$I = \int\limits_0^\pi {\frac{{2(\pi – x)\sin (\pi – x)}}{{3 + \cos 2(\pi – x)}}dx} $
$ = \int\limits_0^\pi {\frac{{2(\pi – x)\sin x}}{{3 + \cos 2x}}dx} $
$2I = \int\limits_0^\pi {\frac{{2x\sin x}}{{3 + \cos 2x}}dx} + \int\limits_0^\pi {\frac{{2(\pi – x)\sin x}}{{3 + \cos 2x}}dx} = \int\limits_0^\pi {\frac{{2\pi \sin x}}{{3 + \cos 2x}}dx} $ Continue reading $I = \int\limits_0^\pi {\frac{{2x\sin x}}{{3 + \cos 2x}}dx} = ?$
We have, $x^2+1-2x=0$
$\Rightarrow (x-1)^2=0$
$\Rightarrow x=1$
$\therefore x^{99}+\frac {1}{x^{99}}=1+1=2$
$R(x)=\frac{{{x^{16}} – 1}}{{{x^8} + 2{x^7}}} = \frac{{({x^2} – 1)({x^2} + 1)({x^4} + 1)({x^8} + 1)}}{{{x^7}(x + 2)}}$
$\because {x^2} – x – 1 = 0,{x^2} – 1 = x$
$R(x) = \frac{{x({x^2} + 1)({x^4} + 1)({x^8} + 1)}}{{{x^7}(x + 2)}} = \frac{{({x^2} + 1)({x^4} + 1)({x^8} + 1)}}{{{x^6}(x + 2)}}$
$\because {x^2} – x – 1 = 0,{x^2} = x + 1$ Continue reading $x^2-x-1=0$
$R(x)=\frac{{{x^{16}} – 1}}{{{x^8} + 2{x^7}}} = ?$
For $x \in \mathbb{R}$, the number of real roots of the equation $$3{x^2} – 4|{x^2} – 1| + x – 1 = 0$$ is ___ . Continue reading Roots of
$3{x^2} – 4|{x^2} – 1| + x – 1 = 0$
(A) $\{ \mp 4, \pm 2, \pm 1\} $
(B) $\{ \pm 4, \mp 2, \pm 1\} $
(C) $\{ \pm 4, \pm 2, \mp 1\} $
(D) $\{ \pm 4, \pm 2, \pm 1\} $ Continue reading $x^2+4y^2+16z^2=48$
$xy+4yz+2zx=24$
$\{x,y,z\}\equiv$
An $\alpha$-particle (mass 4 amu) and a singly charged sulfur ion (mass 32 amu) are initially at rest. They are accelerated through a potential V and then allowed to pass into a region of uniform magnetic field which is normal to the velocities of the particles. Within this region, the $\alpha$-particle and the sulfur ion move in circular orbits of radii $r_{\alpha}$ and $r_S$, respectively. The ratio ($r_S/r_\alpha$) is _____. Continue reading An $\alpha$-particle (mass 4 amu) and a singly ….
The volume V of an enclosure contains a mixture of three gases, 16 g of oxygen, 28 g of nitrogen and 44 g of carbon dioxide at absolute temperature T. Consider R as universal gas constant. The pressure of the mixture of gases is :
(A) $\frac {3RT}{V}$ (B) $\frac {5}{2} \frac {RT}{V}$
(C) $\frac {88RT}{V}$ (D) $\frac {4RT}{V}$ Continue reading The volume V of an enclosure contains a mixture ….
