HSC Science (General) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Evaluate:

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

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.

`int 1/(sin^2x cos^2x)dx` = ______.

Using the statements

p: Seema is fat,

q: Seema is happy,

Write the following statements in symbolic form;

1. Seema is thin and happy.
2. If Seema is fat then she is unhappy.

Evaluate:

`int(cos 2x)/sinx dx`

The perimeter of ΔABC is 20, ∠A = 60°, area of ΔABC = `10sqrt(3)`, then find the values of a, b, c.

Evaluate:

`intsqrt(3 + 4x - 4x^2)  dx`

The sum of the slopes of the lines given by x² – 2λxy – 7y² = 0 is 4 times their product, then the value of λ is ______.

`int (cos4x)/(sin2x + cos2x)dx` = ______.

Write the negation of (p `leftrightarrow` q).

Evaluate:

`int sin^3x cos^3x  dx`

The side of a square is increasing at the rate of 0.5 cm/sec. Find the rate of increase of the perimeter when the side of the square is 10 cm long.

Find the particular solution of the differential equation `x^2 dy/dx + y^2 = xy dy/dx`, if y = 1 when x = 1.

In ΔABC, a = 3, b = 1, cos(A – B) = `2/9`, find c.