Math Chapter Wise

EnglishUP PCS (Pre) 1996

Q01.If (1 + i tanα)1+itanβ has real values only, then one of them is given by:

A2 sec (sec) α β Bsec (sec) β α C2 sec (sec) β α DNone of these

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EnglishUP PCS (Pre) 2003

Q02.If r r r Z cos isin, 2 2 then Z1.Z2.Z3....∞ ∞ ∞ is equal to:

A–1 B1 C– 1 2 D1/2

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EnglishTGT 2013

Q03.The value of 2 2 cosh z sinh z − −:

Acosh(2z) B1 Csinh(2z) D0

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EnglishTGT 2010

Q04.The Real part of sin (x+iy) is:

Asin x Bcos y Csin x cos hy Dsin x sin hy

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EnglishTGT 2009

Q05.Time period of sin hx:

A2πi B–2πi C2π DNone of these

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EnglishBihar (TRE 1.0) -2023

Q06.The function f (z) = |z|2 is

Acontinuous at the origin only Bcontinuous everywhere but nowhere differentiable Ccontinuous everywhere but nowhere differentiable, except at the origin DMore than one of the above ENone of the above

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EnglishNVS PGT 2022

Q07.The analytic function whose real part is ex cosy is:

Aze2z/+ ci Bzez/+ ci Ce2z/+ ci Dez/+ ci

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EnglishNVS PGT 2022

Q08.Match the following: Series Radius of convergence (i) n n 0 nz ∞ = ∑

A(i) - (c), (ii) - (a), (iii) - (b) B(i) - (b), (ii) - (c), (iii) - (a) C(i) - (b), (ii) - (a), (iii) - (c) D(i) - (a), (ii) - (c), (iii) - (b)

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EnglishRPSC PGT 2022

Q09.Which of the following is not a harmonic function?

AU = x2 +y2 Bu = x2 – y2 Cu = sin hx cos y Du = 1 2 2 log(x y) 2 +

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EnglishDSSSB PGT 2021

Q10.Consider the series () ∞ ∑ n 2 n=1 1 z +1 which of the following true?

AThe sum of the series is z2 + 1 BThe sum of the series is 2 1 z CThe series converges if z 1 < DThe sum of the series is z2 – 1

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EnglishDSSSB PGT 2021

Q11.If f(z) = |z|2, then:

Af is continuous but not differentiable at 0 Bf is neither continuous nor differentiable at 0 Cf is continuous and differentiable at 0 Df is discontinuous at 1

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EnglishJSSC PGT 2018

Q12.Which of the following statements is INCORRECT?

AThe composite of two continuous functions is also continuous. BLet f (x + iy) = u (x, y) + iv (x, y). If u and v both have continuous partial derivatives with respect to x and y at point (x0, y0) then f is also differentiable at point (x0, y0). CIf f is analytic at all points interior to and on a simple closed contour C, then ʃ f (z) over C= 0. DIf f is a continuous function defined on domain D, then ʃ f (z) d (z) over (z1, z2), where z1, z2 lie in D, is the same along all contours connecting z1 and z2 and lying entirely in D.

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EnglishJSSC PGT 2018

Q13.If a function f(z) defined on some domain D of the complex plane is differentiable on D, it is said to be:

AAnalytic function BNon-singular function CBounded function DHarmonic function

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EnglishJSSC PGT 2018

Q14.Let f (r cos θ+ irsin θ) = u (r, θ) + iv (r, θ). For f to be differentiable at (r, θ), Cauchy-Riemann equation states that:

Aur = –vr/r, vr = –uθ/r Bur = –vθ/r, vr = –uθ/r Cur = –vθ/r, vθ = –uθ/r DUr = –vθ/r, vr = –uθ/r

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EnglishJSSC PGT 2018

Q15.Select the function f (z) from below that is analytic on the unit sphere z < 1.

Af (z) = ex.e(–iy) Bf (z) = cos (x) cosh (y) – isin (x) sinh (y) Cf (z) = 3x + i3xy Df (z) = 2x + ixy2

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EnglishJSSC PGT 2018

Q16.The below power series expansion represents which analytic function? () () 4n+1 z / 2n! n = 0,1,2,..., z < 100 ∑

Az sin h(z) Bz sin h (z2/) C1/z2 Dz cos h (z2/)

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EnglishJSSC PGT 2018

Q17.The below Taylor series expansion represents which analytic function? ∑ (–1)n z2n + 1 /(2n + 1)! (n = 0, 1, 2,.......)

ATan z BSin z CCos hz DCos z

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EnglishOPSC PGT 2017

Q18.Complex valued function 2 f(z) z, for complex z, is analytic:

ANowhere in complex plane BAt z = 0 only CIn entire complex plane DIn complex plane except at z = 0

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EnglishOPSC PGT 2017

Q19.For analytic f(z) = u + iv, Cauchy-Riemann equations in polar coordinates are given by:

Ar r u v,u v Br r rv u,v ru Cr r u v,u v Dr r ru v,u rv

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EnglishOPSC PGT 2017

Q20.The complex valued function f(z) = u(x,y) + iv(x,y), for z = x + iy, is analytic if and only if:

Av is derivative of u Bv is integral of u Cu and v are harmonic Dv is harmonic conjugate of u

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