The number of solutions of the equation 2sin-1 (√x2 - x + 1) + cos-1 (√x2 - x) = (3π/2) is

Question

The number of solutions of the equation 2sin-1 (√x2 - x + 1) + cos-1 (√x2 - x) = (3π/2) is (A) 0 (B) Infinite (C) 2 (D) 4

Tardigrade Admin · Accepted Answer

sin-1√x, cos-1√x are defined for x le 1 and x ge 0 ∴ √x2 - x + 1 le 1 and √x2 - x ge 0 ⇒ x2 - x le 0 and x2 - x ge 0
⇒ x2 - x = 0
⇒ x = 1, 0 ∴ There are two solutions, both satisfies the equation.

Q. The number of solutions of the equation $2 s i n^{- 1} (x^{2} - x + 1) + co s^{- 1} (x^{2} - x) = \frac{3 π}{2}$ is

$0$

Infinite

$2$

$4$

$s i n^{- 1} x, co s^{- 1} x$ are defined for $x \leq 1$ and $x \geq 0$
$∴ x^{2} - x + 1 \leq 1$ and $x^{2} - x \geq 0$
$\Rightarrow x^{2} - x \leq 0$ and $x^{2} - x \geq 0$
$\Rightarrow x^{2} - x = 0$
$\Rightarrow x = 1, 0$
$∴$ There are two solutions, both satisfies the equation.

Q. The number of solutions of the equation 2sin−1(x2−x+1​)+cos−1(x2−x​)=23π​ is

0

Infinite

2

4

sin−1x​,cos−1x​ are defined for x≤1 and x≥0 ∴x2−x+1​≤1 and x2−x​≥0 ⇒x2−x≤0 and x2−x≥0 ⇒x2−x=0 ⇒x=1,0 ∴ There are two solutions, both satisfies the equation.

Q. The number of solutions of the equation $2 s i n^{- 1} (x^{2} - x + 1) + co s^{- 1} (x^{2} - x) = \frac{3 π}{2}$ is

$0$

$2$

$4$

$s i n^{- 1} x, co s^{- 1} x$ are defined for $x \leq 1$ and $x \geq 0$
$∴ x^{2} - x + 1 \leq 1$ and $x^{2} - x \geq 0$
$\Rightarrow x^{2} - x \leq 0$ and $x^{2} - x \geq 0$
$\Rightarrow x^{2} - x = 0$
$\Rightarrow x = 1, 0$
$∴$ There are two solutions, both satisfies the equation.