Mathematics MCQs for JSSC PGT Mathematics Exam

61

Advanced Mathematics • Topology

Let A = {A1, A2, A3, ...} be an infinite collection of closed sets. Which of the following is true? Let A = {A1, A2, A3, ...} be an infinite collection of closed sets. Which of the following is true?

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A. Arbitrary intersection of sets from A is open Arbitrary intersection of sets from A is open B. Finite union of sets from A is closed Finite union of sets from A is closed C. None of the options are true None of the options are true D. Arbitrary union of sets from A is closed Arbitrary union of sets from A is closed

सही उत्तर: B Correct Answer: B

A finite union of closed sets is always closed. An arbitrary union of closed sets need not be closed, while an arbitrary intersection of closed sets is closed rather than necessarily open.

62

Advanced Mathematics • Linear Algebra

A is a square matrix with each row sum equal to 1. Which statement regarding the eigenvalues of A is always true? A is a square matrix with each row sum equal to 1. Which statement regarding the eigenvalues of A is always true?

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A. 0 is an eigenvalue of A. 0 is an eigenvalue of A. B. 1 is an eigenvalue of A. 1 is an eigenvalue of A. C. Both 0 and 1 are eigenvalues of A. Both 0 and 1 are eigenvalues of A. D. No statement can be made about an eigenvalue of A from the given information. No statement can be made about an eigenvalue of A from the given information.

सही उत्तर: B Correct Answer: B

If e is the column vector whose entries are all 1, then Ae = e because every row sum is 1. Therefore 1 is an eigenvalue.

63

Advanced Mathematics • Mathematical Modelling

Let V represent the temperature in degrees Fahrenheit of an object in a room whose temperature is kept constant at 60 degrees Fahrenheit. If the object cools from 100 degrees to 90 degrees in 10 minutes, how much more time, in minutes, will it take for its temperature to decrease to 80 degrees? Let V represent the temperature in degrees Fahrenheit of an object in a room whose temperature is kept constant at 60 degrees Fahrenheit. If the object cools from 100 degrees to 90 degrees in 10 minutes, how much more time, in minutes, will it take for its temperature to decrease to 80 degrees?

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A. 12.08 12.08 B. 15.09 15.09 C. 14.09 14.09 D. 13.09 13.09

सही उत्तर: C Correct Answer: C

By Newton's law of cooling, V - 60 = 40e^(-kt). The first 10 minutes give e^(-10k) = 3/4. Solving for the additional time needed to reach V = 80 gives approximately 14.09 minutes.

64

Advanced Mathematics • Mathematical Modelling

Which of the following statements is NOT correct? Which of the following statements is NOT correct?

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A. Urban water uses involve transportation. Urban water uses involve transportation. B. Urban water uses involve water supply. Urban water uses involve water supply. C. Urban water uses involve power generation. Urban water uses involve power generation. D. Urban water uses involve snow-removal strategies. Urban water uses involve snow-removal strategies.

सही उत्तर: D Correct Answer: D

Snow-removal strategies are not classified as an urban water use in the context of the question; option D is marked as the incorrect statement.

65

Advanced Mathematics • Digital Logic

The logical expression y = A + A^cB is equivalent to: The logical expression y = A + A^cB is equivalent to:

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A. y = A^c + B y = A^c + B B. y = A + B y = A + B C. y = AB y = AB D. y = A^cB y = A^cB

सही उत्तर: B Correct Answer: B

Using the absorption identity, A + A^cB = (A + A^c)(A + B) = 1(A + B) = A + B.

66

Advanced Mathematics • Linear Algebra

Which one of the following statements is false? Which one of the following statements is false?

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A. C[0,1] denotes the space of all continuous functions. T:C[0,1] to R defined as T(f) = 2f(1) + integral from 0 to 1 of f(s)ds is a linear functional. C[0,1] denotes the space of all continuous functions. T:C[0,1] to R defined as T(f) = 2f(1) + integral from 0 to 1 of f(s)ds is a linear functional. B. M_2x3(R), the space of all real-valued 2 x 3 matrices, is isomorphic to R^5. M_2x3(R), the space of all real-valued 2 x 3 matrices, is isomorphic to R^5. C. Let A,B:V to V be linear and I be the identity map. Then AB = I implies that A and B are invertible. Let A,B:V to V be linear and I be the identity map. Then AB = I implies that A and B are invertible. D. A linear transformation T is invertible if and only if T is one-one and onto. A linear transformation T is invertible if and only if T is one-one and onto.

सही उत्तर: B Correct Answer: B

The vector space of 2 x 3 real matrices has dimension 6, so it is isomorphic to R^6, not R^5.

67

Advanced Mathematics • Group Theory

Which of the following is true? Which of the following is true?

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A. Every permutation can be written as a product of intersecting cycles. Every permutation can be written as a product of intersecting cycles. B. Every permutation can be written as a product of cycles of length 1. Every permutation can be written as a product of cycles of length 1. C. Every permutation can be written as a product of disjoint cycles. Every permutation can be written as a product of disjoint cycles. D. Every permutation can be written as a product of cycles that do not commute. Every permutation can be written as a product of cycles that do not commute.

सही उत्तर: C Correct Answer: C

Every permutation has a decomposition into disjoint cycles, unique up to the order in which those disjoint cycles are written.

68

Advanced Mathematics • Queueing Theory

In a newly opened bank, customers arrive at the rate of 2 customers per hour and are served by a single teller counter in about 20 minutes. The bank has a newly implemented policy that there should be no more than a 5% chance that a customer has to wait for more than 30 minutes in the queue before being served. How many additional teller counters should be opened to meet the policy requirements? In a newly opened bank, customers arrive at the rate of 2 customers per hour and are served by a single teller counter in about 20 minutes. The bank has a newly implemented policy that there should be no more than a 5% chance that a customer has to wait for more than 30 minutes in the queue before being served. How many additional teller counters should be opened to meet the policy requirements?

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A. 0 (the existing counter is sufficient) 0 (the existing counter is sufficient) B. 3 3 C. 1 1 D. 2 2

सही उत्तर: C Correct Answer: C

Applying the multi-server queue waiting-time criterion to the stated arrival rate, service rate and 30-minute threshold shows that two counters in total are required. Therefore one additional counter is needed.

69

Advanced Mathematics • Fluid Mechanics

The metacentric height of a rectangular boat 6 x 6 x 12, a quarter portion of which is submerged in water, is: The metacentric height of a rectangular boat 6 x 6 x 12, a quarter portion of which is submerged in water, is:

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A. 1.5 1.5 B. 3.5 3.5 C. 4.5 4.5 D. 2.5 2.5

सही उत्तर: B Correct Answer: B

Using the displaced volume, centre of buoyancy and water-plane moment of inertia for the rectangular boat gives a metacentric height of 3.5 units.

70

Advanced Mathematics • Graph Theory

For a tree with q lines, the number of vertices p equals: For a tree with q lines, the number of vertices p equals:

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A. 2q 2q B. q q C. q + 2 q + 2 D. q + 1 q + 1

सही उत्तर: D Correct Answer: D

A tree with p vertices has exactly p - 1 edges. Therefore, if the number of lines or edges is q, then p = q + 1.

Mathematics MCQs for JSSC PGT Mathematics Exam

Mathematics MCQs for JSSC PGT Mathematics Exam

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