Download App

Determinants — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDeterminants5 Marks
Question
$\begin{vmatrix}\text{b}+\text{c}&\text{a}&\text{a}\\\text{b}&\text{c}+\text{a}&\text{b}\\\text{c}&\text{c}&\text{a}+\text{b}\end{vmatrix}=4\text{abc}$
✓

Answer

$\text{L.H.S}=\begin{vmatrix}\text{b}+\text{c}&\text{a}&\text{a}\\\text{b}&\text{c}+\text{a}&\text{b}\\\text{c}&\text{c}&\text{a}+\text{b}\end{vmatrix}$
$=\begin{vmatrix}0&-2\text{c}&-2\text{b}\\\text{b}&\text{c}+\text{a}&\text{b}\\\text{c}&\text{c}&\text{a}+\text{b}\end{vmatrix}$ $[$Applying $R_1 → R_1 - (R_2 + R_3)]$
$=\begin{vmatrix}0&-2\text{c}&-2\text{b}\\\text{b}&\text{c}+\text{a}-\text{b}&0\\\text{c}&0&\text{a}+\text{b}-\text{c}\end{vmatrix}$ $[$Applying $C_2 → C_2 - C_1$ and $C_3 → C_3 - C_1]$
$=0\begin{vmatrix}\text{c}+\text{a}-\text{b}&0\\0&\text{a}+\text{b}-\text{c}\end{vmatrix}-(-2\text{c})\begin{vmatrix}\text{b}&0\\\text{c}&\text{a}+\text{b}-\text{c}\end{vmatrix}-2\text{b}\begin{vmatrix}\text{b}&\text{c}+\text{a}-\text{b}\\\text{c}&0\end{vmatrix}$
$=2\text{c}[\text{b}(\text{a}+\text{b}-\text{c})-0]-2\text{b}[0-\text{c}(\text{c}+\text{a}-\text{b})]$
$=2\text{bc}[\text{a}+\text{b}-\text{c}]-2\text{bc}[\text{b}-\text{c}-\text{a}]$
$=2\text{bc}[(\text{a}+\text{b}-\text{c})-(\text{b}-\text{c}-\text{a})]$
$=4\text{abc}$
$=\text{R.H.S}$

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

Evaluate the following integrals:$\int(\log\text{x})^2\cdot\text{x dx}$
→
Find the values of a and b so that the function $\text{f(x)}\begin{cases}\text{x}^2+3\text{x}+\text{a}, & \text{if x}\leq1\\\text{bx}+2, & \text{if x} > 1\end{cases}$ is differentiable at each $\text{x}\in\text{R}.$
→
Find the mean variance and standard deviation of the following probability distribution
$x_i$ a b
$p_i$ p q
Where $p + q = 1$
→
Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}1^2&2^2&3^2&4^2\\2^2&3^2&4^2&5^2\\3^2&4^2&5^2&6^2\\4^2&5^2&6^2&7^2\end{vmatrix}$
→
Find the particular solution of the differential equation $\frac{dy}{dx} = \frac{xy}{x^{2} + y^{2}}$ given that $\text{y - 1, when x = 0.}$
→
Show that the plane whose vector equation is $\vec{\text{r}}\cdot(\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}})=3$ contains the line whose vector equation is $\vec{\text{r}}=\hat{\text{i}}+\hat{\text{j}}+\lambda(2\hat{\text{i}}+\hat{\text{j}}+4\hat{\text{k}}).$
→
Assume that each born child is equally likely to be a boy or a girl. If a family has two children, then what is the constitutional probability that both are girls? Given that.
  1. The youngest is a girl,
  2. At least one is a girl.
→
Solve the following differential equations:$(\text{y + xy})\text{dx}+(\text{x}-\text{xy}^2)\text{dy}=0$
→
Find a point on the curve $y = (x - 3)^2,$ where the tangent is parallel to the chord joining the points $(3, 0)$ and $(4, 1).$
→
One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball is transferred from first bag to the second bag and then a ball is drawn from the second bag. Find the probability that the ball drawn is white.
→