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Determinants

Multiple Choice Questions

131.

$|\begin{matrix} a - b - c & 2 a & 2 a \\ 2 b & b - c - a & 2 b \\ 2 c & 2 c & c - a - b \end{matrix}|$ is equal to

0
a + b + c
(a + b + c)²
(a + b + c)³

132.

$The inverse of the matrix [\begin{matrix} 7 & - 3 & - 3 \\ - 1 & 1 & 0 \\ - 1 & 0 & 1 \end{matrix}] is$

$[\begin{matrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{matrix}]$
$[\begin{matrix} 1 & 3 & 1 \\ 4 & 3 & 8 \\ 3 & 4 & 1 \end{matrix}]$
$[\begin{matrix} 1 & 1 & 1 \\ 3 & 3 & 4 \\ 3 & 4 & 3 \end{matrix}]$
$[\begin{matrix} 1 & 1 & 1 \\ 3 & 4 & 3 \\ 3 & 3 & 4 \end{matrix}]$

133.

If x, y, z are all positive and are the pth, qth and rth terms of a geometric progression respectively, then the value of the determinant

$|\begin{matrix} \log (x) & p & 1 \\ \log (y) & q & 1 \\ \log (z) & r & 1 \end{matrix}|$ equals

log(xyz)
(p - 1)(q - 1)(r - 1)
pqr
0

134.

If $[\begin{matrix} 1 & - 1 & x \\ 1 & x & 1 \\ x & - 1 & 1 \end{matrix}]$ has no inverse, then the real value of x is

135.

$If f (x) |\begin{matrix} 1 & x & x + 1 \\ 2 x & x (x - 1) & x (x + 1) \\ 3 x (x - 1) & x (x - 1) (x - 2) & (x - 1) x (x + 1) \end{matrix}| then f (2012) = ?$

0
1
- 500
500

136.

$|\begin{matrix} x + 2 & x + 3 & x + 5 \\ x + 4 & x + 6 & x + 9 \\ x + 8 & x + 11 & x + 15 \end{matrix}| = ?$

3x² + 4x + 5
x³ + 8x + 2
0
- 2

137.

${(\frac{1 + i}{1 - i})}^{4} + {(\frac{1 - i}{1 - i})}^{4} = ?$

138.

The value of the determinant $|\begin{matrix} b^{2} - ab & b - c & bc - ac \\ ab - a^{2} & a - b & b^{2} - ab \\ bc - ac & c - a & ab - a^{2} \end{matrix}| is$

abc
a + b + c
0
ab + bc + ca

139.
If the lines x + 2ay + a = 0, x + 3by + b = 0, 32x + 4cy + c = 0 are concurrent, then a, b and c are in

arithmetic progression

geometric progression

harmonc progression

arithmetico-geometric progression

harmonc progression

$Three lines L_{1} = a_{1} x + b_{1} y + c_{1} = 0 L_{2} = a_{2} x + b_{2} y + c_{2} = 0 and L_{3} = a_{3} x + b_{3} y + c_{3} = 0 are concurrent, if |\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}| = 0 According to the question, |\begin{matrix} 1 & 2 a & a \\ 1 & 3 b & b \\ 1 & 4 c & c \end{matrix}| = 0 Applying R_{1} \to R_{1} - R_{2}, R_{2} \to R_{2} - R_{3}, |\begin{matrix} 0 & 2 a - 3 b & a - b \\ 10 & 3 b - 4 c & b - c \\ 1 & 4 c & c \end{matrix}| = 0 \Rightarrow 1 [(2 a - 3 b) (b - c) - (3 b - 4 c) (a - b)] = 0 \Rightarrow (2 a - 3 b) (b - c) = (3 b - 4 c) (a - b) \Rightarrow 2 ab - 2 ca - 3 b^{2} + 3 bc = 3 ab - 3 b^{2} - 4 ca + 4 bc \Rightarrow ab + bc = 2 ca \Rightarrow \frac{1}{a} + \frac{1}{c} = \frac{2}{b} Hence, a, b and c are in HP$

140.

If a, b, c are distinct positive real numbers, then the value of the determinant $|\begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix}| is$

$< 0$
$> 0$
0
$\geq 0$

Previous Year Papers

Test Series

Test Yourself

Multiple Choice Questions

139. If the lines x + 2ay + a = 0, x + 3by + b = 0, 32x + 4cy + c = 0 are concurrent, then a, b and c are in arithmetic progression geometric progression harmonc progression arithmetico-geometric progression

139.
If the lines x + 2ay + a = 0, x + 3by + b = 0, 32x + 4cy + c = 0 are concurrent, then a, b and c are in

arithmetic progression

geometric progression

harmonc progression

arithmetico-geometric progression