Let \(y=y(x)\) be the solution curve of the differential equation \(\sin \left(2 x^{2}\right) \log _{c}\left(\tan x^{2}\right) d y+\left(4 x y-4 \sqrt{2} x \sin \left(x^{2}-\frac{\pi}{4}\right)\right) d x=0\) \(0 < x < \sqrt{\frac{\pi}{2}}\), which passes through the point \(\left(\sqrt{\frac{\pi}{6}}, 1\right)\). Then \(\left|y\left(\sqrt{\frac{\pi}{3}}\right)\right|\) is equal to \(.....\)

If the distance between the plane, \(23 \mathrm{x}-10 \mathrm{y}-2 \mathrm{z}+48=0\) and the plane containing the lines \(\frac{x+1}{2}=\frac{y-3}{4}=\frac{z+1}{3}\) and \(\frac{x+3}{2}=\frac{y+2}{6}=\frac{z-1}{\lambda}(\lambda \in R)\) is equal to \(\frac{\mathrm{k}}{\sqrt{633}},\) then \(\mathrm{k}\) is equal to

The area (in sq. units) of the part of circle \(\mathrm{x}^2+\mathrm{y}^2=169\) which is below the line \(5 \mathrm{x}-\mathrm{y}=13\) is \(\frac{\pi \alpha}{2 \beta}-\frac{65}{2}+\frac{\alpha}{\beta} \sin ^{-1}\left(\frac{12}{13}\right)\) where \(\alpha, \beta\) are coprime numbers. Then \(\alpha+\beta\) is equal to ...........

Let the image of the point \(P (1,2,3)\) in the line \(L : \frac{ x -6}{3}=\frac{ y -1}{2}=\frac{ z -2}{3}\) be \(Q .\) let \(R (\alpha, \beta, \gamma)\) be a point that divides internally the line segment \(PQ\) in the ratio \(1: 3\). Then the value of \(22(\alpha+\beta+\gamma)\) is equal to

Suppose that the mean and median of the non-negative numbers \(21, 8, 17, a, 51, 103, b, 13, 67, (a > b)\), are \(40\) and \(21\), respectively. If the mean deviation about the median is \(26\), then \(2a\) is equal to:

Let \(f : (-1, 1) \to R\) be a continuous function. If \(\int\limits_0^{\sin \,x} {f\left( t \right)dt}  = \frac{{\sqrt 3 }}{2}x\) , then \(f\left( {\frac{{\sqrt 3 }}{2}} \right)\) is equal to

Let \(S =\{\sqrt{ n }: 1 \leq n \leq 50\) and \(n\) is odd \(\}\) Let \(a \in S\) and \(A =\left[\begin{array}{ccc}1 & 0 & a \\ -1 & 1 & 0 \\ - a & 0 & 1\end{array}\right]\) If \(\sum_{ a \in S } \operatorname{det}(\operatorname{adj} A )=100 \lambda\), then \(\lambda\) is equal to

If the area of an equilateral triangle inscribed in the circle, \(x^2 + y^2 + 10x + 12y + c = 0\) is \(27\sqrt 3 \,sq.\,units\) then \(c\) is equal to

The area (in sq. units) of the smaller portion enclosed between the curves, \(x^2 + y^2 = 4\) and \(y^2 =3x\), is

For two non-zero complex number \(z_1\) and \(z_2\), if \(\operatorname{Re}\left(z_1 z_2\right)=0\) and \(\operatorname{Re}\left(z_1+z_2\right)=0\), then which of the following are possible ? \((A)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((B)\) \(\operatorname{Im}\left(z_1\right) < 0\) and \(\operatorname{Im}\left(z_2\right) > 0\) \((C)\) \(\operatorname{Im}\left(z_1\right) > 0\) and \(\operatorname{Im}\left(z_2\right) < 0\) \((D)\) \(\operatorname{Im}\left( z _1\right) < 0\) and \(\operatorname{Im}\left( z _2\right) < 0\) Choose the correct answer from the options given below :

Let \(S=\left\{z \in C : z^{2}+\bar{z}=0\right\}\). Then \(\sum \limits_{z \in S}(\operatorname{Re}(z)+\operatorname{Im}(z))\) is equal to\(......\)

If \(\lim _{x \rightarrow 0} \frac{\alpha x e^{x}-\beta \log _{e}(1+x)+\gamma x^{2} e^{-x}}{x \sin ^{2} x}=10, \alpha, \beta, \gamma \in R\), then the value of \(\alpha+\beta+\gamma\) is \(......\)

Let a vector \( \overrightarrow{a}=\sqrt{2i}-\hat{j}+\lambda\hat{k}, \lambda>0, \) make an obtuse angle with the vector \( \overrightarrow{b}=-\lambda^{2}\hat{i}+4\sqrt{2j}+4\sqrt{2}\hat{k} \) and an angle \( \theta, \frac{\pi}{6}<\theta<\frac{\pi}{2} \) with the positive z-axis. If the set of all possible values of \( \lambda \) is \( (\alpha,\beta)-\{\gamma\} \), then \( \alpha+\beta+\gamma \) is equal to ___ .