By using properties of determinants, prove the following: x + 4 &… - Vidyadip

Question

By using properties of determinants, prove the following:$\begin{vmatrix} \text {x + 4} & \text{2x} & \text{2x} \\ \text{ 2x} & \text{x + 4} & \text{2x} \\ \text{2x} & \text{2x} & \text{x + 4} \end{vmatrix} = ( 5\text{x} + 4) (4 -\text{x}) = 1. $

✓

Answer

$C_{1} \rightarrow C_{1} + C_{2} + C_{3} \text{gives Det} = (5\text{x} + 4) \begin{vmatrix} 1 & 2\text{x} & 2\text{x} \\ 1 & \text{x} + 4 & 2\text{x} \\ 1 & 2\text{x} & \text{x} + 4 \end{vmatrix} $$\begin{matrix} R_{2} \rightarrow & R_{2} - & R_{1} \\ R_{3} \rightarrow & R_{3} & R_{1} \\ \end{matrix} \Rightarrow \text{Det} = (5\text{x} + 4) \begin{vmatrix} 1 & 2\text{x} & 2\text{x} \\ 0 & 4 -\text{x} & 0 \\ 0 & 0 & 4 - \text{x} \end{vmatrix} $
$\text{ Expanding by C}_{1} \text{to get Det.} = ( 5\text{x} + 4) (4- \text{x})^{2}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from DETERMINANTS→DETERMINANTS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Determine if $\text{f(x)}=\begin{cases}\text{x}^2\sin\frac{1}{\text{x}},&\text{ x}\neq0\\0,&\text{x}=0\end{cases}$ is a continuous function?

Write the minimum value of $\text{f}(\text{x})=\text{x}^\frac{1}{\text{x}}.$

Evaluate the following integrals:
$\int\frac{1}{\text{x}^4+3\text{x}^2+1}\ \text{dx}$

Solve the following systems of linear equations by cramer's rule:

2x - 3z + w = 1,

x - y + 2w = 1,

-3y + z + w = 1,

If $\text{A}=\begin{bmatrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix},$ find $A^{-1}$ and show that $\text{A}^{-1}=\frac{1}{2}(\text{A}^2-3\text{I}).$

Prove that the semi $-$ vertical angle of the right circular cone of given volume and least curved surface area is $\cot ^{-1} \sqrt{2}$

Evaluate the following integrals:
$\int\text{x}\sqrt{\text{x}^2+\text{x}}\text{dx}$

If $\text{f}(\text{a}+\text{b}-\text{x})=\text{f(x)},$ then prove that $\int\limits^{\text{b}}_\text{a}\text{x}\text{f(x)}\text{dx}=\frac{\text{a}+\text{b}}{2}\int\limits^{\text{b}}_\text{a}\text{f(x)}\text{dx}$

Find the inverse of the following matrices and verify that $A^{-1} A = I_3$.
$\begin{bmatrix}1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}$

Evaluate the following integrals:
$\int\frac{(\text{x}\tan^{-1}\text{x})}{(1+\text{x}^2)^{\frac{3}{2}}}\text{dx}$