If _ r=1 ^ 50 ^ -1 1 2 r^2 =p, then p is - Vidyadip

MCQ

If $\sum_{r=1}^{50} \tan ^{-1} \frac{1}{2 r^2}=p$, then $\tan p$ is

A
$\frac{100}{101}$
B
$\frac{51}{50}$
✓
$\frac{50}{51}$
D
$\frac{101}{102}$

✓

Answer

Correct option: C.

$\frac{50}{51}$

(c) : $\sum_{r=1}^{50} \tan ^{-1} \frac{1}{2 r^2}=\sum_{r=1}^{50} \tan ^{-1}\left(\frac{2}{4 r^2}\right)$
$=\sum_{r=1}^5 \tan ^{-1}\left(\frac{(2 r+1)-(2 r-1)}{1+(2 r+1)(2 r-1)}\right)$
$=\sum_{r=1}^{50}\left[\tan ^{-1}(2 r+1)-\tan ^{-1}(2 r-1)\right]$
$=\tan ^{-1}(101)-\tan ^{-1}(1)$
$=\tan ^{-1}\left[\frac{101-1}{1+101}\right]=\tan ^{-1} \frac{100}{102}=\tan ^{-1} \frac{50}{51}=p$
$\therefore \tan p=\frac{50}{51}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "MCQ" from Trigonometric Functions→Trigonometric Functions — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

The principal solutions of $\sqrt{3} \sec x-2=0$ are

The angle between the lines, whose direction cosines $l, m, n$ satisfy the equations $l+m+n=0$ and $2 l^2+2 m^2-n^2=0$, as

If $\cos 2 \theta=(\sqrt{2}+1)\left(\cos \theta-\frac{1}{\sqrt{2}}\right)$, then the value of $\theta$ is

The area of the region bounded by the ellipse $x^2 / 64+y^2 / 100=1$, is ____ sq.units

A random variable $X$ has the following probability distribution:

$X:$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$P(X):$	$0.15$	$0.23$	$0.12$	$0.10$	$0.20$	$0.08$	$0.07$	$0.05$

Find the events $E = \{X : X$ is a prime number$\}, F\{X : X < 4\},$ the probability $\text{P}(\text{E}\cup\text{F})$ is:

$6 x^2+h x y+12 y^2=0$ represents pair of parallel straight lines, if $h$ is

If $I=\int \frac{\sin x+\sin ^3 x}{\cos 2 x} d x$ $=P \cos x+Q \log \left|\frac{\sqrt{2} \cos x-1}{\sqrt{2} \cos x+1}\right|+c$, (where $c$ is a constant of integration), then the values of $P$ and $Q$ are respectively

The angle between the straight lines $\frac{\text{x}+1}{2}=\frac{\text{y}-2}{5}=\frac{\text{z}+3}{4}$ and $\frac{\text{x}-1}{1}=\frac{\text{y}+2}{2}=\frac{\text{z}-3}{-3}$ is:

If $\sin x$ is the integrating factor (I.F.) of the linear differential equation $\frac{d y}{d x}+P y=Q$, then $P$ is

$\int_0^{\pi / 8} \cos ^3 4 \theta d \theta=$