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

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`

Solve: `tan^-1 ("1 - x"/"1 + x") = 1/2 (tan^-1 "x")`, for x > 0.

Solve: `tan^-1 ("1 - x"/"1 + x") = 1/2 (tan^-1 "x")`, for x > 0.

Choose the correct option from the given alternatives : 

`int (1 + x + sqrt(x + x^2))/(sqrt(x) + sqrt(1 + x))*dx` =

Choose the correct options from the given alternatives :

`int sqrt(cotx)/(sinx*cosx)*dx` =

Choose the correct options from the given alternatives :

`int (e^x(x - 1))/x^2*dx` =

Choose the correct options from the given alternatives :

`int f x^x (1 + log x)*dx`

Choose the correct options from the given alternatives :

`2 int (cos^2x - sin^2x)/(cos^2x + sin^2x)*dx` =

Choose the correct options from the given alternatives : 

`int dx/(cosxsqrt(sin^2x - cos^2x))*dx` =

`int logx/(log ex)^2*dx` = ______.