Assuming $\beta > \alpha $ which of the following option(s) is/are correct?
(A) $y \le \alpha $
(B) $y \ge \beta $
(C) $\alpha \le y \le \beta $
(D) $y \in \mathbb{R}$ Continue reading $y = \frac{{{x^2} – \alpha \beta }}{{2x – \alpha – \beta }}$
$x \in \mathbb{R} – \left\{ {\frac{{\alpha + \beta }}{2}} \right\}$
Using Sine Rule in the left triangle,
$\frac {sin \alpha}{a} = \frac {sin \beta}{b}$
Using Sine Rule in the right triangle,
$\frac {sin 20^\circ}{a} = \frac {sin 140^\circ}{b}$
Dividing the above equations we have,
$\frac {sin \alpha}{sin 20^\circ}=\frac {sin \beta}{sin 140^\circ}$ Continue reading $\alpha = ?$ in the Triangle
The given integral can be split as,
$ \int\limits_{ – \frac{\pi }{6}}^{\frac{\pi }{6}} {\frac{{6{x^5} + \frac{{{x^3}}}{3} + \ln \frac{{1 – x}}{{1 + x}} + \sin x}}{{1 + \cos 2x}}} dx + \int\limits_{ – \frac{\pi }{6}}^{\frac{\pi }{6}} {\frac{1}{{1 + \cos 2x}}} dx$
$\frac{{6{x^5} + \frac{{{x^3}}}{3} + \ln \frac{{1 – x}}{{1 + x}} + \sin x}}{{1 + \cos 2x}}$ is an odd function. Thus, the given integral
$ = 0 + \int\limits_{ – \frac{\pi }{6}}^{\frac{\pi }{6}} {\frac{1}{{2{{\cos }^2}x}}} dx = \frac{1}{2}\int\limits_{ – \frac{\pi }{6}}^{\frac{\pi }{6}} {{{\sec }^2}xdx} = \frac{1}{2}\left. {\tan x} \right|_{ – \frac{\pi }{6}}^{\frac{\pi }{6}} = \frac{1}{{\sqrt 3 }}$
A disc or radius r on top of another disc of radius 4r is made to go around it via pure rolling motion. How many maximum full rotations will the small disc make in this process? Continue reading Disc Rolling on Disc
Consider a frame that is made up of two thin massless rods AB and AC as shown in the figure. A vertical force $\vec P$ of magnitude 100N is applied at point A of the frame. Suppose the force $\vec P$ is resolved parallel to the arms AB and AC of the frame. The magnitude of the resolved component along the arm AC is xN. The value of x, to the nearest integer, is————.
[Given: $\sin 35^\circ = 0.573,{\rm{ }}\cos 35^\circ = 0.819,{\rm{ }}\sin 110^\circ = 0.939,{\rm{ }}\cos 110^\circ = – 0.342$] Continue reading Consider a frame that is made up of two …
Let $f:R \to R$ be a continuous function such that $f(x) + f(x + 1) = 2$, for all $x\in R$. If ${I_1} = \int\limits_0^8 {f(x)dx} $ and ${I_2} = \int\limits_{ – 1}^3 {f(x)dx} $, then the value of $I_1 +2I_2 $ is equal to …… Continue reading $f(x) + f(x + 1) = 2$
$\int\limits_0^8 {f(x)dx + 2\int\limits_{ – 1}^3 {f(x)dx} } = ?$
Consider a 20 kg uniform circular disk of radius 0.2 m. It is pin supported at its centre and is at rest initially. The disk is acted upon by a constant force F = 20 N through a massless string wrapped around its periphery as shown in the figure. Suppose the disk makes n number of revolutions to attain an angular speed of 50 rad/s. The value of n, to the nearest integer, is ———. [Given : In one complete revolution, the disk rotates by 6.28 rad] Continue reading Consider a 20 kg uniform circular disk …
