Q. If $\tan \, \theta + \tan (\theta + \frac{\pi}{3}) + \tan (\theta + = 3,$ then which of the following is equal to $1$ ?

BITSATBITSAT 2008 Report Error

A

$\tan \, 2 \theta$

0%

B

$\tan \, 3 \theta$

100%

C

$\tan^2 \theta$

0%

D

$\tan^3 \theta$

0%

Solution:

Given,
$\tan \theta+\tan \theta+\frac{\pi}{3}+\tan \theta+\frac{2 \pi}{3}=3$
$\Rightarrow \tan \theta+\frac{\tan \theta+\overline{3}}{1-\overline{3} \tan $\theta}+\frac{\tan \theta-\overline{3}}{1+\overline{3} \tan ^{2} \theta}=3$
$\Rightarrow \tan \theta+\frac{8 \tan \theta}{1-3 \tan ^{2} \theta}=3$
$\Rightarrow \frac{9 \tan \theta-3 \tan ^{2} \theta}{1-3 \tan ^{2} \theta}=3$
$\Rightarrow 3 \tan 3 \theta=3$
$\Rightarrow \tan 3 \theta=1$

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Q. If $\tan \, \theta + \tan (\theta + \frac{\pi}{3}) + \tan (\theta + = 3,$ then which of the following is equal to $1$ ?

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$\tan^2 \theta$

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1. The transformed equation of $3x^2 + 3y^2 + 2xy = 2$. when the coordinate axes are rotated through an angle of $45^{\circ}$, is

2. If $I, m, n$ are in arithmetic progression, then the straight line $lx + my + n = 0$ will pass through the point

3. A pair of perpendicular straight lines passes through the origin and also through the point of intersection of the curve $x^2 + y^2 = 4$ with $x + y = a$. The set containing the value of '$a$' is

4. In $\triangle A B C$ the mid points of the sides $A B, B C$ and $C A$ are respectively $(1,0,0),(0$, $m , 0)$ and $(0,0, n )$. Then, $\frac{A B^{2}+B C^{2}+C A^{2}}{l^{2}+m^{2}+n^{2}}$ is equal to

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