If the solution curve of the differential equation \(\left(y-2 \log _e x\right) d x+\left(x \log _e x^2\right) d y=0, x > 1\) passes through the points \(\left(e, \frac{4}{3}\right)\) and \(\left(e^4, \alpha\right)\), then \(\alpha\) is equal to \(................\).

Let \(f : R \rightarrow R\) and \(g : R \rightarrow R\) be defined as \(f(x)=\left\{\begin{array}{ll}x+a, & x<0 \\ |x-1|, & x \geq 0\end{array}\right.\) and \(g(x)=\left\{\begin{array}{cc}x+1, & x<0 \\ (x-1)^{2}+b, & x \geq 0\end{array}\right.\) where \(a , b\) are non-negative real numbers. If \((gof)\,( x )\) is continuous for all \(x \in R\), then \(a + b\) is equal to ...... .

Let \([\mathrm{t}]\) denote the greatest integer \(\leq \mathrm{t}\). Then the value of \(8 \cdot \int \limits_{-\frac{1}{2}}^{1}([2 x]+|x|) \,d x\) is .... .

If the system of linear equations
\(\begin{aligned} & 3 x+y+\beta z=3 \\ & 2 x+\alpha y-z=-3 \\ & x+2 y+z=4\end{aligned}\)
has infinitely many solutions, then the value of \(22 \beta-9 \alpha\) is :

If \(\theta \in\left[-\frac{7 \pi}{6}, \frac{4 \pi}{3}\right]\), then the number of solutions of \(\sqrt{3} \operatorname{cosec}^2 \theta-2(\sqrt{3}-1) \operatorname{cosec} \theta-4=0\), is equal to

If \(\sum_{r=0}^5 \frac{{ }^{11} C_{2r+1}}{2 r+2}=\frac{\mathrm{m}}{\mathrm{n}}, \operatorname{gcd}(\mathrm{m}, \mathrm{n})=1\), then \(\mathrm{m}-\mathrm{n}\) is equal to _______

Let \(\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{d}}=\vec{a} \times \overrightarrow{\mathrm{b}}\). If \(\overrightarrow{\mathrm{c}}\) is a vector such that \(\vec{a} \cdot \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{c}}|\), \(|\overrightarrow{\mathrm{c}}-2 \vec{a}|^2=8\) and the angle between \(\overrightarrow{\mathrm{d}}\) and \(\overrightarrow{\mathrm{c}}\) is \(\frac{\pi}{4}\), then \(|10-3 \overrightarrow{\mathrm{~b}} \cdot \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{d}} \times \overrightarrow{\mathrm{c}}|^2\) is equal to

All the pairs \((x, y)\) that satisfy the inequality \({2^{\sqrt {{{\sin }^2}{\kern 1pt} x - 2\sin {\kern 1pt} x + 5} }}.\frac{1}{{{4^{{{\sin }^2}\,y}}}} \leq 1\) also Satisfy the equation

Let the line \(\mathrm{L}\) be the projection of the line  \(\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-4}{2}\) in the plane \(x-2 y-z=3 .\) If \(d\) is the distance of the point \((0,0,6)\) from \(\mathrm{L}\), then \(\mathrm{d}^{2}\) is equal to .... .

If the system of equations \(2 x+y-z=5\) \(2 x-5 y+\lambda z=\mu\) \(x+2 y-5 z=7\) has infinitely many solutions, then \((\lambda+\mu)^2+(\lambda-\mu)^2\) is equal to

The value of \(\left(\frac{1+\sin \frac{2 \pi}{9}+i \cos \frac{2 \pi}{9}}{1+\sin \frac{2 \pi}{9}-i \cos \frac{2 \pi}{9}}\right)^{3}\) is

For some \(a, b, c \in N\), let \(f(x)=a x-3\) and \(g ( x )= x ^{ b }+ c , x \in R\). If \((\text { fog })^{-1}( x )=\left(\frac{ x -7}{2}\right)^{1 / 3}\) then \((fog) (ac) + (gof) (b)\) is equal to \(..........\)

Let \(P = \left[ {\begin{array}{*{20}{c}}
  1&0&0 \\ 
  3&1&0 \\ 
  9&3&1 
\end{array}} \right]\) and \(Q = [q_{ij}]\) be two \(3\times3\) matrices such that \(Q -P^5 = I_3\). Then \(\frac{{{q_{21}} + {q_{31}}}}{{{q_{32}}}}\) is equal to