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

`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.

If y = `log((x + sqrt(x^2 + a^2))/(sqrt(x^2 + a^2) - x))`, find `dy/dx`.

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` = ______.

If y = `tan^-1((6x - 7)/(6 + 7x))`, then `dy/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.

The p.m.f. of a random variable X is as follows:

P (X = 0) = 5k², P(X = 1) = 1 – 4k, P(X = 2) = 1 – 2k and P(X = x) = 0 for any other value of X. Find k.

Given below is the probability distribution of a discrete random variable x.

Find K and hence find P(2 ≤ x ≤ 3)

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.

If y = f(u) is a differentiable function of u and u = g(x) is a differentiate 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 `dy/dx` if y = log(x² + 5)

Using truth table prove that:

~ (p `leftrightarrow` q) ≡ (p ∧ ~ q) ∨ (q ∧ ~ p)

Solve the differential equation

e^x tan y dx + (1 + e^x) sec² y dy = 0

Form the differential equation of all concentric circles having centre at the origin.