\(\cot \left(\sum_{n=1}^{50} \tan ^{-1}\left(\frac{1}{1+n+n^2}\right)\right)=\)

The angle between the planes \(2 x-y+z=6\) and \(x+y+2 z=3\) is

Assertion (A): \(\int_0^{\frac{\pi}{2}}\left(\sin ^6 x+\cos ^6 x\right) d x\) lies in the interval \(\left(\frac{\pi}{8}, \frac{\pi}{2}\right)\)
Reason (R): \(\sin ^6 x+\cos ^6 x\) is a periodic function with period \(\frac{\pi}{2}\)

If \(\left|z_1\right|=1,\left|z_2\right|=2,\left|z_3\right|=3\) and \(\left|9 z_1 z_2+4 z_1 z_3+z_2 z_3\right|=12\), then the value of \(\left|z_1+z_2+z_3\right|\) is

If \(7 \cos \theta-\sin \theta=5\) and \(\tan \theta>0\), then \(\tan \theta=\)

If \(A=\left[\begin{array}{ccc}1 & 1 & a+1 \\ 1 & a+1 & 1 \\ a+1 & 1 & 1\end{array}\right]\) is not an invertible matrix, then the sum of all the values of \(a\) is

Match the following
List - I
A) The equation of line passing through \((4,3)\) whose \(\mathrm{X}\)-intercept is twice its Y-intercept
B) The equation of the line passing through the centroid and circumcentre of \(\triangle A B C\) with vertices \(\mathrm{A}(1,1), \mathrm{B}(3,3), \mathrm{C}(6,-6)\)
C) The equation of the line whose \(\mathrm{X}\)-intercept is \((-3 / 5)\)
III) \(x+2 y+\sqrt{2}=0\) and is perpendicular to \(x-y+2=0\)
D) The equation of the line whose distance from the origin is 2 and the normal drawn from the origin makes an angle \(45^{\circ}\) with the positive direction of \(\mathrm{X}\)-axis
List - II
I) \(x+y-2 \sqrt{2}=0\)
II) \(7 x+23 y-8=0\)
IV) \(x+2 y-10=0\)
IV) \(x+2 y-10=0\)
V) \(5 x+5 y+3=0\)
The correct answer is

If \(\cos \alpha+3 \cos 3 \beta+5 \cos 5 \gamma=0\), \(\sin \alpha+3 \sin 3 \beta+5 \sin 5 \gamma=0\) and \(\cos 3 \alpha+27 \cos 9 \beta+125 \cos 15 \gamma=\left(\lambda^2-4\right)\) \(\cos (\alpha+3 \beta+5 \gamma)\), then \(\lambda\) is equal to

An object A of mass ' \(m\) ' is located at a point ' \(\mathrm{P}\) ' at distances ' \(\mathrm{r}\) ' and ' \(2 \mathrm{r}\) ' from two planets \(B\) and \(C\) of masses ' \(M\) ' and ' \(6 M\) ' respectively as shown in the figure. If the escape speed of the object A from point ' \(\mathrm{P}\) ' due to the gravitational influence of only planet B is \(5 \mathrm{kms}^{-1}\), then the escape speed of the object A from point ' \(\mathrm{P}\) ' due to the gravitational influence of both the planets is ____ \(\mathrm{kms}^{-1}\).

Identify the correct statement.

If $f\left(x\right)=Max\left\{3-x,3+x,6\right\}$ is not differentiable at $x=a$, and $x=b$, then $\left|a\right|+\left|b\right|=$

At a certain temperature and at infinite dilution, the equivalent conductances of sodium benzoate, hydrochloric acid and sodium chloride are 240, 349 and \(229 \mathrm{ohm}^{-1} \mathrm{~cm}^2\) equiv \(^{-1}\) respectively. The equivalent conductance of benzoic acid in \(\mathrm{ohm}^{-1} \mathrm{~cm}^2\) equiv \(^{-1}\) at the same conditions is

The non-biodegradable waste formed in fertilizer industries is