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

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"):}`

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 = ______.

Prove that:

tan^–1x + tan^–1y = `π + tan^-1((x + y)/(1 - xy))`, provided x > 0, y > 0, xy > 1

If the straight lines `(x - 1)/k = (y - 2)/2 = (z - 3)/3` and `(x - 2)/3 = (y - 3)/k = (z - 1)/2` intersect at a point, then the integer k is equal to ______.

Find k, if the following function is p.d.f. of r.v.X:

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

Evaluate:

`int((1 + sinx)/(1 + cosx))e^x dx`

The value of `tan(cos^-1  4/5 + tan^-1  2/3)` is ______.

Evaluate:

`int_0^(π/2) sin^8x  dx`

Evaluate:

`int_(-π/2)^(π/2) |sinx|dx`

Find the value of `sin(2cos^-1  sqrt(5)/3)`.

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`