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(D) Given the parametric equations $x = \sec^2 t$ and $y = \cot t$.Eliminating $t$ using the identity $\sec^2 t = 1 + 	an^2 t = 1 + \frac{1}{\cot^2 t

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$\frac{3\sqrt{5}}{2}$

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(D) Given the parametric equations $x = \sec^2 t$ and $y = \cot t$.Eliminating $t$ using the identity $\sec^2 t = 1 + 	an^2 t = 1 + \frac{1}{\cot^2 t}$,we get $x = 1 + \frac{1}{y^2}$,which simplifies to $y^2(x - 1) = 1$.At $t = \pi / 4$,$x = \sec^2(\pi / 4) = 2$ and $y = \cot(\pi / 4) = 1$. So,$P = (2, 1)$.Differentiating $y^2(x - 1) = 1$ with respect to $x$: $2y \frac{dy}{dx}(x - 1) + y^2 = 0$.At $P(2, 1)$,$2(1) \frac{dy}{dx}(2 - 1) + 1^2 = 0 \implies 2 \frac{dy}{dx} + 1 = 0 \implies \frac{dy}{dx} = -1/2$.The equation of the tangent at $P(2, 1)$ is $y - 1 = -\frac{1}{2}(x - 2) \implies 2y - 2 = -x + 2 \implies x + 2y = 4$.Substitute $x = 4 - 2y$ into the curve equation $y^2(x - 1) = 1$: $y^2(4 - 2y - 1) = 1 \implies y^2(3 - 2y) = 1 \implies 3y^2 - 2y^3 = 1 \implies 2y^3 - 3y^2 + 1 = 0$.Since $P$ is a point of tangency,$y = 1$ is a double root. Dividing $2y^3 - 3y^2 + 1$ by $(y - 1)^2 = y^2 - 2y + 1$,we get $(y - 1)^2(2y + 1) = 0$.The other root is $y = -1/2$. Then $x = 4 - 2(-1/2) = 4 + 1 = 5$. So,$Q = (5, -1/2)$.The distance $|PQ| = \sqrt{(5 - 2)^2 + (-1/2 - 1)^2} = \sqrt{3^2 + (-3/2)^2} = \sqrt{9 + 9/4} = \sqrt{45/4} = \frac{3\sqrt{5}}{2}$.

List-$I$	List-$II$
$A. \frac{d}{dx}\left(\tan^{-1}\left(\sqrt{\frac{1-\cos x}{1+\cos x}}\right)\right)$	$(i) \log(x+\sqrt{1+x^2})$
$B. \frac{d}{dx}\left(\frac{3+\|x-1\|}{3x+4}\right)$	$(ii) -\frac{4x}{(1+x^2)^2}$
$C. \sinh^{-1} x$	$(iii) \frac{1}{2}$
$D. \frac{d^2}{dx^2}\left(\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right)$	$(iv) \frac{1}{\sqrt{1+x^2}}$
	$(v) \text{not differentiable at } x=1$

$(a)$ $x\|x\|$	$(i)$ continuous in $(-1, 1)$
$(b)$ $\sqrt{\|x\|}$	$(ii)$ differentiable in $(-1, 1)$
$(c)$ $x+[x]$	$(iii)$ strictly increasing in $(-1, 1)$
$(d)$ $\|x-1\|+\|x+1\|$	$(iv)$ not differentiable at,at least one point in $(-1, 1)$

$A$ curve is represented by the equations $x = \sec^2 t$ and $y = \cot t$,where $t$ is a parameter. If the tangent at the point $P$ on the curve where $t = \pi / 4$ meets the curve again at the point $Q$,then $|PQ|$ is equal to

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