Online IIT JEE Tests - Moderate Trigonometry Practice # 1 | IIT JEE 2011 | IIT Papers | IIT Key | IIT Solutions | IIT JEE Cutoff

Question

Q. 1 Given that x is a real number such that $\sec x-\tan x = 2$ . Find $\sec x+\tan x.$

Possible Answers

Selected	Possible Answer
	$\frac {1}{2}$
	3
	7
	2

Question

Q. 2 Let $0^0 < \theta < 45^0$ Arrange

$t_1=(tan \theta)^{tan \theta}, \quad \quad \quad t_2=(tan \theta)^{cot \theta},$
$t_3=(cot \theta)^{tan \theta}, \quad \quad \quad t_4=(cot \theta)^{cot \theta},$

in decreasing order.

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Selected	Possible Answer

Question

Compute
$\sin \frac{ \pi}{12} , \cos \frac {\pi}{12},$ and $\tan \frac {\pi}{12};$

Possible Answers

Selected	Possible Answer
	$3- \sqrt{5}$
	$6- \sqrt{4}$
	$2- \sqrt{3}$
	$4- \sqrt{4}$

Question

Compute
$\cos^4 \frac {\pi}{24} - \sin^4 \frac {\pi}{24}$

Possible Answers

Selected	Possible Answer
	$\frac {\sqrt{2}+ \sqrt{6}}{6}$
	$\frac {\sqrt{4}+ \sqrt{6}}{4}$
	$\frac {\sqrt{7}+ \sqrt{6}}{4}$
	$\frac {\sqrt{2}+ \sqrt{6}}{4}$

Question

Compute
$\cos 36^0 - \cos 72^0$

Possible Answers

Selected	Possible Answer
	$\frac {1}{3}$
	$\frac {1}{5}$
	$\frac {1}{4}$
	$\frac {1}{2}$

Question

Compute

$\sin 10^0 \sin 50^0 \sin 70^0$

Possible Answers

Selected	Possible Answer
	$\frac {6}{9}$
	$\frac {2}{8}$
	$\frac {4}{8}$
	$\frac {1}{8}$

Question

Simplify the expression

$\sqrt{ \sin^4 x + 4 \cos ^2 x } - \sqrt{ \cos^4 x + 4 \sin^2 x }$

Possible Answers

Selected	Possible Answer
	$\cos^2 x -\sin^2 x = \cos 2x$
	$\cos^2 x -\sin^2 x = \cos 5x$
	$\cos^2 x -\sin^2 x = \cos 4x$
	$\cos^2 x -\sin^2 x = \cos 3x$

Question

Q. 8 Region R contains all the points (x, y) such that $x^2 + y^2 \underline < 100$ and $sin(x + y ) \underline >0.$ Find the area of regain of R.

Possible Answers

Selected	Possible Answer
	$70 \pi$
	$10 \pi$
	$30 \pi$
	$50 \pi$

Question

Q. 9 Let I denote the interval $[- \frac {\pi}{4}\frac {\pi}{4}]$ . Determine the funcation f defined on the interval [-1, 1] such that $f(\sin 2x) = \sin x + \cos x$ and simplify $f(\tan^2 x)$ for x in the interval I.

Possible Answers

Selected	Possible Answer
	$f(\tan^2 x) = \sqrt{ 1+ \tan ^2 x }= \cos x$
	$f(\tan^2 x) = \sqrt{ 1+ \tan ^2 x }= \sec x$
	$f(\tan^2 x) = \sqrt{ 1+ \tan ^2 x }= \sin x$
	$f(\tan^2 x) = \sqrt{ 1+ \tan ^2 x }= \tan x$

Question

Compute the Following

$1 - \cot 23^0 *$ $1 - \cot 22^0 = ?$

Possible Answers

Selected	Possible Answer
	3
	4
	5
	2

Question

Q.1 In triangle ABC

$3 \sin A + 4 \cos B =6$ and $4 \sin B + 3 \cos A = 1.$

Find the measure of angle C.

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Q.2 In triangle ABC, $\frac{q}{b} = 2 + \sqrt{3}$ and $\angle C = 60^0$ .

Find the measure of angles A and B.

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Selected	Possible Answer
	and
	and
	and
	and

Question

Q.3 Evaluate

$\cos a \cos 2a \cos 3a\cdots \cos 999a,$

where $a = \frac{2\pi}{1999.}$

Possible Answers

Selected	Possible Answer
	$P = \frac{1}{3^{888}}$
	$P = \frac{1}{2^{999}}$
	$P = \frac{1}{3^{999}}$
	$P = \frac{1}{2^{888}}$

Question

Q.4 Determine the minimum value of

$\frac{sec^4 \alpha}{tan^2 \beta} + \frac{sec^4 \beta}{tan^2 \alpha}$

Overall $\alpha, \beta \not= \frac{k\pi}{2}$ , where k is in Z.

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Selected	Possible Answer

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Q.5 Find all pairs (x, y) of real numbers with $0 < x < \frac{\pi}{2}$ such that

$\frac{(\sin x)^{2y}}{(\cos x)^{\frac{y^2}{2}}} + \frac{(\cos x^){2y}}{(\sin x^){\frac{y^2 }{2}}} = \sin 2x.$

Possible Answers

Selected	Possible Answer
	$\big(\frac{\pi}{3}, 2 \big).$
	$\big(\frac{\pi}{4}, 2 \big).$
	$\big(\frac{\pi}{1}, 2 \big).$
	$\big(\frac{\pi}{2}, 2 \big).$

Question

Q.6 [USAMO 2002 proposal, by Cecil Rousseau] Find the maximum value of

S = (1 - x_1)(1 - y_1) + (1 - x_2)(1 - y_2)

if x_2^1 + x_2^2 = y_2^1 + y_2^2 = c^2

Possible Answers

Selected	Possible Answer
	$\frac{-a\sqrt{3}}{2}$
	$\frac{-b\sqrt{4}}{2}$
	$\frac{-c\sqrt{2}}{2}$
	$\frac{-c\sqrt{1}}{2}$

Question

Q.7 [China 1997] Let x, y, z be real numbers with $x\quad \underline >\quad y\quad \underline >\quad z\quad \underline >\quad \frac{\pi}{12}$ such that $x + y + z = \frac{\pi}{2}$ . Find the maximum and the minimum values of the product $\cos x \sin y \cos z.$

Possible Answers

Selected	Possible Answer
	$x = y = \frac{5\pi}{24}$ and $z = \frac{\pi}{12}$
	$x = y = \frac{2\pi}{24}$ and $z = \frac{\pi}{10}$
	$x = y = \frac{3\pi}{24}$ and $z = \frac{\pi}{12}$
	$x = y = \frac{5\pi}{20}$ and $z = \frac{\pi}{6}$

Question

Q.8 Find the sum of all x in the interval $[0, 2\pi]$ such that

$3 \cot^2 x + 8 \cot x + 3 = 0.$

Possible Answers

Selected	Possible Answer
	$4\pi$
	$3\pi$
	$2\pi$
	$5\pi$

Question

Q.9 [AMC12P 2002] Given that

$(1+\tan 1^0)(1+\tan 2^0)\cdots (1+\tan 45^0)=2^n,$

find n.

Possible Answers

Selected	Possible Answer
	25
	22
	21
	23

Question

Q.10 [AIME 2003] Let A=(0,0) and B=(b,2) be points in the coordinate plane. Let ABCDEF be a convex equilateral hexagon such that $\angle FAB=120^0,AB||DE,BC||EF,$ and CD||FA, and the y coordinates of its vertices are distinct elements of the set $\{0,2,4,6,8,10 \}.$ The area of the hexagon can be written in the form $m\sqrt {n},$ where m and n are positive integers and n is not divisible by the square of any prime. Find m+n.

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