Let \(y=y(x)\) be the solution of the differential equation \(\left(x^2+4\right)^2 d y+\left(2 x^3 y+8 x y-2\right) d x=0 \text {. If } y(0)=0 \text {, }\) then \(y(2)\) is equal to

The sum \(\sum \limits_{n=1}^{\infty} \frac{2 n^2+3 n+4}{(2 n) !}\) is equal to :

Let \(P \left( a _1, b _1\right)\) and \(Q \left( a _2, b _2\right)\) be two distinct points on a circle with center \(C (\sqrt{2}, \sqrt{3})\). Let \(O\) be the origin and \(OC\) be perpendicular to both \(CP\) and \(CQ\). If the area of the triangle \(OCP\) is \(\frac{\sqrt{35}}{2}\), then \(a _1^2+ a _2^2+ b _1^2+ b _2^2\) is equal to \(...........\).

The number of relations, on the set \(\{1,2,3\}\) containing \((1,2)\) and \((2,3)\), which are reflexive and transitive but not symmetric, is

For three unit vectors \(\vec{a}, \vec{b}, \vec{c}\) satisfying \({|\vec{a}-\vec{b}|^{2}}+{|\vec{b}-\vec{c}|^{2}}+{|\vec{c}-\vec{a}|^{2}}=9\) and \({|2\vec{a}+k\vec{b}+k\vec{c}|}=3\), the positive value of k is:

Let L be the line \( \frac{x+1}{2}=\frac{y+1}{3}=\frac{z+3}{6} \) and let S be the set of all points (a, b, c) on L, whose distance from the line \( \frac{x+1}{2}=\frac{y+1}{3}=\frac{z+9}{0} \) along the line L is 7. Then \( \sum_{(a,b,c)\in S}\ (a+b+c) \) is equal to:

If \(\sum_{r=1}^{10} r !\left( r ^{3}+6 r ^{2}+2 r +5\right)=\alpha(11 !),\) then the value of \(\alpha\) is equal to ...... .

Let \(S=\left\{ x : x \in R \text { and }(\sqrt{3}+\sqrt{2})^{ x ^2-4}+(\sqrt{3}-\sqrt{2})^{ x ^2-4}=10\right\} \text {. }\) Then \(n ( S )\) is equal to

Let \(A\) be a symmetric matrix such that \(|A|=2\) and \(\left[\begin{array}{ll}2 & 1 \\ 3 & \frac{3}{2}\end{array}\right] A =\left[\begin{array}{ll}1 & 2 \\ \alpha & \beta\end{array}\right]\). If the sum of the diagonal elements of \(A\) is s, then \(\frac{\beta s}{\alpha^2}\) is equal to \(..........\).

Let \(D\) be the domain of the function \(f(x)=\sin ^{-1}\) \(\left(\log _{3 x}\left(\frac{6+2 \log _3 x}{-5 x}\right)\right)\). If the range of the function \(g: D \rightarrow R\) defined by \(g( x )= x -[ x ],([ x ]\) is the greatest integer function), is ( \(\alpha, \beta)\), then \(\alpha^2+\frac{5}{\beta}\) is equal to

The angle of elevation of a jet plane from a point \(A\) on the ground is \(60^{\circ}\). After a flight of \(20\, seconds\) at the speed of \(432\, km / hour\), the angle of elevation changes to \(30^{\circ}\). If the jet plane is flying at a constant height, then its height is ..... \(m.\)

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 :

If the area of the larger portion bounded between the curves \(x^2+y^2=25\) and \(y=|x-1|\) is \(\frac{1}{4}(b \pi+c), b, c \in N\), then \(b+c\) is equal to