HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

Evaluate: `int_0^(1/2) 1/((1 - 2x^2) sqrt(1 - x^2))  "d"x`

Evaluate: `int_0^(pi/4)  (sec^2x)/(3tan^2x + 4tan x + 1)  "d"x`

Evaluate: `int_(1/sqrt(2))^1  (("e"^(cos^-1x))(sin^-1x))/sqrt(1 - x^2)  "d"x`

Evaluate: `int_0^1 (log(x + 1))/(x^2 + 1)  "d"x`

Evaluate: `int_0^pi x*sinx*cos^2x* "d"x`

Evaluate: `int_(-1)^1 (1 + x^2)/(9 - x^2)  "d"x`

Evaluate: `int_0^1 (1/(1 + x^2)) sin^-1 ((2x)/(1 + x^2))  "d"x`

Evaluate: `int_0^(pi/4)  (cos2x)/(1 + cos 2x + sin 2x)  "d"x`

Evaluate: `int_0^(pi/4) log(1 + tanx)  "d"x`

Evaluate: `int_0^pi 1/(3 + 2sinx + cosx)  "d"x`

Evaluate: `int_0^(π/4) sec^4 x  dx`

Solve the following system of equations by the method of inversion.

x – y + z = 4, 2x + y – 3z = 0, x + y + z = 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 ______.

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

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"):}`