A value of \(\alpha \) such that \(\int\limits_\alpha ^{\alpha  + 1} {\frac{{dx}}{{\left( {x + \alpha } \right)\left( {x + \alpha  + 1} \right)}} = {{\log }_e}\left( {\frac{9}{8}} \right)} \) is

Let \(\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{d}}=\vec{a} \times \overrightarrow{\mathrm{b}}\). If \(\overrightarrow{\mathrm{c}}\) is a vector such that \(\vec{a} \cdot \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{c}}|\), \(|\overrightarrow{\mathrm{c}}-2 \vec{a}|^2=8\) and the angle between \(\overrightarrow{\mathrm{d}}\) and \(\overrightarrow{\mathrm{c}}\) is \(\frac{\pi}{4}\), then \(|10-3 \overrightarrow{\mathrm{~b}} \cdot \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{d}} \times \overrightarrow{\mathrm{c}}|^2\) is equal to

Let \(A,B\) be points on the two half-lines \(x-\sqrt{3}|y|=\alpha\), \(\alpha>0\) at a distance of \(\alpha\) from their point of intersection \(P\). The line segment \(AB\) meets the angle bisector of the given half-lines at the point \(Q\). If \(PQ=\dfrac{9}{2}\) and \(R\) is the radius of the circumcircle of \(\triangle PAB\), then \(\dfrac{\alpha^2}{R}\) is equal to ______

If for some \(x \in  R\), the frequency distribution of the marks obtained by \(20\) students in a test is
Marks                \(2\)                \(3\)                \(5\)                  \(7\)
Frequency     \((x+1)^2\)      \(2x -5\)        \(x^2 -3x\)       \(x\)
Then the mean of the marks is

Area (in sq. units) of the region outside \(\frac{|\mathrm{x}|}{2}+\frac{|\mathrm{y}|}{3}=1\) and inside the ellipse \(\frac{\mathrm{x}^{2}}{4}+\frac{\mathrm{y}^{2}}{9}=1\) is

Let \(\mathrm{Q}\) and \(\mathrm{R}\) be the feet of perpendiculars from the point \(\mathrm{P}(\mathrm{a}, \mathrm{a}, \mathrm{a})\) on the lines \(\mathrm{x}=\mathrm{y}, \mathrm{z}=1\) and \(\mathrm{x}=-\mathrm{y}\), \(\mathrm{z}=-1\) respectively. If \(\angle \mathrm{QPR}\) is a right angle, then \(12 \mathrm{a}^2\) is equal to

\(\lim _{x \rightarrow 0}\left(\frac{(x+2 \cos x)^{3}+2(x+2 \cos x)^{2}+3 \sin (x+2 \cos x)}{(x+2)^{3}+2(x+2)^{2}+3 \sin (x+2)}\right)^{\frac{100}{x}}\)is equal to\(.....\)

Let \(f\) and \(g\) be twice differentiable functions on \(R\) such that \(f^{\prime \prime}(x)=g^{\prime \prime}(x)+6 x\) \(f^{\prime}(1)=4 g^{\prime}(1)-3=9\) \(f(2)=3 g(2)=12\) Then which of the following is NOT true ?

In an increasing geometric progression ol positive terms, the sum of the second and sixth terms is \(\frac{70}{3}\) and the product of the third and fifth terms is \(49\). Then the sum of the \(4^{\text {th }}, 6^{\text {th }}\) and \(8^{\text {th }}\) terms is :-

The number of solutions of the equation \(\cos 2 \theta \cos \frac{\theta}{2}+\cos \frac{5 \theta}{2}=2 \cos ^3 \frac{5 \theta}{2}\) in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) is :

Let \(I(x)=\int \frac{6}{\sin ^2 x(1-\cot x)^2} d x\). If \(I(0)=3\), then \(\mathrm{I}\left(\frac{\pi}{12}\right)\) is equal to :

The number of solutions of \(sin \,3x\, = cos\, 2x\) , in the interval \(\left( {\frac{\pi }{2},\pi } \right)\) is

For the function \(f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right) \text {, where } x \in\left[0, \frac{\pi}{2}\right] \text {, }\) consider the following two statements : (\(I\)) \(\mathrm{f}\) is increasing in \(\left(0, \frac{\pi}{2}\right)\). (\(II\)) \(\mathrm{f}^{\prime}\) is decreasing in \(\left(0, \frac{\pi}{2}\right)\). Between the above two statements,