JEE Advanced Maths PYQ – Determinants: Properties of Determinants & Area of a Triangle

Q 1 :

Let $P = [a_{i j}]$ be a $3 \times 3$ matrix and let $Q = [b_{i j}],$ where $b_{i j} = 2^{i + j} a_{i j}$ for $1 \leq i, j \leq 3$ . If the determinant of $P$ is 2, then the determinant of the matrix $Q$ is [2012]

$2^{10}$
$2^{11}$
$2^{12}$
$2^{13}$

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4

(4)

$| Q |$ $=$ $| \begin{matrix} 2^{2} a_{11} & 2^{3} a_{12} & 2^{4} a_{13} \\ 2^{3} a_{21} & 2^{4} a_{22} & 2^{5} a_{23} \\ 2^{4} a_{31} & 2^{5} a_{32} & 2^{6} a_{33} \end{matrix} |$

$= 2^{2} \cdot 2^{3} \cdot 2^{4}$ $| \begin{matrix} a_{11} & a_{12} & a_{13} \\ 2 a_{21} & 2 a_{22} & 2 a_{23} \\ 2^{2} a_{31} & 2^{2} a_{32} & 2^{2} a_{33} \end{matrix} |$

$= 2^{9} \cdot 2 \cdot 2^{2}$ $| \begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix} |$

$= 2^{12} \times | P | = 2^{12} \times 2 = 2^{13}$

Q 2 :

If $A = [\begin{matrix} α & 2 \\ 2 & α \end{matrix}]$ and $| A^{3} |$ $= 125,$ then the value of $α$ is [2004]

$\pm 1$
$\pm 2$
$\pm 3$
$\pm 5$

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2

3

4

(3)

$A = [\begin{matrix} α & 2 \\ 2 & α \end{matrix}]$ and $| A^{3} |$ $= 125$ $\Rightarrow$ $| A^{3} |$ $= 125$

$∴$ $| A |$ $= α^{2} - 4$

Now, $| A |^{3}$ $= 125$

$\Rightarrow {(α^{2} - 4)}^{3} = 125 = 5^{3}$

$\Rightarrow α^{2} - 4 = 5$

$\Rightarrow α = \pm 3$

Q 3 :

Let $ω = - \frac{1}{2} + i \frac{\sqrt{3}}{2} .$ Then the value of the determinant $| \begin{matrix} 1 & 1 & 1 \\ 1 & - 1 - ω^{2} & ω^{2} \\ 1 & ω^{2} & ω^{4} \end{matrix} |$ is [2002]

$3 ω$
$3 ω (ω - 1)$
$3 ω^{2}$
$3 ω (1 - ω)$

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(2)

Given that $ω = - \frac{1}{2} + i \frac{\sqrt{3}}{2},$ then $ω^{2} = - \frac{1}{2} - i \frac{\sqrt{3}}{2} .$

Also, $1 + ω + ω^{2} = 0$ and $ω^{3} = 1$

Now, $Δ =$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & - 1 - ω^{2} & ω^{2} \\ 1 & ω^{2} & ω^{4} \end{matrix} |$ $=$ $| \begin{matrix} 1 & 1 & 1 \\ 1 & ω & ω^{2} \\ 1 & ω^{2} & ω \end{matrix} |$ $(∵ ω = - 1 - ω^{2} and ω^{3} = 1)$ $[C_{1} \to C_{1} + C_{2} + C_{3}]$

$Δ =$ $| \begin{matrix} 3 & 1 & 1 \\ 0 & ω & ω^{2} \\ 0 & ω^{2} & ω \end{matrix} |$ $(∵ 1 + ω + ω^{2} = 0)$

On expanding along $C_{1}$ , we get

$Δ = 3 (ω^{2} - ω^{4}) = 3 (ω^{2} - ω) = 3 ω (ω - 1)$

Topic Question Set

Q 1 :

Let $P = [a_{i j}]$ be a $3 \times 3$ matrix and let $Q = [b_{i j}],$ where $b_{i j} = 2^{i + j} a_{i j}$ for $1 \leq i, j \leq 3$ . If the determinant of $P$ is 2, then the determinant of the matrix $Q$ is [2012]

Q 2 :

If $A = [\begin{matrix} α & 2 \\ 2 & α \end{matrix}]$ and $| A^{3} |$ $= 125,$ then the value of $α$ is [2004]

Q 3 :

Let $ω = - \frac{1}{2} + i \frac{\sqrt{3}}{2} .$ Then the value of the determinant $| \begin{matrix} 1 & 1 & 1 \\ 1 & - 1 - ω^{2} & ω^{2} \\ 1 & ω^{2} & ω^{4} \end{matrix} |$ is [2002]

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Topic Question Set

Q 1 : Let P=[aij] be a 3×3 matrix and let Q=[bij], where bij=2 i+jaij for 1≤i, j≤3. If the determinant of P is 2, then the determinant of the matrix Q is [2012]

Q 2 : If A=[α22α] and |A3|=125, then the value of α is [2004]

Q 3 : Let ω=-12+i32. Then the value of the determinant |1111-1-ω2ω21ω2ω4| is [2002]

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Q 1 :

Let $P = [a_{i j}]$ be a $3 \times 3$ matrix and let $Q = [b_{i j}],$ where $b_{i j} = 2^{i + j} a_{i j}$ for $1 \leq i, j \leq 3$ . If the determinant of $P$ is 2, then the determinant of the matrix $Q$ is [2012]

Q 2 :

If $A = [\begin{matrix} α & 2 \\ 2 & α \end{matrix}]$ and $| A^{3} |$ $= 125,$ then the value of $α$ is [2004]

Q 3 :

Let $ω = - \frac{1}{2} + i \frac{\sqrt{3}}{2} .$ Then the value of the determinant $| \begin{matrix} 1 & 1 & 1 \\ 1 & - 1 - ω^{2} & ω^{2} \\ 1 & ω^{2} & ω^{4} \end{matrix} |$ is [2002]