A box contains \(15\) green and \(10\) yellow balls. lf \(10\)  balls are randomly drawn, one-by-one, with replacement, then the variance of the number of green balls drawn is :

The set of all \(\alpha  \in R\), for which \(w = \frac{{1 + \left( {1 - 8\alpha } \right)z}}{{1 - z}}\) is a purely imaginary number, for all \(z \in C\) satisfying \(\left| z \right| = 1\) and \({\mathop{\rm Re}\nolimits} \,z \ne 1\),  is

Consider the function \(f(x)=\frac{\mathrm{P}(\mathrm{x})}{\sin (\mathrm{x}-2)}, \quad \mathrm{x} \neq 2\) \(\quad \quad \quad \quad 7, \quad\quad\quad \mathrm{x}=2\) where \(P(x)\) is a polynomial such that \(P^{\prime \prime}(x)\) is always a constant and \(P(3)=9\). If \(f(x)\) is continuous at \(x=2\), then \(P(5)\) is equal to \(.....\)

The number of solutions, of the equation \(\mathrm{e}^{\sin x}-2 e^{-\sin x}=2\) is

The distance of the point \((6,-2 \sqrt{2})\) from the common tangent \(y = mx + c , m > 0\), of the curves \(x =2 y ^2\) and \(x =1+ y ^2\) is

The relation \(R =\{( a , b ): \operatorname{gcd}( a , b )=1,2 a \neq b , a , b \in Z \}\) is:

Let \(A=\{(\alpha, \beta) \in \mathbf{R} \times \mathbf{R}:|\alpha-1| \leq 4 \text { and }|\beta-5| \leq 6\}\) and \(B=\{(\alpha, \beta) \in \mathbf{R} \times\) \(\mathbf{R}: 16(\alpha-2)^2+9(\beta-6)^2 \leq 144\}\)

Let \(f\) be a composite function of \(x\) defined by \(f\left( u \right) = \frac{1}{{{u^2} + u - 2}}\,,\,u\left( x \right) = \frac{1}{{x - 1}}\) . Then the number of points \(x\) where \(f\) is discontinuous is

Let \(\mathrm{f}(\mathrm{x})=\cos \left(2 \tan ^{-1} \sin \left(\cot ^{-1} \sqrt{\frac{1-\mathrm{x}}{\mathrm{x}}}\right)\right)\) \(0<\mathrm{x}<1\). Then :

Let \(\quad \sum \limits_{n=0}^{\infty} \frac{n^3((2 n) !)+(2 n-1)(n !)}{(n !)((2 n) !)}=a e+\frac{b}{e}+c\), where \(a, b, c \in Z\) and \(e=\sum \limits_{n=0}^{\infty} \frac{1}{n!}\) Then \(a^2-b+c\) is equal to \(................\).

Let \(L_1: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}), \lambda \in \mathrm{R}\) \(L_2: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(3 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{p} \hat{\mathrm{k}}), \mu \in \mathrm{R}\) and  \(L_3: \overrightarrow{\mathrm{r}}=\delta(\ell \hat{\mathrm{i}}+\mathrm{m} \hat{\mathrm{j}}+\mathrm{n} \hat{\mathrm{k}}) \delta \in \mathrm{R}\). Be three lines such that \(\mathrm{L}_1\) is perpendicular to \(\mathrm{L}_2\) and \(L_3\) is perpendicular to both \(L_1\) and \(L_2\). Then the point which lies on \(\mathrm{L}_3\) is

Let \(\alpha_1, \alpha_2, \ldots, \alpha_7\) be the roots of the equation \(x^7+\) \(3 x^5-13 x^3-15 x=0\) and \(\left|\alpha_1\right| \geq\left|\alpha_2\right| \geq \ldots \geq\left|\alpha_7\right|\). Then \(\alpha_1 \alpha_2-\alpha_3 \alpha_4+\alpha_5 \alpha_6\) is equal to \(..................\).

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is \(10\) and one of the foci is at \((0, 5\sqrt 3 )\), then the length of its latus rectum is