Download App

DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS3 Marks
Question
Without expanding the determinant, prove that $\begin{vmatrix}a&a^2&bc\\b&b^2&ca\\c&c^2&ab\end{vmatrix}=\begin{vmatrix}1&a^2&a^3\\1&b^2&b^3\\1&b^2&b^3\end{vmatrix}.$
✓

Answer

$\text{L.H.S.}=\begin{vmatrix}a&a^2&bc\\b&b^2&ca\\c&c^2&ab\end{vmatrix}$
$=\frac{1}{abc}\begin{vmatrix}a^2&a^3&abc\\b^2&b^3&abc\\c^2&c^3&abc\end{vmatrix}\ \ \ \ \ \ \bigg[\text{R}_1\rightarrow a\text{R}_1,\text{R}_2\rightarrow b\text{R}_2,\text{and}\ \text{R}_3\rightarrow c\text{R}_3\bigg]$
$=\frac{1}{abc}.abc\begin{vmatrix}a^2&a^3&1\\b^2&b^3&1\\c^2&c^3&1\end{vmatrix}$ [Taking out factor abc from $C_3$]
$=\begin{vmatrix}a^2&a^3&1\\b^2&b^3&1\\c^2&c^3&1\end{vmatrix}$
$=\begin{vmatrix}1&a^2&a^3\\1&b^2&b^3\\1&c^2&c^3\end{vmatrix}$ [Applying $C_1 \leftrightarrow C_3$ and $C_2 \leftrightarrow C_3$]
$=\text{R.H.S.}$
Hence, the given result is 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

Using determinants, find the area of the triangle whose vertices are $(1, 4), (2, 3)$ and $(-5, -3)$. Are the given points collinear?
→
Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\sin^2\text{y}$
→
Show that the relation $R$ defined in the set A of all polygons as $R = \{(P_1, P_2) : P_1$ and $P_2$ have same number of sides$\},$ is an equivalence relation. What is the set of all elements in $A$ related to the right angle triangle T with sides $3, 4,$ and $5?$
→
Two coins are tossed once. Find $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)$ in each of the following:
A = No tail appears,
B = No head appears.
→
Write a value of $\int\frac{\sin\text{x}-\cos\text{x}}{\sqrt{1+\sin2\text{x}}}\text{ dx}$
→
Using determinants show that the following points are collinear:
(2, 3), (-1, -2) and (5, 8)
→
Evaluate the following integrals:
$\int5^{\text{x}+\tan^{-1}\text{x}}.\Big(\frac{\text{x}^2+2}{\text{x}^2+1}\Big)\text{dx}$
→
Determine the value of $\lambda$ for which the following planes are perpendicular to other.
$\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})=7$ and $\vec{\text{r}}\cdot(\lambda\hat{\text{i}}+2\hat{\text{j}}-7\hat{\text{k}})=26$
→
Five defective bolts are acciedently mixed with twenty good ones. If four bolts are drawn at random from this lot, find the probability distribution of the number of defective bolts.
→
Discuss the continuity of the following functions at the indicated point:
$\text{f}\text{(x)}=\begin{cases}(\text{x}-\text{a}){\sin}\Big(\frac{1}{\text{x}-\text{a}}\Big) & \text{x} \neq \text{a}\\\ 0, & \text{ x} = \text{a}\end{cases}\text{at x}=\text{a}$
→