Question

Assertion :Consider a curve C: $y={\cos }^{-1}(2x-1)$ and a straight line $L:2px-4y+2\pi -p=0$.The set of values of $p$ for which the line $L$ intersects the curve at three distinct points is $[-2\pi ,-4]$. Reason: The line $L$ is always passing through point of inflection of the curve $C$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$y={\cos }^{-1}(2x-1)$
$\frac{dy}{dx}=\frac{-2}{\sqrt{1-{(2x-1)}^2}}=\frac{-1}{\sqrt{{x-x}^2}}=-{(x-x^2)}^{\frac{1}{2}}$
$\frac{d^{2}y}{{dx}^2}=\frac{1}{2}\frac{(1-2x)}{{(x-x^2)}^{\frac{3}{2}}}=0\Rightarrow x=\frac{1}{2}$

The point $(\frac{1}{2},\frac{\pi }{2})$ is a point of inflection of the curve
is and it satisfies the line L.

The line L is always passing through point of inflexion of the curve C.

Slope of the tangent to the curve C at $(\frac{1}{2},\frac{\pi }{2})$
$\frac{dy}{dx}=-2$

As the slope decreases from -2, line cuts the curve at three distinct points and minimum slope of the line when it intersects the curve at three distinct points is :

$\frac{\pi -\frac{\pi }{2}}{0-\frac{1}{2}}=-\pi$

$\therefore \frac{p}{2}\in [-\pi .-2)\Rightarrow p\in [-2\pi ,-4)$