Let \(f: \left(-\frac{\pi}{4}, \frac{\pi}{4}\right) \rightarrow \mathrm{R}\) be defined as \(f(x)=(1+|\sin x|)^{\frac{3 a}{\sin x \mid}} ,\quad -\frac{\pi}{4}\,<\,x\,<\,0\) \(\quad\quad\quad\quad\quad\quad b ,\quad\quad\quad\quad\quad x=0\) \(\quad\quad\quad\quad e^{\cot 4 x / \cot 2 x} ,\quad\quad\quad 0\,<\,x\,<\,\frac{\pi}{4}\) If \(\mathrm{f}\) is continuous at \(\mathrm{x}=0\), then the value of \(6 \mathrm{a}+\mathrm{b}^{2}\) is equal to:

If \(\alpha ,\beta \ne 0\) and \(f\left( n \right) = {\alpha ^n} + {\beta ^n}\) and \(\left| {\begin{array}{*{20}{c}}3&{1 + f\left( 1 \right)}&{1 + f\left( 2 \right)}\\{1 + f\left( 1 \right)}&{1 + f\left( 2 \right)}&{1 + f\left( 3 \right)}\\{1 + f\left( 2 \right)}&{1 + f\left( 3 \right)}&{1 + f\left( 4 \right)}\end{array}} \right|\; = K{\left( {1 - \alpha } \right)^2}\) \({\left( {1 - \beta } \right)^2}{\left( {\alpha - \beta } \right)^2}\) ,then \(K=\) . . . . . .

Let \(\alpha, \beta ; \alpha>\beta\), be the roots of the equation \(x^2-\sqrt{2} x-\sqrt{3}=0\). Let \(P_n=\alpha^n-\beta^n, n \in N\). Then \((11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}\) is equal to :

The value of \(\int_{-\pi / 2}^{\pi / 2} \frac{\cos ^{2} x}{1+3^{x}} d x\) is

If \(\lim _{x \rightarrow 0} \frac{3+\alpha \sin x+\beta \cos x+\log _e(1-x)}{3 \tan ^2 x}=\frac{1}{3}\), then \(2 \alpha-\beta\) is equal to :

Let \(n\) be a positive integer. Let  \(A =\sum_{ k =0}^{ n }(-1)^{ k } n _{ C _{ k }}\left[\left(\frac{1}{2}\right)^{ k }+\left(\frac{3}{4}\right)^{ k }+\left(\frac{7}{8}\right)^{ k }+\left(\frac{15}{16}\right)^{ k }+\left(\frac{31}{32}\right)^{ k }\right]\) . If \(63 A =1-\frac{1}{2^{30}},\) then \(n\) is equal to ...... .

The imaginary part of \((3+2 \sqrt{-54})^{1 / 2}-(3-2 \sqrt{-54})^{1 / 2}\) can be

The first of the two samples in a group has \(100\) items with mean \(15\) and standard deviation \(3 .\) If the whole group has \(250\) items with mean \(15.6\) and standard deviation \(\sqrt{13.44}\), then the standard deviation of the second sample is:

Statement \(-1\) : The system of linear equations \(x + \left( {\sin \,\alpha } \right)y + \left( {\cos \,\alpha } \right)z = 0\) \(x + \left( {\cos \,\alpha } \right)y + \left( {\sin \alpha } \right)z = 0\) \(x - \left( {\sin \,\alpha } \right)y - \left( {\cos \alpha } \right)z = 0\) has a non-trivial solution for only one value of \(\alpha \) lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)  Statement \(-2\) : The equation in \(\alpha \) \(\left| {\begin{array}{*{20}{c}}
  {\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha } \\ 
  {\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha } \\ 
  {\cos {\mkern 1mu} \alpha }&{ - \sin {\mkern 1mu} \alpha }&{ - \cos {\mkern 1mu} \alpha } 
\end{array}} \right| = 0\) has only one solution lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero non-coplanar vectors. Let the position vectors of four points \(A, B, \quad C\) and \(D\) be \(\vec{a}-\vec{b}+\vec{c}, \lambda \vec{a}-3 \vec{b}+4 \vec{c}\), \(-\vec{a}+2 \vec{b}-3 \vec{c}\) and \(2 \vec{a}-4 \vec{b}+6 \vec{c}\) respectively. If \(\overrightarrow{A B}\), \(\overline{ AC }\) and \(\overline{ AD }\) are coplanar, then \(\lambda\) is :

The locus of the mid points of the chords of the circle \(C_1:(x-4)^2+(y-5)^2=4\) which subtend an angle \(\theta_i\) at the centre of the circle \(C_1\), is a circle of radius \(r_i\). If \(\theta_1=\frac{\pi}{3}, \theta_3=\frac{2 \pi}{3}\) and \(r_1^2=r_2^2+r_3^2\), then \(\theta_2\) is equal to

Let \(\omega=z \bar{z}+k_1 z+k_2 i z+\lambda(1+i), k_1, k_2 \in R\). Let \(\operatorname{Re}(\omega)=0\) be the circle \(C\) of radius 1 in the first quadrant touching the line \(y=1\) and the \(y\)-axis. If the curve \(\operatorname{Im}(\omega)=0\) intersects \(C\) at \(A\) and \(B\), then \(30(A B)^2\) is equal to \(.......\).

If two vertices of an equilateral triangle are \(A (- a, 0)\) and \(B ( a, 0),\,a > 0,\) and the third vertex \(C\) lies above \(x-\) axis then the equation of the circumcircle of \(\Delta ABC\) is