The system of linear equations x+λy−z=0; λx−y−z=0; x+y�

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 Multiple Choice QuestionsMultiple Choice Questions

1.

If A = open square brackets table row cell 5 straight a end cell cell negative straight b end cell row 3 2 end table close square brackets and A adj A = AAT, then 5a +b is equal to

  • -1

  • 5

  • 4

  • 4

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2.

The system of linear equations x+λy−z=0; λx−y−z=0; x+y−λz=0 has a non-trivial solution for

  • infinitely many values of λ.

  • exactly one value of λ.

  • exactly two values of λ.

  • exactly two values of λ.


D.

exactly two values of λ.

Given system of linear equations is 

x+λy−z=0;
λx−y−z=0;
x+y−λz=0 

Note that, given system will have a non-trivial solution only if the determinant of the coefficient matrix is zero, ie.
open vertical bar table row 1 straight lambda cell negative 1 end cell row straight lambda cell negative 1 end cell cell negative 1 end cell row 1 1 cell negative straight lambda end cell end table close vertical bar space equals 0

rightwards double arrow space 1 space left parenthesis straight lambda space plus 1 right parenthesis minus straight lambda left parenthesis negative straight lambda squared space plus 1 right parenthesis minus 1 left parenthesis straight lambda plus 1 right parenthesis space equals 0
rightwards double arrow space space straight lambda plus 1 space plus straight lambda squared minus straight lambda minus straight lambda minus 1 space equals 0
rightwards double arrow straight lambda cubed minus straight lambda space equals space 0
rightwards double arrow straight lambda left parenthesis straight lambda squared minus 1 right parenthesis space equals 0
straight lambda space equals space 0 space or space straight lambda space plus-or-minus 1

Hence, given system of linear equation has a non-trivial solution for exactly three values of λ.

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3.

A = open square brackets table row 1 2 2 row 2 1 cell negative 2 end cell row straight a 2 straight b end table close square brackets is a matrix satisfying the equation AAT = 9I, Where I is 3 x 3 identity matrix, then the ordered pair (a,b) is equal to

  • (2,-1)

  • (-2,1)

  • (2,1)

  • (2,1)

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4.

The set of all values of λ for which the system of linear equations 

2x1-2x2+x3 = λx1
2x1- 3x2 + 2x3 = λx2
-x1 + 2x2 = λx3
a non- trivial solution.

  • is an empty set

  • is a singleton set

  • contains two elements

  • contains two elements

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5.

If α, β ≠ 0 and f(n) = αn+ βn and 

open vertical bar table row 3 cell 1 plus straight f left parenthesis 1 right parenthesis space space space space end cell cell 1 plus space straight f left parenthesis 2 right parenthesis end cell row cell 1 plus straight f left parenthesis 1 right parenthesis space space space space space end cell cell 1 plus straight f left parenthesis 2 right parenthesis space space space space space space end cell cell 1 plus straight f left parenthesis 3 right parenthesis end cell row cell 1 plus straight f left parenthesis 2 right parenthesis space space space end cell cell 1 plus straight f left parenthesis 3 right parenthesis space space space space space space space end cell cell 1 plus space straight f left parenthesis 4 right parenthesis end cell end table close vertical bar
= K(1-α)2(1-β)2(α- β)2, then K is equal to 

  • αβ 

  • 1/αβ 

  • 1

  • 1

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6.

If A is a 3x3 non- singular matrix such that AAT = ATA, then BBT is equal to

  • l +B
  • l
  • B-1

  • B-1

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7.

Let P and Q be 3 × 3 matrices with P ≠ Q. If P3= Q3 and P2Q = Q2P, then determinant of(P2+ Q2) is equal to

  • -2

  • 1

  • 0

  • 0

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8.

The number of values of k for which the linear equations
4x + ky + 2z = 0
kx + 4y + z = 0
2x + 2y + z = 0
posses a non-zero solution is:

  • 3

  • 2

  • 1

  • 1

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9.

Consider the system of linear equation
x1 + 2x2 + x3 = 3
2x1 + 3x2 + x3 = 3
3x1 + 5x2 + 2x3 = 1
The system has

  • infinite number of solutions

  • exactly 3 solutions

  • a unique solution

  • a unique solution

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10.

Let A be a 2 × 2 matrix with non-zero entries and let A2 = I, where I is 2 × 2 identity matrix. Define Tr(A) = sum of diagonal elements of A and |A| = determinant of matrix A.
Statement-1: Tr(A) = 0.
Statement-2: |A| = 1.

  • Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.

  • Statement-1 is true, Statement-2 is true; statement-2 is not a correct explanation for Statement-1.

  • Statement-1 is true, Statement-2 is false.

  • Statement-1 is true, Statement-2 is false.

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