List of practice Questions

The length of the normal drawn at \( t = \frac{\pi}{4} \) on the curve \( x = 2(\cos 2t + t \sin 2t) \), \( y = 4(\sin 2t + t \cos 2t) \) is: 

A body of mass \( M \) is at equilibrium under the action of four forces \( F_1, F_2, F_3 \), and \( F_4 \). If \( F_1 \) is removed from the body, then the body moves with an acceleration of:
Evaluate the integral: \[ \int \frac{x^2 - 1}{x^4 + 3x^2 + 1} \, dx \]

Steam of mass 60 g at a temperature \( 100^\circ C \) is mixed with water of mass 360 g at a temperature \( 40^\circ C \). The ratio of the masses of steam and water in equilibrium is? 
(Latent heat of steam = 540 cal/g and specific heat capacity of water = 1 cal/g◦C)

Two capacitors of capacitances \( 1\mu F \) and \( 2\mu F \) can separately withstand potentials of \( 6 \) kV and \( 4 \) kV respectively. The total potential, they together can withstand when they are connected in series is: 

The process by which the electrons can come out of the metal in a spark plug is:
At 400 K, an ideal gas is enclosed in a 0.5 m\(^3\) vessel at a pressure of 203 kPa. What is the change in temperature required (in K), if it occupies a volume of 0.2 m\(^3\) under a pressure of 304 kPa? (Nearest integer)
If \( \tan \left( \frac{\pi}{12} + 2x \right) = \cot 3x \), where \( 0<x<\frac{\pi}{2} \), then the value of \( x \) is:

At 400 K, the following graph is obtained for \( x \) moles of an ideal gas. \( x \) is equal to (R = gas constant, P = pressure, V = volume)
 

The functional groups involved in the conversion of glucose to gluconic acid and gluconic acid to saccharic acid respectively are:

If \[ \int e^x (x^3 + x^2 - x + 4) \, dx = e^x f(x) + C, \] then \( f(1) \) is:

If \( (1,1) \) is the vertex and \( x + y + 1 = 0 \) is the directrix of a parabola. If \( (a, b) \) is its focus and \( (c, d) \) is the point of intersection of the directrix and the axis of the parabola, then \( a + b + c + d \) is:
In moving a body of mass \( m \) down a smooth incline of inclination \( \theta \) with velocity \( v \), the power required is (g = acceleration due to gravity):
The ratio of the centripetal accelerations of the electron in two successive orbits of hydrogen is 81:16. Due to a transition between these two states, the angular momentum of the electron changes by:
If the number of electron-hole pairs per cm\(^3\) of an intrinsic Si wafer at temperature 300 K is \(1.1 \times 10^{10}\) and the mobilities of electrons and holes at 300 K are 1500 and 500 cm\(^2\) per volt, second, respectively, then the conductivity of the Si wafer at this temperature (in \(\mu\)mho cm\(^{-1}\)) is nearly:
In a radioactive sample \( 2 \times 10^8 \) nuclei reduce to \( 10^8 \) nuclei in 15 minutes, then the half-life of the sample in minutes is:

Let \( I = \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{\tan^2 x}{1+5^x} \, dx \). Then:

Which of the following statement is correct?
If \( z = x + iy \) and the point \( P \) represents \( z \) in the Argand plane. If the amplitude of \( \frac{2z-i}{z+2i} \) is \( \frac{\pi}{4} \), then the equation of the locus of \( P \) is:
In \( \triangle ABC \), if \( (r_2 - r_1)(r_3 - r_1) = 2r_2r_3 \), then \( 2(r + R) = \):
If 35 is removed from the data 30, 34, 35, 36, 37, 38, 39, 40, then the median increases by
If \(\cos x + \cos y = \tfrac{2}{3}\) and \(\sin x - \sin y = \tfrac{3}{4}\), then \[ \sin(x - y) \;+\; \cos(x - y) \;=\;? \]
If R is the resistance of a conductor of length l, then
Among the oxides SiO\(_2\), SO\(_2\), Al\(_2\)O\(_3\), and P\(_2\)O\(_5\), the correct order of acidic strength is:
The power of a lens of focal length 20 cm is