\(3 \times 7^{22}+2 \times 10^{22}-44\) when divided by \(18\) leaves the remainder .... .

The least positive value of \(a\) for which the equation \(2 \mathrm{x}^{2}+(\mathrm{a}-10) \mathrm{x}+\frac{33}{2}=2 \mathrm{a}\) has real roots is

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 \(\cos \,\left( {\alpha  + \beta } \right) = \frac{3}{5},\,\sin \,\left( {\alpha  - \beta } \right) = \frac{5}{{13}}\) and \(0 < \alpha ,\beta  < \frac{\pi }{4}\) then \(\tan \,\left( {2\alpha } \right)\) is equal to

Two dice are thrown  \(5\) times, and each time the sum of the numbers obtained being  \(5\) is considered a success. If the probability of having at least \(4\) successes is \(\frac{\mathrm{k}}{3^{11}}\), then \(\mathrm{k}\) is equal to

Which of the following points lies on the tangent to the curve \(x^{4} e^{y}+2 \sqrt{y+1}=3\) at the point \((1,0) ?\)

The axis of a parabola is the line \(y=x\) and its vertex and focus are in the first quadrant at distances \(\sqrt{2}\) and \(2 \sqrt{2}\) units from the origin, respectively. If the point \((1, \mathrm{k})\) lies on the parabola, then a possible value of \(k\) is :-

If \(\tan \left(\frac{\pi}{9}\right), x, \tan \left(\frac{7 \pi}{18}\right)\) are in arithmetic progression and \(\tan \left(\frac{\pi}{9}\right), y, \tan \left(\frac{5 \pi}{18}\right)\) are also in arithmetic progression, then \(|x-2 y|\) is equal to:

The line of shortest distance between the lines \(\frac{x-2}{0}=\frac{y-1}{1}=\frac{z}{1}\) and \(\frac{x-3}{2}=\frac{y-5}{2}=\frac{z-1}{1}\) makes an angle of \(\cos ^{-1}\left(\sqrt{\frac{2}{27}}\right)\) with the plane \(P: a x-y-\) \(z=0\), \((a>0)\). If the image of the point \((1,1,-5)\) in the plane \(P\) is \((\alpha, \beta, \gamma)\), then \(\alpha+\beta-\gamma\) is equal to \(........\)

The sum of the solutions of the equation \(\left| {\sqrt x  - 2} \right| + \sqrt x \left( {\sqrt x  - 4} \right) + 2 = 0\left( {x > 0} \right)\) is equal to

Let a variable line of slope \(m>0\) passing through the point \((4,-9)\) intersect the coordinate axes at the points \(A\) and \(B\). the minimum value of the sum of the distances of \(\mathrm{A}\) and \(\mathrm{B}\) from the origin is

Let \(B\) and \(C\) be the two points on the line \(y+x=0\) such that \(B\) and \(C\) are symmetric with respect to the origin. Suppose \(A\) is a point on \(y -2 x =2\) such that \(\triangle ABC\) is an equilateral triangle. Then, the area of the \(\triangle ABC\) is

Let \(f(x)\) and \(g(x)\) be two functions satisfying \(f\left(x^{2}\right)\) \(+g(4-x)=4 x^{3}\) and \(g(4-x)+g(x)=0\), then the value of \(\int_{-4}^{4} f(x)^{2} d x\) is