Download JNU 2019 Mathematics 237 Question Paper

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Question Paper Name: Mathematics 237 30th May 2019 Shift 1 Set1

Subject Name: Mathematics 237

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Creation Date: 2019-05-30 13:50:40

Duration: 180

Total Marks: 100

Display Marks: Yes

Share Answer Key With Delivery Engine: Yes

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Actual Answer Key: Yes

Mathematics 237

Group Number : 1

Group Id : 128206193

Group Maximum Duration : 0

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Group Minimum Duration : 120

Revisit allowed for view?: No

Revisit allowed for edit? : No

Break time: 0

Group Marks: 100

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PART 1

Section Id : 128206325

Section Number : 1

Section type : Online

Mandatory or Optional: Mandatory

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Number of Questions: 11

Number of Questions to be attempted: 11

Section Marks: 55

Display Number Panel: Yes

Group All Questions: No

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Sub-Section Number: 1

Sub-Section Id: 128206524

Question Shuffling Allowed : Yes

Question Number : 1 Question Id : 12820611350 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation : Vertical

Correct Marks : 5 Wrong Marks : 0

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Let B = {(1,1,1), (1,1,0), (1,0,0)}. Let T be a linear transformation from R³ to R³ given by T(x, y, z) := (x + y + z, x - y + z, x + y - z).

Which one of the following is true?

(a) B is not a basis

(b) B is a basis for R³ and the matrix of T with respect to B is $\left(\begin{array}{ccc}0& 2& 4\\ -1& 1& 0\\ 1& 3& 2\end{array}\right)$

(c) B is a basis for R³ and the matrix of T with respect to B is $\left(\begin{array}{ccc}1& 0& 0\\ 1& 2& 0\\ 1& 1& 3\end{array}\right)$

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(d) B is a basis for R³ and the matrix of T with respect to B is $\left(\begin{array}{ccc}3& 0& 0\\ -1& 1& 0\\ 1& 1& 2\end{array}\right)$

Options:

12820644847. A

12820644848. B

12820644849. C

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12820644850. D

Question Number : 2 Question Id : 12820611351 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation : Vertical

Correct Marks : 5 Wrong Marks : 0

Let f,g: [0,8) ?R be continuous functions satisfying the following equations

?₀^x f(t)dt = x²(1+x), ?₀^x²(1+x) g(t)dt = x².

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Which one of the following is necessarily true?

(a) f(x)²g(x)²(1 + g(x)) = 1

(b) f(x²)g(x²(1+x)) = 1

(c) f(x)²/(g(x)²(1 + g(x))) = 1

(d) f(x²) = g(x²(1+x))

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Options:

12820644851. A

12820644852. B

12820644853. C

12820644854. D

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Question Number: 3 Question Id : 12820611352 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation : Vertical

Correct Marks : 5 Wrong Marks : 0

The series ?_n=1⁸ log cos(¹/_n)

(a) converges absolutely

(b) converges conditionally but not absolutely

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(c) diverges to +8

(d) diverges to -8

Options:

12820644855. A

12820644858. D

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Question Number: 4 Question Id : 12820611353 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation : Vertical

Correct Marks : 5 Wrong Marks : 0

Consider the following statements:

[S1] The number of groups of order 15 up to isomorphism is 2.

[S2] The number of groups of order 15 up to isomorphism is 1.

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[S3] All groups of order 15 are Abelian.

[S4] There is a group of order 15 which is non-Abelian.

Which one of the following is necessarily true?

(a) Only [S1] and [S4]

(b) Only [S1] and [S3]

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(c) Only [S2] and [S3]

(d) Only [S3]

Options:

12820644859. A

12820644860. B

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12820644861. C

12820644862. D

Question Number: 5 Question Id : 12820611354 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation : Vertical

Correct Marks : 5 Wrong Marks : 0

Consider the series S1 and S2 given by

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S1 := ?_n=1⁸ (1 - ¹/_vn)

S2 := ?_n=1⁸ (-1)ⁿ log³n / n

Which one of the following is true?

(a) S1 converges and S2 converges absolutely

(b) S1 converges and S2 converges conditionally but not absolutely

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(c) S1 converges and S2 diverges

(d) S1 diverges and S2 converges

Options:

12820644863. A

12820644864. B

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12820644865. C

12820644866. D

Question Number: 6 Question Id : 12820611355 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation : Vertical

Correct Marks : 5 Wrong Marks : 0

?_C ^{x² + y²}/_{(x-2)² + (y-3)²} calculated counterclockwise around the circle C centred at (0,3) and radius 2. Then, the value of I equals

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(a) 0

(b) 2p

(c) -2p

(d) None of the above

Options:

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12820644867. A

12820644868. B

12820644869. C

12820644870. D

Question Number : 7 Question Id : 12820611356 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation : Vertical

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Correct Marks: 5 Wrong Marks : 0

For subsets T1 and T2 of R, we define T1 + T2 := {t1+t2 | t1 ? T1, t2 ? T2}. Let

S1 := (Q? [0,1]) + {v2} and S2 := (Q? [1,2]) + {v3}. Let S = S1 U S2.

Which one of the following is true?

(a) There exists a one to one map from [0, 1] to S

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(b) There exists an onto map from S to [0,1]

(c) #S1 = #S2

(d) None of the above

Options:

12820644871. A

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12820644872. B

12820644873. C

12820644874. D

Question Number: 8 Question Id : 12820611357 Question Type : MCQ Option Shuffling: No Display Question Number: Yes Single Line Question Option: No Option Orientation: Vertical

Correct Marks : 5 Wrong Marks : 0

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Let A = $\left(\begin{array}{ccc}1& 1& 1\\ 1& 1& 1\\ 1& 1& 1\end{array}\right)$ Consider the following statements:

[S1] A^k = 0 for some k= 1.

[S2] A has two distinct real eigenvalues.

[S3] A has rank 1.

Which one of the following is necessarily true?

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(a) Only [S1]

(b) Only [S2]

(c) Only [S2] and [S3]

(d) Only [S3]

Options:

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12820644875. A

12820644876. B

12820644877. C

12820644878. D

Question Number : 9 Question Id : 12820611358 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation: Vertical

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Correct Marks: 5 Wrong Marks : 0

Let f: R ? R be a continuous map and S = {p + ^(-1)n/_2n+1 | n?N}U (0,1). Which one of the following is necessarily true?

(a) f((0,1)) is an open set

(b) f(S) is compact

(c) f(S) is compact

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(d) f(S) is closed but not compact

Options:

12820644879. A

12820644880. B

12820644881. C

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12820644882. D

Question Number : 10 Question Id : 12820611359 Question Type : MCQ Option Shuffling: No Display Question Number: Yes Single Line Question Option : No Option Orientation : Vertical

Correct Marks: 5 Wrong Marks : 0

Let f: R² ? R be a map that sends (x,y) ? R² to f(x, y). Which one of the following is necessarily true?

(a) If ^?f/_?x and ^?f/_?y exist at (0,0), then f is continuous at (0,0)

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(b) If ^?f/_?x exists at (0,0), then f is continuous separately in both variables x and y at (0,0)

(c) If f is continuous at (0,0), then ^?f/_?x and ^?f/_?y exist at (0,0)

(d) None the above

Options:

12820644883. A

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12820644884. B

12820644885. C

12820644886. D

Question Number: 11 Question Id : 12820611360 Question Type : MCQ Option Shuffling : No Display Question Number: Yes Single Line Question Option : No Option Orientation : Vertical

Correct Marks: 5 Wrong Marks : 0

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Let p be a prime number. Let ?_i=1^{p-1 1}/_i = ^a/_ß, where a, ß are positive integers and are relatively prime. Which one of the following is necessarily true?

(a) p|a, if p = 3,5,7 and p ? a, if p= 11

(b) p|a for all odd prime numbers p

(c) ß = -1 mod p, for all p

(d) None of the above

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Options:

12820644887. A

12820644888. B

12820644889. C

12820644890. D

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PART 2

Section Id : 128206326

Section Number : 2

Section type : Online

Mandatory or Optional: Mandatory

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Number of Questions: 15

Number of Questions to be attempted: 15

Section Marks: 45

Display Number Panel: Yes

Group All Questions: No

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Sub-Section Number: 1

Sub-Section Id: 128206525

Question Shuffling Allowed : Yes

Question Number : 12 Question Id : 12820611361 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation : Vertical

Correct Marks: 3 Wrong Marks : 0

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Consider the permutation s = (1 2)(3 4)(4 5 6) (1 2 3) in S7. Then, the order of s equals

(a) 2

(b) 3

(c) 5

(d) 6

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Options:

12820644891. A

12820644892. B

12820644893. C

12820644894. D

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Question Number: 13 Question Id : 12820611362 Question Type: MCQ Option Shuffling: No Display Question Number: Yes Single Line Question Option: No Option Orientation: Vertical

Correct Marks: 3 Wrong Marks : 0

What is the volume of the region in the positive octant in R³ bounded by the planes x = 0, y = 0, z = 0, x = 3 and the parabolic cylinder z = 4 - y²?

(a) 4

(b) 8

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(c) 16

(d) 8

Options:

12820644895. A

12820644896. B

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12820644897. C

12820644898. D

Question Number : 14 Question Id : 12820611363 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation Vertical

Correct Marks: 3 Wrong Marks : 0

How many real roots does the polynomial 3X⁵ + 9X +7 have?

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(a) 1

(b) 3

(c) 5

(d) 0

Options:

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12820644899. A

12820644900. B

12820644901. C

12820644902. D

Question Number: 15 Question Id : 12820611364 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation : Vertical

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Correct Marks: 3 Wrong Marks : 0

The value of lim_x?0+ (1+x²)^2/x is

(a) 0

(b) 1

(c) e²

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(d) +8

Options:

12820644903. A

12820644904. B

12820644905. C

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12820644906. D

Question Number : 16 Question Id : 12820611365 Question Type : MCQ Option Shuffling : No Display Question Number : Yes Single Line Question Option : No Option Orientation : Vertical

Correct Marks : 3 Wrong Marks : 0

Let m = (20)¹⁸(mod 7). Then, m mod 7 is equal to

(a) 6

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(b) 2

(c) 1

(d) None of these

Options:

12820644907. A

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12820644908. B

12820644909. C

12820644910. D

Question Number : 17 Question Id : 12820611366 Question Type : MCQ Option Shuffling: No Display Question Number: Yes Single Line Question Option : No Option Orientation: Vertical

Correct Marks : 3 Wrong Marks : 0

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S = {2p + (-1)ⁿ tan(¹/_2n+1) | n?N} ? {3sinx | x ? (0, p/2)} ? {4e^5x | x ? (-8,0)}.

Then, the set of limit points of S is

(a) (0,4)

(b) (0,4] ? {2p}

(c) [0,4] ? {2p}

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(d) (-8, 4] ? {2p}

Options:

12820644911. A

12820644912. B

12820644913. C

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12820644914. D

Question Number : 18 Question Id : 12820611367 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation : Vertical

Correct Marks: 3 Wrong Marks : 0

Let a map f: (-1,1) ? R be continuous at 0. Consider the following statements:

[S1] lim_x?0+ f(x) = f(0) = lim_x?0- f(x)

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[S2] sup{f(x) | x ? (-1,0)} = f(0) = inf {f(x) | x ? (0,1)}

[S3] lim_n?8 f (¹/_2n+1) exists and equals f(0).

[S4] sup{f(¹/_n) | n ? N} = f(0) = inf{f(¹/_n) | n?N}

Which of the above statements are necessarily true?

(a) Only [S1]

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(b) Only [S1] and [S3]

(c) Only [S1], [S2] and [S3]

(d) Only [S1], [S3] and [S4]

Options:

12820644915. A

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12820644916. B

12820644917. C

12820644918. D

Question Number : 19 Question Id : 12820611368 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation : Vertical

Correct Marks : 3 Wrong Marks : 0

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Let f: (-1,1) ? R be a differentiable function. Which one of the following is necessarily true?

(a) If f'(0) > 0, then f(x) > f(0) for all x ? (0,1).

(b) If f'(0) < 0, then f(x) < f(0) for all x ? (0, d) for some d > 0.

(c) If f'(0) = 0, then f(x) = f(0) for all x ? (-d, d) for some d > 0.

(d) If f'(0) > 0, then f(0) > 0.

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Options:

12820644919. A

Question Number: 20 Question Id : 12820611369 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation : Vertical

Correct Marks: 3 Wrong Marks : 0

Let S = Q? [0.1]. Let f be a bijection from N to S. Which one of the following is necessarily true?

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(a) ¹/_v2 is a limit point but ¹/_v2 is not a limit point of the sequence {f(n)}n=1

(b) ¹/₂ is a limit point but ¹/₂ is not a limit point of the sequence {f(n)}n=1

(c) We can choose a bijection ƒ from N to S such that ¹/_v2 is not a limit point of of the sequence {f(n)}n=1

(d) Both ¹/₂ and ¹/_v2 are limit points of the sequence {f(n)}n=1

Options:

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12820644923. A

12820644924. B

12820644925. C

12820644926. D

Question Number : 21 Question Id : 12820611370 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation : Vertical

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Correct Marks : 3 Wrong Marks : 0

Which one of the following is true?

(a) The number of groups of order 14 up to isomorphism is 2

(b) The number of groups of order 14 up to isomorphism is 3

(c) The number of groups of order 26 up to isomorphism is 1

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(d) The number of groups of order 14 up to isomorphism is 1

Options:

12820644927. A

12820644928. B

12820644929. C

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12820644930. D

Question Number : 22 Question Id : 12820611371 Question Type : MCQ Option Shuffling: No Display Question Number : Yes Single Line Question Option : No Option Orientation : Vertical

Correct Marks : 3 Wrong Marks : 0

Which one of the following is necessarily true?

(a) If f

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