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1

### AIEEE 2009

Let A and B denote the statements

A: $$\cos \alpha + \cos \beta + \cos \gamma = 0$$

B: $$\sin \alpha + \sin \beta + \sin \gamma = 0$$

If $$\cos \left( {\beta - \gamma } \right) + \cos \left( {\gamma - \alpha } \right) + \cos \left( {\alpha - \beta } \right) = - {3 \over 2},$$ then:

A
A is false and B is true
B
both A and B are true
C
both A and B are false
D
A is true and B is false

## Explanation 2

### AIEEE 2006

If $$0 < x < \pi$$ and $$\cos x + \sin x = {1 \over 2},$$ then $$\tan x$$ is
A
$${{\left( {1 - \sqrt 7 } \right)} \over 4}$$
B
$${{\left( {4 - \sqrt 7 } \right)} \over 3}$$
C
$$- {{\left( {4 + \sqrt 7 } \right)} \over 3}$$
D
$${{\left( {1 + \sqrt 7 } \right)} \over 4}$$

## Explanation

$$\cos x + \sin x = {1 \over 2}$$

$$\Rightarrow {\left( {\cos x + {\mathop{\rm sinx}\nolimits} } \right)^2} = {1 \over 4}$$

$$\Rightarrow {\cos ^2}x + {\sin ^2}x + 2\cos x\sin x = {1 \over 4}$$
$$\left[ \because {{{\cos }^2}x + {{\sin }^2}x = 1\, \,and \,\,2\cos x\sin x = \sin 2x} \right]$$

$$\Rightarrow 1 + \sin 2x = {1 \over 4}$$

$$\Rightarrow \sin 2x = - {3 \over 4},$$ so $$x$$ is obtuse and

$${{2\tan x} \over {1 + {{\tan }^2}x}} = - {3 \over 4}$$

$$\Rightarrow 3{\tan ^2}x + 8\tan x + 3 = 0$$

$$\therefore$$ $$\tan x = {{ - 8 \pm \sqrt {64 - 36} } \over 6}$$

$$= {{ - 4 \pm \sqrt 7 } \over 3}$$

as $$\tan x < 0\,$$

$$\therefore$$ $$\tan x = {{ - 4 - \sqrt 7 } \over 3}$$
3

### AIEEE 2006

The number of values of $$x$$ in the interval $$\left[ {0,3\pi } \right]\,$$ satisfying the equation $$2{\sin ^2}x + 5\sin x - 3 = 0$$ is
A
4
B
6
C
1
D
2

## Explanation $$2{\sin ^2}x + 5\sin x - 3 = 0$$

$$\Rightarrow \left( {\sin x + 3} \right)\left( {2\sin x - 1} \right) = 0$$

$$\sin x = {1 \over 2}$$ and $$\,\,\sin x \ne - 3$$

Given that $$x \in \left[ {0,3\pi } \right]$$

So possible values of x are $$30^\circ$$, $$150^\circ$$, $$390^\circ$$, $$510^\circ$$. That means x have 4 values.
4

### AIEEE 2004

A line makes the same angle $$\theta$$, with each of the $$x$$ and $$z$$ axis.
If the angle $$\beta \,$$, which it makes with y-axis, is such that $$\,{\sin ^2}\beta = 3{\sin ^2}\theta ,$$ then $${\cos ^2}\theta$$ equals
A
$${2 \over 5}$$
B
$${1 \over 5}$$
C
$${3 \over 5}$$
D
$${2 \over 3}$$

## Explanation

Concept : If a line makes the angle $$\alpha ,\beta ,\gamma$$ with x, y, z axis respectively then $${\cos ^2}\alpha + {\cos ^2}\beta + {\cos ^2}\gamma = 1$$\$

In this question given that the line makes angle θ with x and z-axis and β with y−axis.

$$\therefore\: cos^2\theta+cos^2\beta+cos^2\theta=1$$

$$\Rightarrow\:2cos^2\theta=1-cos^2\beta$$

$$\Rightarrow 2{\cos ^2}\theta = {\sin ^2}\beta$$

But given that $$sin^2\beta=3sin^2\theta$$

$$\therefore$$ $$2{\cos ^2}\theta = 3{\sin ^2}\theta$$

$$\Rightarrow 2{\cos ^2}\theta = 3\left( {1 - {{\cos }^2}\theta } \right)$$

$$\Rightarrow 2{\cos ^2}\theta = 3 - 3{\cos ^2}\theta$$

$$\Rightarrow 5{\cos ^2}\theta = 3$$

$$\Rightarrow {\cos ^2}\theta = {3 \over 5}$$

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