List of practice Questions

If a resistor of resistance \( 4 \Omega \), a capacitor of capacitive reactance \( 6 \Omega \) and an inductor of inductive reactance \( 9 \Omega \) are connected in series with an AC source, then the impedance of the circuit is:

The ratio of the magnitudes of the electric field and \( 10^8 \) times the magnetic field of a plane electromagnetic wave is:

The capacitance of a spherical capacitor is \( 100 \) pF. If the spacing between the two spheres is \( 1 \) cm, then the radius of the inner sphere of the capacitor is:

If a charged particle enters a uniform magnetic field normally with certain velocity, then the time period of revolution of the particle:

The internal energy of one mole of a rigid diatomic gas at absolute temperature \( T \) is:

In a closed organ pipe, the number of nodes formed in fifth and ninth harmonics are respectively:

A body of mass \( 4 \) kg attached to a spring of force constant \( 64 \) N/m executes simple harmonic motion on a frictionless horizontal surface. The time period of oscillation is:

A mass of \( 6 \times 10^{24} \) kg is to be compressed in the form of a solid sphere such that the escape velocity from its surface is \( 3 \times 10^5 \) m/s. The radius of the sphere is:

A body of mass \( M \) is moving with a uniform speed \( v \) on a frictionless horizontal surface under the influence of two forces \( F_1 \) and \( F_2 \) as shown in the figure. The net power of the system is:

Evaluate the integral: \[ I = \int_{\pi/6}^{\pi/3} \cos^{-4} x \, dx \]

Evaluate the integral: \[ I = \int_0^{3\pi/2} \frac{\cos^5 x}{\cos^3 x+\sin^3 x}dx \]

The differential equation for which \( y^2 = 4a(x + a) \) (where \( a \) is a parameter) is the general solution is:

If the lengths of the tangent, subtangent, normal, and subnormal for the curve \( y = x^2 + x - 1 \) at the point \( (1,1) \) are \( a, b, c, \) and \( d \) respectively, then their increasing order is:

Evaluate the integral: \[ \int \frac{x+1}{x^3 - 1}dx \]

Evaluate the integral: \[ \int \frac{x^4-16x^2+2x+8}{x^3-4x^2+2}dx \]

Evaluate the integral: \[ \int \frac{\sec^2 x}{(\sec x+\tan x)^{5/2}}dx \]

Evaluate the integral: \[ \int \operatorname{Cos}^{-1} \left( \frac{1-x^2}{1+x^2} \right) dx \]

If \( y = \log(\sec(\tan^{-1}x)) \) for \( x>0 \), then \( \frac{dy}{dx} \) at \( x = 1 \) is:

If \( x = \sqrt{2}e^t(\sin t - \cos t) \) and \( y = \sqrt{2}e^t(\sin t + \cos t) \), then \( \left[ \frac{d^2y}{dx^2} \right]_{t=\frac{\pi}{4}} \) is:

P and Q are the ends of a diameter of the circle \( x^2+y^2=a^2(a>\frac{1}{\sqrt{2}}) \). \( s \) and \( t \) are the lengths of the perpendiculars drawn from P and Q onto the line \( x+y=1 \) respectively. When the product \( st \) is maximum, the greater value among \( s, t \) is:

If \( \theta \) is the acute angle between the tangents drawn from the point \( (1,1) \) to the hyperbola \( 4x^2-5y^2-20=0 \), then \( \tan\theta \) is:

If \( A(2,-1,1) \), \( B(2,5,1) \) and \( C(0,-2,3) \) are the vertices of a triangle, and \( D \) is the point of intersection of the side \( BC \) and the internal angular bisector of angle \( A \), then \( AD = \):

A plane \( \pi \) given by \( ax+by+11z+d = 0 \) is perpendicular to the planes \( 2x-3y+z=4 \), \( 3x+y-z=5 \), and the perpendicular distance from the origin to the plane \( \pi \) is \( \sqrt{6} \) units. If all the intercepts made by the plane \( \pi \) on the coordinate axes are positive, then \( d = \):

If the equation of the polar of the point \( (\alpha, -1) \) with respect to the circle \( x^2+y^2-4x-6y-12=0 \) is \( y = \beta \), then \( 4(\alpha+\beta) = \):

If \( \theta \) is the angle between the tangents drawn from the point \( (-1, -1) \) to the circle \( x^2+y^2-4x-6y+c=0 \) and \( \cos\theta = -\frac{7}{25} \), then the radius of the circle is: