Let \(A\) be a square matrix of order \(2\) such that \(|A|=2\) and the sum of its diagonal elements is \(-3\) . If the points \((x, y)\) satisfying \(A^2+x A+y I=0\) lie on a hyperbola, whose transverse axis is parallel to the x-axis, eccentricity is e and the length of the latus rectum is \(\ell\), then \(\mathrm{e}^4+\ell^4\) is equal to ...........

the tangent at the point \(\left(x_{1}, y_{1}\right)\) on the curve \(y=x^{3}+3 x^{2}+5\) passes through the origin, then \(\left(x_{1}, y_{1}\right)\) does NOT lie on the curve

Let the equation of the plane \(P\) containing the line \(x+10=\frac{8-y}{2}=z\) be \(a x+b y+3 z=2(a+b)\) and the distance of the plane \(P\) from the point \((1,27,7)\) be c. Then \(a^2+b^2+c^2\) is equal to \(.............\).

Let \(A B C\) be an isosceles triangle in which \(A\) is at \((-1,0), \angle A=\frac{2 \pi}{3}, A B=A C\) and \(B\) is on the positive \(\mathrm{x}\)-axis. If \(\mathrm{BC}=4 \sqrt{3}\) and the line \(\mathrm{BC}\) intersects the line \(y=x+3\) at \((\alpha, \beta)\), then \(\frac{\beta^4}{\alpha^2}\) is :

Let \(f(x)\) be a real differentiable function such that \(f(0)=1\) and \(f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y)\) for all \(x, y \in \mathbf{R}\). Then \(\sum_{\mathrm{n}=1}^{100} \log _{\mathrm{e}} f(\mathrm{n})\) is equal to :

Let \(\alpha\) and \(\beta\) be the roots of the equation \(x^{2}+(2 i -\) \(1)=0\). Then, the value of \(\left|\alpha^{8}+\beta^{8}\right|\) is equal to

The angle of elevation of a cloud \(C\) from a point \(P , 200 m\) above a still lake is \(30^{\circ} .\) If the angle of depression of the image of \(C\) in the lake from the point \(P\) is \(60^{\circ},\) then \(P C\) (in \(m\) ) is equal to

In a survey of \(220\) students of a higher secondary school, it was found that at least \(125\) and at most \(130\) students studied Mathematics; at least \(85\) and at most \(95\) studied Physics; at least \(75\) and at most \(90\) studied Chemistry; \(30\) studied both Physics and Chemistry; \(50\) studied both Chemistry and Mathematics; \(40\) studied both Mathematics and Physics and \(10\) studied none of these subjects. Let \(\mathrm{m}\) and \(\mathrm{n}\) respectively be the least and the most number of students who studied all the three subjects. Then \(\mathrm{m}+\mathrm{n}\) is equal to ...........

Let \(f(x)=x^5+2 x^3+3 x+1, x \in R\), and \(g(x)\) be a function such that \(g(f(x))=x\) for all \(x \in R\). Then \(\frac{g(7)}{g^{\prime}(7)}\) is equal to :

Consider the following frequency distribution :

Class:	\(0-6\)	\(6-12\)	\(12-18\)	\(18-24\)	\(24-30\)
Frequency :	\(a\)	\(b\)	\(12\)	\(9\)	\(5\)

If mean \(=\frac{309}{22}\) and median \(=14\), than value \((a-b)^{2}\) is equal to \(.....\)

If the coefficient of \(x ^{15}\) in the expansion of \(\left(a x^3+\frac{1}{b x^{\frac{1}{3}}}\right)^{15}\) is equal to the coefficient of \(x^{-15}\) in the expansion of \(\left(a x^{\frac{1}{3}}-\frac{1}{b x^3}\right)^{15}\), where \(a\) and \(b\) are positive real numbers, then for each such ordered pair \((a, b) :\)

Let a function \(f: N \rightarrow N\) be defined by \(f ( n )=\left[\begin{array}{ll}2 n , \,\,\, \,\,\,\,\,\,n =2,4,6,8, \ldots . \\ n -1,\,\,\,  n =3,7,11,15, \ldots . \\ \frac{ n +1}{2}, \,\,\, \,\,\,n =1,5,9,13, \ldots \ldots\end{array}\right.\) then, \(f\) is

Let the circles \(C_1 : |z| = r\) and \(C_2 : |z - 3 - 4i| = 5\), \(z \in \mathbb{C}\), be such that \(C_2\) lies within \(C_1\). If \(z_1\) moves on \(C_1\), \(z_2\) moves on \(C_2\) and \(\min |z_1 - z_2| = 2\), then \(\max |z_1 - z_2|\) is equal to: