HSC Science Class 12 - Tamil Nadu Board of Secondary Education Question Bank Solutions for Mathematics

Using Mean Value Theorem prove that for, a > 0, b > 0, |e^–a – e^–b| < |a – b|

Choose the correct alternative:

The number given by the Rolle’s theorem for the function x³ – 3x², x ∈ [0, 3] is

Choose the correct alternative:

The number given by the Mean value theorem for the function `1/x`, x ∈ [1, 9] is

Express the following physical statements in the form of differential equation.

Radium decays at a rate proportional to the amount Q present

Express the following physical statements in the form of differential equation.

The population P of a city increases at a rate proportional to the product of population and to the difference between 5,00,000 and the population

Express the following physical statements in the form of differential equation.

For a certain substance, the rate of change of vapor pressure P with respect to temperature T is proportional to the vapor pressure and inversely proportional to the square of the temperature

Express the following physical statements in the form of differential equation.

A saving amount pays 8% interest per year compound continuously. In addition, the income from another investment is credited to the amount continuously at the rate of ₹ 400 per year.

Assume that a spherical rain drop evaporates at a rate proportional to its surface area. Form a differential equation involving the rate of change of the radius of the rain drop

Choose the correct alternative:

The differential equation representing the family of curves y = A cos (x + B), where A and B are parameters, is

Choose the correct alternative:

The differential equation of the family of curves y = Ae^x + Be^-x, where A and B are arbitrary constants is

Three fair coins are tossed simultaneously. Find the probability mass function for a number of heads that occurred

A six sided die is marked ‘1’ on one face, ‘3’ on two of its faces, and ‘5’ on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find the probability mass function

A six sided die is marked ‘1’ on one face, ‘3’ on two of its faces, and ‘5’ on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find the cumulative distribution function

A six sided die is marked ‘1’ on one face, ‘3’ on two of its faces, and ‘5’ on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find P(4 ≤ X < 10)

A six sided die is marked ‘1’ on one face, ‘3’ on two of its faces, and ‘5’ on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find P(X ≥ 6)

Find the probability mass function and cumulative distribution function of a number of girl children in families with 4 children, assuming equal probabilities for boys and girls

Suppose a discrete random variable can only take the values 0, 1, and 2. The probability mass function is defined by 
`f(x) = {{:((x^2 + 1)/k","  "for"  x = 0","  1","  2),(0","  "otherwise"):}` 
Find the value of k

Suppose a discrete random variable can only take the values 0, 1, and 2. The probability mass function is defined by 
`f(x) = {{:((x^2 + 1)/k","  "for"  x = 0","  1","  2),(0","  "otherwise"):}` 
Find cumulative distribution function

Suppose a discrete random variable can only take the values 0, 1, and 2. The probability mass function is defined by 
`f(x) = {{:((x^2 + 1)/k","  "for"  x = 0","  1","  2),(0","  "otherwise"):}` 
Find P(X ≥ 1)

The cumulative distribution function of a discrete random variable is given by
F(x) = `{{:(0,  - oo < x < - 1),(0.15, - 1 ≤ x < 0),(0.35, 0 ≤ x < 1),(0.60, 1 ≤ x < 2),(0.85, 2 ≤ x < 3),(1, 3 ≤ x < oo):}`
Find the probability mass function