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Consider a spherical gaseous cloud of mass density ρ(r) in free space where r is the radial distance from its center. The gaseous cloud is made of particles of equal mass m moving in circular orbits about the common center with the same kinetic energy K. The force acting on the particles is their mutual gravitational force. If ρ(r) is constant in time, the particle number density n(r) = ρ(r)/m is [G is universal gravitational constant]

a) 3K/πr2m2G b) K/2πr2m2G c) K/6πr2m2G d) K/πr2m2G Answer: Solution:b) K/2πr2m2G Gravitational force = Centripetal force FG = FC GMm/r2 = mv2/r GMm/r2 = 2... View Article

A non-homogenous sphere of radius R has the following density variation.

A non-homogenous sphere of radius R R has the following density variation. \(\begin{array}{l}\rho \left\{\begin{matrix}\rho _{0}; &r \leq R/3 \\ \rho... View Article

What is the probability of getting any number when we roll a single die for once?

Solution: Possible outcomes are 1, 2, 3, 4, 5, 6. Number of outcomes = 6 The probability of getting any... View Article

Find the intervals on which f(x) = x⁴ – 8x² is increasing and decreasing.

Solution: Given f(x) = x4 – 8x2 f’(x) = 4x3 – 16x f’(x) = 0 => 4x3 – 16x =... View Article

The LCM of two numbers is 25 times their HCF. the product of the two numbers is 2025. Find the HCF.

(a) 81 (b) 45 (c) 9 (d) 18 Solution: Given LCM = 25HCF Product of two numbers = 2025 We... View Article

Two students appeared at an examination. One of them secured 9 marks more than the other and his marks were 56% of the sum of their marks. What are the marks obtained by both of them?

(a) 25 and 78 (b) 58 and 77 (c) 33 and 42 (d) 56 and 63 Solution: Let x be... View Article

A bag contains 1 Rupee 2 Rupees and 5 Rupees coins in the ratio 4:8:5. The total amount was Rs.90. The number of 5 Rupee coins equal

(a) 8 (b) 16 (c) 15 (d) 10 Solution: Given ratio = 4:8:5 Total amount = 90 Let number of... View Article

Set = {1, 2, 2, 3, 3, 3, 4, 4, 4, 4}. Evaluate the following expression for the given set of numbers. 3(mean)×2(mode) – 6(median).

Solution: Given set is {1, 2, 2, 3, 3, 3, 4, 4, 4, 4}. Number of observations, n = 10... View Article

If x and y are two co-primes, then their LCM is?

(a) xy (b) x+y (c) x/y (d) 1 Solution: Co-prime numbers are the numbers whose common factor is only 1.... View Article

Find the mode and median of the following frequency distribution:

x 10 11 12 13 14 15 f 1 4 7 5 9 8 Solution: x f c.f 10 1... View Article

On selling 17 balls at Rs. 720, there is a loss equal to the cost price of 5 balls. The cost price of a ball is:

(a) 45 (b) 50 (c) 55 (d) 60 Solution: Given that loss = cost price of 5 balls. Cost price... View Article

The number of permutations of letters a, b, c, d, e, f, g so that neither the pattern beg nor cad appears is

(a) 7!/3!3! (b) 7!/2!3!3! (c) 4806 (d) none of these Solution: Given letters are a, b, c, d, e, f,... View Article

If f(x) be a polynomial function satisfying f(x).f(1/x) = f(x) + f(1/x) and f(4) = 65. Then find f(6).

(a) 65 (b) 217 (c) 63 (d) none of these Solution: Let f(x) = a0xn + a1xn-1+…an-1x + an be... View Article

The point on the curve 3y = 6x – 5x³, the normal at which passes through the origin is

(a) (1, 1/3) (b) (1/3, 1) (c) (2, -28/3) (d) none of these Solution: Let P(x1, y1) be the required... View Article

If the equation x⁴ – 4x³ + ax² + bx + 1 = 0 has four positive roots, then the value of |a – b| is

(a) 2 (b) 1 (c) 10 (d) 3 Solution: Given x4 – 4x3 + ax2 + bx + 1 =... View Article

Convert 25 radian to degree.

Solution: We know π radian = 1800 1 radian = 1800/π 25 radian = (180/π)25 = 1433.120

Look at this series 1, 1, 2, 3, 5, 8, 13 … what number should come next?

Solution: Given series is 1, 1, 2, 3, 5, 8, 13,… We got the third number by adding first and second... View Article

The decimal equivalent of the binary number 11010101 is

Solution: The given number is 11010101. To convert a binary number to a decimal number, write the powers of 2... View Article

radon is obtained from radium due to

Radon is obtained from radium by the alpha decay process. In the alpha decay process an atomic nucleus emits an... View Article

if 0.224l of H2 gas is formed at the cathode, the volume of O2 gas formed at the anode under identical conditions is:

The reaction is H2O → H2 + ½ O2 Here H2 gas form at cathode and O2 gas formed at... View Article

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