Let \(f\) be a function such that \(f(x)+3 f\left(\frac{24}{x}\right)\) \(=4 x, x \neq 0\). Then \(f(3)+f(8)\) is equal to

Let \(O\) be the origin and the position vector of \(A\) and \(B\) be \(2 \hat{i}+2 \hat{j}+\hat{k}\) and \(2 \hat{i}+4 \hat{j}+4 \hat{k}\) respectively. If the internal bisector of \(\angle A O B\) meets the line \(A B\) at \(\mathrm{C}\), then the length of \(\mathrm{OC}\) is

If \(\frac{d y}{d x}=\frac{2^{x} y+2^{y} \cdot 2^{x}}{2^{x}+2^{x+y} \log _{e} 2}, y(0)=0\), then for \(y=1\) the value of \(x\) lies in the interval:

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

The equations of the sides \(AB\) and \(AC\) of a triangle \(ABC\) are \((\lambda+1) x +\lambda y =4 \text { and } \lambda x +(1-\lambda) y +\lambda=0\) respectively. Its vertex \(A\) is on the \(y\)-axis and its orthocentre is \((1,2)\). The length of the tangent from the point \(C\) to the part of the parabola \(y^2=6 x\) in the first quadrant is

Let \(f: R \rightarrow R\) satisfy the equation \(f(x+y)=f(x) \cdot f(y)\) for all \(x, y \in R\) and
\(f ( x ) \neq 0\) for any \(x \in R .\) If Ihe function \(f\) is differentiable at \(x =0\) and \(f^{\prime}(0)=3,\) then \(\lim _{h \rightarrow 0} \frac{1}{h}(f(h)-1)\) is equal to ....... .

A plane containing the point \((3, 2, 0)\) and the line \(\frac{{x - 1}}{1} = \frac{{y - 2}}{5} = \frac{{z - 3}}{4}\) also contains the point

\(\int \limits_{\frac{3 \sqrt{2}}{4}}^{\frac{3 \sqrt{3}}{4}} \frac{48}{\sqrt{9-4 x^2}} d x\) is equal to

If \(f(x)=\frac{2^x}{2^x+\sqrt{2}}, \mathrm{x} \in \mathbb{R}\), then \(\sum_{\mathrm{k}=1}^{81} f\left(\frac{\mathrm{k}}{82}\right)\) is equal to

A number is called a palindrome if it reads the same backward as well as forward. For example \(285582\) is a six digit palindrome. The number of six digit palindromes, which are divisible by \(55\), is ...... .

Let \(k\) be a non-zero real number  If \(f(x) = {\rm{ }}\left\{ {\begin{array}{*{20}{c}}
{\frac{{\left( {{e^x} - 1} \right)^2}}{{\sin {\mkern 1mu} \left( {\frac{x}{k}} \right){\mkern 1mu} \log {\mkern 1mu} \left( {1 + \frac{x}{4}} \right)}}{\mkern 1mu} ,{\mkern 1mu} x \ne 0}\\
{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} 12{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} ,x{\mkern 1mu} {\mkern 1mu}  = 0{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} }
\end{array}} \right.\)  is a continuous function then the value of \(k\) is

Let \(x = 9\) be a directrix of an ellipse \(E\), whose centre is at the origin and eccentricity is \(\dfrac{1}{3}\). Let \(P(\alpha, 0)\), \(\alpha > 0\), be a focus of \(E\) and \(AB\) be a chord passing through \(P\). Then the locus of the mid point of \(AB\) is :

If \(z_1, z_2, z_3 \in C\) are the vertices of an equilateral triangle, whose centroid is \(\mathrm{z}_0\), then \(\sum_{\mathrm{k}=1}^3\left(\mathrm{z}_{\mathrm{k}}-\mathrm{z}_0\right)^2\) is equal to