If a random variable x has the probability distribution

x	0	1	2	3	4	5	6	7
p(x)	0	2k	k	3k	\(2 k ^2\)	2k	\(k ^2+ k\)	\(7 k ^2\)

then \(P (3< x \leq 6)\) is equal to

Let \(\alpha x=\exp \left(x^\beta y^\gamma\right)\) be the solution of the differential equation \(2 x^2 y d y-\left(1-x y^2\right) d x=0\), \(x >0, y (2)=\sqrt{\log _e 2}\). Then \(\alpha+\beta-\gamma\) equals :

A circle touching the \(x-\) axis at \((3, 0)\) and making an intercept of length \(8\) on the \(y-\) axis passes through the point 

A value of \(x\) satisfying the equation \(\sin \left[ {{{\cot }^{ - 1}}\left( {1 + x} \right)} \right] = \cos \left[ {{{\tan }^{ - 1}}\,x} \right]\) , is

Let \(f\) be a twice differentiable function such that \(f(x)=\int_{0}^{x}\tan(t-x)dt-\int_{0}^{x}f(t)\tan t\,dt\), \(x \in \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)\). Then \(f''\left(\dfrac{\pi}{6}\right)+12f'\left(-\dfrac{\pi}{6}\right)+f\left(\dfrac{\pi}{6}\right)\) is equal to ______

A board has 16 squares as shown in the figure:

Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is :

If \(0\, \le \,x\, < \frac{\pi }{2},\) then the number of values of \(x\) for which \(sin\,x -sin\,2x + sin\,3x=0,\) is

Let the mean and variance of 7 observations 2, 4, 10, x, 12, 14, y, \( x>y \) be 8 and 16 respectively. Two numbers are chosen from {1, 2, 3, x-4, y, 5} one after another without replacement, then the probability, that the smaller number among the two chosen numbers is less than 4, is:

Let \(P\) be a point on the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\). Let the line passing through \(P\) and parallel to \(y\)-axis meet the circle \(x^2+y^2=9\) at point \(Q\) such that \(P\) and \(Q\) are on the same side of the \(x\)-axis. Then, the eccentricity of the locus of the point \(R\) on \(P Q\) such that \(P R: R Q=4: 3\) as \(P\) moves on the ellipse, is :

Let \(AD\) and \(BC\) be two vertical poles at \(A\) and \(B\) respectively on a horizontal ground. If \(AD =8 m , BC =11 m\) and \(AB =10 m ;\) then the distance (in meters) of a point \(M\) on \(AB\) from the point \(A\) such that \(M D^{2}+M C^{2}\) is minimum is

If \(\mathop \sum \limits_{i = 1}^9 \left( {{x_i} - 5} \right) = 9\) and \(\mathop \sum \limits_{i = 1}^9 {\left( {{x_i} - 5} \right)^2} = 45,\) then the standard deviation of the \(9\) items  \({x_1},{x_2},\;.\;.\;.\;,{x_9}\) is :

If both the means and the standard deviation of \(50\) observations \(x_1, x_2, ………, x_{50}\) are equal to \(16\) , then the mean of \((x_1 - 4)^2, (x_2 - 4)^2, …., (x_{50} - 4)^2\) is

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 :