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Determinants

Multiple Choice Questions

461.

If A is an invertible matrix of order n, then the determinant of adj A is equal to :

${|A|}^{n}$
${|A|}^{n + 1}$
${|A|}^{n - 1}$
${|A|}^{n + 2}$

462.

$If A is a square matrix such that A (adjA) = [\begin{matrix} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{matrix}], Then \det (adjA) = ?$

463.

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

0
a + b + c
(a + b + c)²
(a + b + c)³

464.

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

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

465.
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

$Given, |\begin{matrix} \log (x) & p & 1 \\ \log (y) & q & 1 \\ \log (z) & r & 1 \end{matrix}| Let a and R be the first term and common ratio of a GP . T_{p} = {aR}^{p - 1} = x T_{q} = {aR}^{q - 1} = y and T_{r} = {aR}^{r - 1} = z \Rightarrow \log (x) = \log (a) + (p - 1) \log (R) \log (y) = \log (a) + (q - 1) \log (R) ∴ ||\begin{matrix} \log (x) & p & 1 \\ \log (y) & q & 1 \\ \log (z) & r & 1 \end{matrix}|| = |\begin{matrix} \log (a) + (p - 1) \log (R) & p & 1 \\ \log (a) + (q - 1) \log (R) & q & 1 \\ \log (a) + (r - 1) \log (R) & r & 1 \end{matrix}| = |\begin{matrix} \log (a) & p & 1 \\ \log (a) & q & 1 \\ \log (a) & r & 1 \end{matrix}| + |\begin{matrix} (p - 1) \log (R) & p & 1 \\ (q - 1) \log (R) & q & 1 \\ (r - 1) \log (R) & r & 1 \end{matrix}| = |\begin{matrix} 1 & p & 1 \\ 1 & q & 1 \\ 1 & r & 1 \end{matrix}| + \log (R) |\begin{matrix} p - 1 & p - 1 & 1 \\ q - 1 & q - 1 & 1 \\ r - 1 & r - 1 & 1 \end{matrix}| [∵ C_{2} \to C_{2} - C_{3}] = 0 + 0 = 0 [∵ two columns are identical]$

466.

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

467.

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

0
1
- 500
500

468.

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

3x² + 4x + 5
x³ + 8x + 2
0
- 2

469.

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

470.

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

Previous Year Papers

Test Series

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Multiple Choice Questions

465. 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 determinantlogxp1logyq1logzr1 equals log(xyz) (p - 1)(q - 1)(r - 1) pqr 0

465.
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