If \(g(x)=x^{2}+x-1\) and \((\operatorname{gof})(\mathrm{x})=4 \mathrm{x}^{2}-10 \mathrm{x}+5,\) then \(f\left(\frac{5}{4}\right)\) is equal to

Let \(y = y(x)\) be the solution of the differential equation \(x\sin\left(\dfrac{y}{x}\right)dy = \left(y\sin\left(\dfrac{y}{x}\right) - x\right)dx\), \(y(1) = \dfrac{\pi}{2}\) and let \(\alpha = \cos\left(\dfrac{y(e^{12})}{e^{12}}\right)\). Then the number of integral values of \(p\), for which the equation \(x^2 + y^2 - 2px + 2py + \alpha + 2 = 0\) represents a circle of radius \(r \leq 6\), is __________.

The greatest value of \(c \in R\) for which the system of linear equations \(x - cy - cz = 0 \,\,;\,\, cx - y + cz = 0 \,\,;\,\, cx + cy - z = 0 \) has a non -trivial solution, is

Let \(f\) be a differentiable function satisfying \(f ( x )=\frac{2}{\sqrt{3}} \int_{0}^{\sqrt{3}} f \left(\frac{\lambda^{2} x }{3}\right) d \lambda, x >0\) and \(f (1)=\sqrt{3}\). If \(y=f(x)\) passes through the point \((\alpha, 6)\), then \(\alpha\) is equal to \(.........\)

A coin is tossed three times. Let \(X\) denote the number of times a tail follows a head. If \(\mu\) and \(\sigma^2\) denote the mean and variance of \(X\), then the value of \(64\left(\mu+\sigma^2\right)\) is :

The equation \(\arg \left(\frac{\mathrm{z}-1}{\mathrm{z}+1}\right)=\frac{\pi}{4}\) represents a circle with:

Suppose \(\sum \limits_{ r =0}^{2023} r ^{20023} C _{ r }=2023 \times \alpha \times 2^{2022}\). Then the value of \(\alpha\) is \(............\)

Let \(f(x)=\log _{\mathrm{e}} x\) and \(g(x)=\frac{x^4-2 x^3+3 x^2-2 x+2}{2 x^2-2 x+1}\). Then the domain of \(f \circ g\) is

Suppose the vectors \(x_{1}, x_{2}\) and \(x_{3}\) are the solutions of the system of linear equations, \(Ax = b\) when the vector \(b\) on the right side is equal to \(b _{1}, b _{2}\) and \(b _{3}\) respectively. If \(x =\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], x _{2}=\left[\begin{array}{l}0 \\ 2 \\ 1\end{array}\right], x _{3}=\left[\begin{array}{l}0 \\ 0 \\ 1\end{array}\right], b _{1}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]\) \(b _{2}=\left[\begin{array}{l}0 \\ 2 \\ 0\end{array}\right]\) and \(b _{3}=\left[\begin{array}{l}0 \\ 0 \\ 2\end{array}\right],\) then the determinant of \(A\) is equal to

A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let \(X\) denote the number of defective pens. Then the variance of X is

Let \(R =\{( P , Q ) \mid P\) and \(Q\) are at the same distance from the origin \(\}\) be a relation, then the equivalence class of \((1,-1)\) is the set

lf the mean deviation of the numbers \(1, 1 + d, . . . ,1 + 100d\) from their mean is \(255,\)  then a value of \(d\) is

The integral \(\int\limits_{7\pi /4}^{7\pi /3} {\sqrt {{{\tan }^2}\,x}\,dx } \) is equal to