HSC Arts (English Medium) इयत्ता १२ वी - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

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).

Evaluate:

`int sqrt((a - x)/x) dx`

If in ΔABC, `sin  A/2 * sin  C/2 = sin  B/2` and 2s is the perimeter of the triangle, then s = ______.

Evaluate:

`int(sqrt(tanx) + sqrt(cotx))dx`

If x = Φ(t) is a differentiable function of t, then prove that:

`int f(x)dx = int f[Φ(t)]*Φ^'(t)dt`

Hence, find `int(logx)^n/x dx`.

Write the contrapositive of the inverse of the statement:

‘If two numbers are not equal, then their squares are not equal’.

From the following set of statements, select two statements which have similar meaning.

1. If a man is judge, then he is honest.
2. If a man is not a judge, then he is not honest.
3. If a man is honest, then he is a judge.
4. If a man is not honest, then he is not a judge.

The distance of the point (2, – 3, 1) from the line `(x + 1)/2 = (y - 3)/3 = (z + 1)/-1` is ______.

`int "cosec"^4x  dx` = ______.

The differential equation for a²y = log x + b, is ______.

Evaluate:

`int sin^2(x/2)dx`

Find the area common to the parabola y² = x – 3 and the line x = 5.

Find the area bounded by the lines y = 5x – 10, X-axis and x = 5.

Solve the differential equation

cos²(x – 2y) = `1 - 2dy/dx`

Two kinds of foods A and B are being considered to form a weekly diet. The minimum weekly requirements of fats, Carbohydrates and proteins are 12, 16 and 15 units respectively. One kg of food A has 2, 8 and 5 units respectively of these ingredients and one kg of food B has 6, 2 and 3 units respectively. The price of food A is Rs. 4 per kg and that of food B is Rs. 3 per kg. Formulate the L.P.P. and find the minimum cost.

If 2x = `y^(1/m) + y^(-1/m)`, then show that `(x^2 - 1) (dy/dx)^2` = m²y²