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 :

Let \(P Q R\) be a triangle with \(R(-1,4,2)\). Suppose \(M(2,1,2)\) is the mid point of \(PQ\). The distance of the centroid of \(\triangle \mathrm{PQR}\) from the point of intersection of the line \(\frac{x-2}{0}=\frac{y}{2}=\frac{z+3}{-1}\) and \(\frac{x-1}{1}=\frac{y+3}{-3}=\frac{z+1}{1}\) is

Let \(\mathrm{ABCD}\) and \(\mathrm{AEFG}\) be squares of side \(4\) and \(2\) units, respectively. The point \(\mathrm{E}\) is on the line segment \(\mathrm{AB}\) and the point \(\mathrm{F}\) is on the diagonal \(\mathrm{AC}\). Then the radius \(r\) of the circle passing through the point \(\mathrm{F}\) and touching the line segments \(\mathrm{BC}\) and \(\mathrm{CD}\) satisfies :

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 :

The number of elements in the set \(S=\left\{x \in R : 2 \cos \left(\frac{x^{2}+x}{6}\right)=4^{x}+4^{-x}\right\}\) is\(.....\)

Let the mean and the variance of \(20\) observations \(x_{1}, x_{2}, \ldots x_{20}\) be \(15\) and \(9 ,\) respectively. For \(\alpha \in R\), if the mean of \(\left( x _{1}+\alpha\right)^{2},\left( x _{2}+\alpha\right)^{2}, \ldots,\left( x _{20}+\alpha\right)^{2}\) is \(178 ,\) then the square of the maximum value of \(\alpha\) is equal to \(...........\)

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

If the coefficients of \(x^{-2}\) and \(x^{-4}\) in the expansion of \({\left( {{x^{\frac{1}{3}}} + \frac{1}{{2{x^{\frac{1}{3}}}}}} \right)^{18}}\,,\,\left( {x > 0} \right),\) are \(m\) and \(n\)  respectively, then \(\frac{m}{n}\) is equal to 

A bag contains \(30\) white balls and \(10\) red balls. \(16\) balls are drawn one by one randomly from the bag with replacement. If \(X\) be the number of white balls drawn, then \(\left( {\frac{{{\rm{mean\, of\, X}}}}{{{\rm{standard\, deviation\, of\, X}}}}} \right)\) is equal to

Let \(\mathrm{S}=\{1,2,3,4,5,6,9\} .\) Then the number of elements in the set \(\mathrm{T}=\{\mathrm{A} \subseteq \mathrm{S}: \mathrm{A} \neq \phi\) and the sum of all the elements of \(\mathrm{A}\) is not a multiple of 3\(\}\) is ..... .

If \(\smallint f\left( x \right)\;dx = \varphi \left( x \right)\), then \(\smallint {x^5}\;f\left( {{x^3}} \right)\;dx = \)

Let \(\mathrm{A}(4,-2), \mathrm{B}(1,1)\) and \(\mathrm{C}(9,-3)\) be the vertices of a triangle \(A B C\). Then the maximum area of the parallelogram AFDE , formed with vertices \(\mathrm{D}, \mathrm{E}\) and F on the sides \(\mathrm{BC}, \mathrm{CA}\) and AB of the triangle ABC respectively, is ______ .

The number of points, where the curve \(f(x)=e^{8 x}-e^{6 x}-3 e^{4 x}-e^{2 x}+1, x \in R\) cuts \(x\)-axis, is equal to