Mathematics Tes...

Find the value of\(\tan \frac{1}{2}\left[\sin ^{-1} \frac{2 x}{1+x^{2}}+\cos ^{-1} \frac{1-y^{2}}{1+y^{2}}\right]\).

If \(\sin ^{4} x+2 \cos ^{4} x=\frac{2}{3}\), then what is the value of \(\sec ^{2} x ?\)

Let \(\vec{b}=4 \hat{i}+3 \hat{j}\) and \(\vec{c}\) be two vector perpendicular to each other in the \(x y-\) plane. Then a vector in the same plane having projections 1 and 2 along \(\vec{b}\) and \(\vec{c}\), respectively, is:

\(\lim _{x \rightarrow 0} \frac{\sin \left(\pi \cos ^{2} x\right)}{x^{2}}\) equals:

The curve represented by the equations

\(x=3(\cos t+\sin t)\)

\(y=4(\cos t-\sin t)\) is:

The volume of a spherical balloon is increasing at the rate of \(2 \mathrm{~cm}^3 / \mathrm{sec}\). When its radius is \(4 \mathrm{~cm}\), the rate of change of its surface area (in \(\mathrm{cm}^2 / \mathrm{sec}\) ) is:

Find the minimum value of sin x + cos 2x

Find the general solution of \(\frac{d y}{d x}+y \tan x=2 \sin x\).

Find the area under the curve \(y=\cos x\) in the interval \(0

If \(\tan ^{-1} \frac{x-1}{x-2}+\tan ^{-1} \frac{x+1}{x+2}=\frac{\pi}{4},\) then find the value of \(x\).

The equation \(x^{2}-6 x-3 y+21=0\) represents a/an:

If \(f(x)=\left\{\begin{array}{ll}\frac{\sin 3 x}{e^{2 x}-1}, & x \neq 0 \\ k-2, & x=0\end{array}\right.\) is continuous at \(x=0\), then find out the value of \(k\).

\(2^{3 n}-7 n-1\) is divisible by: