List of practice Questions

If the maximum value of the term independent of t in the expansion of (t2x\(^{\frac{1}{5}}\)+\(\frac{(1−x)^{\frac{1}{10}}}{t}\)),x≥0 is K, then 8 K is equal to__________.
If \(x=∑_{n=0}^∞a^n,y=∑_{n=0}^∞b^n,z=∑_{n=0}^∞c^n\), where a, b, c are in A.P. and |a| < 1, |b| < 1, |c| < 1, abc≠ 0, then :
Let a1 = b1 = 1, an = an – 1 + 2 and bn = aa + bn – 1 for every natural number n ≥ 2. Then\(\sum_{n=1}^{15}\) \(a_n⋅b_n\) is equal to ______.
Let a be an integer such that  \(lim _{x→7} \frac{18−[1−x]}{[x−3a]}\) exists, where [t] is greatest integer \(≤ t\). Then \(a\) is equal to :
The number of distinct real roots of \(x^4 – 4x + 1 = 0\) is :
The lengths of the sides of a triangle are 10 + x2, 10 + x2 and 20 – 2x2. If for x = k, the area of the triangle is maximum, then 3k2 is equal to :

Let the equation of two diameters of a circle x2 + y2 – 2x + 2fy + 1 = 0 be 2px – y = 1 and 2x + py = 4p. Then the slope m ∈ (0, ∞) of the tangent to the hyperbola 3x2 – y2 = 3 passing through the centre of the circle is equal to _______.

The line of the shortest distance between the lines \(\frac{x-2}{0}=\frac{y-1}{1}=\frac{z}{1}\) and \(\frac{x-3}{2}=\frac{y-5}{2}=\frac{z-1}{1}\) makes an angle of cos−1⁡\(\sqrt{\frac{2}{27}}\) with the plane P: ax – y – z = 0, (a> 0). If the image of the point (1, 1, –5) in the plane P is (α, β, γ), then α + β – γ is equal to ________.
Let the points on the plane P be equidistant from the points (–4, 2, 1) and (2, –2, 3). Then the acute angle between the plane P and the plane 2x + y + 3z = 1 is
If the shortest distance between the lines \(\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{λ}\; and\; \frac{x-2}{1 }= \frac{y-4}{4 }= \frac{z-5}{5 }\;is\; \frac{1}{\sqrt3}\), then the sum of all possible values of \(λ\) is :
The number of solutions of the equation \(cos \bigg(x+\frac{π}{3}\bigg)cos\bigg(\frac{π}{3-x}\bigg)=\frac{1}{4}cos^22x,x ∈ [-3π, 3π]\) is:
An EM wave propagating in x-direction has a wavelength of 8 mm. The electric field vibrating y-direction has maximum magnitude of 60 Vm–1. Choose the correct equations for electric and magnetic field if the EM wave is propagating in vacuum:
In Young’s double slit experiment performed using a monochromatic light of wavelength λ, when a glass plate (μ = 1.5) of thickness xλ is introduced in the path of the one of the interfering beams, the intensity at the position where the central maximum occurred previously remains unchanged. The value of x will be:
A cylinder of fixed capacity of 44.8 litres contains helium gas at standard temperature and pressure. The amount of heat needed to raise the temperature of gas in the cylinder by 20.0°C will be (Given gas constant R = 8.3 JK–1-mol–1)
A wire of length L is hanging from a fixed support. The length changes to L1 and L2 when masses 1 kg and 2 kg are suspended, respectively, from its free end. Then the value of L is equal to
Let K1 and K2 be the maximum kinetic energies of photo-electrons emitted when two monochromatic beams of wavelength λ1 and λ2, respectively are incident on a metallic surface. If λ1 = 3λ2 then:
A person moved from A to B on a circular path as shown in figure. If the distance travelled by him is 60 m, then the magnitude of displacement would be
(Given cos135°= –0.7)
Following statements related to radioactivity are given below:
(A) Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions.
(B) The number of un-decayed nuclei in the radioactive sample decays exponentially with time.
(C) Slope of the graph of loge (no. of undecayed nuclei) Vs. time represents the reciprocal of mean life time (τ).
(D) Product of decay constant (λ) and half-life time (T1/2) is not constant.
Choose the most appropriate answer from the options given below:

In the given circuit the input voltage Vin is shown in figure. The cut-in voltage of p–n junction diode
(D1 or D2) is 0.6 V. Which of the following output voltage (V0) waveform across the diode is correct?

Given below are two statements : one is labelled as AssertionA and the other is labelled as ReasonR.

Assertion A : The photoelectric effect does not takes place, if the energy of the incident radiation is less than the work function of a metal.

Reason R :Kinetic energy of the photoelectrons is zero, if the energy of the incident radiation is equal to the work function of a metal.

In the light of the above statements, choose the most appropriate answer from the options given below.

A particle of mass 500 gm is moving in a straight line with velocity v = bx5/2. The work done by the net force during its displacement from x = 0 to x = 4 m is : (Take b = 0.25 m–3/2s–1).
A charge particle moves along circular path in a uniform magnetic field in a cyclotron. The kinetic energy of the charge particle increases to 4 times its initial value. What will be the ratio of new radius to the original radius of circular path of the charge particle

For a series LCR circuit, I vs ω curve is shown :

(a) To the left of ωr, the circuit is mainly capacitive.

(b) To the left of ωr, the circuit is mainly inductive.

(c) At ωr, impedance of the circuit is equal to the resistance of the circuit.

(d) At ωr, impedance of the circuit is 0.

To the left of ωr, the circuit is mainly capacitive.

Choose the most appropriate answer from the options given below.

Amplitude modulated wave is represented by VAM = 10[1+0.4cos(2π×104t] cos(2π×107t). The total bandwidth of the amplitude modulated wave is:
A body of mass 0.5 kg travels on straight line path with velocity v = (3x2 + 4) m/s. The network done by the force during its displacement from x = 0 to x = 2 m is