List of practice Questions

Let a b, and c be three non-zero non-coplanar vectors. Let the position vectors of four points A, B, C and D be \(a-b+c\), \(λa-3b+4c\), \(-a+2b-3c\) and \(2a-4b+6c\) respectively. If AB AC , and AD are coplanar, then λ is equal to ____.

Let the coefficients of three consecutive terms in the binomial expansion of (1 + 2x)n be the ratio 2 : 5 : 8. Then the coefficient of the term, which is in the middle of these three terms, is _______.

Suppose f is a function satisfying \(f(x + y) + f(x) + f(y)\) for all \(x, y ∈N\) and \(f(1)=\frac{1}{5}\). If \(∑_{n=1}^{m} \frac{f(n)}{n(n+1)(n+2)}=\frac{1}{2}\), then m is equal to _______.

If all the six digit numbers x₁ x₂ x₃ x₄ x₅ x₆ with \(0<x_1<x_2<x_3<x_4<x_5<x_6\) are arranged in the increasing order, then the sum of the digits in the 72^th number is _____________.

If the co-efficient of x⁹ in \((αx^3+\frac{1}{βx})^{11}\) and the co-efficient of x^-9 in \((αx-\frac{1}{βx^3})^{11}\) are equal, then \((αβ)^2\) is equal to ________.

By what percentage will the transmission range of a TV tower be affected when the height of the tower is increased by 21% ?

A small ball of mass M and density \(\rho\) is dropped in a viscous liquid of density \(\rho_0\). After some time, the ball falls with a constant velocity. What is the viscous force on the ball ?

The energy levels of an hydrogen atom are shown below. The transition corresponding to emission of shortest wavelength is

The kinetic energy of an electron, α-particle and a proton are given as 4K, 2K and K respectively. The de-Broglie wavelength associated with electron (λe) α-particle (λα) and the proton (λp) are as follows :

A monochromatic light wave with wavelength \(λ_1\) and frequency \(v_1\) in air enters another medium. If the angle of incidence and angle of refraction at the interface are 45° and 30° respectively, then the wavelength \(λ_2\) and frequency \(v_2\) of the refracted wave are : 

For the plane electromagnetic wave given by \(E = E_o sin (ωt-kx)\) and \(B = B_o sin ( ωt-kx)\), the ratio of average electric energy density to average magnetic energy density is

The resistivity (\(\rho\)) of semiconductor varies with temperature. Which of the following curve represents the correct behavior

For a uniformly charged thin spherical shell, the electric potential (V) radially away from the entre (O) of shell can be graphically represented as –

A mass $m$ is attached to two springs as shown in the figure. The spring constants of the two springs are $K_1$ and $K_2$. For the frictionless surface, the time period of oscillation of mass $m$ is:

A source supplies heat to a system at the rate of 1000 W. If the system performs work at a rate of 200 W. The rate at which internal energy of the system increase is

The number of air molecules per cm³ increased from 3 × 10¹⁹ to 12 × 10¹⁹. The ratio of collision frequency of air molecules before and after the increase in the number respectively is :

Two resistances are given as \(R_1=(10 ±0.5) Ω\) and \(R_2= (15±0.5) Ω\). The percentage error in the measurement of equivalent resistance when they are connected in parallel is -

A small block of mass 100 g is tied to a spring of spring constant 7.5 N/m and length 20 cm. The other end of spring is fixed at a particular point A. If the block moves in a circular path on a smooth horizontal surface with constant angular velocity 5 rad/s about point A, then tension in the spring is-

A pole is vertically submerged in swimming pool, such that it gives a length of shadow 2.15 m within water when sunlight is incident at an angle of 30o with the surface of water. If swimming pool is filled to a height of 1.5 m, then the height of the pole above the water surface in centimeters is (n-w = 4/3) ____________

The radius of fifth orbit of the Li++ is __________ × 10^-12 m.
Take : radius of hydrogen atom = 0.51 Å

An ideal transformer with purely resistive load operates at 12 kV on the primary side. It supplies electrical energy to a number of nearby houses at 120 V. The average rate of energy consumption in the houses served by the transformer is 60 kW. The value of resistive load (Rs) required in the secondary circuit will be ________ mΩ.

The length of a metallic wire is increased by 20% and its area of cross section is reduced by 4%. The percentage change in resistance of the metallic wire is ______

A parallel plate capacitor with plate area A and plate separation d is filed with a dielectric material of dielectric constant K = 4. The thickness of the dielectric material is x, where \(x < d\).

Let C₁ and C₂ be the capacitance of the system for \(x=\frac{1}{3d}\) and \(x=\frac{2d}{3}\), respectively. If \(C_1 = 2μF\), the value of C₂ is __________μF.

Two identical circular wires of radius 20 cm and carrying current \(\sqrt2 \)A are placed in perpendicular planes as shown in figure. The net magnetic field at the centre of the circular wires is _________ × 10^{-8} T.
 

A steel rod bas a radius of 20 mm and a length of 2.0 m. A force of 62.8 kN stretches it along its length. Young's modulus of steel is 2.0 × 10^{11} N/m^2 . The longitudinal strain produced in the wire is _______ × 10^{-5}