A swimmer can swim in still water with speed \(v\) and the river flowing with velocity \(\frac{v}{2}\).
To cross the river in shortest time, he should swim making angel \(\theta\) with the upstream. What is the ratio of the time taken to swim across in the shortest time to that in swimming across over shortest distance?

A metal tape is calibrated at $25°\mathrm{C}$. On a cold day when the temperature is $-15°\mathrm{C}$, the percentage error in the measurement of length is
(Coefficient of linear expansion of metal $=1\times {10}^{-5}{}^{\circ }\mathrm{C}^{-1}$)

When a sphere is taken to the bottom of a sea of depth 1 km, it contracts in volume by \(0.01 \%\), then the bulk modulus of the material of the sphere is (Acceleration due to gravity \(=10 \mathrm{~m} \mathrm{~s}^{-2}\) )

A fully charged capacitor has a capacitance \(C\). It is discharged through a small coil of resistance wire, embedded in a block of specific heat \(s\) and mass \(m\) under thermally isolated conditions. If the temperature of the block is raised by \(\Delta T\), the potential difference \(V\) across the capacitor initially is

A particle of mass \(10 \mathrm{~g}\) is moving towards east with a velocity of \(10 \mathrm{~ms}^{-1}\) and another particle of mass \(15 \mathrm{~g}\) is moving towards north with a velocity of \(5 \mathrm{~ms}^{-1}\). The magnitude of the velocity of the centre of mass of the system of the two particles is

The radius ' \(r\) ' of a wire is given by \(r=\sqrt{\frac{64 \mathrm{LA}}{\pi \mathrm{Bv}}}\), where \(\mathrm{I}\) is electric current, \(\mathrm{B}\) is magnetic field and ' \(v\) ' is velocity. Then the parameter A in the formula represents ____.

The minimum force required to move a body up an inclined plane is three times the minimum force required to prevent it from sliding down the plane. If the coefficient of friction between the body and the inclined plane is \(\frac{1}{2 \sqrt{3}}\), then the angle of the inclined plane 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

In the interval \([0,3]\), the function \(f(x)=|x-1|+|x-2|\) is

A short magnet oscillates with a frequency \(10 \mathrm{~Hz}\) where horizontal component of Earth's magnetic field is \(12 \mu \mathrm{T}\). A downward current of \(15 \mathrm{~A}\) is established in a vertical wire placed \(20 \mathrm{~cm}\) west of the magnet. The new frequency of the magnet is

In a triangle, if the angles are in the ratio \(3: 2: 1\), then the ratio of its sides is

If \(\frac{\tan 3 A}{\tan A}=a\), then \(\frac{\sin 3 A}{\sin A}\) is equal to

The equations of the lines perpendicular to \(x^2-5 x y+4 y^2=0\) and passing through \((2,1)\) is