Four fair dice are thrown independently \(27\) times. Then the expected number of times, at least two dice show up a three or a five, is

Let \(y=y(x)\) be the solution of the differential equation \(x\frac{dy}{dx}-sin~2y=x^{3}(2-x^{3})cos^{2}y,\) \(x\ne0.\) If \(y(2)=x,\) then \(tan(y(1))\) is equal to

Consider the following two statements : Statement \(I\) : For any two non-zero complex numbers \(\mathrm{z}_1, \mathrm{z}_2\) \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)\) and Statement \(II\) : If \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are three distinct complex numbers and a, b, c are three positive real numbers such that \(\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}\), then \(\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1\) Between the above two statements,

Let the line \(\ell: x =\frac{1- y }{-2}=\frac{ z -3}{\lambda}, \lambda \in R\) meet the plane \(P : x +2 y +3 z =4\) at the point \((\alpha, \beta, \gamma)\). If the angle between the line \(\ell\) and the plane \(P\) is \(\cos ^{-1}\left(\sqrt{\frac{5}{14}}\right)\), then \(\alpha+2 \beta+6 \gamma\) is equal to

Let \(p\) and \(q\) be two positive numbers such that \(p + q =2\) and \(p ^{4}+ q ^{4}=272 .\) Then \(p\) and \(q\) are roots of the equation

Let the area enclosed between the curves \(|y|=1-x^2\) and \(x^2+y^2=1\) be \(\alpha\). If \(9 \alpha=\beta \pi+\gamma ; \beta, \gamma\) are integers, then the value of \(|\beta-\gamma|\) equals.

Let \(y=y(x)\) be a solution of the differential equation \((x \cos x) d y+(x y \sin x+y \cos x-1) d x=0\), \(0 < x < \frac{\pi}{2}\). If \(\frac{\pi}{3} y\left(\frac{\pi}{3}\right)=\sqrt{3}\), then \(\left|\frac{\pi}{6} y^{\prime \prime}\left(\frac{\pi}{6}\right)+2 y^{\prime}\left(\frac{\pi}{6}\right)\right|\) is equal to \(.........\).

The sum of all the elements of the set \(\{\alpha \in\{1,2, \ldots, 100\}: \operatorname{HCF}(\alpha, 24)=1\}\) is

The distance of the point \((1, 0, 2)\) from the point of intersection of the line \(\frac{{x - 2}}{3} = \frac{{y + 1}}{4} = \frac{{z - 2}}{{12}}\)   and the plane \(x - y + z = 16\) is

Let the set of all \(a \in R\) such that the equation \(\cos 2 x+a \sin x=2 a-7\) has a solution be \([p, q]\) and \(\mathrm{r}=\tan 9^{\circ}-\tan 27^{\circ}-\frac{1}{\cot 63^{\circ}}+\tan 81^{\circ}\), then \(pqr\) is equal to ...........

The set of values of \(a\) for which \(\lim _{x \rightarrow a}([x-5]-[2 x+2])=0\), where, \([\zeta]\) denotes the greatest integer less than or equal to \(\zeta\) is equal to

Let \(\vec a = \hat i - \hat j,\) \(\vec b = \hat i + \hat j + \hat k\) and \(\vec c\) be a vector such that \(\vec a \times \vec c + \vec b = 0\) and \(\vec a.\vec c = 4\), then \({\left| {\vec c} \right|^2}\) is equal to

Let \(S=\left\{x \in R: 0 < x < 1\right.\) and \(\left.2 \tan ^{-1}\left(\frac{1-x}{1+x}\right)=\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right\}\). If \(n ( S )\) denotes the number of elements in \(S\) then: