Let \(y=y(x)\) be the solution of the differential equation \(x\frac{dy}{dx}-y=x^{2}\cot x, x\in(0,\pi)\). If \(y(\frac{\pi}{2})=\frac{\pi}{2}\), then \(6y(\frac{\pi}{6})-8y(\frac{\pi}{4})\) is equal to :

With the usual notation, in \(\Delta ABC\), if \(\angle A + \angle B = {120^o}\), \(a = \sqrt 3 - 1\), then the ratio \(\angle A : \angle B\), is

The greatest value of \(c \in R\) for which the system of linear equations \(x - cy - cz = 0 \,\,;\,\, cx - y + cz = 0 \,\,;\,\, cx + cy - z = 0 \) has a non -trivial solution, is

If the inverse trigonometric functions take principal values, then \(\cos ^{-1}\left(\frac{3}{10} \cos \left(\tan ^{-1}\left(\frac{4}{3}\right)\right)+\frac{2}{5} \sin \left(\tan ^{-1}\left(\frac{4}{3}\right)\right)\right)\) is equal to

Fractional part of the number \(\frac{4^{2022}}{15}\) is equal to

If \(0 < a , b < 1,\) and \(\tan ^{-1} a +\tan ^{-1} b =\frac{\pi}{4},\) then the value of \((a+b)-\left(\frac{a^{2}+b^{2}}{2}\right)+\left(\frac{a^{3}+b^{3}}{3}\right)-\left(\frac{a^{4}+b^{4}}{4}\right)+\ldots\) is ..... .

Let the mean and the standard deviation of the probability distribution

\(X\)	\(\alpha\)	\(1\)	\(0\)	\(-3\)
\(P(X)\)	\(\frac{1}{3}\)	\(K\)	\(\frac{1}{6}\)	\(\frac{1}{4}\)

be \(\mu\) and \(\sigma\), respectively. If \(\sigma-\mu=2\), then \(\sigma+\mu\) is equal to ...........

Suppose for a differentiable function \(h, h(0)=0\), \(\mathrm{h}(1)=1\) and \(\mathrm{h}^{\prime}(0)=\mathrm{h}^{\prime}(1)=2\). If \(\mathrm{g}(\mathrm{x})=\mathrm{h}\left(\mathrm{e}^{\mathrm{x}}\right) \mathrm{e}^{\mathrm{h}(\mathrm{x})}\), then \(g^{\prime}(0)\) is equal to :

Let \(3, a, b, c\) be in \(A.P.\) and \(3, a-1, b+1, c+9\) be in \(G.P.\) Then, the arithmetic mean of \(a, b\) and \(c\) is :

Let \( \frac{\pi}{2}<\theta<\pi \) and \( cot\theta=-\frac{1}{2\sqrt{2}} \). Then the value of \( sin(\frac{15\theta}{2})(cos8\theta+sin8\theta)+cos(\frac{15\theta}{2})(cos8\theta-sin8\theta) \) is equal to:

If \( f(x)=\left\{\begin{array}{cc} \frac{a|x|+x^2-2(\sin |x|)(\cos |x|)}{x} & , x \neq 0 \\ b & , x=0 \end{array}\right. \) is continuous at \( x=0 \) then \( a+b \) is equal to:

A circle with centre \((2,3)\) and radius \(4\) intersects the line \(x + y =3\) at the points \(P\) and \(Q\). If the tangents at \(P\) and \(Q\) intersect at the point \(S(\alpha, \beta)\), then \(4 \alpha-7 \beta\) is equal to \(........\).

For two non-zero complex number \(z_1\) and \(z_2\), if \(\operatorname{Re}\left(z_1 z_2\right)=0\) and \(\operatorname{Re}\left(z_1+z_2\right)=0\), then which of the following are possible ? \((A)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((B)\) \(\operatorname{Im}\left(z_1\right) < 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((C)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) < 0\) \((D)\) \(\operatorname{Im}\left( z _1\right) < 0\) and \(\operatorname{Im}\left( z _2\right) < 0\) Choose the correct answer from the options given below :