Download App

Determinants — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDeterminants5 Marks
Question
Prove that: $\begin{vmatrix} 1&\text{a}&\text{bc}\\1&\text{b}&\text{ca}\\1&\text{c}&\text{ab}\end{vmatrix}=\begin{vmatrix} 1&\text{a}&\text{a}^2\\1&\text{b}&\text{b}^2\\1&\text{c}&\text{c}^2\end{vmatrix}$
✓

Answer

$\begin{vmatrix} 1&\text{a}&\text{bc}\\1&\text{b}&\text{ca}\\1&\text{c}&\text{ab}\end{vmatrix}$
Apply $R_1 \rightarrow R_1a, R_2 \rightarrow R_2b, R_3 \rightarrow R_3c$
$=\frac{1}{\text{abc}}\begin{vmatrix} \text{a}&\text{a}^2&\text{abc}\\\text{b}&\text{b}^2&\text{cab}\\\text{c}&\text{c}^2&\text{abc}\end{vmatrix}$
$=\frac{\text{abc}}{\text{abc}}\begin{vmatrix} \text{a}&\text{a}^2&1\\\text{b}&\text{b}^2&1\\\text{c}&\text{c}^2&1\end{vmatrix}$
$=-\begin{vmatrix} \text{a}&1&\text{a}^2\\\text{b}&1&\text{b}^2\\\text{c}&1&\text{c}^2\end{vmatrix}$
$=\begin{vmatrix} 1&\text{a}&\text{a}^2\\1&\text{b}&\text{b}^2\\1&\text{c}&\text{c}^2\end{vmatrix}$

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→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

$\text{If f(x)} = \begin{vmatrix} a & -1 & 0 \\ ax & a & -1 \\ ax^{2} & ax & a \end{vmatrix} , $ using properties of determinants find the value of $\text{f}(2x) - \text{f}(x).$
→
Evaluate the following integrals:
$\int\frac{\text{x}}{(\text{x}-3)\sqrt{\text{x}+1}}\text{ dx}$
→
Find $\text{y}=\text{Ae}^{-\text{kt}}\cos\text({pt}+\text{c})$prove that $\frac{\text{d}^2\text{y}}{\text{dt}^2}+2\text{k}\frac{\text{dy}}{\text{dt}}+\text{n}^2\text{y}=0,$Where $\text{n}^2=\text{p}^2+\text{k}^2.$
→
$\int\frac{\text{x}^2+5\text{x}+2}{\text{x}+2}\text{dx}$
→
If $y^x + x^y + x^x = a^b,$ find $\frac{\text{dy}}{\text{dx}}$
→
Find the area of the region in the first quadrant enclosed by the $x-$axis, the line $y = x$ and the circle $x^2+ y^2 = 32.$
→
Evaluate the following definite integrals:
$\int_{0}^\limits{\frac{\pi}{2}}\text{x}^2\sin\text{x}\text{ dx}$
→
Differentiate the functions given in Exercise:
$\text{x}^\text{x}-2^{\sin\text{x}}$
→
Show that the following set of curves intersect orthogonally. $x^3 - 3xy^2 = -2$ and $3x^2y - y^3 = 2$
→
Prove that the given vectors are non-coplanar:
$\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}},\ 2\hat{\text{i}}+\hat{\text{j}}+3\hat{\text{k}}$ and $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$
→