Download App

MATRICES — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsMATRICES5 Marks
Question
If $\text{P}(\text{x})=\begin{bmatrix}\cos\text{x}&\sin\text{x}\\-\sin\text{x}&\cos\text{x}\end{bmatrix},$ then show that P(x).P(y) = P(x + y) = P(y)P(x).
✓

Answer

We have, $\text{P}(\text{x})=\begin{bmatrix}\cos\text{x}&\sin\text{x}\\-\sin\text{x}&\cos\text{x}\end{bmatrix}$ $\Rightarrow\ \text{P}(\text{y})=\begin{bmatrix}\cos\text{y}&\sin\text{y}\\-\sin\text{y}&\cos\text{y}\end{bmatrix}$ Consider, P(x).P(y) = P(x + y)LHS = P(x).P(y)
$=\begin{bmatrix}\cos\text{x}&\sin\text{x}\\-\sin\text{x}&\cos\text{x}\end{bmatrix}\begin{bmatrix}\cos\text{y}&\sin\text{y}\\-\sin\text{y}&\cos\text{y}\end{bmatrix}$
$=\begin{bmatrix}\cos\text{x}.\cos\text{y}-\sin\text{x}.\sin\text{y}&\cos\text{x}.\sin\text{y}+\sin\text{x}.\cos\text{y}\\-\sin\text{x}.\cos\text{y}-\cos\text{x}.\sin\text{y}&-\sin\text{x}.\sin\text{y}+\cos\text{x}.\sin\text{y}\end{bmatrix}$
$=\begin{bmatrix}\cos(\text{x}+\text{y})&\sin(\text{x}+\text{y})\\-\sin(\text{x}+\text{y})&\cos(\text{x}+\text{y})\end{bmatrix}$
$\begin{bmatrix}\because\ \cos(\text{x}+\text{y})=\cos\text{x}.\cos\text{y}-\sin\text{x}.\sin\text{y}\\\text{and }\sin(\text{x}+\text{y})=\sin\text{x}.\cos\text{y}+\cos\text{x}.\sin\text{y}\end{bmatrix}$
$=\text{P}(\text{x}+\text{y})$
= RHS Hence 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 "5 Marks Questions" from MATRICES→MATRICES — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Maximum Z = 3x + 4y
Subject to
$2\text{x}+2\text{y}\leq80$
$2\text{x}+4\text{y}\leq120$
→
Show that the matrix, $A=\left[\begin{array}{ccc}1 & 0 & -2 \\ -2 & -1 & 2 \\ 3 & 4 & 1\end{array}\right]$ satisfies the equation,
$A ^3- A ^2-3 A- I _3= O$. Hence, find $A ^{-1}$
→
Maximum Z = 5x + 3y Subject to $2\text{x}+\text{y}\geq10$ $\text{x}+3\text{y}\geq15$ $\text{x}\leq10$$\text{y}\leq8$
$\text{x},\text{y}\geq0$
→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\frac{\text{y}}{\text{x}}\{\log\text{y}-\log\text{x}+1\}$
→
Find the general solution of the differential equation $(1+\text{y}^2)+(\text{x}-\text{e}^{{\tan^{-1}\text{y}}})\frac{\text{dy}}{\text{dx}}=0.$
→
Evaluate the following intregals:
$\int\frac{\text{x}^2+\text{x}-1}{(\text{x}+1)^2(\text{x}+2)}\ \text{dx}$
→
Differentiate $\sin^{-1}\Big(2\text{x}\sqrt{1-\text{x}^2}\Big)$ with respect to $\sec^{-1}\Big(\frac{1}{\sqrt{1+\text{x}^2}}\Big),$ if:
$\text{x}\in\Big(0,\frac{1}{\sqrt{2}}\Big)$
→
If $\hat{\text{a}}$ and $\hat{\text{b}}$ are unit vectors inclined at an angle $\theta$, prove that$\cos\frac{\theta}{2}=\frac{1}{2}\big|\hat{\text{a}}+\hat{\text{b}}\big|$
→
Discuss the continuity and differentiability of the f(x) = |x| + |x - 1| in the interval (-1, 2).
→
Find the position vector of the foot of perpendicular and the perpendicular distance from the point P with position vector $2\hat{i} + 3\hat{j} + 4\hat{k}$ to the plane $\overrightarrow{\text{r}}. (2\hat{i} + \hat{j} + 3\hat{k}) - 26 = 0.$ Also find image of P in the plane.
→