## Matrices and Determinants Questions and Answers Part-17

1. If $pqr\neq0$  and the system of equations
(p + a)x + by + cz = 0
ax + (q + b)y + cz = 0
ax + by + (r + c)z = 0
has a non-trivial solution, then value of $\frac{a}{p}+\frac{b}{q}+\frac{c}{r}$     is
a) -1
b) 0
c) 1
d) 2

Explanation:

2. Let $a = 2^{A}, b = 2^{B}, C = 2^{C}$     the system of equations
ax + by + (a $\alpha$ + b)z = 0
bx + cy + (b $\alpha$ + c)z = 0
(a $\alpha$ + b)x + (b $\alpha$ + c)y = 0
has a non-zero solutions if A, B, C are in
a) A.P
b) G.P
c) H.P
d) none of these

Explanation:

3. If the system of equations
ax + ay – z = 0
bx – y + bz = 0
– x + cy + cz = 0
(where $a, b, c \neq – 1$   ) has a non-trivial solution, then value of $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}$     is
a) 2
b) -1
c) -2
d) 0

Explanation:

4. All the values of $\lambda$ for which the system of equations
$\left(\lambda + 5\right)x + \left(\lambda – 4\right)y + z = 0$
$\left(\lambda - 2\right)x + \left(\lambda + 3\right)y + z = 0$
$\lambda x+\lambda y+z=0$
has a non-trivial solution lie in the set
a) {– 1, 2}
b) {0, – 1}
c) {0}
d) R

Explanation:

5. Given the system of equations
(b + c) (y + z) – ax = b – c
(c + a) (z + x) – by = c – a
(a + b) (x + y) – cz = a – b
(where $a + b + c \neq 0$   ); then x : y : z is given by
a) b – c : c – a : a – b
b) b + c : c + a : a + b
c) a : b : c
d) $\frac{a}{b}:\frac{b}{c}:\frac{c}{a}$

6. Number of real values of $\lambda$ for which the system of equations
$\left(\lambda + 3\right)x + \left(\lambda + 2\right)y + z = 0$
$3x + \left(\lambda + 3\right)y + z = 0$
2x + 3y + z = 0
has a non-trivial solutions is
a) 0
b) 1
c) 2
d) Infinite

Explanation:

7. The value of $\lambda$ for which the system of equations
2x – y – 2z = 2
x – 2y + z = – 4
$x+y+\lambda z=4$
has no solution is
a) 3
b) -3
c) 2
d) -2

Explanation:

8. The determinant $\begin{bmatrix}\sin\alpha & \cos\alpha & 1 \\\sin\beta & \cos\beta & 1 \\\sin\gamma & \cos\gamma & 1\end{bmatrix}$         is equals to
a) $-4\sin\frac{\alpha-\beta}{2}\sin\frac{\beta-\gamma}{2}\sin\frac{\gamma-\alpha}{2}$
b) $\sin\alpha+\sin\beta+\sin\gamma$
c) $\sin\left(\alpha-\beta\right)+\sin\left(\beta-\gamma\right)+\sin\left(\gamma-\alpha\right)$
d) Both a and c

Explanation:

9. If a, b, c > 0 and x, y, $z\epsilon R$ , then the determinant $\begin{bmatrix}\left(a^{x}+a^{-x}\right)^{2} & \left(a^{x}-a^{-x}\right)^{2} & 1 \\\left(b^{y}+b^{-y}\right)^{2} & \left(b^{y}-b^{-y}\right)^{2} & 1 \\\left(c^{z}+c^{-z}\right)^{2} & \left(c^{z}-c^{-z}\right)^{2} & 1\end{bmatrix}$
is indepenent of
a) a, b, c
b) x, y, z
c) a, b, c, x, y, z
d) All of the above

Explanation: Use C1 $\rightarrow$ C1 - C2 - 4C3
10. If $a\neq b$  , the equation $\begin{bmatrix}x & a & a \\a & x & a \\a & a & x\end{bmatrix}+\begin{bmatrix}b & b & x \\b & x & b \\x & b & b\end{bmatrix}=0$
d) $\frac{2}{3}\frac{a^{2}+ab+b^{2}}{a+b}$