical Response Sheet (ORS) from this Question Booklet without breaking the seal and read the instructions printed on the ORS carefully. 3. On the right half of the ORS, using ONLY a black ink ball point pen, (i) darken the bubble corresponding to your test paper code and the appropriate bubble under each digit of your registration number and (ii) write your registration number, your name and name of the examination centre and put your signature at the specified location. 4. This Question Booklet contains 20 pages including blank pages for rough work. After you are permitted to open the seal, please check all pages and report discrepancies, if any, to the invigilator. 5. There are a total of 65 questions carrying 100 marks. All these questions are of objective type. Each question has only one correct answer. Questions must be answered on the left hand side of the ORS by darkening the appropriate bubble (marked A, B, C, D) using ONLY a black ink ball point pen against the question number. For each question darken the bubble of the correct answer. More than one answer bubbled against a question will be treated as an incorrect response. 6. Since bubbles darkened by the black ink ball point pen cannot be erased, candidates should darken the bubbles in the ORS very carefully. 7. Questions Q.1 – Q.25 carry 1 mark each. Questions Q.26 – Q.55 carry 2 marks each. The 2 marks questions include two pairs of common data questions and two pairs of linked answer questions. The answer to the second question of the linked answer questions depends on the answer to the first question of the pair. If the first question in the linked pair is wrongly answered or is unattempted, then the answer to the second question in the pair will not be evaluated. 8. Questions Q.56 – Q.65 belong to General Aptitude (GA) section and carry a total of 15 marks. Questions Q.56 – Q.60 carry 1 mark each, and questions Q.61 – Q.65 carry 2 marks each. 9. Unattempted questions will result in zero mark and wrong answers will result in NEGATIVE marks. For all 1 mark questions, ? mark will be deducted for each wrong answer. For all 2 marks questions, ? mark will be deducted for each wrong answer. However, in the case of the linked answer question pair, there will be negative marks only for wrong answer to the first question and no negative marks for wrong answer to the second question. 10. Calculator is allowed whereas charts, graph sheets or tables are NOT allowed in the examination hall. 11. Rough work can be done on the question paper itself. Blank pages are provided at the end of the question paper for rough work. 12. Before the start of the examination, write your name and registration number in the space provided below using a black ink ball point pen. Name Registration Number MA 2012 MATHEMATICS – MA MA 2/20 Notations and Symbols used R : Set of all real numbers C : Set of all complex numbers Z : Set of all integers F : A field C n : The set of all n-tuples of complex numbers n F : The set of all n-tuples over F 12 ... n R R R ? ? ? : Cartesian product of rings 12 , ,..., n R R R ( , ) x D f x y : Partial derivative with respect to x. 2 ( , ) N ?? : Normal distribution with mean ? and variance 2 ? () EX : Expectation of X ( , ) Cov X Y : Covariance between X and Y S n : The group of all permutations on n symbols n P : The set of all polynomials of degree at most n C n : Cyclic Group of Order n Z(G) : Centre of the Group G 1 i?? 2012 MATHEMATICS – MA MA 3/20 Q. 1 – Q. 25 carry one mark each. Q.1 The straight lines 12 : 0, : 0 L x L y ?? and 3 :1 L x y ?? are mapped by the transformation 2 wz ? into the curves 12 , CC and 3 C respectively. The angle of intersection between the curves at 0 w ? is (A) 0 (B) /4 ? (C) /2 ? (D) ? Q.2 In a topological space, which of the following statements is NOT always true : (A) Union of any finite family of compact sets is compact. (B) Union of any family of closed sets is closed. (C) Union of any family of connected sets having a non empty intersection is connected. (D) Union of any family of dense subsets is dense. Q.3 Consider the following statements: P: The family of subsets 11 , , 1,2,... n An nn ?? ?? ? ? ? ?? ?? ?? ?? satisfies the finite intersection property. Q: On an infinite set X , a metric : d X X R ?? is defined as 0, ( , ) 1, xy d x y xy ? ? ? ? ? ? . The metric space ( , ) Xd is compact. R: In a Frechet ( 1 T ) topological space, every finite set is closed. S: If : f R X ? is continuous, where R is given the usual topology and ( , ) X ? is a Hausdorff ( 2 T ) space, then f is a one-one function. Which of the above statements are correct? (A) P and R (B) P and S (C) R and S (D) Q and S Q.4 Let H be a Hilbert space and S ? denote the orthogonal complement of a set SH ? . Which of the following is INCORRECT? (A) For 1 2 1 2 1 2 ,; S S H S S S S ?? ? ? ? ? (B) () SS ?? ? (C) {0} H ? ? (D) S ? is always closed. Q.5 Let H be a complex Hilbert space, : T H H ? be a bounded linear operator and let * T denote the adjoint of T . Which of the following statements are always TRUE? P: , , , , * x y H Tx y x T y ? ? ? Q: , , , * , x y H x Ty T x y ? ? ? R: , , , , * x y H x Ty x T y ? ? ? S: , , , * , * x y H Tx Ty T x T y ? ? ? (A) P and Q (B) P and R (C) Q and S (D) P and S Q.6 Let { , , } X a b c ? and let ? ? ,{ },{ },{ , }, a b a b X ? ?? be a topology defined on X . Then which of the following statements are TRUE? P: ( , ) X ? is a Hausdorff space. Q: ( , ) X ? is a regular space. R: ( , ) X ? is a normal space. S: ( , ) X ? is a connected space. (A) P and Q (B) Q and R (C) R and S (D) P and S 2012 MATHEMATICS – MA MA 4/20 Q.7 Consider the statements P: If X is a normed linear space and MX ? is a subspace, then the closure M is also a subspace of . X Q: If X is a Banach space and n x ? is an absolutely convergent series in X , then n x ? is convergent. R: Let 1 M and 2 M be subspaces of an inner product space such that 12 {0} MM ?? .Then 2 2 2 1 1 2 2 1 2 1 2 ,; m M m M m m m m ? ? ? ? ? ? . S: Let : f X Y ? be a linear transformation from the Banach Space X into the Banach space Y . If f is continuous, then the graph of f is always compact. The correct statements amongst the above are: (A) P and R only (B) Q and R only (C) P and Q only (D) R and S only Q.8 A continuous random variable X has the probability density function 3 5 3 ,0 () 5 0, 0. x ex fx x ? ? ? ? ? ? ? ? ? The probability density function of 32 YX ?? is (A) 1 ( 2) 5 1 ,2 () 5 0, y 2 y ey fy ?? ? ? ? ? ? ? ? ? (B) 2 ( 2) 5 2 ,2 () 5 0, y 2 y ey fy ?? ? ? ? ? ? ? ? ? (C) 3 ( 2) 5 3 ,2 () 5 0, y 2 y ey fy ?? ? ? ? ? ? ? ? ? (D) 4 ( 2) 5 4 ,2 () 5 0, y 2 y ey fy ?? ? ? ? ? ? ? ? ? Q.9 A simple random sample of size 10 from 2 ( , ) N ?? gives 98% confidence interval (20.49, 23.51). Then the null hypothesis 0 : 20.5 H ? ? against : 20.5 A H ? ? (A) can be rejected at 2% level of significance (B) cannot be rejected at 5% level of significance (C) can be rejected at 10% level of significance (D) cannot be rejected at any level of significance Q.10 For the linear programming problem Maximize 1 2 3 4 2 3 4 z x x x x ? ? ? ? Subject to 1 2 3 4 2 3 15 x x x x ? ? ? ? 1 2 3 4 6 3 21 x x x x ? ? ? ? 1 2 3 4 8 2 3 4 30 x x x x ? ? ? ? 1 2 3 4 , , , 0 x x x x ? , 1 2 3 4 4, 3, 0, 2 x x x x ? ? ? ? is (A) an optimal solution (B) a degenerate basic feasible solution (C) a non-degenerate basic feasible solution (D) a non-basic feasible solution Page 5 2012 MATHEMATICS – MA MA 1/20 MA : MATHEMATICS Duration: Three Hours Maximum Marks: 100 Read the following instructions carefully. 1. Do not open the seal of the Question Booklet until you are asked to do so by the invigilator. 2. Take out the Optical Response Sheet (ORS) from this Question Booklet without breaking the seal and read the instructions printed on the ORS carefully. 3. On the right half of the ORS, using ONLY a black ink ball point pen, (i) darken the bubble corresponding to your test paper code and the appropriate bubble under each digit of your registration number and (ii) write your registration number, your name and name of the examination centre and put your signature at the specified location. 4. This Question Booklet contains 20 pages including blank pages for rough work. After you are permitted to open the seal, please check all pages and report discrepancies, if any, to the invigilator. 5. There are a total of 65 questions carrying 100 marks. All these questions are of objective type. Each question has only one correct answer. Questions must be answered on the left hand side of the ORS by darkening the appropriate bubble (marked A, B, C, D) using ONLY a black ink ball point pen against the question number. For each question darken the bubble of the correct answer. More than one answer bubbled against a question will be treated as an incorrect response. 6. Since bubbles darkened by the black ink ball point pen cannot be erased, candidates should darken the bubbles in the ORS very carefully. 7. Questions Q.1 – Q.25 carry 1 mark each. Questions Q.26 – Q.55 carry 2 marks each. The 2 marks questions include two pairs of common data questions and two pairs of linked answer questions. The answer to the second question of the linked answer questions depends on the answer to the first question of the pair. If the first question in the linked pair is wrongly answered or is unattempted, then the answer to the second question in the pair will not be evaluated. 8. Questions Q.56 – Q.65 belong to General Aptitude (GA) section and carry a total of 15 marks. Questions Q.56 – Q.60 carry 1 mark each, and questions Q.61 – Q.65 carry 2 marks each. 9. Unattempted questions will result in zero mark and wrong answers will result in NEGATIVE marks. For all 1 mark questions, ? mark will be deducted for each wrong answer. For all 2 marks questions, ? mark will be deducted for each wrong answer. However, in the case of the linked answer question pair, there will be negative marks only for wrong answer to the first question and no negative marks for wrong answer to the second question. 10. Calculator is allowed whereas charts, graph sheets or tables are NOT allowed in the examination hall. 11. Rough work can be done on the question paper itself. Blank pages are provided at the end of the question paper for rough work. 12. Before the start of the examination, write your name and registration number in the space provided below using a black ink ball point pen. Name Registration Number MA 2012 MATHEMATICS – MA MA 2/20 Notations and Symbols used R : Set of all real numbers C : Set of all complex numbers Z : Set of all integers F : A field C n : The set of all n-tuples of complex numbers n F : The set of all n-tuples over F 12 ... n R R R ? ? ? : Cartesian product of rings 12 , ,..., n R R R ( , ) x D f x y : Partial derivative with respect to x. 2 ( , ) N ?? : Normal distribution with mean ? and variance 2 ? () EX : Expectation of X ( , ) Cov X Y : Covariance between X and Y S n : The group of all permutations on n symbols n P : The set of all polynomials of degree at most n C n : Cyclic Group of Order n Z(G) : Centre of the Group G 1 i?? 2012 MATHEMATICS – MA MA 3/20 Q. 1 – Q. 25 carry one mark each. Q.1 The straight lines 12 : 0, : 0 L x L y ?? and 3 :1 L x y ?? are mapped by the transformation 2 wz ? into the curves 12 , CC and 3 C respectively. The angle of intersection between the curves at 0 w ? is (A) 0 (B) /4 ? (C) /2 ? (D) ? Q.2 In a topological space, which of the following statements is NOT always true : (A) Union of any finite family of compact sets is compact. (B) Union of any family of closed sets is closed. (C) Union of any family of connected sets having a non empty intersection is connected. (D) Union of any family of dense subsets is dense. Q.3 Consider the following statements: P: The family of subsets 11 , , 1,2,... n An nn ?? ?? ? ? ? ?? ?? ?? ?? satisfies the finite intersection property. Q: On an infinite set X , a metric : d X X R ?? is defined as 0, ( , ) 1, xy d x y xy ? ? ? ? ? ? . The metric space ( , ) Xd is compact. R: In a Frechet ( 1 T ) topological space, every finite set is closed. S: If : f R X ? is continuous, where R is given the usual topology and ( , ) X ? is a Hausdorff ( 2 T ) space, then f is a one-one function. Which of the above statements are correct? (A) P and R (B) P and S (C) R and S (D) Q and S Q.4 Let H be a Hilbert space and S ? denote the orthogonal complement of a set SH ? . Which of the following is INCORRECT? (A) For 1 2 1 2 1 2 ,; S S H S S S S ?? ? ? ? ? (B) () SS ?? ? (C) {0} H ? ? (D) S ? is always closed. Q.5 Let H be a complex Hilbert space, : T H H ? be a bounded linear operator and let * T denote the adjoint of T . Which of the following statements are always TRUE? P: , , , , * x y H Tx y x T y ? ? ? Q: , , , * , x y H x Ty T x y ? ? ? R: , , , , * x y H x Ty x T y ? ? ? S: , , , * , * x y H Tx Ty T x T y ? ? ? (A) P and Q (B) P and R (C) Q and S (D) P and S Q.6 Let { , , } X a b c ? and let ? ? ,{ },{ },{ , }, a b a b X ? ?? be a topology defined on X . Then which of the following statements are TRUE? P: ( , ) X ? is a Hausdorff space. Q: ( , ) X ? is a regular space. R: ( , ) X ? is a normal space. S: ( , ) X ? is a connected space. (A) P and Q (B) Q and R (C) R and S (D) P and S 2012 MATHEMATICS – MA MA 4/20 Q.7 Consider the statements P: If X is a normed linear space and MX ? is a subspace, then the closure M is also a subspace of . X Q: If X is a Banach space and n x ? is an absolutely convergent series in X , then n x ? is convergent. R: Let 1 M and 2 M be subspaces of an inner product space such that 12 {0} MM ?? .Then 2 2 2 1 1 2 2 1 2 1 2 ,; m M m M m m m m ? ? ? ? ? ? . S: Let : f X Y ? be a linear transformation from the Banach Space X into the Banach space Y . If f is continuous, then the graph of f is always compact. The correct statements amongst the above are: (A) P and R only (B) Q and R only (C) P and Q only (D) R and S only Q.8 A continuous random variable X has the probability density function 3 5 3 ,0 () 5 0, 0. x ex fx x ? ? ? ? ? ? ? ? ? The probability density function of 32 YX ?? is (A) 1 ( 2) 5 1 ,2 () 5 0, y 2 y ey fy ?? ? ? ? ? ? ? ? ? (B) 2 ( 2) 5 2 ,2 () 5 0, y 2 y ey fy ?? ? ? ? ? ? ? ? ? (C) 3 ( 2) 5 3 ,2 () 5 0, y 2 y ey fy ?? ? ? ? ? ? ? ? ? (D) 4 ( 2) 5 4 ,2 () 5 0, y 2 y ey fy ?? ? ? ? ? ? ? ? ? Q.9 A simple random sample of size 10 from 2 ( , ) N ?? gives 98% confidence interval (20.49, 23.51). Then the null hypothesis 0 : 20.5 H ? ? against : 20.5 A H ? ? (A) can be rejected at 2% level of significance (B) cannot be rejected at 5% level of significance (C) can be rejected at 10% level of significance (D) cannot be rejected at any level of significance Q.10 For the linear programming problem Maximize 1 2 3 4 2 3 4 z x x x x ? ? ? ? Subject to 1 2 3 4 2 3 15 x x x x ? ? ? ? 1 2 3 4 6 3 21 x x x x ? ? ? ? 1 2 3 4 8 2 3 4 30 x x x x ? ? ? ? 1 2 3 4 , , , 0 x x x x ? , 1 2 3 4 4, 3, 0, 2 x x x x ? ? ? ? is (A) an optimal solution (B) a degenerate basic feasible solution (C) a non-degenerate basic feasible solution (D) a non-basic feasible solution 2012 MATHEMATICS – MA MA 5/20 Q.11 Which one of the following statements is TRUE? (A) A convex set cannot have infinite many extreme points. (B) A linear programming problem can have infinite many extreme points. (C) A linear programming problem can have exactly two different optimal solutions. (D) A linear programming problem can have a non-basic optimal solution. Q.12 Let 2 /5 i e ? ? ? and the matrix 234 234 234 34 4 1 0 00 000 0 0 0 0 M ? ? ? ? ? ? ? ? ? ? ? ?? ? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? . Then the trace of the matrix 2 I M M ?? is (A) 5 ? (B) 0 (C) 3 (D) 5 Q.13 Let V = C 2 be the vector space over the field of complex numbers and {(1, ),( ,1)} B i i ? be a given ordered basis of V. Then for which of the following, 12 * { , } B f f ? is a dual basis of B over C? (A) 1 1 2 1 2 1 ( , ) ( ) 2 f z z z iz ?? , 2 1 2 1 2 1 ( , ) ( ) 2 f z z z iz ?? (B) 1 1 2 1 2 1 ( , ) ( ) 2 f z z z iz ?? , 2 1 2 1 2 1 ( , ) ( ) 2 f z z iz z ?? (C) 1 1 2 1 2 1 ( , ) ( ) 2 f z z z iz ?? , 2 1 2 1 2 1 ( , ) ( ) 2 f z z iz z ? ? ? (D) 1 1 2 1 2 1 ( , ) ( ) 2 f z z z iz ?? , 2 1 2 1 2 1 ( , ) ( ) 2 f z z iz z ? ? ? Q.14 Let R = Z ? Z ? Z and I = Z ? Z ?{0}. Then which of the following statement is correct? (A) I is a maximal ideal but not a prime ideal of R . (B) I is a prime ideal but not a maximal ideal of R . (C) I is both maximal ideal as well as a prime ideal of R . (D) I is neither a maximal ideal nor a prime ideal of R . Q.15 The function ( , ) ur ? satisfying the Laplace equation 22 2 2 2 2 11 0, u u u e r e r r r r ? ? ? ? ? ? ? ? ? ? ? ? subject to the conditions 2 ( , ) 1, ( , ) 0 u e u e ?? ?? is (A) ln( / ) er (B) 2 ln( / ) er (C) 2 ln( / ) er (D) 2 2 1 sin n re n ee ? ? ? ?? ? ?? ? ?? ? Q.16 The functional ? ? 1 22 0 ( 2 ) y y y y kxyy y dx ? ? ? ? ? ? ? ? ? ? , (0) 0, (1) 1, (0) 2, (1) 3 y y y y ?? ? ? ? ? is path independent if k equals (A) 1 (B) 2 (C) 3 (D) 4Read More

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