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

In ΔABC prove that `sin  "A"/(2). sin  "B"/(2). sin  "C"/(2) = ["A(ΔABC)"]^2/"abcs"`

Find the principal value of the following: `sin^-1 (1/2)`

Find the principal value of the following: cosec^{- 1}(2)

Find the principal value of the following: tan^-1(– 1)

Find the principal value of the following: tan^{- 1}( - √3)

Find the principal value of the following: sin^-1 `(1/sqrt(2))`

Find the principal value of the following: cos^{- 1}`(-1/2)`

Evaluate the following:

`tan^-1(1) + cos^-1(1/2) + sin^-1(1/2)`

Evaluate the following:

`cos^-1(1/2) + 2sin^-1(1/2)`

Evaluate the following:

`tan^-1 sqrt(3) - sec^-1 (-2)`

Evaluate the following:

`"cosec"^-1(-sqrt(2)) + cot^-1(sqrt(3))`

Prove the following: 

`sin^-1(1/sqrt(2)) -3sin^-1(sqrt(3)/2) = -(3π)/(4)`

Prove the following:

`sin^-1(-1/2) + cos^-1(-sqrt(3)/2) = cos^-1(-1/2)`

Prove the following:

`sin^-1(3/5) + cos^-1(12/13) = sin^-1(56/65)`

Prove the following:

`cos^-1(3/5) + cos^-1(4/5) = pi/(2)`

Prove the following:

`tan^-1(1/2) + tan^-1(1/3) = pi/(4)`

Prove the following: 

`2tan^-1(1/3) = tan^-1(3/4)`

Prove the following:

`tan^-1["cosθ + sinθ"/"cosθ - sinθ"] = pi/(4) + θ, if θ ∈ (- pi/4, pi/4)`

Prove the following:

`tan^-1[sqrt((1 - cosθ)/(1 + cosθ))] = θ/(2)`, if θ ∈ (– π, π).

A table of values of f, g, f' and g' is given :

If r(x) =f [g(x)] find r' (2).