2 , ^2 + 4 , , ,( + ) , , , , , + ,2 ,( + ) = - Vidyadip

MCQ

$2\,{\sin ^2}\beta + 4\,\,\cos \,(\alpha + \beta )\,\,\sin \,\alpha \,\sin \,\beta + \cos \,2\,(\alpha + \beta ) = $

A
$\sin \,\,2\alpha $
B
$\cos \,\,2\beta $
✓
$\cos \,\,2\alpha $
D
$\sin \,\,2\beta $

✓

Answer

Correct option: C.

$\cos \,\,2\alpha $

c
(c) $\cos 2(\alpha + \beta ) = 2{\cos ^2}(\alpha + \beta ) - 1,$

$2{\sin ^2}\beta = 1 - \cos 2\beta $

$L.H.S.$ $ = - \cos 2\beta + 2\cos (\alpha + \beta )\,[2\sin \alpha \sin \beta + \cos (\alpha + \beta )]$

$ = - \cos 2\beta + 2\cos (\alpha + \beta )\cos (\alpha - \beta )$

$ = - \cos 2\beta + (\cos 2\alpha + \cos 2\beta ) = \cos 2\alpha $.

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let the relations $R_1$ and $R_2$ on the set
$X =\{1,2,3, \ldots, 20\}$ be given by
$R _1=\{( x , y ): 2 x -3 y =2\}$ and
$R_2=\{(x, y):-5 x+4 y=0\}$. If $M$ and $N$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the relations symmetric, then $M + N$ equals

The range of the function,

$\mathrm{f}(\mathrm{x})=\log _{\sqrt{5}}(3+\cos \left(\frac{3 \pi}{4}+\mathrm{x}\right)+\cos \left(\frac{\pi}{4}+\mathrm{x}\right)+\cos \left(\frac{\pi}{4}-\mathrm{x}\right)$

$-\cos \left(\frac{3 \pi}{4}-\mathrm{x}\right))$ is :

The line $2x + 3y = 12$ meets the $x -$ axis at $A$ and the $y -$ axis at $B$ . The line through $(5, 5)$ perpendicular to $AB$ meets the $x -$ axis, $y -$ axis $\&$ the line $AB$ at $C, D, E$ respectively. If $O$ is the origin, then the area of the $OCEB$ is :

If the vectors $3i+\lambda \,j+k$ and $2i-j+8k$ are perpendicular, then $\lambda $ is

Let $a , b , c$ be in arithmetic progression. Let the centroid of the triangle with vertices $( a , c ),(2, b)$ and $(a, b)$ be $\left(\frac{10}{3}, \frac{7}{3}\right)$. If $\alpha, \beta$ are the roots of the equation $ax ^{2}+ bx +1=0$, then the value of $\alpha^{2}+\beta^{2}-\alpha \beta$ is ....... .

If $z = {\sin ^{ - 1}}\left( {{{x + y} \over {\sqrt x + \sqrt y }}} \right)$, then $x{{\partial z} \over {\partial x}} + y{{\partial z} \over {\partial y}}$ is equal to

The line through $i + 3j + 2k$ and perpendicular to the lines $r = (i + 2j - k) + \lambda (2i + j + k)$ and $r = (2i + 6j + k) + \mu (i + 2j + 3k)$ is

Let $a$ and $ẞ$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^2+|z|=0, z \in C$ Then $4\left(\alpha^2+\beta^2\right)$ is equal to:

If $A = \left[ {\begin{array}{*{20}{c}}1&{ - 1}\\2&{ - 1}\end{array}} \right],\,\,B = \left[ {\begin{array}{*{20}{c}}a&1\\b&{ - 1}\end{array}} \right]$ and ${(A + B)^2} = {A^2} + {B^2}$, then the value of $a$ and $b$ are

If $A = [x:f(x) = 0]$ and $B = [x:g(x) = 0]$, then $A \cap B$ will be