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Question 1

If the system of equations

\begin{aligned} &x+y+z=6 \\ &2 x+5 y+\alpha z=\beta \\ &x+2 y+3 z=14 \end{aligned}

has infinitely many solutions, then $\alpha+\beta$ is equal to

Question 2

1.

If $A = \left[ \begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{matrix} \right]$ and $M = A + A^{2} + A^{3} + \ldots + A^{20}$ , then the sum of all the elements of the matrix $M$ is equal to _____________.

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Question 3

If the system of linear equations

x + ky + 3z = 0
3x + ky - 2z = 0
2x + 4y - 3z = 0

has a non-zero solution (x, y, z), then ${{xz} \over {{y^2}}}$ is equal to

Question 4

Let $A = [{a_{ij}}]$ be a square matrix of order 3 such that ${a_{ij}} = {2^{j - i}}$ , for all i, j = 1, 2, 3. Then, the matrix A² + A³ + ...... + A¹⁰ is equal to :

Question 5

For the system of equations $x+y+z=6$ $x+2 y+\alpha z=10$ $x+3 y+5 z=\beta$ , which one of the following is NOT true?

Question 6

For the system of linear equations:

$x - 2y = 1,x - y + kz = - 2,ky + 4z = 6,k \in R$ ,

consider the following statements :

(A) The system has unique solution if $k \ne 2,k \ne - 2$ .

(B) The system has unique solution if k = $-$ 2

(C) The system has unique solution if k = 2

(D) The system has no solution if k = 2

(E) The system has infinite number of solutions if k $\ne$ $-$ 2.

Which of the following statements are correct?

Question 7

If $\alpha ,\beta \ne 0,$ and $f\left( n \right) = {\alpha ^n} + {\beta ^n}$ and

3 &amp; {1 + f\left( 1 \right)} &amp; {1 + f\left( 2 \right)} \cr {1 + f\left( 1 \right)} &amp; {1 + f\left( 2 \right)} &amp; {1 + f\left( 3 \right)} \cr {1 + f\left( 2 \right)} &amp; {1 + f\left( 3 \right)} &amp; {1 + f\left( 4 \right)} \cr } } \right|$$$ <br> $$ = K{\left( {1 - \alpha } \right)^2}{\left( {1 - \beta } \right)^2}{\left( {\alpha - \beta } \right)^2},$$ then $$K$$ is equal to :

Question 8

Let $A=\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right]$ . If $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} 2 A))|=(16)^{n}$ , then $n$ is equal to :

Question 9

If $${\Delta _1} = \left| {\matrix{ x & {\sin \theta } & {\cos \theta } \cr { - \sin \theta } & { - x} & 1 \cr {\cos \theta } & 1 & x \cr

} } \right| $and <br>$ {\Delta _2} = \left| {\matrix{ x & {\sin 2\theta } & {\cos 2\theta } \cr { - \sin 2\theta } & { - x} & 1 \cr {\cos 2\theta } & 1 & x \cr

} } \right| $,$ x \ne 0 $; <br><br> then for all$ \theta \in \left( {0,{\pi \over 2}} \right)$$ :

Question 10

1.

Let $$A = \left( {\matrix{ 2 & { - 2} \cr 1 & { - 1} \cr

} } \right) $and$ B = \left( {\matrix{ { - 1} & 2 \cr { - 1} & 2 \cr

} } \right) $. Then the number of elements in the set {(n, m) : n, m$ \in$$ {1, 2, .........., 10} and nAⁿ + mB^m = I} is ____________.

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Matrices And Determinants

Matrices And Determinants