Download App

DETERMINANTS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDETERMINANTS2 Marks
Question
Evaluate $\begin{vmatrix}\cos\alpha\cos\beta&\cos\alpha\sin\beta&-\sin\alpha\\-\sin\beta&\cos\beta&0\\\sin\alpha\cos\beta&\sin\alpha\sin\beta&\cos\alpha\end{vmatrix}.$
✓

Answer

$\triangle=\begin{vmatrix}\cos\alpha\cos\beta&\cos\alpha\sin\beta&-\sin\alpha\\-\sin\beta&\cos\beta&0\\\sin\alpha\cos\beta&\sin\alpha\sin\beta&\cos\alpha\end{vmatrix}$
Expanding along $C_3$, we have:
$\triangle= -\sin\alpha(-\sin\alpha\sin^2\beta-\cos^2\beta\sin\alpha)+\cos\alpha(\cos\alpha\cos^2\beta+\cos\alpha\sin^2\beta)$
$=\sin^2\alpha(\sin^2\beta+\cos^2\beta)+\cos^2\alpha(\cos^2\beta+\sin^2\beta)$
$=\sin^2\alpha(1)+\cos^2\alpha(1)$
$= 1$

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 "2 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

If $A = [A_{ij}]$ is a $3 \times 3$ scalar matrix such that $a_{11} = 2$, then write the value of $|A|$.
→
If the relative error in measuring the radius of a circular plane is $\alpha,$ find the relative error in measuring its area.
→
Write the angle between the line $\frac{\text{x}-1}{2}=\frac{\text{y}-2}{1}=\frac{\text{z}+3}{-\text{2}}$ and the plane x + y + 4 = 0.
→
Find the interval in which the function $f(x)=x^2-6 x+5$ is strictly increasing.
→
Write a unit vector in the direction of $\overrightarrow{\text{PQ}}$, where P and Q are the points (1, 3, 0) and (4, 5, 6) respectively.
→
Write the value of $k$ for which the planes $x - 2y + kz = 4$ and $2x + 5y - z = 9$ are perpendicular.
→
If $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}},\ \vec{\text{b}}=\hat{\text{j}}+\hat{\text{k}},\ \vec{\text{c}}=\hat{\text{k}}+\hat{\text{i}}$, find the unit vector in the direction of $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}$.
→
Find the second order derivatives of the function given in Exercise:
$\text{e}^\text{x}\sin5\text{x}$
→
Find the angle between the vectora $\vec{\text{a}}=\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{b}}=\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}.$
→
What are the direction cosines of Y-axis?
→