Let \(f(x)\) be a real differentiable function such that \(f(0)=1\) and \(f(x+y)=f(x) f^{\prime}(y)+f^{\prime}(x) f(y)\) for all \(x, y \in \mathbf{R}\). Then \(\sum_{\mathrm{n}=1}^{100} \log _{\mathrm{e}} f(\mathrm{n})\) is equal to :

Line \(L_1\) of slope 2 and line \(L_2\) of slope \(\frac{1}{2}\) intersect at the origin O . In the first quadrant, \(\mathrm{P}_1, \mathrm{P}_2, \ldots . \mathrm{P}_{12}\) are 12 points on line \(L_1\) and \(Q_1, Q_2, \ldots . . Q_9\) are 9 points on line \(L_2\). Then the total number of triangles, that can be formed having vertices at three of the 22 points \(\mathrm{O}, \mathrm{P}_1, \mathrm{P}_2, \ldots \mathrm{P}_{12}\), \(\mathrm{Q}_1, \mathrm{Q}_2, \ldots . \mathrm{Q}_9\), is:

From a lot of \(10\) items, which include \(3\) defective items, a sample of \(5\) items is drawn at random. Let the random variable \(\mathrm{X}\) denote the number of defective items in the sample. If the variance of \(X\) is \(\sigma^2\), then \(96 \sigma^2\) is equal to ....................

Consider the parabola \(P : y^2 = 4kx\) and the ellipse \(E : \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1\). Let the line segment joining the points of intersection of \(P\) and \(E\), be their latus rectums. If the eccentricity of \(E\) is \(e\), then \(e^2 + 2\sqrt{2}\) is equal to _____.

Let \(\mathrm{f}: R \rightarrow R\) and \(\mathrm{g}: R \rightarrow R\) be defined as \(f(x)=\left\{\begin{array}{lll}\log _e x & , & x>0 \\ e^{-x} & , & x \leq 0\end{array}\right.\) and \(g(x)=\left\{\begin{array}{lll} x & , & x \geq 0 \\ e^{x} & , & x <  0\end{array}\right.\) Then \(gof:R \to R\) is ...........

If two lines \(L_1\) and \(L_2\) in space, are defined by \({L_1} = \{ x = \sqrt \lambda  y + \left( {\sqrt \lambda   - 1} \right),z = \left( {\sqrt \lambda   - 1} \right)y + \sqrt \lambda  \} \) and \({L_2} = \{ x = \sqrt \mu  y + \left( {1 - \sqrt \mu  } \right),z = \left( {1 - \sqrt \mu  } \right)y + \sqrt \mu  \} \) then \(L_1\) is perpendicular to \(L_2\), for all non-negative reals \(\lambda \) and \( \mu \), such that

Let \(\vec{a}=\hat{i}+\hat{j}-\hat{k}\) and \(\vec{c}=2 \hat{i}-3 \hat{j}+2 \hat{k}\). Then the number of vectors \(\vec{b}\) such that \(\vec{b} \times \vec{c}=\vec{a}\) and \(|\vec{b}| \in\{1,2, \ldots ., 10\}\) is

Let \(\int_\alpha^{\log _e^4} \frac{\mathrm{dx}}{\sqrt{\mathrm{e}^{\mathrm{x}}-1}}=\frac{\pi}{6}\). Then \(\mathrm{e}^\alpha\) and \(\mathrm{e}^{-\alpha}\) are the roots of the equation :

Consider the following two statements : Statement \(I\) : For any two non-zero complex numbers \(\mathrm{z}_1, \mathrm{z}_2\) \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)\) and Statement \(II\) : If \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are three distinct complex numbers and a, b, c are three positive real numbers such that \(\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}\), then \(\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1\) Between the above two statements,

Considering only the principal values of the inverse trigonometric functions, the domain of the function \(f(x)=\cos ^{-1}\left(\frac{x^{2}-4 x+2}{x^{2}+3}\right)\) is.

The line \(y=x+1\) meets the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{2}=1\) at two points \(P\) and \(Q\). If \(r\) is the radius of the circle with \(PQ\) as diameter then \((3 r )^{2}\) is equal to

Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}, \quad \vec{b}=-\hat{i}-8 \hat{j}+2 \hat{k} \quad\) and \(\overrightarrow{\mathrm{c}}=4 \hat{\mathrm{i}}+\mathrm{c}_2 \hat{\mathrm{j}}+\mathrm{c}_3 \hat{\mathrm{k}}\) be three vectors such that \(\vec{b} \times \vec{a}=\vec{c} \times \vec{a}\). If the angle between the vector \(\vec{c}\) and the vector \(3 \hat{i}+4 \hat{j}+\hat{k}\) is \(\theta\), then the greatest integer less than or equal to \(\tan ^2 \theta\) is :

Let a relation \(R\) on \(\mathbb{N} \times \mathbb{N}\) be defined as : \(\left(\mathrm{x}_1, \mathrm{y}_1\right) \mathrm{R}\left(\mathrm{x}_2, \mathrm{y}_2\right)\) if and only if \(\mathrm{x}_1 \leq \mathrm{x}_2\) or \(\mathrm{y}_1 \leq \mathrm{y}_2\) Consider the two statements : (\(I\)) \(\mathrm{R}\) is reflexive but not symmetric. (\(II\)) \(\mathrm{R}\) is transitive Then which one of the following is true?