Let the eccentricity of the hyperbola \(H : \frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1\) be \(\sqrt{\frac{5}{2}}\) and length of its latus rectum be \(6 \sqrt{2}\), If \(y =2 x + c\) is a tangent to the hyperbola \(H\), then the value of \(c ^{2}\) is equal to

If the domain of the function \( f(x)=\sin^{-1}(\frac{2}{x^{2}-2x-2}) \) is \( (-\infty,\alpha]\cup[\beta,\gamma]\cup[\delta,\infty) \), then \( \alpha+\beta+\gamma+\delta \) is equal to

Equation of the tangent to the circle, at the point \((1 , -1)\) whose centre is the point of intersection of the straight lines \(x - y = 1\) and \(2x + y= 3\) is

Let two vertices of triangle \(ABC\) be \((2,4,6)\) and \((0,-2,-5)\), and its centroid be \((2,1,-1)\). If the image of third vertex in the plane \(x+2 y+4 z=11\) is \((\alpha, \beta, \gamma)\), then \(\alpha \beta+\beta \gamma+\gamma \alpha\) is equal to

Let \(\mathrm{ABC}\) be a triangle of area \(15 \sqrt{2}\) and the vectors \(\overrightarrow{\mathrm{AB}}=\hat{i}+2 \hat{j}-7 \hat{k}, \quad \overrightarrow{B C}=a \hat{i}+b \hat{j}+c \hat{k}\) and \(\overrightarrow{\mathrm{AC}}=6 \hat{\mathrm{i}}+\mathrm{d} \hat{\mathrm{j}}-2 \hat{\mathrm{k}}, \mathrm{d}>0\). Then the square of the length of the largest side of the triangle \(\mathrm{ABC}\) is ...........

The temperature \(\mathrm{T}(\mathrm{t})\) of a body at time \(\mathrm{t}=0\) is \(160^{\circ}\) \(\mathrm{F}\) and it decreases continuously as per the differential equation \(\frac{\mathrm{dT}}{\mathrm{dt}}=-\mathrm{K}(\mathrm{T}-80)\), where \(\mathrm{K}\) is positive constant. If \(\mathrm{T}(15)=120^{\circ} \mathrm{F}\), then \(\mathrm{T}(45)\) is equal to ...........

If the vector \(\vec b = 3\hat j + 4\hat k\) is written as the sum of a vector \({\vec {b_1}}\) , parallel to \(\vec a = \hat i + \hat j\) and a vector \({\vec {b_2}}\) , perpendicular to \(\vec a\) , then \({\vec {b_1}} \times {\vec {b_2}}\) is equal to

A rod of length eight units moves such that its ends \(A\) and \(B\) always lie on the lines \(x-y+2=0\) and \(y+2=0\), respectively. If the locus of the point \(P\), that divides the rod \(A B\) internally in the ratio \(2: 1\) is \(9\left(x^2+\alpha y^2+\beta x y+\gamma x+28 y\right)-76=0\), then \(\alpha-\beta-\gamma\) is equal to :

Let \(A_1, A_2, A_3\) be the three A.P. with the same common difference \(d\) and having their first terms as \(A , A +1, A +2\), respectively. Let \(a , b , c\) be the \(7^{\text {th }}, 9^{\text {th }}, 17^{\text {th }}\) terms of \(A_1, A_2, A_3\), respectively such that \(\left|\begin{array}{lll} a & 7 & 1 \\ 2 b & 17 & 1 \\ c & 17 & 1\end{array}\right|+70=0\) If \(a=29\), then the sum of first \(20\) terms of an \(AP\) whose first term is \(c - a - b\) and common difference is \(\frac{ d }{12}\), is equal to \(........\).

If \(\sum_{r=1}^n T_r=\frac{(2 n-1)(2 n+1)(2 n+3)(2 n+5)}{64}\), then \(\lim _{n \rightarrow \infty} \sum_{r=1}^n\left(\frac{1}{T_r}\right)\) is equal to :

Statement \(-1\) : The system of linear equations \(x + \left( {\sin \,\alpha } \right)y + \left( {\cos \,\alpha } \right)z = 0\) \(x + \left( {\cos \,\alpha } \right)y + \left( {\sin \alpha } \right)z = 0\) \(x - \left( {\sin \,\alpha } \right)y - \left( {\cos \alpha } \right)z = 0\) has a non-trivial solution for only one value of \(\alpha \) lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)  Statement \(-2\) : The equation in \(\alpha \) \(\left| {\begin{array}{*{20}{c}}
  {\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha } \\ 
  {\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha } \\ 
  {\cos {\mkern 1mu} \alpha }&{ - \sin {\mkern 1mu} \alpha }&{ - \cos {\mkern 1mu} \alpha } 
\end{array}} \right| = 0\) has only one solution lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)

Let \(\mathrm{M}\) denote the median of the following frequency distribution then \(20\) \(M\) is equal to :

Class	\(0-4\)	\(4-8\)	\(8-12\)	\(12-16\)	\(16-20\)
Freq	\(3\)	\(9\)	\(10\)	\(8\)	\(6\)

A function \(f\) is defined on \([-3,3]\) as \(f(x)=\left\{\begin{array}{cc}\min \left\{|x|, 2-x^{2}\right\} & , \quad-2 \leq x \leq 2 \\ {[|x|]} & , \quad 2<|x| \leq 3\end{array}\right.\) where \([x]\) denotes the greatest integer \(\leq x .\) The number of points, where \(f\) is not differentiable in \((-3,3)\) is