PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions

Solve the following equation for x:

`cos^-1((x^2-1)/(x^2+1))+1/2tan^-1((2x)/(1-x^2))=(2x)/3`

Solve the following equation for x:

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

Prove that `2tan^-1(sqrt((a-b)/(a+b))tan  theta/2)=cos^-1((a costheta+b)/(a+b costheta))`

Prove that:

`tan^-1  (2ab)/(a^2-b^2)+tan^-1  (2xy)/(x^2-y^2)=tan^-1  (2alphabeta)/(alpha^2-beta^2),`   where `alpha=ax-by  and  beta=ay+bx.`

For any a, b, x, y > 0, prove that:

`2/3tan^-1((3ab^2-a^3)/(b^3-3a^2b))+2/3tan^-1((3xy^2-x^3)/(y^3-3x^2y))=tan^-1  (2alphabeta)/(alpha^2-beta^2)`

`where  alpha =-ax+by, beta=bx+ay`

Write the value of `sin^-1((-sqrt3)/2)+cos^-1((-1)/2)`

Write the difference between maximum and minimum values of  sin⁻¹ x for x ∈ [− 1, 1].

If `sin^-1x+sin^-1y+sin^-1z=(3pi)/2,`  then write the value of x + y + z.

If x > 1, then write the value of sin−1 `((2x)/(1+x^2))` in terms of tan⁻¹ x.

If x < 0, then write the value of cos⁻¹ `((1-x^2)/(1+x^2))` in terms of tan⁻¹ x.

Write the value of tan⁻¹x + tan⁻¹ `(1/x)`for x > 0.

Write the value of tan⁻¹ x + tan⁻¹ `(1/x)`  for x < 0.

What is the value of cos⁻¹ `(cos  (2x)/3)+sin^-1(sin  (2x)/3)?`

If −1 < x < 0, then write the value of `sin^-1((2x)/(1+x^2))+cos^-1((1-x^2)/(1+x^2))`

Write the value of sin (cot⁻¹ x).

Write the value of

\[\cos^{- 1} \left( \frac{1}{2} \right) + 2 \sin^{- 1} \left( \frac{1}{2} \right)\].

Write the value of cos⁻¹ (cos 1540°).

Write the value of sin⁻¹

\[\left( \sin( -{600}°) \right)\].

Write the value of cos\[\left( 2 \sin^{- 1} \frac{1}{3} \right)\]

Write the value of sin⁻¹ (sin 1550°).