By using properties of determinants, show that: y+k&y&y &y+k&y &y… - Vidyadip

Question

By using properties of determinants, show that:
$\begin{vmatrix}y+k&y&y\\y&y+k&y\\y&y&y+k\end{vmatrix}=k^2(3y+k)$

✓

Answer

$\text{L.H.S.}=\begin{vmatrix}y+k&y&y\\y&y+k&y\\y&y&y+k\end{vmatrix}$
$=\begin{vmatrix}3y+k&y&y\\3y+k&y+k&y\\3y+k&y&y+k\end{vmatrix}\ \left[\text{C}_1\rightarrow\text{C}_1+\text{C}_2+\text{C}_3\right]$

$=(3y+k)\begin{vmatrix}1&y&y\\1&y+k&y\\1&y&y+k\end{vmatrix}$

$=(3y+k)\begin{vmatrix}1&y&y\\0&k&0\\0&0&k\end{vmatrix}\ \left[\text{operating}\ \text{R}_2\rightarrow\text{R}_2-\text{R}_1\ \text{and R}_3\rightarrow\text{R}_3-\text{R}_1\right]$

$ =(3y+k).1\begin{vmatrix}k&0\\0&k\end{vmatrix}=(3y+k)k^2=k^2(3y+k)=\text{R.H.S.}$ Proved.

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 "3 Marks Question" 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

Discuss the applicability of the Rolle's theorem for the following function on the indicated interval
$\text{f}(\text{x})=[\text{x}]\text{ for }-1\leq\text{x}\leq1,$ where [x] denotes the greatest integer not exceeding x.

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum?

Prove that $\int_0^{\pi / 2} \frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}} d x=\frac{\pi}{4}$

Prove the following results:
$\tan^{-1}\frac{2}{3}=\frac{1}{2}\tan^{-1}\frac{12}{5}$

A factory has two machines $A$ and $B$. Past records show that the machine $A$ produced $60\%$ of the items of output and machine $B$ produced $40\%$ of the items. Further $2\%$ of the items produced by machine $A$ were defective and $1\%$ produced by machine $B$ were defective. If an item is drawn at random$,$ what is the probability that it is defective?

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

Using determinants, find the value of k so that the points $(k, 2 - 2k), (-k + 1, 2k)$ and $(-4 - k, 6 - 2k)$ may be collinear.

Prove the following results
$\tan\Big(\sin^{-1}\frac{15}{13}+\cos^{-1}\frac{3}{5}\Big)=\frac{63}{16}$

If $\text{y}=\text{e}^{-\text{x}}\cos\text{x},$ show that $\frac{\text{d}^2\text{y}}{\text{dx}^2}=2\text{e}^{-\text{x}}\sin\text{x}.$

Solve: $\cos\big(\sin^{-1}\text{x}\big)=\frac{1}{6}$