HSC Science (General) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Choose correct alternatives :

The equation of the plane in which the line `(x - 5)/(4) = (y - 7)/(4) = (z + 3)/(-5) and (x - 8)/(7) = (y - 4)/(1) = (z - 5)/(3)` lie, is

Choose correct alternatives :

The foot of perpendicular drawn from the point (0,0,0) to the plane is (4, -2, -5) then the equation of the plane is

Solve the following :

Find the perpendicular distance of the origin from the plane 6x + 2y + 3z - 7 = 0

Solve the following :

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x + 3y + 6z = 49.

Solve the following :

Reduce the equation `bar"r".(6hat"i" + 8hat"j" + 24hat"k")` = 13 normal form and hence find
(i) the length of the perpendicular from the origin to the plane.
(ii) direction cosines of the normal.

Check the validity of the Rolle’s theorem for the following functions : f(x) = x² – 4x + 3, x ∈ [1, 3]

Check the validity of the Rolle’s theorem for the following functions : f(x) = e^–x sin x, x ∈ [0, π].

Check the validity of the Rolle’s theorem for the following functions : f(x) = 2x² – 5x + 3, x ∈ [1, 3].

Check the validity of the Rolle’s theorem for the following functions : f(x) = sin x – cos x + 3, x ∈ [0, 2π].

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].