The integral \(\int_0^\pi \frac{8 x d x}{4 \cos ^2 x+\sin ^2 x}\) is equal to

Let \(f:[1, \infty) \rightarrow[2, \infty)\) be a differentiable function, If \(10 \int_1^{\mathrm{x}} f(\mathrm{t}) \mathrm{dt}=5 \mathrm{x} f(\mathrm{x})-\mathrm{x}^5-9\) for all \(\mathrm{x} \geq 1\), then the value of \(f(3)\) is :

Let \(f\left( x \right) = x\left| x \right|\,,\,g\left( x \right) = \sin \,x\) and \(h\left( x \right) = \left( {gof} \right)\left( x \right)\). Then

The intercepts on \(x-\) axis made by tangents to the curve \(y = \mathop \smallint \limits_0^x \left| t \right|dt,x \in R\) which is parallel to the line \(y = 2x\) are equal to ::

Let \(\quad \overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{b}}=-\hat{\mathrm{i}}+\hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{c}}=\beta \hat{\mathrm{j}}-\hat{\mathrm{k}}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\) for which the area of the parallelogram of diagonals \(\vec{a}+\vec{b}\) and \(\vec{b}+\vec{c}\) is \(\frac{\sqrt{21}}{2}\), be \(\left(\alpha_1, \beta_1\right)\) and \(\left(\alpha_2, \beta_2\right)\). Then \(\alpha_1^2+\beta_1^2-\alpha_2 \beta_2\) is equal to

The angle between the straight lines, whose direction cosines are given by the equations \(2 l+2 \mathrm{~m}-\mathrm{n}=0\) and \(\mathrm{mn}+\mathrm{n} l+l \mathrm{~m}=0\), is :

Let \(A=\{1,2,3, \ldots ,100\}\). Let \(R\) be a relation on A defined by \((x, y) \in R\) if and only if \(2 x=3 y\). Let \(R_1\) be a symmetric relation on \(A\) such that \(\mathrm{R} \subset \mathrm{R}_1\) and the number of elements in \(\mathrm{R}_1\) is \(\mathrm{n}\). Then, the minimum value of \(n\) is ...........

Let \(p, q\) and \(r\)  be real numbers \((p \ne q,r \ne 0),\) such that the roots of the equation \(\frac{1}{{x + p}} + \frac{1}{{x + q}} = \frac{1}{r}\) are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to .

The number of times the digit \(3\) will be written when listing the integers from \(1\) to \(1000\) is

Let \(S=\left\{n \in N \mid\left(\begin{array}{ll}0 & i \\ 1 & 0\end{array}\right)^{n}\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \forall a, b, c, d \in R\right\}\), where \(i=\sqrt{-1} .\) Then the number of \(2 -\) digit numbers in the set \(\mathrm{S}\) is \(......\)

\(\mathop {{\rm{lim}}}\limits_{x \to 0} \frac{{{\rm{sin}}\left( {\pi {{\cos }^2}x} \right)}}{{{x^2}}} = \)

If \(x\,{\log _e}({\log _e}\,\,x)\, - \,{x^2} + {y^2} = 4\,(y\, > \,0),\) then \(\frac{{dy}}{{dx}}\) at \(x = e\) is equal to

Equation of the line of the shortest distance between the lines \(\frac{x}{1} = \frac{y}{{ - 1}} = \frac{z}{1}\) and \(\frac{{x - 1}}{0} = \frac{{y + 1}}{{ - 2}} = \frac{z}{1}\) is