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

Solve the following system of equations by the method of inversion.

x – y + z = 4, 2x + y – 3z = 0, x + y + z = 2

The area bounded by the curve y = x³, the X-axis and the Lines x = –2 and x = 1 is ______.

Solve the following differential equation:

`xsin(y/x)dy = [ysin(y/x) - x]dx`

(x² – y²)dx + 2xy dy = 0

`int cos^3x  dx` = ______.

Write `int cotx  dx`.

Find the area of the region bounded by the curve y² = 4x, the X-axis and the lines x = 1, x = 4 for y ≥ 0.

Find the area of the region bounded by the curve y = x² and the line y = 4.

Find the length of the perpendicular drawn from the point P(3, 2, 1) to the line `overliner = (7hati + 7hatj + 6hatk) + λ(-2hati + 2hatj + 3hatk)`

Find separate equations of lines represented by x² – y² + x + y = 0.

Express the following compound statement symbolically:

Delhi is in India but Dhaka is not in Sri Lanka

Express the following compound statement symbolically:

3 + 8 ≥ 12 if and only if 5 × 4 ≤ 25

In any ΔABC, prove that:

(b + c) cos A + (c + a) cos B + (a + b) cos C = a + b + c.

Prove that:

`int 1/sqrt(x^2 - a^2) dx = log |x + sqrt(x^2 - a^2)| + c`.

Find `dy/dx` if ,

`x= e^(3t) , y = e^(4t+5)`

Evaluate:

`int 1/(1 + cosα . cosx)dx`

Show that the joint equation of a pair of straight lines through the origin is a homogeneous equation of second degree in x and y.

lf y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, such that the composite function y = f[g(x)] is a differentiable function of x, then prove that:

`dy/dx = dy/(du) xx (du)/dx`

Hence, find `d/dx[log(x^5 + 4)]`.

Find the area of the region lying in the first quadrant and bounded by y = 4x², x = 0, y = 2 and y = 4.

If f(x) = `sqrt(7*g(x) - 3)`, g(3) = 4 and g'(3) = 5, find f'(3).