#### Matrices and Determinants Two Marks Question

11th Standard

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Maths

Time : 00:45:00 Hrs
Total Marks : 30
15 x 2 = 30
1. If A=$\begin{bmatrix} 0 &c &b \\ c & 0 &a \\ b & a & 0 \end{bmatrix}$,compute A2

2. Construct an m × n matrix A= [aij], where a ij is given by
$a_{ij}={(i-2j)^2\over 2}with \ m=2,n=3$

3. Determine the value of x + y if $\begin{bmatrix} 2x+y & 4x \\ 5x-7 & 4x \end{bmatrix}=\begin{bmatrix} 7 & 7y-13 \\ y & x+6 \end{bmatrix}$

4. Determine the matrices A and B if they satisfy
$2A-B+\begin{bmatrix} 6 & -6 & 0\\ -4 & 2 & 1\end{bmatrix}=0 \ and \ A-2B=\begin{bmatrix} 3 & 2&8 \\ -2 & 1&-7 \end{bmatrix}$

5. Evaluate :$\begin{vmatrix} 2 & 4 \\ -1 & 2 \end{vmatrix}$

6. Compute |A| using Sarrus rule if A=$\begin{bmatrix} 3& 4 & 1 \\ 0 &-1 &2 \\ 5 & -2 & 6 \end{bmatrix}$ .

7. Find the value of x if $\begin{vmatrix} x-1 & x & x-2 \\ 0 &x-2 & x-3 \\ 0 & 0 & x-3 \end{vmatrix}=0$

8. Prove that $\begin{vmatrix} a^2 & bc & ac+c^2 \\ a^2+ab & b^2 & ac \\ ab & b^2+bc & c^2 \end{vmatrix}=4a^2b^2c^2$

9. Show that $\begin{vmatrix} x+2a & y+2b & z+2c \\ x & y & z \\ a & b & c \end{vmatrix}=0$ .

10. If the area of the triangle with vertices (- 3, 0), (3, 0) and (0, k) is 9 square units, find the values of k.

11. Show that the points (a, b + c), (b, c + a), and (c, a + b) are collinear

12. Find the value of x such that [1 x 1]$\left[ \begin{matrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{matrix} \right] \left[ \begin{matrix} 1 \\ 2 \\ x \end{matrix} \right] =0$

13. Show that all positive integral powers of a symmetric are symmetric.

14. For what value of x the matrix A =$\left[ \begin{matrix} 1 & -2 & 3 \\ 1 & 2 & 1 \\ x & 2 & -3 \end{matrix} \right]$  is singular.

15. Prove that $\left| \begin{matrix} 1 & 1+p & 1+p+q \\ 2 & 3+2p & 4+4p+2q \\ 3 & 6+3p & 10+6p+3q \end{matrix} \right| =1$