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\( \Large 3 \left(\sin x - \cos x\right)^{4} + 6 \left(\sin x + \cos x\right)^{2} + 4 \left(\sin^{6}x + \cos^{6}x\right) \) is equal to:

A) 11
B) 12
C) 13
D) 14

Correct Answer:



C) 13

Description for Correct answer:

\( \Large 3 \left(\sin x - \cos x\right)^{4}+6 \left(\sin x + \cos x\right)^{2} + 4 \left(\sin^{6}x+\cos^{6}x\right) \)

= \( \Large 3 \left(1- \sin 2x\right)^{2} + 6 \left(1+ \sin 2x\right) \) \( \Large + 4 \{ \left(\sin^{2}x+\cos^{2}x\right)^{3} \) \( \Large -3 \sin^{2}x \cos^{2}x \left(\sin^{2}x+\cos^{2}x\right) \}\)

\(= \Large 3 \left(1-\sin 2x + \sin^{2}2x\right)+6+6 \sin 2x + 4\{ 1-3 \sin^{2}x \cos^{2}x \} \)

= \( \Large 3 \left(1-2 \sin 2x + \sin^{2}2x+2+2 \sin 2x\right)+4\{ 1 - \frac{3}{4}\sin^{2} 2x \} \)

\( \Large 13 + 3 \sin^{2} 2x - 3 \sin^{2} 2x = 13 \)

Part of solved Trigonometric ratio questions and answers : >> Elementary Mathematics >> Trigonometric ratio

Comments

Similar Questions

1). If \( \Large \cos \theta = \cos \alpha \cos \beta .\ then\ \tan \frac{ \theta + \alpha }{2} \tan \frac{ \theta - \alpha }{2} \) is equal to:

A). \( \Large \tan^{2}\frac{ \alpha }{2} \)

B). \( \Large \tan^{2}\frac{ \beta }{2} \)

C). \( \Large \tan^{2}\frac{ \theta }{2} \)

D). \( \Large \cot^{2}\frac{ \beta }{2} \)

2). The greatest value of \( \Large \cos \theta \) for which \( \Large \cos 5 \theta = 0 \), is:

A). 0

B). \( \Large \frac{1+\sqrt{5}}{4} \)

C). \( \Large \sqrt{\frac{5+\sqrt{5}}{8}} \)

D). \( \Large \sqrt{\frac{5+\sqrt{1}}{4}} \)

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D). \( \Large \frac{12}{13} \)

5). The distance of the point (-2, 3) from the line x -y = 5 is:

A). \( \Large 5\sqrt{2} \)

B). \( \Large 2\sqrt{5} \)

C). \( \Large 3\sqrt{5} \)

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6). The equation of the straight line joining the origin to the point of intersection of \( \Large y-x+7=0\ and\ y+2x-2=0 \)is:

A). \( \Large 3x+4y=0 \)

B). \( \Large 3x-4y=0 \)

C). \( \Large 4x-3y=0 \)

D). \( \Large 4x+3y=0 \)

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A). \( \Large x-y = 5 \)

B). \( \Large x+y = 5 \)

C). \( \Large x+y = 1 \)

D). \( \Large x-y = 1 \)

8). The angle between the straight lines \( \Large x-y\sqrt{3}=5\ and\ \sqrt{3}x+y=7 \) is:

A). \( \Large 90 ^{\circ} \)

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

C). \( \Large 75 ^{\circ} \)

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

9). The three straight lines \( \Large ax+by=c,\ bx+cy=a\ and\ cx+ay=b \) are collinear if:

A). \( \Large a+b+c = 0 \)

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A). (6, 1)

B). (8, 2)

C). (2, 8)

D). (4, 2)