Let \(y=y(x)\) be the solution curve of the differential equation \(\frac{d y}{d x}=\frac{y}{x}\left(1+x y^2\left(1+\log _e x\right)\right)\) \(x > 0, y(1)=3\). Then \(\frac{y^2(x)}{9}\) is equal to :

Let \(f (x) = a^x (a > 0)\) be written as \(f( x) = f_1( x) + f_2( x)\) , where \(f_1( x)\) is an even function and \(f_2( x)\) is an odd function. Then \(f_1( x + y) + f_1( x - y )\) equals

The number of real roots of the equation \(\tan ^{-1} \sqrt{x(x+1)}+\sin ^{-1} \sqrt{x^{2}+x+1}=\frac{\pi}{4}\) is:

Let \(y=f(x)\) represent a parabola with focus \(\left(-\frac{1}{2}, 0\right)\) and directrix \(y =-\frac{1}{2}\). Then \(S=\left\{x \in R : \tan ^{-1}\left(\sqrt{f(x)}+\sin ^{-1}(\sqrt{f(x)+1})\right)=\frac{\pi}{2}\right\}:\)

The square of the distance of the point of intersection of the lines \(\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(a\hat{i} - \hat{j})\), \(a \neq 0\) and \(\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + a\hat{k})\) from the origin is:

Let the lines \(3 x-4 y-\alpha=0,8 x-11 y-33=0\), and \(2 x-3 y+\lambda=0\) be concurrent. If the image of the point
\((1,2)\) in the line \(2 x-3 y+\lambda=0\) is \(\left(\frac{57}{13}, \frac{-40}{13}\right)\), then \(|\alpha \lambda|\) is equal to

If \(z = x + iy\) satisfies \(|z|-2=0\) and \(|z-i|-|z+5 i|=0\), then

The roots of the quadratic equation \(3 x^2-\mathrm{p} x+\mathrm{q}=0\) are \(10^{\text {th }}\) and \(11^{\text {th }}\) terms of an arithmetic progression with common difference \(\frac{3}{2}\). If the sum of the first 11 terms of this arithmetic progression is 88 , then \(q-2 p\) is equal to \(\qquad\) -.

\(8-\) digit numbers are formed using the digits \(1, 1, 2, 2, 2, 3, 4, 4.\) The number of such numbers in which the odd digits do no occupy odd places, 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 \(y=f(x)\) be the solution of the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{x y}{x^2-1}=\frac{x^6+4 x}{\sqrt{1-x^2}},-1 \lt x \lt 1\) such that \(f(0)=0\). If \(6 \int_{-1 / 2}^{1 / 2} f(x) \mathrm{d} x=2 \pi-\alpha\) then \(\alpha^2\) is equal to _______ .

If \(\alpha, \beta\) are the roots of the equation, \(x^2-x-1=0\) and \(S_n=2023 \alpha^n+2024 \beta^n\), then

If \(a _{1}(>0), a _{2}, a _{3}, a _{4}, a _{5}\) are in a G.P., \(a _{2}+ a _{4}=2 a _{3}+1\) and \(3 a _{2}+ a _{3}=2 a _{4}\), then \(a _{2}+ a _{4}+2 a _{5}\) is equal to