If \(P(\alpha, \beta)\) be a point on the line \(3 x+y=0\) such that the point \(P\) and the point \(Q(1,1)\) lie on either side of the line \(3 x=4 y+8\), then

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

The sum of the coefficients of \(x^{32}\) and \(x^{-31}\) in \(\left(x^4-\frac{1}{x^3}\right)^{15}\) is \(\left(x^4-\frac{1}{x^3}\right)^{15}\)

The probability of a non-leap year having 53 Mondays is .........

\(\sin \mathrm{h}^{-1}\left(2^{3 / 2}\right)\) is equal to :

A is a point on the circle with radius 8 and centre at \(O\). A particle P is moving on the circumference of the circle starting from A. M. is the foot of the perpendicular from \(P\) on \(O A\) and \(\angle \mathrm{POM}=\theta\). When \(O M=4\) and \(\frac{d \theta}{d t}=6\) radians \(/ \mathrm{sec}\), then the rate of change of PM is (in units/sec)

An urn contains 3 black and 5 red balls. If 3 balis are drawn at random from the urn, the mean of the probability distribution of the number of red balls drawn is

If \(\left|\begin{array}{ccc}\cos (A+B) & -\sin (A+B) & \cos 2 B \\ \sin A & \cos A & \sin B \\ -\cos A & \sin A & \cos B\end{array}\right|=0\)
then \(B\) is equal to

In a meter bridge experiment, when a nichrome wire is in the right gap, the balancing length is \(60 \mathrm{~cm}\). When the nichrome wire is uniformly stretched to increase its length by \(20 \%\) and again connected in the right gap, the new balancing length is nearly

\(\lim _{n \rightarrow \infty}\left(\frac{1}{1^2+n^2}+\frac{2}{2^2+n^2}+\frac{3}{3^2+n^2}+\ldots+\frac{n}{n^2+n^2}\right)=\)

The principle quantum number ' \(n\) ' corresponding to the exited state of \(\mathrm{He}^{+}\)ion, if on transition to the ground state two photons in succession with wavelength \(1026 \mathrm{~A}^{\circ}\) and \(304 \mathrm{~A}^{\circ}\) are emitted ( \(\mathrm{R}=1.097 \times 10^7 \mathrm{~m}^{-1}\) )

If the angle between the tangents drawn to the circle \(x^2+y^2-12 x-16 y=0\) at the points where the line \(5 y=5 x+k\) cut the circle is \(60^{\circ}\), then the value of \(k\) is

Total number of hydroxyls $(-\mathrm{OH})$ groups in saccharic acid are