_ x 4 _2^ ^2 x f(t) d t x^2- ^2 16 equals

MCQ

$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}}$ equals

✓

Answer

Correct option: A.

$\frac{8}{\pi} f(2)$

a
$\lim _{x \rightarrow \frac{\pi}{4}} \frac{\int_2^{\sec ^2 x} f(t) d t}{x^2-\frac{\pi^2}{16}} \quad\left(\frac{0}{0} \text { form }\right)$

Let $L=\lim _{x \rightarrow \frac{\pi}{4}} \frac{f\left(\sec ^2 x\right) 2 \sec x \sec x \tan x}{2 x}$

$\therefore \mathrm{L}=\frac{2 \mathrm{f}(2)}{\pi / 4}=\frac{8 \mathrm{f}(2)}{\pi} \text {. }$

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Statement-$2$ : Two lines are skew lines if there exists no plane passing through them.

List $I$	List $II$
$P.\quad$ Volume of parallelopiped determined by vectors $\vec{a}, \vec{b}$ and $\overrightarrow{ c }$ is $2$ . Then the volume of the parallelepiped determined by vectors $2(\vec{a} \times \vec{b}), 3(\vec{b} \times \vec{c})$ and $(\vec{c} \times \vec{a})$ is	$1.\quad$ $100$
$Q.\quad$ Volume of parallelepiped determined by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $5$ . Then the volume of the parallelepiped determined by vectors $3(\overrightarrow{ a }+\overrightarrow{ b }),(\overrightarrow{ b }+\overrightarrow{ c })$ and $2(\overrightarrow{ c }+\overrightarrow{ a })$ is	$2.\quad$ $30$
$R.\quad$ Area of a triangle with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $20$ . Then the area of the triangle with adjacent sides determined by vectors $(2 \vec{a}+3 \vec{b})$ and $(\vec{a}-\vec{b})$ is	$3.\quad$ $24$
$S.\quad$ Area of a paralelogram with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $30$ . Then the area of the parallelogram with adjacent sides determined by vectors $(\vec{a}+\vec{b})$ and $\vec{a}$ is	$4.\quad$ $60$

Codes: $ \quad P \quad Q \quad R \quad S $