The normal to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{9}=1\) at the point \((8,3 \sqrt{3})\) on it passes through the point

Let the function \(f :[0,2] \rightarrow R\) be defined as \(f(x)=\left\{\begin{array}{cc}e^{\min \left[x^2, x-[x]\right\}}, & x \in[0,1) \\e^{\left[x-\log _e x\right]}, & x \in[1,2]\end{array}\right.\) where [t] denotes the greatest integer less than or equal to \(t\). Then the value of the integral \(\int \limits_0^2 x f(x) d x\) is

The interval in which the function \(\mathrm{f}(\mathrm{x})=\mathrm{x}^{\mathrm{x}}, \mathrm{x}>0\), is strictly increasing is

Let \(0 < \theta  < \frac{\pi }{2}\). If the eccentricity of the hyperbola \(\frac{{{x^2}}}{{{{\cos }^2}\,\theta }} - \frac{{{y^2}}}{{{{\sin }^2}\,\theta }} = 1\) is greater than \(2\), then the length of its latus rectum lies in the interval

Consider the relations \(R_1\) and \(R_2\) defined as \(a R_1 b\) \(\Leftrightarrow a^2+b^2=1\) for all \(a, b, \in R\) and \((a, b) R_2(c, d)\) \(\Leftrightarrow a+d=b+c\) for all \((a, b),(c, d) \in N \times N\). Then

If \(y = f(x)\) is the solution of the differential equation \(\frac{{dy}}{{dx}} = \left( {\tan \,x - y} \right){\sec ^2}\,x,\,x \in \left( { - \frac{\pi }{2},\frac{\pi }{2}} \right)\), such that \(y(0) = 0\), then \(y\left( { - \frac{\pi }{4}} \right)\) is equal to

If \(\sum_{ k =1}^{10} K ^{2}\left(10_{ C _{ K }}\right)^{2}=22000 L\), then \(L\) is equal to \(.....\)

If the system of linear equations \(7 x+11 y+\alpha z=13\) \(5 x+4 y+7 z=\beta\) \(175 x+194 y+57 z=361\) has infinitely many solutions, then \(\alpha+\beta+2\) is equal to

The function \(f(x)=\left|x^{2}-2 x-3\right| \cdot e^{\left|9 x^{2}-12 x+4\right|}\) is not differentiable at exactly :

Let \(z \in C\) with \(Im(z) = 10\) and it satisfies \(\frac{{2z - n}}{{2z + n}} = 2i - 1\) for some natural number \(n\). Then

Let \(\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{d}}=\vec{a} \times \overrightarrow{\mathrm{b}}\). If \(\overrightarrow{\mathrm{c}}\) is a vector such that \(\vec{a} \cdot \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{c}}|\), \(|\overrightarrow{\mathrm{c}}-2 \vec{a}|^2=8\) and the angle between \(\overrightarrow{\mathrm{d}}\) and \(\overrightarrow{\mathrm{c}}\) is \(\frac{\pi}{4}\), then \(|10-3 \overrightarrow{\mathrm{~b}} \cdot \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{d}} \times \overrightarrow{\mathrm{c}}|^2\) is equal to

If the number of five digit numbers with distinct digits and \(2\) at the \(10^{\text {th }}\) place is \(336 \mathrm{k}\), then \(\mathrm{k}\) is equal to

The distance of the point \((1,-5, 9)\) from the plane \(x - y + z = 5\) measured along the line \(x = y = z\) is