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Elementary Mathematics

61). The equation \( \Large cos^{2} \theta =\frac{(x+y)^{2}}{4xy} \) is only possible when

A). x > y

B). x = y

C). x < y

D). x = -y

View Answer

Correct Answer: x = y

The equation \( \Large cos^{2} \theta =\frac{(x+y)^{2}}{4 \ xy} \) is possible,

only when x = y

On putting x = y in the equation,

\( \Large cos^{2} \theta =\frac{(2x)^{2}}{4x^{2}} \)

=\( \Large \frac{4x^{2}}{4x^{2}}=1 \)

\( \Large => \ \theta =0 ^{\circ} \)

62). The value of \( \Large sin^{2}1 ^{\circ} +sin^{2}3 ^{\circ} +sin^{2}45 ^{\circ}\)
\( \Large +.....+sin^{2}87 ^{\circ} +sin^{2}89 ^{\circ} \) is

A). 22

B). \( \Large 22\frac{1}{2} \)

C). 23

D). \( \Large 22\frac{1}{4} \)

View Answer

Correct Answer: \( \Large 22\frac{1}{2} \)

\( \Large sin^{2}1 ^{\circ} +sin^{2}3 ^{\circ} +.....+sin^{2}45 ^{\circ} +\)

\( \Large .....+sin^{2}(90-3)+sin^{2}(90 ^{\circ} -1 ^{\circ} ) \)

= \( \Large sin^{2}1 ^{\circ} +sin^{2}3 ^{\circ} +.....+sin^{2}45 ^{\circ}\)

\( \Large +....+cos^{2}3 ^{\circ} +cos^{2}1 ^{\circ} \)

= \( \Large (1 \times 22) \times \left(\frac{1}{\sqrt{2}}\right)^{2} \)

= \( \Large 22 +\frac{1}{2}=22\frac{1}{2} \)

63). If \( \Large \angle A \) and \( \Large \angle B \) are complementary to each other, then the value of \( \Large sec^{2} \ A+sec^{2} \ B-sec^{2} \ A \ sec^{2} \ B \) is

A). 1

B). -1

C). 2

D). 0

View Answer

Correct Answer: 0

\( \Large A+B=90 ^{\circ} \)

\( \Large => \ B=90 ^{\circ} -A \)

\( \Large sec^{2}A+sec^{2}B-sec^{2}A.sec^{2}B \)

=\( \Large sec^{2}A+sec^{2}(90 ^{\circ} -A)-sec^{2}Asec^{2}(90 ^{\circ} -A) \)

=\( \Large sec^{2}A+cosec^{2}A-sec^{2}A.cosec^{2}A \)

=\( \Large \frac{1}{cos^{2}A}+\frac{1}{sin^{2}A}-\frac{1}{sin^{2}A.cos^{2}A} \)

=\( \Large \frac{sin^{2}A+cos^{2}A-1}{sin^{2}A.cos^{2}A} \)

=\( \Large \frac{1-1}{sin^{2}A.cos^{2}A}=0 \)

64). The value of \( \Large sin^{2}5 ^{\circ} +sin^{2}10 ^{\circ} \) \( \Large +sin^{2}15 ^{\circ} +.....+sin^{2}85 ^{\circ} +sin^{2}90 ^{\circ} \) is

A). \( \Large 7\frac{1}{2} \)

B). \( \Large 8\frac{1}{2} \)

C). \( \Large 10\frac{1}{2} \)

D). \( \Large 9\frac{1}{2} \)

View Answer

Correct Answer: \( \Large 9\frac{1}{2} \)

\( \Large sin^{2}5 ^{\circ} +sin^{2}10 ^{\circ} +.....+sin^{2}45 ^{\circ}\)

\( \Large +....+sin^{2}85 ^{\circ} +sin^{2}90 ^{\circ} \)

= \( \Large sin^{2}5 ^{\circ} +sin^{2}10 ^{\circ} +....+sin^{2}45 ^{\circ} +\)

\( \Large ....+sin^{2}(90 ^{\circ} -5)+sin^{2}90 ^{\circ} \)

= \( \Large 8 \times [sin^{2}5 ^{\circ} +cos^{2}5 ^{\circ}... ]+sin^{2}45 ^{\circ} +sin^{2}90 ^{\circ} \)

= \( \Large 8+ \left(\frac{1}{\sqrt{2}}\right)^{2}+1=8+1+\frac{1}{2} =4\frac{1}{2} \)

65). If \( \Large sin(3x-20 ^{\circ} ) \) = cos (3y + 20 \( \Large ^{\circ} \)), then the value of (x + y) is

A). 20 \( \Large ^{\circ} \)

B). 30 \( \Large ^{\circ} \)

C). 40 \( \Large ^{\circ} \)

D). 45 \( \Large ^{\circ} \)

View Answer

Correct Answer: 30 \( \Large ^{\circ} \)

\( \Large sin(3x-20 ^{\circ} )=cos(3y+20 ^{\circ} ) \)

\( \Large => \ sin(3x-20 ^{\circ} )=sin(90 ^{\circ} -(3y+20 ^{\circ} )) \)

\( \Large sin(3x-20 ^{\circ} )=sin(70 ^{\circ} -3y) \)

\( \Large => \ 3x-20 ^{\circ} =70 ^{\circ} -3y \)

\( \Large => \ 3x+3y=90 ^{\circ} \)

\( \Large => \ 3(x+y)=90 ^{\circ} \)

\( \Large => \ x+y=30 ^{\circ} \)

66). If \( \Large \frac{sin \theta +cos \theta }{sin \theta -cos \theta }=\frac{5}{4} \), the value of \( \Large \frac{tan^{2} \theta +1 }{tan^{2} \theta -1 } \) is

A). \( \Large \frac{25}{16} \)

B). \( \Large \frac{41}{9} \)

C). \( \Large \frac{41}{40} \)

D). \( \Large \frac{40}{41} \)

View Answer

Correct Answer: \( \Large \frac{41}{40} \)

Given, \( \Large \frac{sin \theta +cos \theta }{sin \theta -cos \theta }=\frac{5}{4} \)

\( \Large => \ \frac{cos \theta \left(\frac{sin \theta }{cos \theta }+1\right) }{cos \theta \left(\frac{sin \theta }{cos \theta }-1\right)} =\frac{5}{4} \)

\( \Large => \ \frac{tan \theta +1}{tan \theta -1}=\frac{5}{4} \)

\( \Large => \ 4tan \theta +4=5tan \theta -5 \)

\( \Large tan \theta =9 \)

Putting volume of \( \Large tan \theta \)

\( \Large \frac{tan^{2} \theta +1}{tan^{2} \theta -1}=\frac{(9)^{2}+1}{(9)^{2}-1} \)

=\( \Large \frac{81+1}{81-1}=\frac{82}{80}=\frac{41}{40} \)

67). The value of \( \Large \frac{sin39 ^{\circ} }{cos51 ^{\circ} }+2 \) tan 11 \( \Large ^{\circ} \)tan 31 \( \Large ^{\circ} \)tan 45 \( \Large ^{\circ} \)tan 59 \( \Large ^{\circ} \)tan 79 \( \Large ^{\circ} \)- 3 \( \Large (sin^{2}21 ^{\circ} +sin^{2}69 ^{\circ} ) \) is

A). 2

B). -1

C). 1

D). 0

View Answer

Correct Answer: 0

\( \Large \frac{sin \ 39 ^{\circ} }{cos \ 51 ^{\circ} }+2 \ tan11 ^{\circ} .tan79 ^{\circ} . \)

\( \Large tan31 ^{\circ} .tan59 ^{\circ} .tan45 ^{\circ} \)

\( \Large -3(sin^{2}21+sin^{2}69 ^{\circ} ) \)

= \( \Large \frac{sin39 ^{\circ} }{cos(90 ^{\circ} -39 ^{\circ} )}+2 \ tan11 ^{\circ} .tan(90 ^{\circ} -11 ^{\circ}) \)

\( \Large tan31 ^{\circ} .tan(90 ^{\circ} -31 ^{\circ} ). \)

\( \Large 1-3(sin^{2}21 ^{\circ} +sin^{2}(90 ^{\circ} -21 ^{\circ} )) \)

\( \Large [tan(90 ^{\circ} - \theta )=cot \theta ,cos(90 ^{\circ} - \theta )=sin \theta ] \)

= \( \Large \frac{sin39 ^{\circ} }{sin39 ^{\circ} }+2 \ tan11 ^{\circ} .cot11 ^{\circ} .tan31 ^{\circ}. \)

\( \Large cot31 ^{\circ} -3(sin^{2}21 ^{\circ} +cos^{2} 21 ^{\circ} ) \)

=1+2-3=0

\( \Large [tan \theta .cot \theta =1 \ and \ sin^{2} \theta +cos^{2} \theta =1] \)

68). If \( \Large \frac{cos^{2} \theta }{cot^{2} \theta -cos^{2} \theta }=3 \) and \( \Large 0 ^{\circ} < \theta <90 ^{\circ} \), then the value of \( \Large \theta \) is

A). 30 \( \Large ^{\circ} \)

B). 45 \( \Large ^{\circ} \)

C). 60 \( \Large ^{\circ} \)

D). None of these

View Answer

Correct Answer: 60 \( \Large ^{\circ} \)

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

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

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

\( \Large => \ 4 \ cos^{2} \theta -\frac{3cos^{2} \theta }{sin^{2} \theta }=0 \)

\( \Large => \ cos^{2} \theta \left(4-\frac{3}{sin^{2} \theta }\right)=0 \)

\( \Large 4-\frac{3}{sin^{2} \theta }=0 \)

\( \Large => \ 4 \ sin^{2} \theta -3=0 \)

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

\( \Large => \ sin \theta =\frac{\sqrt{3}}{2}=sin60 ^{\circ} \)

\( \Large => \ \theta =60 ^{\circ} \)

69). The numerical value of \( \Large cot \ 18 ^{\circ} \left(cot \ 72 ^{\circ} \ cos^{2}22 ^{\circ}+\frac{1 }{tan\ 72 ^{\circ} \ sec^{2}68 ^{\circ} }\right) \)is

A). 1

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

C). 3

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

View Answer

Correct Answer: 1

\( \Large cot \ 18 ^{\circ} \left(cot \ 72 ^{\circ} .cos^{2}22 ^{\circ} +\frac{1}{tan72 ^{\circ} .sec^{2}68 ^{\circ} }\right) \)

= \( \Large cot \ 18 ^{\circ} .cot \ 72 ^{\circ} .cos^{2}22 ^{\circ} +\frac{cot18 ^{\circ} }{tan72 ^{\circ} .sec^{2}68 ^{\circ} } \)

= \( \Large cot \ 18 ^{\circ} .tan \ 18 ^{\circ} .cos^{2}22 ^{\circ} \)

\( \Large +\frac{cot \ 18 ^{\circ} }{cot \ 18 ^{\circ} }.cos^{2}68 ^{\circ} \)

=\( \Large cos^{2}22 ^{\circ} +cos^{2}68 ^{\circ} \)

=\( \Large cos^{2}22 ^{\circ} +sin^{2}22 ^{\circ} =1 \)

70). If \( \Large \theta \) be an acute angle and \( \Large 7 \ sin^{2} \theta +3 \ cos^{2} \theta=4 \), then the value of \( \Large tan \theta \) is

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

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

C). 1

D). 0

View Answer

Correct Answer: \( \Large \frac{1}{\sqrt{3}} \)

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

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

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

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

\( \Large => \ 7 \ tan^{2} \theta -4-4 \ tan^{2} \theta +3=0 \)

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

\( \Large => \ tan^{2} \theta =\frac{1}{3} \)

\( \Large => \ tan \theta =\frac{1}{\sqrt{3}} \)

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