#### Matrices and Determinants Three Marks Questions

11th Standard

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Maths

Time : 00:45:00 Hrs
Total Marks : 30
10 x 3 = 30
1. Prove that $\begin{vmatrix} 1& a & a^2-bc \\1 &b &b^2-ca \\ 1 & c & c^2-ab \end{vmatrix}=0.$

2. If a, b, c are pth, qth and rth terms of an A.P, find the value of$\begin{vmatrix} a & b & c \\ p & q & r \\ 1& 1 &1 \end{vmatrix}$

3. Solve the following problems by using Factor Theorem :
Solve $\begin{vmatrix} x+a &b &c \\ a & x+b & c \\ a & b &x+c \end{vmatrix}=0$

4. Identify the singular and non-singular matrices:$\begin{bmatrix} 1&2 &3 \\ 4 & 5 &6 \\ 7 & 8 & 9 \end{bmatrix}$

5. Identify the singular and non-singular matrices:$\begin{bmatrix} 2&-3 &5 \\ 6 & 0 &4 \\ 1 & 5 & -7 \end{bmatrix}$

6. Find non-Zero values of x satisfying the matrix equation, $x\left[ \begin{matrix} 2x & 2 \\ 3 & x \end{matrix} \right] +2\left[ \begin{matrix} 8 & 5x \\ 4 & 4x \end{matrix} \right] =\left[ \begin{matrix} { x }^{ 2 }+8 & 24 \\ 10 & 6x \end{matrix} \right]$

7. If A=$\left[ \begin{matrix} \alpha & 0 \\ 1 & 1 \end{matrix} \right]$ and B=$\left[ \begin{matrix} 1 & 0 \\ 5 & 1 \end{matrix} \right]$ find the values of $\alpha$ for which A2=B.

8. Under what condition is the matrix equation A2-B2 =(A-B)(A+B) is true?

9. Prove that$\left| \begin{matrix} -2a & a+b & a+c \\ b+a & -2b & b+c \\ c+a & c+b & -2c \end{matrix} \right|$ =4(a+b)(b+c)(c+a). Using factor theorem.

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