If the probability of hitting a target by a shooter, in any shot, is \(\frac{1}{3}\), then the minimum number of independent shots at the target required by him so that the probability of hitting the target at least once is greater than \(\frac{5}{6}\), is

If \(f\left( {\frac{{3x - 4}}{{3x + 4}}} \right) = x + 2,\,x \ne -\frac{4}{3}\) , and \(\int {f\left( x \right)dx = A\,\log \left| {1 - x} \right| + Bx + C} \) , then the ordered pair \((A,B) \) is equal to : (where \(C\) is a constant of integration)

Let the mirror image of the point \((a, b\), c) with respect to the plane \(3 x-4 y+12 z+19=0\) be (a- \(6, \beta, \gamma)\). If \(a+b+c=5\), then \(7 \beta-9 \gamma\) is equal to

Let \(y=y(x)\) be the solution of the differential equation \( \frac{d y}{d x}=\frac{(\tan x)+y}{\sin x(\sec x-\sin x \tan x)}\), \(x \in\left(0, \frac{\pi}{2}\right)\) satisfying the condition \(y\left(\frac{\pi}{4}\right)=2\). Then, \(y\left(\frac{\pi}{3}\right)\) is

Let the foci of a hyperbola be \((1,14)\) and \((1,-12)\). If it passes through the point \((1,6)\), then the length of its latus-rectum is :

Consider a matrix \(A =\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta\end{array}\right]\), where \(\alpha, \beta, \gamma\) are three distinct natural numbers. If \(\frac{\operatorname{det}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))))}{(\alpha-\beta)^{16}(\beta-\gamma)^{16}(\gamma-\alpha)^{16}}=2^{32} \times 3^{16}\), then the number of such \(3 -\) tuples \((\alpha, \beta, \gamma)\) is \(.....\)

If the constant term in the expansion of \(\left(\frac{\sqrt[5]{3}}{x}+\frac{2 x}{\sqrt[3]{5}}\right)^{12}, x \neq 0\), is \(\alpha \times 2^8 \times \sqrt[5]{3}\), then \(25 \alpha\) is equal to :

The acute angle between the pair of tangents drawn to the ellipse \(2 x^{2}+3 y^{2}=5\) from the point \((1,3)\) is.

An angle between the lines whose direction cosines are given by the equations, \(l+ 3m + 5n\, = 0\) and \(5lm -2mn + 6nl = 0\) , is

If \(S_1\) and \(S_2\) are respectively the sets of local minimum and local maximum points of the function. \(f(x) = 9{x^4} + 12{x^3} - 36{x^2} + 25,x \in R\), then

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 mean of \(6\) distinct observations is \(6.5\) and their variance is \(10.25\). If \(4\) out of \(6\) observations are \(2,4,5\) and \(7 ,\) then the remaining two observations are:

The maximum value of the function \(f\,(x)\, = 3{x^3} - 18{x^2} + 27x\,\, - 40\) on the set  \(S = \{ x\, \in \,R\,:\,{x^2}\, + \,30\, \le \,11x\} \) is