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

The perpendicular distance of the plane 2x + 3y – z = k from the origin is `sqrt(14)` units, the value of k is ______.

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The equation of the plane passing through (2, -1, 3) and making equal intercepts on the coordinate axes is

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The direction cosines of the normal to the plane 2x – y + 2z = 3 are ______ 

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The equation of the plane passing through the points (1, −1, 1), (3, 2, 4) and parallel to the Y-axis is ______  

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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

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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.