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

Find the principal value of `cot^-1 ((-1)/sqrt(3))`

If f'(x) = x^–1, then find f(x)

Using truth table verify that:

(p ∧ q)∨ ∼ q ≡ p∨ ∼ q

Evaluate: `int_0^(π/4) sec^4 x  dx`

Find the distance between the parallel lines `x/2 = y/-1 = z/2` and `(x - 1)/2 = (y - 1)/-1 = (z - 1)/2`

If u and v ore differentiable functions of x. then prove that:

`int uv  dx = u intv  dx - int [(du)/(d) intv  dx]dx`

Hence evaluate `intlog x  dx`

`int_0^(π/2) sin^6x cos^2x.dx` = ______.

If `int_2^e [1/logx - 1/(logx)^2].dx = a + b/log2`, then ______.

Let the probability mass function (p.m.f.) of a random variable X be P(X = x) = `""^4C_x (5/9)^x xx (4/9)^(4 - x)`, for x = 0, 1, 2, 3, 4 then E(X) is equal to ______.

If ax² + 2hxy + by² = 0 represents a pair of lines and h² = ab ≠ 0 then find the ratio of their slopes.

If –1 ≤ x ≤ 1, the prove that sin^–1 x + cos^–1 x = `π/2`

If θ is the acute angle between the lines represented by ax² + 2hxy + by² = 0 then prove that tan θ = `|(2sqrt(h^2 - ab))/(a + b)|`

Find the approximate value of sin (30° 30′). Give that 1° = 0.0175^c and cos 30° = 0.866

For the following probability density function of a random variable X, find P(X < 1).

`{:(f(x) = (x + 2)/18,";"  "for" -2 < x < 4),(               = 0,","  "otherwise"):}`

For the following probability density function of a random variable X, find P(|X| < 1).

`{:(f(x) = (x + 2)/18,";"  "for" -2 < x < 4),(               = 0,","  "otherwise"):}`

Examine whether the following statement pattern is a tautology or a contradiction or a contingency.

(p ∧ q) → (q ∨ p)

Evaluate:

`int_(π/4)^(π/2) cot^2x  dx`.

Evaluate: `int_0^1 tan^-1(x/sqrt(1 - x^2))dx`.

Evaluate:

`intcos^-1(sqrt(x))dx`

If sin^–1x – cos^–1x = `π/6`, then x = ______.