Math Chapter Wise

English

Q41.Which of the following series converges? I. n n 1 3 cosn e ∞ = + ∑ ∑ II. n 1 1 cos n ∞ = ∑

AI only BII only CI and II both Dneither I nor II

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EnglishUP TGT-2016 (08-03-2019)

Q42.The sequence n 1 n sin 2 n ∞ = π converges to

A0 B1 Cπ D-1

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EnglishKVS PGT 23-12-2018

Q43.The series p p p p 1 1 1 1.... 1 2 3 4 − + − + − + − + − + − + − + − +:

Aconvergesforp>0 Bconvergesforp<0 Cdiverges for p > 0 Ddiverges for p < 0

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EnglishOPSC PGT 2017

Q44.If a function f is measureable then:

Af is always measurable Bf is bounded but not measurable Cf may be measurable subject to some conditions DThen f should be a limit to sequence of functions

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EnglishOPSC PGT 2017

Q45.If f is a non-negative measurable function and S fdm 0 then f is:

AA constant BZero everywhere CA periodic function DZero, almost everywhere

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EnglishOPSC PGT 2017

Q46.A real valued function defined on a measurable space is called a simple function if:

AThe domain of the function is finite BThe range of the function is finite CMeasurable space is a vector space DFunction is a contraction map

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EnglishOPSC PGT 2017

Q47.If {fn} is a monotone increasing sequence of non-negative measurable functions from S to R then n s s n fdm = lim f dm →∞ ∫ ∫ ∫ ∫. This theorem is known as:

ABounded Convergence Theorem BDominated Convergence Theorem CMonotone Convergence Theorem DMonotone Measure Theorem

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EnglishOPSC PGT 2017

Q48.According to Beach's criterion, a normed vector space X is complete if and only if every:

AAbsolutely convergen, series in X is convergent BConvergent series in X is uniformly convergent CSeries in X is uniformly convergent DSeries in X is absolutely convergent

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EnglishOPSC PGT 2017

Q49.Let E and F are Banach spaces. T∈B(E,F) becomes an open map When it is:

ABijective BInjective CSurjective DNeither injective nor surjective

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EnglishOPSC PGT 2017

Q50.Closed-graph theorem is used to given a proof of:

AOpen-mapping lemma BThe principle of uniform boundedness CUrysohns lemma DParseval's identity

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EnglishOPSC PGT 2017

Q51.Let (H, <..>) be an inner product space, then for a, b∈H, the relation 2 2 2 2 x y x y 2 x 2 y is known as:

APythagorean theorem BLaw of convexity CReisz-Fischer theorem DParallelogram Law

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EnglishOPSC PGT 2017

Q52.Let H be a separable Hilbert space. All orthonormal bases of H are:

ACountable BDense in H CProper closed subspaces DSeparable

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EnglishOPSC PGT 2017

Q53.For an orthonormal subset β of H, which of the following are equivalent? 1. β is a basis 2. β is complete 3. Span β = H

A1 and 2 B2 and 3 C1 and 3 DAll of these

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EnglishOPSC PGT 2017

Q54.Linear operator A on a finite-dimensional vector space X is one-to-one if and only if:

AThe range of A is all of X BThe domain of A is subset of X CThe domain and range of A is subset of X DThe domain of A is all of X

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EnglishOPSC PGT 2017

Q55.If A is a n × n non singular matrix, then adj(adj A) is equal to:

An 2/A Bn 1/A A Cn 1/A Dn 2/A A

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EnglishOPSC PGT 2017

Q56.A square matrix A is singular if and only if its:

AColumns are linearly independent BRows are linearly independent CColumns are linearly dependent DEigenvalues are non-zero

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EnglishOPSC PGT 2017

Q57.If α is an eigenvalues of a nonsingular matrix A then corresponding eigenvalues of adjoint of A will be:

AA BA / CA D1/A

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EnglishOPSC PGT 2017

Q58.Of a square matrix, the product of its eigenvalues is equal to:

ASum of its diagonal elements BProduct of its diagonal elements CIts determinant DDeterminant of its adjoint

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EnglishOPSC PGT 2017

Q59.What value of k makes the vectors (1, –1, 3), (1,2,-2), (k,0,1) linearly dependent?

A3/4 B1/2 C1/4 D–3/4

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EnglishOPSC PGT 2017

Q60.Which of the following maps are linear transformations? 1. T: R2 → →R defined by T(x, y) = 2x 3y 2. T: R2 → →R defined by T(x, y) = xy 3. T: R2 → →R2 defined by T(x, y, z) = (z, x + y)

AAll of these B1 and 2 only C3 only D2 and 3 only

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