If x + 1 x = 2 , , then x^3 + 1 x^3 = - Vidyadip

MCQ

If $x + \frac{1}{x} = 2\,\cos \theta ,$ then ${x^3} + \frac{1}{{{x^3}}} = $

A
$\cos \,\,3\theta $
✓
$2\,\cos \,3\theta $
C
$\frac{1}{2}\cos \,3\theta $
D
$\frac{1}{3}\cos \,3\theta $

✓

Answer

Correct option: B.

$2\,\cos \,3\theta $

b
(b) We have $x + \frac{1}{x} = 2\cos \theta $,

Now ${x^3} + \frac{1}{{{x^3}}} = {\left( {x + \frac{1}{x}} \right)^3} - 3x\frac{1}{x}\left( {x + \frac{1}{x}} \right)$

$= {(2\cos \theta )^3} - 3(2\cos \theta ) = 8{\cos ^3}\theta - 6\cos \theta $

$= 2(4{\cos ^3}\theta - 3\cos \theta ) = 2\cos 3\theta $.

Trick : Put $x = 1$ $ \Rightarrow $ $\theta = {0^\circ }$.

Then ${x^3} + \frac{1}{{{x^3}}} = 2 = 2\cos 3\theta $.

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

$\int_{}^{} {\frac{{1 + {{\cos }^2}x}}{{{{\sin }^2}x}}dx} = $

If $\mathop {\lim }\limits_{x \to 0} kx\,{\rm{cosec}}\,x = \mathop {\lim }\limits_{x \to 0} x\,{\rm{cosec}}\;kx$, then $k = $

If $\vec x$ is a unit vector such that $\vec x \times \left( {\hat i - 2\hat j + \hat k} \right) = - \hat i + \hat k$ , then $\vec x$ is

If the locus of the point, whose distances from the point $(2,1)$ and $(1,3)$ are in the ratio $5: 4,$ is $a x^2+b y^2+c x y+d x+e y+170=0,$ then the value of $a^2+2 b+3 c+4 d+e$ is equal to :

If $\mathop {\lim }\limits_{n \to \infty } \frac{{1 - {{(10)}^n}}}{{1 + {{(10)}^{n + 1}}}} = \frac{{ - \alpha }}{{10}}$, then give the value of $\alpha $ is

Let a curve $y=f(x)$ pass through the points $(0,5)$ and $\left(\log _c 2, k\right)$. If the curve satisfies the differential equation $2(3+y) e^{2 x} d x-\left(7+e^{2 x}\right) d y=0$, then $k$ is equal to

The integrating factor of the differential equation $(x\log x)\frac{{dy}}{{dx}} + y = 2\log x$ is

If $A + B + C = \frac{\pi }{2}$ ,then value of $tanA\,\, tanB + tanB\,\, tanC + tanC\,\, tanA$ is

$\mathop {\lim }\limits_{x \to \infty } \frac{{{{(x + 1)}^{10}} + {{(x + 2)}^{10}} + ..... + {{(x + 100)}^{10}}}}{{{x^{10}} + {{10}^{10}}}}$ is equal to

Consider $f(x) =$ $\left[ {\frac{{2\,\,\left( {\sin \,x\,\, - \,\,{{\sin }^3}\,x} \right)\,\, + \,\,\left| {\sin \,x\,\, - \,\,{{\sin }^3}\,x} \right|}}{{2\,\,\left( {\sin \,x\,\, - \,\,{{\sin }^3}\,x} \right)\,\, - \,\,\left| {\sin \,x\,\, - \,\,{{\sin }^3}\,x} \right|}}} \right]$ , $x \ne \,\frac{\pi}{2} \,for\, x \in (0, \pi ) f(\pi /2) = 3$

where [ ] denotes the greatest integer function then,