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Elementary Mathematics

21). If \( \Large \frac{cos \ x}{1+cosec \ x}+\frac{cosx}{cosec\ x-1}=2 \) ,which one of the following is one of the value of x ?

A). \( \Large \frac{ \pi }{2} \)

B). \( \Large \frac{ \pi }{3} \)

C). \( \Large \frac{ \pi }{4} \)

D). \( \Large \frac{ \pi }{6} \)

View Answer

Correct Answer: \( \Large \frac{ \pi }{4} \)

\( \Large \frac{cosx}{cosecx+1}+\frac{cosx}{cosecx-1}=2 \)

=> \( \Large \frac{cosx(cosecx-1)+cosx(cosecx+1)}{(cosecx+1)(cosec-1)}=2 \)

=> \( \Large \frac{cosx \ cosecx-cosx+cosx \ cosecx+cosx }{cosec^{2}x-1}=2 \)

=> \( \Large \frac{2cosx.\frac{1}{sinx}}{cot^{2}x}=2 \) => \( \Large \frac{2cotx}{cot^{2}x}=2 \)

\( \Large cotx=1=cot \ 45 ^{\circ} \)

\( \Large x=45 ^{\circ} =\frac{ \pi }{4} \)

22). \( \Large \frac{cos \theta }{1-sin \theta }-\frac{cos \theta }{1+sin \theta }=2 \),is satisfied by which one of the following values of \( \Large \theta \)?

A). \( \Large \frac{ \pi }{2} \)

B). \( \Large \frac{ \pi }{3} \)

C). \( \Large \frac{ \pi }{4} \)

D). \( \Large \frac{ \pi }{6} \)

View Answer

Correct Answer: \( \Large \frac{ \pi }{4} \)

\( \Large \frac{cos \theta }{1-sin \theta }-\frac{cos \theta }{1+sin \theta }=2 \)

=> \( \Large \frac{cos \theta (1+sin \theta )-cos \theta (1-sin \theta )}{(1-sin \theta )(1+sin \theta )}=2 \)

=> \( \Large \frac{cos \theta +cos \theta sin \theta -cos \theta +cos \theta sin \theta}{1-sin^{2} \theta }=2 \)

=> \( \Large \frac{2 \ cos \theta sin \theta }{cos^{2} \theta }=2 \)

\( \Large 2tan \theta =2 \)

=> \( \Large tan \theta =1=tan \ 45 ^{\circ} \)

\( \Large \theta =45 ^{\circ} =\frac{ \pi }{4} \)

23). What is \( \Large \sqrt{\frac{1+sin \theta }{1-sin \theta }} \) equal to?

A). \( \Large sec \theta -tan \theta \)

B). \( \Large sec \theta +tan \theta \)

C). \( \Large cosec \theta +cot \theta \)

D). \( \Large cosec \theta -cot \theta \)

View Answer

Correct Answer: \( \Large sec \theta +tan \theta \)

\( \Large \sqrt{\frac{1+sin \theta }{1-sin \theta }} \ = \sqrt{\frac{1+sin \theta }{1-sin \theta } \times \frac{1+sin \theta }{1+sin \theta }} \)

= \( \Large \sqrt{\frac{(1+sin \theta )^{2}}{(1-sin \theta )(1+sin \theta )}} \)

= \( \Large \sqrt{\frac{(1+sin \theta )^{2}}{1-sin^{2} \theta }}=\sqrt{\frac{(1+sin \theta )^{2}}{cos^{2} \theta }} \)

= \( \Large \sqrt{ \left(\frac{1+sin \theta }{cos \theta }\right)^{2} }=\frac{1+sin \theta }{cos \theta } \)

= \( \Large \frac{1}{cos \theta }+\frac{sin \theta }{cos \theta } \)

= \( \Large sec \theta +tan \theta \)

24). If \( \Large 7cos^{2} \theta+3sin^{2} \theta =4 \ and \ 0 \ < \ \theta \ < \frac{ \pi }{2} \), what is the value of \( \Large tan \theta \)?

A). \( \Large \sqrt{7} \)

B). \( \Large \frac{7}{3} \)

C). 3

D). \( \Large \sqrt{3} \)

View Answer

Correct Answer: \( \Large \sqrt{3} \)

\( \Large 7 \ cos^{2} \theta +3 \ sin^{2} \theta =4 \)

Dividing by \( \Large cos^{2} \theta \) both sides,we get,

\( \Large \frac{7 \ cos^{2} \theta }{cos^{2} \theta }+\frac{3 \ sin^{2} \theta }{cos^{2} \theta }=\frac{4}{cos^{2} \theta } \)

=> \( \Large 7+3 \ tan^{2} \theta =4 \ sec^{2} \theta \)

=> \( \Large 7+3 \ tan^{2} \theta =4(1+tan^{2} \theta ) \)

=> \( \Large 7+3 \ tan^{2} \theta =4+4 \ tan^{2} \theta \)

=> \( \Large 3=tan^{2} \theta \)

=> \( \Large tan \theta =\sqrt{3} \)

25). The value of \( \Large \frac{tan \ 27 ^{\circ} +cot \ 63 ^{\circ} }{tan \ 27 ^{\circ} (sin \ 25 ^{\circ} +cos \ 65 ^{\circ} )} \)

A). cosec 25 \(^{\circ}\)

B). 2 tan27\(^{\circ}\)

C). sin 25 \(^{\circ}\)

D). tan 65 \(^{\circ}\)

View Answer

Correct Answer: cosec 25 \(^{\circ}\)

\( \Large \frac{tan27 ^{\circ} +cot63 ^{\circ} }{tan27 ^{\circ} (sin25 ^{\circ} +cos65 ^{\circ} )} \)

= \( \Large \frac{tan27 ^{\circ} +cot(90 ^{\circ} -27 ^{\circ} )}

{tan27 ^{\circ} (sin25 ^{\circ} +cos(90 ^{\circ} -25 ^{\circ} ))} \)

= \( \Large \frac{tan27 ^{\circ} +tan27 ^{\circ} }{tan27 ^{\circ} [sin25 ^{\circ} +sin25 ^{\circ} ]} \)

= \( \Large \frac{2}{2sin25 ^{\circ} }=cosec25 ^{\circ} \)

26). If \( \Large cos \theta \ge \frac{1}{2} \) in the first quadrant, which one of the following is correct?

A). \( \Large \theta \le \frac{ \pi }{3} \)

B). \( \Large \theta \ge \frac{ \pi }{3} \)

C). \( \Large \theta \le \frac{ \pi }{6} \)

D). \( \Large \theta \ge \frac{ \pi }{6} \)

View Answer

Correct Answer: \( \Large \theta \le \frac{ \pi }{3} \)

\( \Large cos \theta\ge \frac{1}{2} \) means the value of \( \Large \theta \) lies between \( \Large 0 ^{\circ} \) and \( \Large \frac{ \pi }{3} \)

\( \Large \theta \) is less than or equal to \( \Large \frac{ \pi }{3} \), i.e., \( \Large \theta \le \frac{ \pi }{3} \)

27). If x=\( \Large a \ sec \theta \ cos \theta ,\ y=b \ sec \theta \ sin \theta \) and z=\( \Large c \ tan \theta \ \), then the value of \( \Large \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} \) is

A). 9

B). 0

C). 1

D). 4

View Answer

Correct Answer: 1

Given,

x=\( \Large a \ sec \theta cos \theta \)

y=\( \Large b \ sec \theta . sin \theta\)

z=\( \Large c \ tan \theta \)

Now, \( \Large \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}- \frac{z^{2}}{c^{2}} \)

On putting the values of x, y and z,we get

= \( \Large \frac{a^{2}sec^{2} \theta cos^{2} \theta}{a^{2}}+\frac{b^{2}sec^{2} \theta sin^{2} \theta}{b^{2}}

-\frac{c^{2}tan^{2} \theta }{c^{2}} \)

= \( \Large sec^{2} \theta [cos^{2} \theta +sin^{2} \theta ]-tan^{2} \theta \)

= \( \Large sec^{2} \theta -tan^{2} \theta \)

= 1

28). If \( \Large cos^{4} \theta -sin^{4} \theta =\frac{2}{3} \), then the value of \( \Large 1-2 \ sin^{2} \theta \) is

A). 0

B). \( \Large \frac{2}{3} \)

C). \( \Large \frac{1}{3} \)

D). \( \Large \frac{4}{3} \)

View Answer

Correct Answer: \( \Large \frac{2}{3} \)

\( \Large cos^{4} \theta -sin^{4} \theta =\frac{2}{3} \)

=> \( \Large (cos^{2} \theta )^{2}-(sin^{2} \theta )^{2}=\frac{2}{3} \)

\( \Large (cos^{2} \theta -sin^{2} \theta )(cos^{2} \theta +sin^{2} \theta )=\frac{2}{3} \)

\( \Large cos^{2} \theta -sin^{2} \theta =\frac{2}{3} \) => \( \Large cos \ 2 \theta =\frac{2}{3} \)

\( \Large 1-2sin^{2} \theta =\frac{2}{3} \)

29). If \( \Large 7 sin^{2} \theta +3cos^{2} \theta =4 \) , then the value of \( \Large cos \theta (0 ^{\circ} \le \theta \le 90 ^{\circ} ) \) is

A). \( \Large \frac{\sqrt{6}}{2} \)

B). \( \Large \frac{\sqrt{3}}{2} \)

C). \( \Large \frac{\sqrt{2}}{2} \)

D). \( \Large \frac{\sqrt{5}}{2} \)

View Answer

Correct Answer: \( \Large \frac{\sqrt{3}}{2} \)

\( \Large 7 sin^{2} \theta +3 cos^{2} \theta=4 \)

=> \( \Large 4sin^{2} \theta +3sin^{2} \theta +3cos^{2} \theta =4 \)

=> \( \Large 4sin^{2} \theta +3(sin^{2} \theta +cos^{2} \theta )=4 \)

=> \( \Large 4sin^{2} \theta +3=4 \)

=> \( \Large 4sin^{2} \theta =1 \)

=> \( \Large sin \theta =\frac{1}{2} \)

=> \( \Large \theta =30 ^{\circ} \)

=> \( \Large cos \theta =cos30 ^{\circ} =\frac{\sqrt{3}}{2} \)

30). If \( \Large tan \ 8 \ \theta =cot \ 2 \ \theta \) where \( \Large 0< 8 \ \theta <\frac{ \pi }{2} \), then what is the value of \( \Large tan \ 5 \theta \) ?

A). \( \Large \frac{1}{\sqrt{3}} \)

B). 1

C). \( \Large \sqrt{3} \)

D). 0

View Answer

Correct Answer: 1

\( \Large tan \ 8 \theta =cot \ 2 \theta \)

\( \Large tan \ 8 \theta =tan(90 ^{\circ} -2 \theta ) \)

\( \Large 8 \theta =90 ^{\circ} -2 \theta \ => \ \theta =90 ^{\circ} \)

\( \Large tan5 \theta \ => \ tan45 ^{\circ} =1 \)

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