Let \(g(x)=f(x)+f(1-x)\) and \(f^{\prime \prime}(x) > 0, x \in(0,1)\). If \(g\) is decreasing in the interval \((0, \alpha)\) and increasing in the interval \((\alpha, 1)\), then \(\tan ^1(2 \alpha)+\tan ^{-1}\left(\frac{1}{\alpha}\right)+\tan ^{-1}\left(\frac{\alpha+1}{\alpha}\right)\) is equal to :

Let \(f(x) = \int \left(\dfrac{16x + 24}{x^2 + 2x - 15}\right) dx\). If \(f(4) = 14 \log_e(3)\) and \(f(7) = \log_e(2^{\alpha} \cdot 3^{\beta})\), \(\alpha, \beta \in \mathbb{N}\), then \(\alpha + \beta\) is equal to:

Let \(A = \begin{bmatrix} -1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix}\) satisfy \(A^2 + \alpha(adj(adj(A))) + \beta(adj(A)(adj(adj(A)))) = \begin{bmatrix} 2 & -2 & 2 \\ -2 & 0 & -1 \\ 0 & 0 & -1 \end{bmatrix}\) for some \(\alpha, \beta \in \mathbb{R}\). Then \((\alpha - \beta)^2\) is equal to _______

A point \(\mathrm{z}\) moves in the complex plane such that \(\arg \left(\frac{\mathrm{z}-2}{\mathrm{z}+2}\right)=\frac{\pi}{4}\), then the minimum value of \(|z-9 \sqrt{2}-2 i|^{2}\) is equal to ..... .

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous function satisfying \(f(0)=1\) and \(f(2 \mathrm{x})-f(\mathrm{x})=\mathrm{x}\) for all \(\mathrm{x} \in \mathbb{R}\). If \(\lim _{n \rightarrow \infty}\left\{f(x)-f\left(\frac{x}{2^n}\right)\right\}=G(x)\), then \(\sum_{r=1}^{10} G\left(r^2\right)\) is equal to

The integral \(\int_{0}^{\frac{\pi}{2}} \frac{1}{3+2 \sin x+\cos x} d x\) is equal to.

If \(19^{th}\) terms of non -zero \(A.P.\) is zero, then its (\(49^{th}\) term) : (\(29^{th}\) term) is

Let \(a_1=8, a_2, a_3, \ldots a_n\) be an \(A.P.\) If the sum of its first four terms is \(50\) and the sum of its last four terms is \(170\) , then the product of its middle two terms is

Three urns \(A\), \(B\) and \(C\) contain \(7\) red, \(5\) black; \(5\) red, \(7\) black and \(6\) red, \(6\) black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn \(\mathrm{A}\) is :

Let the area of the triangle formed by the lines \(x+2=y-1=z, \frac{x-3}{5}=\frac{y}{-1}=\frac{z-1}{1}\) and \(\frac{x}{-3}=\frac{y-3}{3}=\frac{z-2}{1}\) be \(A\). Then \(A^2\) is equal to ________

Let \(A\) be the area bounded by the curve \(y=x|x-3|\), the \(x\)-axis and the ordinates \(x=-1\) and \(x=2\). Then \(12\,A\) 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

Let the first term \(a\) and the common ratio \(r\) of a geometric progression be positive integers. If the sum of its squares of first three terms is \(33033\), then the sum of these three terms is equal to