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

Evaluate:

`int_(-π/2)^(π/2) |sinx|dx`

Find the combined equation of the pair of lines through the origin and making an angle of 30° with the line 2x – y = 5

Evaluate:

`int e^(ax)*cos(bx + c)dx`

Evaluate `int_(π/6)^(π/3) cos^2x  dx`

Evaluate:

`int_-4^5 |x + 3|dx`

The value of `int_2^(π/2) sin^3x  dx` = ______.

Evaluate:

`int_(π/6)^(π/3) (root(3)(sinx))/(root(3)(sinx) + root(3)(cosx))dx`

Evaluate:

`int_0^(π/2) (sin 2x)/(1 + sin^4x)dx`

Evaluate:

`inte^x sinx  dx`

`int_0^1 x^2/(1 + x^2)dx` = ______.

Evaluate:

`int e^(logcosx)dx`

If θ is the acute angle between the lines given by 3x² – 4xy + by² = 0 and tan θ = `1/2`, find b.

Find the value of ‘a’ if `int_2^a (x + 1)dx = 7/2`

Evaluate:

`int (logx)^2 dx`

Evaluate:

`int_0^(π/2) sinx/(1 + cosx)^3 dx`

`int (sin^-1 sqrt(x) + cos^-1 sqrt(x))dx` = ______.

If the p.d.f. of X is

f(x) = `x^2/18,   - 3 < x < 3`

      = 0,        otherwise

Then P(X < 1) is ______.

Prove that `int sqrt(x^2 - a^2)dx = x/2 sqrt(x^2 - a^2) - a^2/2 log(x + sqrt(x^2 - a^2)) + c`

Find the c.d.f. F(x) associated with the following p.d.f. f(x)

f(x) = `{{:(3(1 - 2x^2)",", 0 < x < 1),(0",", "otherwise"):}`

Find `P(1/4 < x < 1/3)` by using p.d.f. and c.d.f.

Prove that: `int_0^1 logx/sqrt(1 - x^2)dx = π/2 log(1/2)`