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JEE Mathematics Solved Question Paper 2004

Multiple Choice Questions

21.

If n $\in$ N and the period of $\frac{\cos (nπ)}{\sin (\frac{x}{n})}$ is 4 $π$ , then n is equal to

22.

$The expression for \tan (9 °) - \tan (27 °) - \tan (63 °) + \tan (81 °) is equal to$

23.

$In ∆ ABC, \cos (\frac{B + 2 C + 3 A}{2}) + \cos (\frac{A - B}{2}) is$

24.

$The value of series \cos (12 °) + \cos (84 °) + \cos (132 °) + \cos (156 °) is$

$\frac{1}{2}$
$\frac{1}{4}$
$- \frac{1}{4}$
$- \frac{1}{2}$

25.

$For x \in IR, 3 \cos (4 x - 5) + 4 lies in the interval$

[1, 7]
[4, 7]
[0, 7]
[2, 7]

26.

If $x = \log [\cot (\frac{π}{4} + θ)]$ , then the value of $\sin (hx)$ is

$\tan (2 θ)$
$- \tan (2 θ)$
$\cot (2 θ)$
$- \cot (2 θ)$

27.
If in a $∆$ ABC, r₃ = r₁ + r₂ + r, then $∠ A + ∠ B$ is equal to

120°

100°

90°

80°

90°

$We know that, r = 4 Rsin (\frac{A}{2}) \sin (\frac{B}{2}) \sin (\frac{C}{2}) r_{1} = 4 Rsin (\frac{A}{2}) \cos (\frac{B}{2}) \cos (\frac{C}{2}) r_{2} = 4 Rsin (\frac{B}{2}) \cos (\frac{A}{2}) \cos (\frac{C}{2}) r_{3} = 4 Rsin (\frac{C}{2}) \cos (\frac{A}{2}) \cos (\frac{B}{2})$

$Given that, r_{3} = r_{1} + r_{2} + r \Rightarrow r_{3} - r = r_{1} + r_{2} \Rightarrow 4 Rsin (\frac{C}{2}) (\cos (\frac{A}{2}) \cos (\frac{B}{2}) - \sin (\frac{B}{2}) \sin (\frac{A}{2})) = 4 Rcos (\frac{C}{2}) [\sin (\frac{A}{2}) \cos (\frac{B}{2}) + \cos (\frac{A}{2}) \sin (\frac{B}{2})] \Rightarrow \sin (\frac{C}{2}) (\cos (\frac{A + B}{2})) = \cos (\frac{C}{2}) (\sin (\frac{A + B}{2})) \Rightarrow \sin (\frac{C}{2}) \cos (\frac{π}{2} - \frac{C}{2}) = \cos (\frac{C}{2}) (\sin (\frac{π}{2} - \frac{C}{2})) [∵ A + B + C = π \Rightarrow \frac{A}{2} + \frac{B}{2} = \frac{π}{2} - C]$

$\Rightarrow \sin^{2} (\frac{A}{2}) = \cos^{2} (\frac{C}{2}) \Rightarrow \tan (\frac{C}{2}) = 1$

$\Rightarrow \frac{C}{2} = \frac{π}{4} \Rightarrow C = \frac{π}{2} We know that, A + B C = π \Rightarrow A + B = π - \frac{π}{2} \Rightarrow A + B = \frac{π}{2}$

28.

In a $∆ ABC$ ${(a - b)}^{2} \cos^{2} (\frac{C}{2}) + {(a + b)}^{2} \sin^{2} (\frac{C}{2})$ is equal to

a²
c²
b²
a² + b²

29.

In a $∆ ABC$ , the correct formulae among the following are

$I . r = 4 Rsin (\frac{A}{2}) \sin (\frac{B}{2}) \sin (\frac{C}{2}) II . r_{1} = (s - a) \tan (\frac{A}{2}) III . r_{3} = \frac{∆}{s - c}$

only I, II
only II, III
only I, III
I, II, III

30.

An aeroplane flying with uniform speed horizontally one km above the ground is observed at an elevation of 60°. After 10 s if the elevation is observed to be 30°, then the speed of the plane (in km/h) is