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

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

`int x^2/sqrt(1 - x^6)dx` = ______.

If p, q are true statements and r, s are false statements, then write the truth value of the compound statement

(p `→` ∼ r) `→` (q ∧ s)

Form the differential equation whose general solution is y = a cos 2x + b sin 2x.

Food F₁ contains 2, 6, 1 units and food F₂ contains 1, 1, 3 units of proteins, carbohydrates, fats respectively per kg. 8, 12 and 9 units of proteins, carbohydrates and fats is the weekly minimum requirement for a person. The cost of food F₁ is Rs. 85 and food F₂ is Rs. 40 per kg. Formulate the L.P.P. to minimize the cost.

If y = `sqrt((1 - x)/(1 + x))`, then `(1 - x^2) dy/dx + y` = ______.