HSC Science (General) इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Find the equations of tangents and normals to the following curve at the indicated points on them:

x = sin θ and y = cos 2θ at θ = `pi/(6)`

Find the equations of tangents and normals to the following curves at the indicated points on them : `x = sqrt(t), y = t  - (1)/sqrt(t)` at = 4.

Find the point on the curve y = `sqrt(x - 3)` where the tangent is perpendicular to the line 6x + 3y – 5 = 0.

Find the points on the curve y = x³ – 2x² – x where the tangents are parllel to 3x – y + 1 = 0.

Find the equation of the tangents to the curve x² + y² – 2x – 4y + 1 =0 which a parallel to the X-axis.

Find the equations of the normals to the curve 3x² – y² = 8, which are parallel to the line x + 3y = 4.

If the line y = 4x – 5 touches the curves y² = ax³ + b at the point (2, 3), find a and b.

A particle moves along the curve 6y = x³ + 2. Find the points on the curve at which y-coordinate is changing 8 times as fast as the x-coordinate.

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 ______.