HSC Arts Class 12 - Tamil Nadu Board of Secondary Education Question Bank Solutions

Using the Lagrange’s mean value theorem determine the values of x at which the tangent is parallel to the secant line at the end points of the given interval:

`f(x) = (x - 2)(x - 7), x ∈ [3, 11]`

Show that the value in the conclusion of the mean value theorem for `f(x) = 1/x` on a closed interval of positive numbers [a, b] is `sqrt("ab")`

Show that the value in the conclusion of the mean value theorem for `f(x) = "A"x^2 + "B"x + "C"` on any interval [a, b] is `("a" + "b")/2`

A race car driver is kilometer stone 20. If his speed never exceeds 150 km/hr, what is the maximum kilometer he can reach in the next two hours

Suppose that for a function f(x), f'(x) ≤ 1 for all 1 ≤ x ≤ 4. Show that f(4) – f(1) ≤ 3

Does there exist a differentiable function f(x) such that f(0) = – 1, f(2) = 4 and f(x) ≤ 2 for all x. Justify your answer

Show that there lies a point on the curve `f(x) = x(x + 3)e^(pi/2), -3 ≤ x ≤ 0` where tangent drawn is parallel to the x-axis

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