National Testing Agency
| Question Paper Name : | Fourier Analysis and its Applica |
| Subject Name : | Fourier Analysis and its Applicat |
| Creation Date : | 2020-09-15 13:26:34 |
| Duration : | 180 |
| Total Marks : | 100 |
| Display Marks: | Yes |
| Share Answer Key With Delivery Engine : | Yes |
| Actual Answer Key : | Yes |
Fourier Analysis and its Applications
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| Group Number : | 1 |
| Group Id : | 89951411 |
| Group Maximum Duration : | 0 |
| Group Minimum Duration : | 120 |
| Show Attended Group? : | No |
| Edit Attended Group? : | No |
| Break time : | 0 |
| Group Marks : | 100 |
| Is this Group for Examiner? : | No |
Fourier Analysis and its Applications
| Section Id : | 89951411 |
| Section Number : | 1 |
| Section type : | Online |
| Mandatory or Optional : | Mandatory |
| Number of Questions : | 70 |
| Number of Questions to be attempted : | 50 |
| Section Marks : | 100 |
| Display Number Panel : | Yes |
| Group All Questions : | Yes |
| Mark As Answered Required? : | Yes |
| Sub-Section Number : | 1 |
| Sub-Section Id : | 89951420 |
| Question Shuffling Allowed : | Yes |
Question Number : 1 Question Id : 899514871 Question Type : MCQ Option Shuffling : No Display Questi Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
The constant term in the Fourier series expansion of f(x) = |x|3 on the interval [-p, p] equals:
| A | p3 |
| B | p3/2 |
| C | p3/3 |
| D | p3/4 |
Options:
8995143461.1
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8995143462.2
8995143463.3
8995143464.4
Question Number : 2 Question Id : 899514872 Question Type : MCQ Option Shuffling : No Display Questi Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
The value of 1 + 1/22 + 1/32 + ... equals:
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| ? | p2/6 |
| ? | p2/8 |
| C | p2/16 |
| D | p2/18 |
Options:
8995143465. 1
8995143466. 2
8995143467.3
8995143468.4
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Question Number : 3 Question Id : 899514873 Question Type : MCQ Option Shuffling : No Display Questi Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
Supply the constant b in the following formula:
1/2p ?p-p|f(x)|2 dx = |a0|2 + b ?81(|an|2 + |bn|2)
| A | 1 |
| B | 1/2 |
| C | p |
| D | 2 |
Options:
8995143469.1
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8995143470.2
8995143471.3
8995143472. 4
Question Number : 4 Question Id : 899514874 Question Type : MCQ Option Shuffling : No Display Questi Mandatory : No Single Line Question Option : No Option Orientation : Vertical
Correct Marks : 2 Wrong Marks : 0
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The n th Legendre polynomial is Pn(x). Then the value of Pn(-1) equals:
| A | 1 |
| B | (-1)n |
| C | 1/(2nn!) |
| D | 2nn! |
Options:
8995143473.1
8995143474.2
8995143475.3
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8995143476.4
Question Number : 5 Question Id : 899514875 Question Type : MCQ Option Shuffling : No Display Questi Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
What is the value of lim n?8 nvn! / n ?
| A | e |
| B | 1/e |
| C | 1 |
| D | ve |
Options:
8995143477.1
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8995143478.2
8995143479.3
8995143480.4
Question Number : 6 Question Id : 899514876 Question Type : MCQ Option Shuffling : No Display Questi Mandatory : No Single Line Question Option : No Option Orientation : Vertical
Correct Marks : 2 Wrong Marks : 0
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Which of the following is Fejer's kernel?
| A | 1/(2(n+1)) (sin((n+1)?/2) / sin(?/2))-2 |
| B | 1/(n+1) (sin((n+1)?/2) / sin(?/2))-2 |
| C | 1/(2p) (sin((n+1)?/2) / sin(?/2))-2 |
| D | 1/(2(n+1)p) (sin((n+1)?/2) / sin(?/2))-2 |
Options:
8995143481.1
8995143482.2
8995143483.3
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8995143484.4
Question Number : 7 Question Id : 899514877 Question Type : MCQ Option Shuffling : No Display Questi Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
The behavior of the Fejer kernel Kn(t) near t = 0 is
| A | (n+1)/2 |
| B | 1/(n+1) |
| C | (n+1)/(2p) |
| D | 1/(2p) |
Options:
8995143485.1
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8995143486. 2
8995143487.3
8995143488.4
Question Number : 8 Question Id : 899514878 Question Type : MCQ Option Shuffling : No Display Questi Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
Weyl's equi-distribution theorem is a sharpening of
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| A | Weierstrass's approximation theorem |
| B | Kronecker's theorem |
| C | Riemann Lebesgue Lemma |
| D | Parseval's theorem |
Options:
8995143489.1
8995143490.2
8995143491.3
8995143492.4
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Question Number : 9 Question Id : 899514879 Question Type : MCQ Option Shuffling : No Display Questi Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
Which of the following is FALSE ?
| ? | {n + mp: m, n ? Z} is dense in R |
| ? | {n - [np]: n ? N} is dense in [0, 1] |
| C | {n+22m/7: m, n ? Z} is dense in R |
| D | {q-[qn]: n? N} is dense in [0, 1], where q is any irrational number |
Options:
8995143493.1
8995143494.2
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8995143495.3
8995143496.4
Question Number : 10 Question Id : 899514880 Question Type : MCQ Option Shuffling : No Display Quest Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
Which of the following functions is not in the Schwartz's class?
| A | 1/(cosh x) |
| B | x/ sinh x |
| C | sinh x/cosh x |
| D | x100/cosh x |
Options:
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8995143497.1
8995143498.2
8995143499.3
8995143500.4
Question Number : 11 Question Id : 899514881 Question Type : MCQ Option Shuffling : No Display Quest Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
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The function f(x) equals 1 on the interval [-1,1] and is zero outside this interval. The Fourier transform of f(x) equals
| A | sin?/? |
| ? | 2sin?/? |
| C | cos ?/? |
| D | sin ? |
Options:
8995143501. 1
8995143502. 2
8995143503.3
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8995143504.4
Question Number : 12 Question Id : 899514882 Question Type : MCQ Option Shuffling : No Display Quest Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
Fourier transform of e-x2 is vpe-bx2. Then the value of b equals
| A | ¼ |
| B | 4 |
| C | 1 |
| D | ½ |
Options:
8995143505. 1
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8995143506.2
8995143507.3
8995143508.4
Question Number : 13 Question Id : 899514883 Question Type : MCQ Option Shuffling : No Display Quest Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
Supply the value of the constant c in the following formula (where f(x) is in the Schwartz's class) f(x) = cf(-x)
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| ? | 2p |
| B | p |
| C | p/2 |
| D | 1 |
Options:
8995143509. 1
8995143510. 2
8995143511.3
8995143512. 4
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Question Number : 14 Question Id : 899514884 Question Type : MCQ Option Shuffling : No Display Quest Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
Which of the following is NOT the Fourier transform of a function in L² (R)
| A | |?|/(1 + cosh?) |
| ? | |?|/(1 + |?|2) |
| C | sin?/|?| |
| D | tanh ? |
Options:
8995143513. 1
8995143514.2
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8995143515.3
8995143516.4
Question Number : 15 Question Id : 899514885 Question Type : MCQ Option Shuffling : No Display Quest Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
Given that J'0(x) = Jp(x) (where Jp(x) denotes the Bessel function of order p, the value of p equals:
| A | -1 |
| B | 1 |
| C | 2 |
| D | -3 |
Options:
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8995143517. 1
8995143518.2
8995143519.3
8995143520.4
Question Number : 16 Question Id : 899514886 Question Type : MCQ Option Shuffling : No Display Quest Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
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Which of the following differential equations is invariant under the Fourier transform?
| A | Airy's equation |
| B | Hermite's equation |
| C | Legendre's equation |
| D | Tchebychev' |
Options:
8995143521.1
8995143522.2
8995143523.3
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8995143524.4
Question Number : 17 Question Id : 899514887 Question Type : MCQ Option Shuffling : No Display Quest Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
If (fn(x)) is an orthonormal sequence of functions in L2 [0, 1] then which of the following is always true?
| A | The sequence converges in norm |
| B | The sequence converges weakly |
| C | The sequence converges pointwise |
| D | The sequence converges uniformly |
Options:
8995143525. 1
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8995143526. 2
8995143527.3
8995143528.4
Question Number : 18 Question Id : 899514888 Question Type : MCQ Option Shuffling : No Display Questi Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
Consider the function f(x) = 1 - cos x + 1/3 (sin 3x - cos 3x). The value of ?p-p|f(x)|2dx equals:
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| ? | 20p/9 |
| B | 23p/9 |
| C | 26p/9 |
| D | 29p/9 |
Options:
8995143529.1
8995143530.2
8995143531.3
8995143532.4
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Question Number : 19 Question Id: 899514889 Question Type : MCQ Option Shuffling : No Display Questi
Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
Given that J0(x) = 1/p ?p0 cos(x sin?) d?. Which of the following is true?
| A | J0(x) = ?80 1/n!2 (x/2)2n |
| B | J0(x) = ?80 1/n!2 (x2/2)2n |
| C | J0(x) = ?80 1/n! (x2/2)2n |
| D | J0(x) = ?80 (-1)n/n! (x/2)2n |
Options:
8995143533.1
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8995143534.2
8995143535.3
8995143536.4
Question Number : 20 Question Id : 899514890 Question Type : MCQ Option Shuffling : No Display Questi Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
Let X be the Banach space of continuous functions f(x) on [-p, p] such that f(p) = f(-p) endowed with the supremum norm and S be the subset of X consisting of all the functions in X whose Fourier series converges pointwise everywhere. Then
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| A | S equals X. |
| B | S is not equal to X but S is a countable intersection of dense open sets in X. |
| C | X – S is dense in X |
| D | X – S is a countable union of closed sets each having empty interior in X |
Options:
8995143537. 1
8995143538.2
8995143539.3
8995143540.4
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Question Number : 21 Question Id : 899514891 Question Type : MCQ Option Shuffling : No Display Questi Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
Which of the following represents the Poisson Kernel?
| A | 1/(2p) (1-r2) / (1 + r2 - 2r cos ?) |
| B | p (1-r2) / (1 + r2 - 2r cos ?) |
| C | 1/(2p) (1 + r2) / (1 + r2 - 2r cos ?) |
| D | p (1 + r2) / (1+r2 - 2 cos ?) |
Options:
8995143541. 1
8995143542. 2
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8995143543.3
8995143544.4
Question Number : 22 Question Id : 899514892 Question Type : MCQ Option Shuffling : No Display Questi Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
The Lebesgue constants are the numbers Ln = ?p0 | Dn(t)|dt where Dn(t) is the Dirichlet kernel. Which of the following is true?
| A | ?80 Ln converges. |
| B | Ln~c log n for some constant c and for large n |
| C | Ln? O as n ? 8 |
| D | Lncvn logn for large n |
Options:
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8995143545. 1
8995143546. 2
8995143547.3
8995143548.4
Question Number : 23 Question Id : 899514893 Question Type : MCQ Option Shuffling : No Display Questi Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
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Consider the solution u(x, y) of ?u = 0 on {(x, y): x2 + y2 < 1} and u(cos?, sin?) = | sin ?|. Then the value of u(0, 0) equals:
| A | p/2 |
| B | 1/p |
| C | 2/p |
| D | p |
Options:
8995143549. 1
8995143550.2
8995143551.3
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8995143552.4
Question Number : 24 Question Id : 899514894 Question Type : MCQ Option Shuffling : No Display Questi Mandatory : No Single Line Question Option : No Option Orientation : Vertical Correct Marks : 2 Wrong Marks : 0
Consider the functions xn on L2 (-1,1)(n = 0, 1, 2, 3, ...). When these are subjected to the Gram-Schmidt process the resulting functions are fn(x). Then which of the following is true?
| A | The fn(x) are the Legendre polynomials |
| B | The fn(x) are the Tchebycheff's polynomials |
| C | The fn(x) are the Hermite polynomials |
| D | The fn(x) are the Laguerre polynomials |
Options:
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