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

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)`

The joint equation of the angle bisectors of the angles between the lines 4x² – 16xy + 7y² = 0 is ______.

The value of `int e^x((1 + sinx)/(1 + cosx))dx` is ______.

Evaluate `int_(-π/2)^(π/2) sinx/(1 + cos^2x)dx`

If the lines represented by 5x² – 3xy + ky² = 0 are perpendicular to each other, find the value of k.

If `int_0^π f(sinx)dx = kint_0^π f(sinx)dx`, then find the value of k.

Evaluate `int tan^-1x  dx`

Evaluate:

`int (sin(x - a))/(sin(x + a))dx`

If u and v are two differentiable functions of x, then prove that `intu*v*dx = u*intv  dx - int(d/dx u)(intv  dx)dx`. Hence evaluate: `intx cos x  dx`

Find feasible solution for the following system of linear inequation graphically.

3x + 4y ≥ 12, 4x + 7y ≤ 28, x ≥ 0, y ≥ 0

Evaluate:

`int1/(x^2 + 25)dx`

Find the approximate value of tan⁻¹ (1.002).
[Given: π = 3.1416]

Solve the following system of equations by the method of reduction:

x + y + z = 6, y + 3z = 11, x + z = 2y.

Prove that the acute angle θ between the lines represented by the equation ax² + 2hxy+ by² = 0 is tanθ = `|(2sqrt(h^2 - ab))/(a + b)|` Hence find the condition that the lines are coincident.

If `y=cos^-1(2xsqrt(1-x^2))`, find dy/dx

Find `dy/dx if y=cos^-1(sqrt(x))`

find dy/dx if `y=tan^-1((6x)/(1-5x^2))`