If \(\mathrm{P}\left(\frac{\pi}{4}\right)\) and \(\mathrm{Q}\left(\frac{3 \pi}{4}\right)\) are two points on the hyperbola \(4 x^2-y^2-8 x-2 y-13=0\) in parametric form, then the distance between \(\mathrm{P}\) and \(\mathrm{Q}\) is

If \(a, b\) and \(c\) form a geometric progression with common ratio \(r\), then the sum of the ordinates of the points of intersection of the line \(a x+b y+c=0\) and the curve \(x+2 y^2=0\) is

If \(\int \frac{5 \tan \mathrm{x}}{\tan \mathrm{x}-2} \mathrm{dx}=\mathrm{ax}+\mathrm{b} \log |\sin \mathrm{x}-2 \cos \mathrm{x}|+\mathrm{c}\), then \(\mathrm{a}+\mathrm{b}=\)

The number of integer solutions of the equation ${\left|1-i\right|}^x=2^x$ is

The quotient when \(3 x^5-4 x^4+5 x^3-3 x^2+6 x-8\) is divided by \(x^2+x-3\) is

If \(3 f(x)-2 f\left(\frac{1}{x}\right)=x\), then \(f^{\prime}(2)\) is

The equation of tangent to the circle \(x^2+y^2=1\), which is perpendicular to the line \(y=m x+1\); is

Radiation of wavelength $300 \mathrm{nm}$ and intensity $100 \mathrm{W} {\mathrm{m}}^{-2}$ falls on the surface of a photosensitive material. If $2\%$ of the incident photons produce photo electron, the number of photoelectrons emitted from an area of $2{ \mathrm{cm}}^2$ of the surface is nearly

A car moving with a certain velocity, jumps from an inclined plane placed at one bank of a river and reaches the other bank by attained a maximum height of \(80 \mathrm{~m}\). If the same car, moving with the same velocity jumps from another inclined plane having different angle of inclination and reaches the same point on the other bank by attaining maximum height of \(45 \mathrm{~m}\), then the width of the river is

If a message signal of frequency \(10 \mathrm{kHz}\) and peak voltage \(12 \mathrm{~V}\) is used to modulate a carrier wave of frequency \(1 \mathrm{MHz}\), the modulation index is 0.6 . To make the modulation index 0.75 , the carrier peak voltage should be

Match the item given in Column-I (method used for determining colligative property) and Column-II (for the corresponding colligative property) and find the correct order.
\(\begin{array}{|c|c|c|}
\hline & \text { Column I } & \text { Column II } \\
\hline \text { A. } & \text { Beckmann method } & \text { 1. Osmotic pressure } \\
\hline \text { B. } & \begin{array}{l}
\text { Ostwald-Walker method }
\end{array} & \text { 2. Elevation in B.P } \\
\hline \text { C. } & \begin{array}{l}
\text { Berkeley-Hartley method }
\end{array} & \text { 3. Depression in F.P. } \\
\hline \text { D. } & \text { Landsberger method } & \begin{array}{l}
\text { 4. Relative lowering of vapour pressure }
\end{array} \\
\hline
\end{array}\)

Area of the triangle formed by the lines \(3 x^2-4 x y+y^2=0,2 x-y=6\) is

Three identical spheres each of diameter \(2 \sqrt{3} \mathrm{~m}\) are kept on a horizontal surface such that each sphere touches the other two spheres. If one of the sphere is removed, then the shift in the position of the centre of mass of the system is