HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

If each side of an equilateral triangle increases at the rate of `(sqrt(2)"cm")/sec`, find the rate of increase of its area when its side of length 3 cm.

If water is poured into an inverted hollow cone whose semi-vertical angle is 30°, so that its depth (measured along the axis) increases at the rate of`( 1"cm")/sec`. Find the rate at which the volume of water increasing when the depth is 2 cm.

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Let f(x) and g(x) be differentiable for 0 ≤ x ≤ 1 such that f(0) = 0, g(0), f(1) = 6. Let there exist a real number c in (0, 1) such that f'(c) = 2g'(c), then the value of g(1) must be ______.

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If x = –1 and x = 2 are the extreme points of y = αlogx + βx² + x`, then ______.

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The normal to the curve x² + 2xy – 3y² = 0 at (1, 1)

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The equation of the tangent to the curve y = `1 - e^(x/2)` at the point of intersection with Y-axis is

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If the tangent at (1, 1) on y² = x(2 – x)² meets the curve again at P, then P is

Solve the following : If the curves ax² + by² = 1 and a'x² + b'y² = 1, intersect orthogonally, then prove that `(1)/a - (1)/b = (1)/a' - (1)/b'`.

Solve the following : Determine the area of the triangle formed by the tangent to the graph of the function y = 3 – x² drawn at the point (1, 2) and the coordinate axes.

Solve the following : Find the equation of the tangent and normal drawn to the curve y⁴ – 4x⁴ – 6xy = 0 at the point M (1, 2).

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The principal solutions of equation sin θ = `- 1/2` are ______.

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The principal solutions of equation cot θ = `sqrt3` are ______.

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The general solution of sec x = `sqrt(2)` is ______.

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If cos pθ = cos qθ, p ≠ q, then ______.

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If polar coordinates of a point are `(2, pi/4)`, then its cartesian coordinates are

If `sqrt3`cos x - sin x = 1, then general value of x is ______.

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In Δ ABC if ∠A = 45°, ∠B = 30°, then the ratio of its sides are

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In ΔABC if c² + a² – b² = ac, then ∠B = ____.

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In ΔABC, ac cos B - bc cos A = _______

If in a triangle, the angles are in A.P. and b: c = `sqrt3: sqrt2`, then A is equal to