If \(\displaystyle\int_{\pi/6}^{\pi/4}\left(\cot\left(x-\dfrac{\pi}{3}\right)\cot\left(x+\dfrac{\pi}{3}\right)+1\right)dx = \alpha\log_e(\sqrt{3}-1)\), then \(9\alpha^2\) is equal to ________.

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 ::

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,

If \(1+\left(2+{ }^{49} C _{1}+{ }^{49} C _{2}+\ldots .+{ }^{49} C _{49}\right)\left({ }^{50} C _{2}+{ }^{50} C _{4}+\right.\) \(\ldots . .+{ }^{50} C _{ so }\) ) is equal to \(2^{ n } . m\), where \(m\) is odd, then \(n\) \(+m\) is equal to.

The minimum distance of a point on the curve \(y = x^2 - 4\) from the origin is

Let \(A =\left(\begin{array}{cc}2 & -1 \\ 0 & 2\end{array}\right)\). If \(B = I -{ }^{5} C _{1} (\operatorname{adj} A )+{ }^{5} C _{2}\) \((\operatorname{adjA})^{2}-\ldots-{ }^{5} C _{5} (\operatorname{adj} A )^{5}\), then the sum of all elements of the matrix \(B\) is

Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is :

Let the mean and variance of \(8\) numbers \(x , y , 10\), \(12,6,12,4,8\), be \(9\) and \(9.25\) respectively. If \(x > y\), then \(3 x-2 y\) is equal to \(...........\).

Let \(S =\{ z \in C :| z -2| \leq 1, z (1+ i )+\overline{ z }(1-\) i) \(\leq 2\}\). Let \(|z-4 i|\) attains minimum and maximum values, respectively, at \(z _{1} \in S\) and \(z _{2} \in S\). If \(5\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\right)=\alpha+\beta \sqrt{5}\), where \(\alpha\) and \(\beta\) are integers, then the value of \(\alpha+\beta\) is equal to

If \(\operatorname{gcd}( m , n )=1\) and \(1^2-2^2+3^2-4^2+\ldots \ldots\) \(+(2021)^2-(2022)^2+(2023)^2=1012 m ^2 n\), then \(m ^2- n ^2\) is equal to

If \(\alpha\) and \(\beta\) are the roots of the equation \(2 x (2 x +1)=1,\) then \(\beta\) is equal to

Let \(\frac{d y}{d x}=\frac{a x-b y+a}{b x+c y+a}\), where \(a , b , c\) are constants, represent a circle passing through the point \((2,5)\). Then the shortest distance of the point \((11,6)\) from this circle is

If \(f(x)=\int \frac{5 x^{8}+7 x^{6}}{\left(x^{2}+1+2 x^{7}\right)^{2}} d x,(x \geq 0), f(0)=0\) and \(f(1)=\frac{1}{K},\) then the value of \(K\) is