Math Chapter Wise

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

Q81.Let T: 3 IR → → → 3 IR be the linear transformation defined by T(x,y,z) (x y,y z,z x) = + + + = + + + = + + + = + + + for all 3 (x,y,z) IR then–

Arank(T) 0,nullity(T) 3 Brank(T) 1,nullity(T) 2 Crank(T) 2,nullity(T) 1 Drank(T) 3,nullity(T) 0

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

Q82.Consider the subspace W= 1 2 20 {(x,x.....,x) 3 n n 1 n 2 IR x x x − − ∈ = + ∈ = + ∈ = + ∈ = + for 3 n 20} ≤ of the vector space IR20. The dimension of W is–

A2 B3 C9 D10

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

Q83.Which of the following define a metric on R?

A() 2 2 d x, y x y = − B() d x, y x 2y = − C() x y d x, y 1 x y − = + − Dnone of these

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

Q84.For a continuous function f: R R, → → let (())) ((()) { { } } Z f x R:f x 0, = ∈ = ∈ = ∈ = ∈ = is always–

Acompact Bclosed Copen Dconnected

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

Q85.Any countable set of R has–

Ano measure Bmeasure zero Cmeasure one Dnone of these

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

Q86.The general linear group ((()) 2 P GL F is of order–

AP BP2 CP(P+1) DP(P+1)(P–1)2

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

Q87.All the units in the ring of Gause integers are–

A1/± Bi/± C1, i ± ± Dnone of these

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EnglishPGT 2013

Q88.The vectors (1,–2,1) (2, 1, –1) and (7, 4, –1) in IR3 are:

Alinearly dependent Blinearly independent Clinearly dependent and linearly independent DNone of the above

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EnglishPGT 2010

Q89.If V is vector space of 2×2 matrices over a field F, then dimension of V is

A2 B3 C4 DNone of these

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EnglishPGT 2009

Q90.If T: R3 →R2 is a linear transformation given by T (a, b, c) = (a, b) ∀ (a, b, c) ∈R3 then kernel of T is

A(0, 0, c) B(a, 0, 0) C(0, b, 0) DNone of these

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EnglishPGT 2009

Q91.The eigen values of matrix − 1 2 0 2 1 0 0 0 1 are

A1, 5 ± B1, –1, 1 C1, 2, 0 D0, 0, 1

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EnglishPGT 2009

Q92.For what value of m, the vector (m, 3, 1) is a linear combination of vectors e1 = (3, 2, 1) and e2 = (2, 1, 0)?

A5 B2 C3 D4

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EnglishPGT 2009

Q93.If T and S are invertible linear operators on a vector space V(K), then (TS)–1 is

AST BT–1/S–1 CS–1/T–1 D(ST)–1

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EnglishPGT 2005

Q94.The dimension of the subspace W, of the vector space V of all 2× × ×2 matrices over the real numbers, defined by:,, = ∈ + + a b W a b c R c a b c is

A1 B2 C3 D4

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EnglishPGT 2005

Q95.If the dimensions of subspaces W1 and W2 of a vector space V are respectively 3 and 4, and dim(W1∩ ∩ ∩W2)= 1then dim(W1+W2) is

A5 B6 C7 D8

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EnglishPGT 2005

Q96.Which of the following functions T: R2 → →R2 is a linear transformation?

AT(x1, x2) = (1+x1, x2) BT(x1, x2) = (1+ 2 1 x, x2) CT(x1, x2) = (Sin x1, x2) DT(x1, x2) = (x2, x1)

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EnglishPGT 2005

Q97.The matrix of T given by in the standard ordered basis of R3 is

A3 2 1 0 1 2 1 0 4 − − B3 1 4 0 1 2 1 2 0 − C3 0 1 2 1 0 1 2 4 − D3 0 1 2 1 1 1 2 0 −

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EnglishPGT 2005

Q98.If T is a linear transformation of a vector space V to another vector space W and dimV= 4, dimW=5, nullity T= 1, then rank of T is

A1 B5 C4 D3

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EnglishPGT 2005

Q99.If T is a linear transform on V3(R) defined by T(a,b,c) = (3a, a–b, 2a+b+c)∀ ∀ ∀(a,b,c)∈V3(R), find T–1

A1 (,,),, 3 3 p q T p q r p r p q − = − − + B1 (,,),, 3 3 p p T p q r p r p q − = − − + C1 (,,),, 3 3 p p T p q r q p q r − = − + − D1 (,,),, 3 3 p p T p q r q p q r − = − + +

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EnglishRajasthan TGT 2015

Q100.For non-degenerate BFS in transportation problem with m rows and n columns, total empty cells should be

A(m + 1) (n – 1) Bm + n – 1 C(m – 1) (n – 1) D(m – 1) (n + 1)

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