Math Chapter Wise

EnglishUP PGT 2021

Q21.Which of the following functions T: R2 → → R2 is a linear transformation?

AT(x, y) = (x + 1, y) BT(x, y) = (x, y + 1) CT(x, y) = (x + y, 0) DT(x, y) = (x – 1, y)

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EnglishUPPSC GIC 2021 UPPSC Ashram Paddhati 2021

Q22.The rank and nullity of the linear transformation T: R2 → → R3 defined by– T (a, b) = (a – b, b – a, –a) are respectively–

A1, 1 B2, 0 C0, 2 D2, 1

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EnglishUPPSC GIC 2021

Q23.If W1 and W2 are subspaces of a finite dimensional vector space V, then dim W1 + dim W2 is equal to-

Adim (W1 + W2) Bdim (W1 ∩ W2) Cdim (W1 + W2)- dim (W1∩W2) Ddim (W1 + W2)+ dim (W1∩W2)

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EnglishUPPSC GIC 2021

Q24.Let V and W be two vector spaces over a field F. If T: V → → → W is a linear transformation such that dim V = 5 and nullity (T) = 2, then rank (T) is -

A0 B2 C3 D1

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EnglishUPPSC GIC 2021

Q25.What can be said about the solution of the following system of linear equations? x + y + 2z = 4 2x – y + 3z = 6 x – y – z = 1

AIt has a unique non-zero solution BIt has infinitely many solutions CIt has no solution DNone of the above is true

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EnglishUP Higher Asst. Prof. 2021

Q26.How many linearly independent solution may exist, of a linear homogeneous equation of order n?

AMore than n Bn +1 only Cn – 1 only Dless than or equals to n

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EnglishUP Higher Asst. Prof. 2021

Q27.Orthonormal set of vectors in an inner product space is

ALinearly dependent BLinearly independent CNeither dependent nor independent DNone of these

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EnglishUP Higher Asst. Prof. 2021

Q28.If {α,β} is an orthonormal set in an inner product space then distance between α and β is

A0 B1 C2 D2

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EnglishUP Higher Asst. Prof. 2021

Q29.If [ ] () C a,b is the space of continuous real valued function defined on the closed interval [a,b] of real line then () () b a L x f x dx = ∫ defined the.

Afunction L on [ ] () C a,b Blinear function L on [ ] () C a,b Clinear functional L on [ ] () C a,b Dtransformation L on [ ] () C a,b

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EnglishUP Higher Asst. Prof. 2021 UPPSC Ashram Paddhati 2021

Q30.If V(F) and W(F) are finite dimensional vector spaces of dimensions n and m respectively. Then the space L(V,W) is finite dimensional of the dimension

Am + n Bm × n Cm – n Dm/n

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EnglishUP Higher Asst. Prof. 2021

Q31.The system of linear equations x – y + z = 2 x + y – z = 0 6x – 4y + 4z = 11

Ahas trivial solution Bhas unique solution Cis consistent Dis inconsistent

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EnglishUPPSC Ashram Paddhati 2021

Q32.Let M(R) be the vector space of 2×2 matrices over R and ∈ 1 x y W =: x,y R and 0 x ∈ 2 x y W =: x,y,z R be two z 0 subspaces of M(R), then dim (W1 ∩ ∩ ∩ W2) is

A0 B1 C2 D3

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EnglishUPPSC Ashram Paddhati 2021

Q33.Let A = 1 1 1 0 0 1 x y z and V = {(x, y, z) ∈ R3: |A| = 0}, then the dimension of V is

A0 B1 C2 D3

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EnglishDSSSB TGT 04-09-2021 Shift-I (Male)

Q34.Let A be a complex n × × × n matrix. Let λ1, λ2, λ3 be three distinct Eigen values of A, with corresponding eigenvectors z1, z2, z3. Then, which of the following statements is false?

Az1+ z2, z1 – z2, z3 are linearly independent. Bz1,z2, z3 are linearly independent Cz1, z2 z3 are linearly independent if and only if A is diagonalizable Dz1, z1 + z2, z1+ z2 + z3 are linearly independent

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EnglishDSSSB TGT 04-09-2021 Shift-I (Male)

Q35.If A is a 9× 10 matrix of rank 3. Which of the following is equal to the dimension of the space of A?

A3 B10 C7 D9

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EnglishDSSSB TGT 04-09-2021 Shift-I (Male)

Q36.If A be a n× n real matrix. Then, which of the following statements is satisfied conditions for the matrix?

A= r, then A has r non-zero eigenvalue BIf the Eigenvalues of A are 1 λ,...., n λ, then A is similar to a diagonal matrix with λ1,.... λn along the diagonal. CIf rank DIf A has a repeated eigenvalue, then A is not diagonalizable

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EnglishHaryana PGT 2020

Q37.The system of equations –2x + y + z = a; x – 2y + z = b; x + y –2z = c has unique solution, if:

Aa + b + c = 0 Ba = b = c Ca = 2, b = 2, c = –2 DUnique solution is not possible

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EnglishUKPSC GIC 2018

Q38.The characteristic roots of two matrices A and BAB–1 are:

Athe same Bdifferent Calways zero DNone of these

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English(GIC) 2015

Q39.If T:R2 → →R2 is a linear transformation and it defined by T(x, y) = (y, 2x–y), T–1 (a, b)

Aa + b, a 2 Ba b, 2 2 Cb a, 2 Da - b, b 2

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English(GIC) 2015

Q40.V(F) is a vector space and M is its subspace. If dim (V) = n and V dim =r, M then dim M is

An/r Br–n Cn–r Dn+1

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