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

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`

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 ______.

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"):}`

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 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`

Evaluate:

`int_0^(π/2) sin^8x  dx`