Question
$\mathop {{\rm{lim}}\,}\limits_{n \to \infty } \left[ {\frac{1}{{{n^2}}}{{\sec }^2}\frac{1}{{{n^2}}} + \frac{2}{{{n^2}}}{{\sec }^2}\frac{4}{{{n^2}}} + ..... + \frac{1}{n}{{\sec }^2}1} \right] = $
$\mathop {\lim }\limits_{n \to \infty } \sum\limits_{r = 1}^n {\frac{r}{{{n^2}}}{{\sec }^2}\frac{{{r^2}}}{{{n^2}}}} $
$= \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{r = 1}^n {\frac{r}{n}{{\sec }^2}\frac{{{r^2}}}{{{n^2}}}} $
$= \int_0^1 {x{{\sec }^2}{x^2}dx} $
$= \frac{1}{2}\int_0^1 {2x{{\sec }^2}{x^2}dx} = \frac{1}{2}\int_0^1 {{{\sec }^2}t\;dt} $, [ ${x^2} = t$]
$ = \frac{1}{2}[\tan \;t]_0^1 = \frac{1}{2}\tan 1$.
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$(A)$ रेखाखंड (line segment) $O A_1$ की लंबाई $4 \sqrt{3}$ है
$(B)$ रेखाखंड $A_1 B_1$ की लंबाई 16 है
$(C)$ त्रिभुज (triangle) $A_1 B_1 C_1$ का लंबकेंद्र (orthocenter) $(0,0)$ है
$(D)$ त्रिभुज $A_1 B_1 C_1$ का लंबकेंद्र $(1,0)$ है