Math Chapter Wise

EnglishBihar (TRE 1.0) -2023

Q01.For the function () -z 1-e f z = z the point z = 0 is a/an

Aessential singularity Bpole of order one Cremovable singularity DMore than one of the above ENone of the above

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EnglishBihar (TRE 1.0) -2023

Q02.Let g be any positively oriented circle whose centre is at the origin. Then the value of the integral ∫ 2 γ cosz dz z is

A2πi B0 C–2πi DMore than one of the above ENone of the above

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EnglishBihar (TRE 1.0) -2023

Q03.Let a-2 be the coefficient of 2 1 z in the Laurent series of f(z) = () 3 2 z +1 z z +1 around z = 0 in the region 0 < |z| < 1. Then a–2 is equal to

A1 B0 C–1 DMore than one of the above ENone of the above

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EnglishBihar (TRE 1.0) -2023

Q04.Let f: C → → → C be an entire function with f (0) = 1 and f (1) = 0. Then f maps

Aopen sets in C into open sets in C Bclosed sets in C into closed sets in C Copen sets in C into closed sets in C DMore than one of the above ENone of the above

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EnglishBihar (TRE 1.0) -2023

Q05.A non-constant entire function

Ahas at least one zero in c Bcannot have finite number of real zeros Ccannot have uncountable number of zeros in C DMore than one of the above ENone of the above

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EnglishBihar (TRE 1.0) -2023

Q06.f is an entire function. If f satisfies the following two equations f (z +1) = f (z) and f (z + i) = f (z) for every z in c then

Af = constant Bf (z) ∈ R∀z Cf is a non-constant polynomial DMore than one of the above ENone of the above

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EnglishNVS PGT 2022

Q07.The residue at z = ∞ for the function f(z) = z3 cos 1 z is:

A−1/24 B–1 C0 D1/12

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EnglishNVS PGT 2022

Q08.The value of 2 c z z 1 dz, z 1 − + − ∫ where c is the circle z = 1, is:

A0 B2πi C–2πi Dπi

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EnglishKVS PGT 2022

Q09.Evaluation of 2 |z| 10.5 z cot zdz = π ∫ is equal to:

A1240 i B220 i C1540 i D12100 i

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EnglishRPSC PGT 2022

Q10.A semi-circular disc with centre at origin and radius 2 unit is in Z – plane, its image in w– plane under the transformation w2 = Z is -

Acircular disc of radius 2 units Bsemi-circular disc of radius 2 units Cquarter circular disc of radius 2 units Dquarter circular disc of radius 4 units

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EnglishDSSSB PGT 2021

Q11.Consider the integrals I1 = () c, ∫ z 4 e dz z +1 where c is a circle z = 2 And I2 = c, ∫ 2 sin3z dz z, where c is circle z = 1 Then:

A1 2 ie I, I 2 i 3 π = π B1 2 i I, I –6 i 3e π = π C1 2 ie I, I 0 3 π = D1 2 i I, I 6 i 3e π = π

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EnglishDSSSB PGT 2021

Q12.Consider the bilinear transformer z –1 z 1 ω = +. Which of the following is true?

AThe fixed point are z = i ± and the transformer is hyperbolic BThe fixed point are z = 1, 2 and the transformation is elliptic CThe fixed point are z = ± i and the transformation is elliptic DThe fixed point z = ± 1 and the transformation is hyperbolic

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EnglishDSSSB PGT 2021

Q13.Let T: C ∪ ∪ ∪ {∞ ∞ ∞} ⇒ ⇒ ⇒ C ∪ ∪ ∪ {∞ ∞ ∞} be the bilinear transformation such that T(i) = 0, T(0) = –1, T(–i) = ∞ ∞ ∞. Then the image of the region, {z ∈ C: I m (z) = 0} under the mapping ω = T(z) is:

A{ } C: 1 ω∈ ω > B{ } C: 1 ω∈ ω < C{ } C: 1 ω∈ ω = D() () { } C: Re 0,Im 0 ω∈ ω = ω =

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EnglishDSSSB PGT 2021

Q14.Let f(z) = () cos z 1, z 1 − then,

Af(z) has isolated essential singularity at z = 1 Bf(z) has simple pole at z = 1 CResidue of f at z =1 is undefined DResidue of f at z =1 is not equal to 1

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EnglishDSSSB PGT 2021

Q15.The function f(z) = z e 2 2 1 − has

AAn essential singularity at z = 0 BA pole at z = 0 CAn essential singularity at z = ∞ DNo singularity

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EnglishDSSSB PGT 2021

Q16.The residue of the function f(x) = 1 x e − at x = 0

Ais 1 BDoes not exist, since, 0 is not a singular point Cis 0 DDoes not exist, since 0 is not a non-isolated singular point

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EnglishDSSSB PGT 2021

Q17.Let f(z) = () 2 z +1 z z -1 then the product of the residues of f at poles is:

A1 B0 C2 D–2

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EnglishJSSC PGT 2018

Q18.Using Residue Theorem, evaluate the integral ʃ (5z – 2)

A6πi B10πi C5πi D8πi

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EnglishJSSC PGT 2018

Q19.If C is a positively oriented simple closed contour within and on which function f is analytic, except for a finite number of singular points zk (k = 1, 2,.... n) interior to C, then Residue Theorem states that:

A() () () C k k f z dz e i Resf z / k! = π ∑ ∫ Bn k k 1 c f(z)dz 2 i Res[f(z);z ] = π ∑ ∫ C() () () C k k f z dz 2 i Resf z / k! = π ∑ ∫ D() () () C k k f z dz Resf z = ∑ ∫

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EnglishJSSC PGT 2018

Q20.Let C denote the boundary of circle with radius 3 around the centre z1 = 2 + i. Calculate the integral () ∫ az I = e / z – 2 – i dz over contour C.

A0 B() a 2–i 2 i e π C() a 2 i 2 i e + π D() a 2 2i 2 i e + π

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