Math Chapter Wise

EnglishBihar (TRE 1.0) -2023

Q01.Let A be a real matrix which haracteristic polynomial (X – 1)3. Pick the correct statement from below:

AA is necessarily diagonalizable BIf the minimal polynomial of A is (X – 1)3, the A is diagonalizable CCharacteristic polynomial of A2 is (X – 1)3 DMore than one of the above ENone of the above

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EnglishBihar (TRE 1.0) -2023

Q02.Let T: Rn → → Rn be a linear map that satisfies T2 = T – In. Then which of the following are true?

AT is invertible BT - In is not invertible CT has a real eigenvalue DMore than one of the above ENone of the above

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EnglishBihar (TRE 1.0) -2023

Q03.The trace of the matrix 20 2 1 0 0 2 0 0 0 3 is

A7020 B220/+ 320 C2.220/+ 320 DMore than one of the above ENone of the above

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EnglishBihar (TRE 1.0) -2023

Q04.Which of the following real quadratic forms on R2 is positive definite?

AQ(X,Y) = XY BQ(X,Y) = X2 + 2XY + Y2 CQ(X,Y) =X2 – XY + Y2 DMore than one of the above ENone of the above

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EnglishBihar (TRE 1.0) -2023

Q05.Let V be an inner product space and let v1, v2, v3 ∈V be an orthogonal set of vectors. Which of the following statements is true?

AThe vectors 5v1 + 6v2 + 7v3, 5v2 + v3. v2 + 3v3 can be extended to a basis of V BThe vectors 4v1 + v2 + 2v3, v2 + 3v3 v2 + 3v3 can be extended to an orthogonal basis of V CThe vectors v1 + v2 + 2v3, v2 + v3, 2v1 + v2 + 3v3 can be extended to a basis of V DMore than one of the above ENone of the above

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EnglishBihar (TRE 1.0) -2023

Q06.Consider the vector space Pn of real polynomial in x of degree less than or equal to n. Defined T: P2 → → → P3 by (Tf) (x) = () () ∫ x 0 f t dt + f' x Then the matrix representation of T with respect to the bases {1, x, x2 } and {1, x, x2, x3 } is

A0 1 0 0 1 1 0 0 2 1 0 2 0 2 B0 1 0 1 0 2 1 0 0 2 1 0 0 3 C0 1 0 0 1 0 2 0 1 1 0 0 2 3 DMore than one of the above ENone of the above

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EnglishUPPSC GDC 2021

Q07.Let B = 1 2 0 0 and T: V → V be the linear map defined by T

A1, 3 B3, 1 C2, 2 D4, 0

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EnglishUPPSC GDC 2021

Q08.Which of the following statements is true for a linear transformation T: R3 →R3 defined by T (a, b, c) = (a – b, b – c, c – a)? I: T is one-one II: T is onto

AOnly I BOnly II CEither I or II DNeither I nor II

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EnglishUPPSC GDC 2021

Q09.If T: R2 (R) → R2 (R) is defined by T(2, 3) = (4, 5) and T (1, 0) = (0, 0), then T (x, y) will be

A4y 5y, 3 3 B5y 4y, 3 3 C4y 5y x, 3 3 − D3y 4y 3y x, 3 3 − −

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EnglishUPPSC GDC 2021

Q10.Which of the following set of vectors is a basis for vector space R3?

A() () () { } 1, 1,1, 1,0,2, 2, 1,3 − − B() () () { } 1, 1,1, 1,0,2, 2,1,1 − C() () () { } 1, 1,1, 1,0,2, 0,1,1 − D() () () { } 1, 1,1, 1,0,2, 1, 2,0 − −

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EnglishUPPSC Polytechnic Lecturer 2021(II)

Q11.Let 'A' and 'B' be square matrices of order 'n' over the field 'F'. Then the matrix 'B' is said to be similar to the matrix 'A', if there exists an invertible square matrix 'C' of order 'n' such that 'B' equal to -

AA–1/BA BB–1/AB CC–1/AC DNone of these

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EnglishUP PGT 2021

Q12.Let T be a linear transformation from R3 → → R2, defined by T (x, y, z) = (x + y, y – z) then the matrix T with respect to the ordered basis {(1, 1, 1), (1, –1, 0), (0, 1, 0)} and {(1, 1), (1, 0)} is –

A0 –1 1/2 1 0 B–2 0 1/1 1 –1 C2 1 0 –1 1 1 D0 2 –1 1 1 0

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EnglishUP PGT 2021

Q13.If the nullity of the matrix is 1, then the value of K is – K 1 2 1 –1 –2 1 1 4

A0 B1 C2 D–1

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EnglishUP PGT 2021

Q14.If V is a n-dimensional vector space and T is a linear transformation on V such that rank and nullity of T are identical then –

An is even Bn is odd Csome times even some times odd Dnone of the above

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EnglishUP PGT 2021

Q15.The Eigen values of the matrix A = 5 4 1 2 are

A6, 0 B3, 2 C6, 1 D1, 2

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EnglishUP PGT 2021

Q16.If W = {(x, y, z) ∈ R3: x + y – z = 0} is a subspace of the vector space R3, then dim W is –

A0 B1 C2 D3

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EnglishUP PGT 2021

Q17.The distinct eigen values of the matrix 1 1 0 1 1 0 0 0 0 are –

A0, 1 B1, –1 C0, 2 D1, 2

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EnglishUP PGT 2021

Q18.The dimension of the vector space of all 3×3 real symmetric matrices is –

A3 B6 C3n D9

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EnglishUP PGT 2021

Q19.Let V(F) be a finite dimensional vector space over the field F and W be a subspace of V. If dim V = 5, dim W = 3 then dim Wº is–

A2 B3 C1 D8

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EnglishUP PGT 2021

Q20.The dimension of the vector space C(R) of the complex number over real numbers is –

A1 B2 C3 D4

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