Math Chapter Wise

EnglishUP PGT-2016 (02-02-2019)

Q61.If 2 3 T: V (R) V (R) → → be defined as T(a,b)= (a+b, a–b, b) is a linear transformation, then nullity T is

A0 B1 C2 D3

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EnglishUP PGT-2016 (02-02-2019)

Q62.If ((()) { { } } 3 3 V R,W a,0,0 R, = ∈ = ∈ = ∈ = ∈ then dimension of vector space V/W is

A1 B2 C3 D4

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EnglishUP PGT-2016 (02-02-2019)

Q63.If W is a subspace of R4 spanned by {(1,0,–1,0), (0,1,0,1)}, then which of the following is in W?

A(1, 2, 1, 2) B(1, 2, –1, 2) C(2, 2, 3, 2) D(0, 1, 1, 1)

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EnglishDSSSB 23 SEP 2018 (9 AM-11 AM)

Q64.Consider four sets of vectors A, B, C and D in 3. Out of these sets of vectors, one set in linearly dependent in 3, identity this linearly dependent set.

AA = {[1,1,1], [1,1,0], [1,0,0]} BC = {[1,1,3], [0,2,1], [0,0,0]} CB = {[1,0,0], [0,1,1], [1,0,1]} DD = {[1,1,1], [0,1,1], [2,0,1]}

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EnglishDSSSB 23 SEP 2018 (9 AM-11 AM)

Q65.Consider a non-homogeneous system of 5 linear equation in 4 variables, such that there is at least one variable with a non-zero coefficient in each equation. Then the possible smallest and largest rank of the corresponding augmented matrix are.

ASmallest =1, Largest=5 BSmallest =1, Largest=4 CSmallest =2, Largest=4 DSmallest =2, Largest=5

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EnglishDSSSB 23 SEP 2018 (9 AM-11 AM)

Q66.A transformation T: 2 → → →2 first reflects points through the vertical axis (x2-axis) and then reflects points through the line x2=x1 then the standard matrix of T is.

A−1 0/0 1 B−1 0/0 1 C−0 1/1 0 D−0 1/1 0

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EnglishDSSSB TGT (SECTION-B) 28 DEC 2014

Q67.Which of the following set is a basis of 3 ℝ ℝ?

A1 1 1 0, 2, 2 0 0 3 B1 12 0 2, 24, 0 0 7 1 C1 1 4 2, 3, 9 0 2 5 D1 2 4 1, 3, 5 2 1 5

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EnglishDSSSB TGT (SECTION-B) 28 DEC 2014

Q68.Let 2 3 T: ℝ ℝ ℝ ℝ ℝ ℝ be a linear transformation defined by T(x,y) (x y,x y,y) Then the rank of T is–

A3 B2 C0 DNone of these

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EnglishSECTION B PGT TIER-I 31.11.2014

Q69.If S = {(1,1,0), (2,1,3)} is a subset of 3 R then which one of the following vectors 3 R is not in the linear span of S?

A(0,0,0) B(3,2,3) C(1,2,3) D(4/ 3,1,1)

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EnglishSECTION B PGT TIER-I 31.11.2014

Q70.Let T: 3 R → → → 2 R be a linear transformation defined by T(x,y,z) (x y,y z). Then dimension of nullspace of T is–

A1 B2 C0 DNone of these

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EnglishSECTION B PGT TIER-I 31.11.2014

Q71.Dimension of a subspace W {(a,b,c) a b c 0}of 3 ℝ ℝ is –

A4 B3 C1 D2

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EnglishSECTION B PGT TIER-I 31.11.2014

Q72.Let 3 n T: R R be a linear transformation, Which of the following statements implies that T is bijective?

ANullity (T) = n BRank (T) = Nullity (T) = n CRank (T) + Nullity (T) = n DRank (T) – Nullity (T) = n

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EnglishSECTION B PGT TIER-I 31.11.2014

Q73.A linear transformation T rotates each vector in IR2 clockwise through 900 The matrix T relative to the standard ordered basis 1 0. 0 1 is–

A0 1/1 0 B0 1/1 0 C0 1/1 0 D0 1/1 0

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EnglishJDD-75-PGT TIER II-X-15 28.06.2015

Q74.Given that xy-plane and yz-plane in 3 R are the subspace of 3 R, The dimension of the intersection of these subpaces is

A0 B1 C2 DNone of these

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EnglishJDD-75-PGT TIER II-X-15 28.06.2015

Q75.Let T: V → → →V be a linear operator on V. If W is a subspace of V, then W is invariant subspace under T if–

AT(W)⊂W BT(W)=W CW⊃T(W) DNone of these

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EnglishJDD-75-PGT TIER II-X-15 28.06.2015

Q76.The matrix representation of the linear transformation T: 2 IR → → → 3 IR defined T(x,y)=(3x–y, 2x+4y, 5x–6y) with respect to standard basis is–

A1 3 4 2 6 5 − B3 1 2 4 5 6 − C3 2 5 1 4 6 − − DNone of these

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EnglishJDD-75-PGT TIER II-X-15 28.06.2015

Q77.P3 is a vector space of polynomials in x of degree three or less and D(p(x)) is the derivative of p(x) is a transformation from P3 to P2, then–

Athe polynomial 2x+1 is the Kernel of D Bthe rank of D is 3 Cthe Kernel of D is all those constant polynomials is P3 Dnone of these

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EnglishJDD-75-PGT TIER II-X-15 28.06.2015

Q78.The null space of A is the solution set of the equation

AAx = 0 BAx = b CAx ≥ 0 DNone of these

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EnglishJDD-75-PGT TIER II-X-15 28.06.2015

Q79.Which of the following is not a linear transformation from IR3 to IR3?

AT(x, y,z) (x,2y,3x y) = − BT(x, y,z) (x y,0, y z) = − − CT(x, y,z) (0,0,0) = DT(x, y,z) (1,x,z) =

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EnglishJDD-75-PGT TIER II-X-15 28.06.2015

Q80.Let V be a 3-dimensional vector space over the field 3 Z F 3Z = of 3 elements. The number of distinct 1- dimensional subspaces of V is–

A26 B9 C13 D15

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