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Elementary Mathematics

31). If \( \Large sin \theta -cos \theta =0 \), then what is \( \Large sin^{4} \theta +cos^{4} \theta \) equal to?

A). 1

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

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

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

View Answer

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

\( \Large sin \theta -cos \theta =0 \)

\( \Large sin \theta =cos \theta \)

Since, \( \Large sin \theta \) and \( \Large cos \theta \) are equal for \( \Large \theta \) = 45 degree

So, \( \Large sin^{4} \theta +cos^{4} \theta =(sin45 ^{\circ} )^{4}+(cos45 ^{\circ} )^{4} \)

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

= \( \Large \frac{1}{4}+\frac{1}{4}=\frac{1+1}{4}=\frac{2}{4}=\frac{1}{2} \)

32). The expression \( \Large sin^{2}x+cos^{2}x-1=0 \) is satisfied by how many values of x?

A). Only one value of x

B). Two values of x

C). Infinite values of x

D). No value of x

View Answer

Correct Answer: Infinite values of x

Given that, \( \Large sin^{2}x-cos^{2}x-1=0 \)

\( \Large => \ sin^{2}x+cos^{2}x=1 \)

Which is an identity of trigonometric ratio and always true for every real value of x.

The equation have an infinite solution.

33). Consider the following

\( \Large \frac{cot \ 30 ^{\circ} +1}{cot \ 30 ^{\circ} -1} \) = 2(cos 300 \(^{\circ}\)+ 1)

\( \Large 2 sin 45 ^{\circ} cos45 ^{\circ}-tan45 ^{\circ}cot45 ^{\circ}= 0\)

Which of the above identities is/are correct?

A). Only I

B). Only II

C). Both I and II

D). Neither I nor II

View Answer

Correct Answer: Both I and II

Statement I

\( \Large \frac{cot30 ^{\circ} +1}{cot30 ^{\circ} -1}=2(cos30 ^{\circ} +1) \)

\( \Large \frac{\sqrt{3}+1}{\sqrt{3}-1}=2 \left(\frac{\sqrt{3}}{2}+1\right) \)

\( \Large => \ \frac{\sqrt{3}+1}{\sqrt{3}-1} \times \frac{\sqrt{3}+1}{\sqrt{3}+1}=2 \left(\frac{\sqrt{3}+2}{2}\right) \)

\( \Large => \ \frac{3+1+2\sqrt{3}}{3-1}=\sqrt{3}+2 \)

\( \Large => \ \frac{4+2\sqrt{3}}{2}=\sqrt{3}+2 \)

\( \Large => \ \frac{2(2+\sqrt{3})}{2}=\sqrt{3}+2 \)

\( \Large => \ \sqrt{3}+2=\sqrt{3}+2 \)

It is true

Statement II

\( \Large 2sin 45 ^{\circ} cos45 ^{\circ} - tan45 ^{\circ} cot45 ^{\circ} = 0 \)

\( \Large => \ 2 \times \left(\frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}}\right)-1 \times 1=0 \)

\( \Large => \ 2 \times \frac{1}{2}-1 \times 1=0 \)

1 -1 = 0

Both Statements I and II are true.

34). What is the value of sin A cos A tan A + cos A sin A cot A ?

A). \( \Large sin^{2} \ A+cos \ A \)

B). \( \Large sin^{2} \ A+tan^{2} \ A \)

C). \( \Large sin^{2} \ A+cot^{2} \ A \)

D). \( \Large cosec^{2} \ A-cot^{2} \ A \)

View Answer

Correct Answer: \( \Large cosec^{2} \ A-cot^{2} \ A \)

sinA.cosA.tanA + cosA.sinA.cotA

=\( \Large sinA.cosA.\frac{sinA}{cosA}+cosA.sinA.\frac{cosA}{sinA} \)

= \( \Large sin^{2}A+cos^{2}A \)

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

= \( \Large cosec^{2}A-cot^{2}A \)

[\( \Large 1+cot^{2} \theta =cosec^{2} \theta \)]

35). If \( \Large 2(cos^{2} \theta -sin^{2} \theta )=1 \), (\( \Large \theta \) is a positive acute angle), then \( \Large cot \theta \)is equal to

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

B). 1

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

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

View Answer

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

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

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

We know that,

\( \Large cos^{2} \theta +sin^{2} \theta =1 \) ......(ii)

On solving Eqs. (i) and (ii), we get

\( \Large 2cos^{2} \theta =\frac{3}{2} \ => \ cos^{2} \theta =\frac{3}{4},sin^{2} \theta =\frac{1}{4} \)

\( \Large cot^{2} \theta =\frac{cos^{2} \theta }{sin^{2} \theta }=\frac{\frac{3}{4}}{\frac{1}{4}}=3 \)

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

36). The minimum value of \( \Large cos^{2} \theta +sec^{2} \theta \) is

A). 0

B). 1

C). 2

D). 3

View Answer

37). If cos x + cos y = 2, then the value of sin x + Sin y is

A). 0

B). 1

C). 2

D). -1

View Answer

38). If \( \Large sec^{2} \theta +tan^{2} \theta =\frac{7}{12} \),then \( \Large sec^{4} \theta -tan^{4} \theta \) is equal to

A). \( \Large \frac{7}{12} \)

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

C). \( \Large \frac{5}{12} \)

D). 1

View Answer

Correct Answer: \( \Large \frac{7}{12} \)

\( \Large sec^{2} \theta -tan^{2} \theta =1 \)

\( \Large sec^{2} \theta +tan^{2} \theta =\frac{7}{12} \)

Value of \( \Large sec^{4} \theta -tan^{4} \theta \)

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

= \( \Large 1 \times \frac{7}{12}=\frac{7}{12} \)

39). The numerical value of \( \Large \left(\frac{1}{cos \theta }+\frac{1}{cot \theta }\right) \) \( \Large \left(\frac{1}{cos \theta }-\frac{1}{cot \theta }\right) \) is

A). 0

B). -1

C). 1

D). 2

View Answer

Correct Answer: 1

\( \Large \left(\frac{1}{cos \theta }+\frac{1}{cot \theta }\right) \left(\frac{1}{cos \theta }-\frac{1}{cot \theta }\right) \)

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

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

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

40). If A = tan 11 \( \Large ^{\circ} \) tan 29 \( \Large ^{\circ} \) and B = 2 cot 61 \( \Large ^{\circ} \) cot 79 \( \Large ^{\circ} \), then

A). B=2A

B). A=-2B

C). 2A=B

D). 2A= -B

View Answer

Correct Answer: B=2A

A = tan 11\(^{\circ} \).tan 29\(^{\circ} \)

B = 2 cot 61\(^{\circ} \).cot 79\(^{\circ} \)

= 2 cot (90\(^{\circ} \) - 29 \(^{\circ} \)) cot (90\(^{\circ} \)-11 \(^{\circ} \))

= 2 tan 29 \(^{\circ} \).tan 11 \(^{\circ} \)

B = 2A

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