The value of $2\,\,\cos \,\,x - \cos \,\,3x - \cos \,\,5x{\rm{\;}}$ is equal to

$16\,\,{\sin ^3}\,x\,\,{\rm{co}}{{\rm{s}}^2}\,x$

$4\,\,{\cos ^3}\,x\,\,{\sin ^2}\,x$

$4\,\,{\sin ^3}\,x\,\,{\cos ^2}\,x$

If $A + B + C\; = \frac{{3\,\pi }}{2}$, then $\cos \,\,2A + \cos \,\,2B + \cos \,\,2C$ is equal to

$1 - 4\;\cos A\,\,\cos B\,\,\cos C$

$1 + 2\cos A\,\,\cos B\,\,\cos C$

The sum ${\rm{\;S}} = {\rm{sin}}\,\,{\rm{\theta }} + \sin \,\,2{\rm{\theta }} + ... + \sin \,\,n{\rm{\theta \;}}$ equals

$\sin \,\,\frac{1}{2}\,\,\left( {n + 1} \right)\,\,{\rm{\theta }}\,\,\sin \,\,\frac{{n{\rm{\theta }}}}{2}/\sin \,\,\frac{{\rm{\theta }}}{2}$

$\cos \,\,\frac{1}{2}\,\,\left( {n + 1} \right)\,\,{\rm{\theta }}\,\,\sin \,\,\frac{{n{\rm{\theta }}}}{2}/\sin \,\,\frac{{\rm{\theta }}}{2}$

$\sin \,\,\frac{1}{2}\,\,\left( {n + 1} \right){\rm{\theta }}\,\,\cos \,\,\frac{{n{\rm{\theta }}}}{2}\,\,/\,\,\sin \,\,\frac{{\rm{\theta }}}{2}$

$\cos \,\,\frac{1}{2}\,\,\left( {n + 1} \right){\rm{\theta }}\,\,\cos \,\,\frac{{n{\rm{\theta }}}}{2}\,\,/\,\,\sin \,\,\frac{{\rm{\theta }}}{2}$

If $A + B + C = \pi $, then $\sin \,\,2A + \sin \,\,2B + \sin \,\,2C$ is equal to

$4\,\,\cos A\,\,\cos B\,\,\cos C$

$2\,\,\cos A\,\,\cos B\,\,\cos C$

$2\,\,\sin A\,\,\;{\rm{sin}}\;B\,\,\sin C$

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JEE • MATHEMATICS • TRANSFORMATIONS

Transformations practice · Mixed difficulty

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Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 151-165 of 165 questions

151 Easy

151 of 165

Application +4 | -1

\;\frac{{\sin \,\,A - \sin \,\,C}}{{\cos \,\,C - \cos \,\,A}} = \cot \,\,B,\,\;\mathrm{then}\;\,A,\,B,C

are in

View answer and explanation

Correct option: A

Option A (AP) is correct because the given condition implies that

B

is the arithmetic mean of

A

and

C

, which is the definition of an arithmetic progression.

152 Easy

152 of 165

Application +4 | -1

16\sin \left( {20^\circ } \right)\sin \left( {40^\circ } \right)\sin \left( {80^\circ } \right)

is equal to:

View answer and explanation

Correct option: B

2\sqrt 3

Option B (

2\sqrt 3

) is correct because the expression simplifies using the trigonometric identity

4\sin \theta \sin(60°-\theta)\sin(60°+\theta) = \sin 3\theta

153 Easy

153 of 165

Application +4 | -1

The value of

2\,\,\cos \,\,x - \cos \,\,3x - \cos \,\,5x\mathrm{\;}

is equal to

View answer and explanation

Correct option: A

16\,\,{\cos ^3}\,x\,\,{\sin ^2}\,x

Option A (

16\,\,{\cos ^3}\,x\,\,{\sin ^2}\,x

) is correct because by using trigonometric identities and factoring, the expression simplifies to

16\,\,{\cos ^3}\,x\,\,{\sin ^2}\,x

154 Easy

154 of 165

Application +4 | -1

The least value of

\mathrm{cose}{\mathrm{c}^2}\,x + 25\,\,{\sec ^2}\,x

View answer and explanation

Correct option: D

Option D (38) is correct because the minimum value of the expression is obtained by completing the square and finding the minimum value of the squared term.

155 Easy

155 of 165

Understanding +4 | -1

The function

f\left( x \right) = \left| {\cos x} \right|

is periodic with period

View answer and explanation

Correct option: B

\pi

Option B (

\pi

) is correct because the period of

|\cos x|

\pi

156 Medium

156 of 165

Application +4 | -1

f:R \to R

and is defined by

f\left( x \right) = \frac{1}{{2 - \cos 3x}}

for each

x \in R

, then the range

f

View answer and explanation

Correct option: B

\left[ {1/3,\mathrm{ }1} \right]

Option B

\left[ {1/3,\mathrm{ }1} \right]

is correct because the range is found by considering the bounds of the cosine function and its effect on the given function.

157 Easy

157 of 165

Analyzing +4 | -1

In a

\Delta \,ABC,\,\,\cos \left( {\frac{{B + 2C + 3A}}{2}} \right) + \cos \left( {\frac{{A - B}}{2}} \right)

is equal to

View answer and explanation

Correct option: B

Option B (0) is correct because by using the identity

A+B+C = \pi

and simplifying the expression, we can show that the sum of the cosines equals 0.

158 Medium

158 of 165

Application +4 | -1

A + B + C\; = \frac{{3\,\pi }}{2}

, then

\cos \,\,2A + \cos \,\,2B + \cos \,\,2C

is equal to

View answer and explanation

Correct option: D

1 - 4\sin A\,\,\sin B\,\,\sin C

Option D (

1 - 4\sin A\sin B\sin C

) is correct because, with the given condition, the expression simplifies to this form using trigonometric identities.

159 Medium

159 of 165

Analyzing +4 | -1

y = {\cos ^2}x + {\sec ^2}x

, then

View answer and explanation

Correct option: C

y \ge 2

Option C (

y \ge 2

) is correct because

{\cos ^2}x + {\sec ^2}x

can be rewritten as

(\cos x - \sec x)^2 + 2

, which is always greater than or equal to 2.

160 Easy

160 of 165

Application +4 | -1

The sum

\mathrm{\;S} = \mathrm{sin}\,\,\mathrm{\theta } + \sin \,\,2\mathrm{\theta } + ... + \sin \,\,n\mathrm{\theta \;}

equals

View answer and explanation

Correct option: A

\sin \,\,\frac{1}{2}\,\,\left( {n + 1} \right)\,\,\mathrm{\theta }\,\,\sin \,\,\frac{{n\mathrm{\theta }}}{2}/\sin \,\,\frac{\mathrm{\theta }}{2}

Option A (

\sin \frac{1}{2}\left( {n + 1} \right)\theta \sin \frac{{n\theta }}{2}/\sin \frac{{\theta }}{2}

) is correct because it represents the sum of a sine series using the appropriate trigonometric formula.

161 Easy

161 of 165

Application +4 | -1

The maximum value of

{\cos ^2}A + {\cos ^2}B - {\cos ^2}C

, is

View answer and explanation

Correct option: D

Option D (2) is correct because the maximum value is attained when cos²A and cos²B are maximized (equal to 1) and cos²C is minimized (equal to 0).

162 Medium

162 of 165

Analyzing +4 | -1

\;A + B + C = \pi \,\,\;\mathrm{and}\,\,\cos \,\,A = \cos \,\,B\,\,\cos \,\,C

then

\tan \,\,B\,\,\tan \,\,C

is equal to

View answer and explanation

Correct option: B

\mathrm{1}

Option B (1) is correct because by using the given condition

A + B + C = \pi

and the trigonometric identity, we can derive that

\tan B \tan C = 1

163 Medium

163 of 165

Analyzing +4 | -1

If the equation

\sin \,\,\theta \left( {\sin \,\,\theta + 2\,\,\cos \,\,\theta } \right) = a

has a real solution, then the shortest interval containing

'a'

View answer and explanation

Correct option: A

\left[ {\frac{{1 - \sqrt 5 }}{2},\,\,\frac{{1 + \sqrt 5 }}{2}} \right]

Option A is correct because the range of 'a' is determined by finding the maximum and minimum values of the trigonometric expression.

164 Easy

164 of 165

Analyzing +4 | -1

The equation

{\sin ^4}\theta + {\cos ^4}\theta = a

has a real solution if

View answer and explanation

Correct option: A

a \in \left[ {1/2,1} \right]

Option A is correct because it finds the range of 'a' for which the equation has real solutions by using trigonometric identities and the range of cosine function.

165 Medium

165 of 165

Application +4 | -1

A + B + C = \pi

, then

\sin \,\,2A + \sin \,\,2B + \sin \,\,2C

is equal to

View answer and explanation

Correct option: A

4\,\,\sin A\,\,\sin B\,\,\sin C

Option A (

4\,\,\sin A\,\,\sin B\,\,\sin C

) is correct because it is the result of applying trigonometric identities to simplify the expression

\sin 2A + \sin 2B + \sin 2C

when

A + B + C = \pi

Explore Other Chapters in MATHEMATICS

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SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

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