If \(X\) has a binomial distribution, \(B( n, p)\) with parameters \(n\) and \(p\) such that \(P(X\, = 2)\, = P (X\, = 3)\), then \(E(X)\), the mean of variable \(X\), is

A line is a common tangent to the circle \((x-3)^{2}+y^{2}=9\) and the parabola \(y^{2}=4 x.\) If the two points of contact \(( a , b )\) and \(( c , d )\) are distinct and lie in the first quadrant, then \(2(a+c)\) is equal to ........ .

The locus of the mid points of the chords of the circle \(C_1:(x-4)^2+(y-5)^2=4\) which subtend an angle \(\theta_i\) at the centre of the circle \(C_1\), is a circle of radius \(r_i\). If \(\theta_1=\frac{\pi}{3}, \theta_3=\frac{2 \pi}{3}\) and \(r_1^2=r_2^2+r_3^2\), then \(\theta_2\) is equal to

Let the acute angle bisector of the two planes \(x-2 y-2 z+1=0\) and \(2 x-3 y-6 z+1=0\) be the plane \(\mathrm{P}\). Then which of the following points lies on \(\mathrm{P} ?\)

Let \(A (-3, 2)\) and \(B (-2, 1)\) be the vertices of a triangle \(ABC\). If the centroid of this triangle lies on the line \(3x + 4y + 2 = 0\), then the vertex \(C\) lies on the line

The total number of two digit numbers \('n',\) such that \(3^{n}+7^{n}\) is a multiple of \(10 ,\) is ..... .

The area (in sq. units) of the region bounded by the parabola, \(y = x^2 + 2\) and the lines, \(y = x + 1, x = 0\) and \(x = 3\), 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:

For the function \(f:[1,\infty) \rightarrow [1,\infty)\) defined by \(f(x)=(x-1)^4+1\), among the two statements:
(I) The set \(S=\{x \in [1,\infty): f(x)=f^{-1}(x)\}\) contains exactly two elements, and
(II) The set \(S=\{x \in [1,\infty): f(x)=f^{-1}(x+1)\}\) is an empty set,

The sum of all the local minimum values of the twice differentiable function \(\mathrm{F}: \mathrm{R} \rightarrow \mathrm{R}\) defined by \(f(x)=x^{3}-3 x^{2}-\frac{3 f^{\prime \prime}(2)}{2} x+f^{\prime \prime}(1)\) is:

Let \(S\) be the set of all real values of \(k\) for which the system oflinear equations \(x +y + z = 2\) ; \(2x +y - z = 3\) ; \(3x + 2y + kz = 4\) has a unique solution. Then \(S\) is

Let \(\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k} \quad a_{i}>0, i=1,2,3\) be \(a\) vector which makes equal angles with the coordinates axes OX, OY and OZ. Also, let the projection of \(\vec{a}\) on the vector \(3 \hat{i}+4 \hat{j}\) be \(7\) . Let \(\vec{b}\) be a vector obtained by rotating \(\vec{a}\) with \(90^{\circ}\). If \(\vec{a}, \vec{b}\) and \(x\)-axis are coplanar, then projection of a vector \(\vec{b}\) on \(3 \hat{i}+4 \hat{j}\) is equal to

Consider two vectors \(\overrightarrow{\mathrm{u}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}\) and \(\overrightarrow{\mathrm{v}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\lambda \hat{\mathrm{k}}, \lambda \gt 0\). The angle between them is given by \(\cos ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)\). Let \(\overrightarrow{\mathrm{v}}=\overrightarrow{\mathrm{v}}_1+\overrightarrow{\mathrm{v}}_2\), where \(\overrightarrow{\mathrm{v}}_1\) is parallel to \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}_2\) is perpendicular to \(\overrightarrow{\mathrm{u}}\). Then the value \(\left|\overrightarrow{\mathrm{v}}_1\right|^2+\left|\overrightarrow{\mathrm{v}}_2\right|^2\) is equal to