HSC Arts (English Medium) इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Check the validity of the Rolle’s theorem for the following function:

f(x) = x², if 0 ≤ x ≤ 2

= 6 – x, if 2 < x ≤ 6.

Check the validity of the Rolle’s theorem for the following function:

f(x) = `x^(2/3), x ∈ [ - 1, 1]`

Given an interval [a, b] that satisfies hypothesis of Rolle's theorem for the function f(x) = x⁴ + x² – 2. It is known that a = – 1. Find the value of b.

Verify Rolle’s theorem for the following functions:

f(x) = sin x + cos x + 7, x ∈ [0, 2π]

Verify Rolle’s theorem for the following functions  : f(x) = `sin(x/2), x ∈ [0, 2pi]` 

Verify Rolle’s theorem for the following functions : f(x) = x² – 5x + 9, x ∈ 1, 4].

If Rolle's theorem holds for the function f(x) = x³ + px² + qx + 5, x ∈ [1, 3] with c = `2 + (1)/sqrt(3)`, find the values of p and q.

If Rolle’s theorem holds for the function f(x) = (x –2) log x, x ∈ [1, 2], show that the equation x log x = 2 – x is satisfied by at least one value of x in (1, 2).

The function f(x) = `x(x + 3)e^(-(x)/2)` satisfies all the conditions of Rolle's theorem on [– 3, 0]. Find the value of c such that f'(c) = 0.

Choose the correct option from the given alternatives :  

If the function f(x) = ax³ + bx² + 11x – 6 satisfies conditions of Rolle's theoreem in [1, 3] and  `f'(2 + 1/sqrt(3))` = 0, then values of a and b are respectively

Verify Rolle’s theorem for the function f(x)  `(2)/(e^x + e^-x)` on [– 1, 1].

Integrate the following w.r.t. x : `(x^2 + 2)/((x - 1)(x + 2)(x + 3)`

Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`

Integrate the following w.r.t. x : `(12x + 3)/(6x^2 + 13x - 63)`

Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`

Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`

Integrate the following w.r.t. x:

`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`

Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`

Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`

Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`