HSC Science (Computer Science) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Evaluate the following integrals : `int (3x + 4)/(x^2 + 6x + 5).dx`

In Δ ABC, prove that a² (cos² B - cos² C) + b² (cos² C - cos² A) + c² (cos² A - cos² B) = 0.

Evaluate the following integrals:

`int (2x + 1)/(x^2 + 4x - 5).dx`

Evaluate the following integrals : `int (2x + 3)/(2x^2 + 3x - 1).dx`

Evaluate the following integrals :  `int (3x + 4)/sqrt(2x^2 + 2x + 1).dx`

Evaluate the following integrals:

`int (7x + 3)/sqrt(3 + 2x - x^2).dx`

Evaluate the following integrals : `int sqrt((x - 7)/(x - 9)).dx`

Evaluate the following integrals : `int sqrt((9 - x)/x).dx`

Evaluate the following integral:

`int (3cosx)/(4sin^2x + 4sinx - 1).dx`

Evaluate the following integrals : `int sqrt((e^(3x) - e^(2x))/(e^x + 1)).dx`

With the usual notations, show that
(c² − a² + b²) tan A = (a² − b² + c²) tan B = (b² − c² + a²) tan C

In Δ ABC, if a cos² `"C"/2 + "c cos"^2 "A"/2 = "3b"/2`, then prove that a, b, c are in A.P.

Show that `2 sin^-1 (3/5) = tan^-1(24/7)`

Show that

`tan^-1(1/5) + tan^-1(1/7) + tan^-1(1/3) + tan^-1 (1/8) = pi/4.`

Prove that `tan^-1 sqrt"x" = 1/2 cos^-1 ((1 - "x")/(1 + "x"))`, if x ∈ [0, 1]

Evaluate the following : `int (logx)2.dx`

Show that `(9pi)/8 - 9/4 sin^-1 (1/3) = 9/4 sin^-1 ((2sqrt2)/3)`.

If sin `(sin^-1  1/5 + cos^-1 x) = 1`, then find the value of x.

If `tan^-1 (("x" - 1)/("x" - 2)) + tan^-1 (("x" + 1)/("x" + 2)) = pi/4`, find the value of x.

State whether the following equation has a solution or not?

cos 2θ = `1/3`