If $\;\cos \,\,x + \cos \,\,y + \cos \,\,\alpha = 0\;\,{\rm{and}}\,\sin x + \sin \,\,y + \sin \,\,\alpha = 0$ then $\cot \left( {\frac{{x + y}}{2}} \right)\;$ is equal to

$\sin \,\,\left( {\frac{{x + y}}{2}} \right)$

$\cos \,\,2{\rm{\theta }} + 2\,\,\cos \,\,{\rm{\theta \;}}$ is always

Less than or equal to $\frac{3}{2}$

Consider the following statements: 1.If $\sin A = \sin B$ then we have $\sin \,\,2A = \sin \,\,2B\;$ always 2.The value of ${\rm{\;cos}}\,\,\frac{\pi }{7}\,\,\cos \,\,\frac{{4\,\pi }}{7}\,\,\cos \,\,\frac{{5\,\pi }}{7}{\rm{\;is}}\,\;\frac{1}{4}$ Which of the statements given above is/are correct?

Neither $\left( {\rm{1}} \right)$ nor $\left( {\rm{2}} \right)$

Both $\left( {\rm{1}} \right)$ and $\left( {\rm{2}} \right)$

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JEE • MATHEMATICS • TRANSFORMATIONS

Transformations practice · Easy

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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 1-30 of 112 questions

1 Easy

1 of 112

Application +4 | -1

The value of

\cos \,\,\frac{\pi }{{11}} + \cos \,\,\frac{{3\,\pi }}{{11}} + \cos \,\,\frac{{5\,\pi }}{{11}} + \cos \,\,\frac{{7\pi }}{{11}} + \cos \,\,\frac{{9\,\pi }}{{11}}

, is

View answer and explanation

Correct option: C

\frac{1}{2}

Option C (

\frac{1}{2}

) is correct because the given sum of cosines simplifies to 1/2 using trigonometric identities and sum-to-product formulas.

2 Easy

2 of 112

Analyzing +4 | -1

\;\cos \,\,x + \cos \,\,y + \cos \,\,\alpha = 0\;\,\mathrm{and}\,\sin x + \sin \,\,y + \sin \,\,\alpha = 0

then

\cot \left( {\frac{{x + y}}{2}} \right)\;

is equal to

View answer and explanation

Correct option: C

\cot \,\,\alpha

Option C (

\cot \alpha

) is correct because by using trigonometric identities and dividing the equations, we can derive that

\cot \left( {\frac{{x + y}}{2}} \right) = \cot \alpha

3 Easy

3 of 112

Application +4 | -1

The period of the function

f\left( x \right) = \sin \left( {\frac{{2x + 3}}{{6\;\pi }}} \right),

View answer and explanation

Correct option: C

6\;{\pi ^2}

Option C (

6{\pi ^2}

) is correct because the period of sin(ax + b) is 2π/|a|, and in this case, a = 1/(3π).

4 Easy

4 of 112

Analyzing +4 | -1

0 < x,\mathrm{ }y < \pi

and

\cos x + \cos y - \cos (x + y) = \frac{3}{2}

, then

\sin x + \cos y

is equal to

View answer and explanation

Correct option: D

\frac{{1 + \sqrt 3 }}{2}

Option D (

\frac{{1 + \sqrt 3 }}{2}

) is correct because solving the given trigonometric equation leads to

\cos x = \frac{1}{2}

and

\sin x = \frac{{\sqrt 3 }}{2}

, allowing us to find the value of

\sin x + \cos y

5 Easy

5 of 112

Application +4 | -1

The value of

\mathrm{\;sin}\,\,36^\circ \,\,\sin \,\,72^\circ \,\,\sin \,\,108^\circ \,\,\sin \,\,144^\circ \;

is equal to

View answer and explanation

Correct option: B

\frac{1}{16}

Option D (

\frac{5}{16}

) is correct because by using the properties of sine function and trigonometric identities, the expression simplifies to

\frac{5}{16}

6 Easy

6 of 112

Application +4 | -1

The value of

\cos \,\,12^\circ \,\,\cos \,\,24^\circ \,\,\cos \,\,36^\circ \,\,\cos \,\,48^\circ \,\,\cos \,\,72^\circ \,\,\cos \,\,84^\circ

, is

View answer and explanation

Correct option: A

1/64

Option A (1/64) is correct because it is the result of applying trigonometric identities to simplify the product of cosines.

7 Easy

7 of 112

Analyzing +4 | -1

\cos \,\,2\mathrm{\theta } + 2\,\,\cos \,\,\mathrm{\theta \;}

is always

View answer and explanation

Correct option: C

Greater than or equal to

- \frac{3}{2}

and less than or equal to

3

Option C (Greater than or equal to

- \frac{3}{2}

and less than or equal to

3

) is correct because the expression can be rewritten as a quadratic in

\cos \theta

, and its range can be determined.

8 Easy

8 of 112

Analyzing +4 | -1

y\tan \left( {A + B + C} \right) = x\tan \left( {A + B - C} \right) = \lambda

, then the

2C =

View answer and explanation

Correct option: D

\frac{{\lambda \left( {x - y} \right)}}{{xy + {\lambda ^2}}}

Option D is correct because it uses the tangent subtraction formula to find tan(2C) and then substitutes the given values to arrive at the correct expression.

9 Easy

9 of 112

Application +4 | -1

The number of integral values of

k

for which the equation

7\cos \,\,\theta + 5\sin \theta = 2k + 1

has a solution is

View answer and explanation

Correct option: B

Option B (8) is correct because it represents the number of integral values of k for which the given trigonometric equation has a solution, determined by the range of the expression on the left-hand side.

10 Easy

10 of 112

Application +4 | -1

If in a

\triangle A B C, C=3 B

and

C-B=90^{\circ}

, then

\tan B

equals

View answer and explanation

Correct option: B

\frac{1}{3}

Option B (

\frac{1}{3}

) is correct because given the conditions

C=3B

and

C-B=90^{\circ}

, we can deduce that

B = 45^{\circ}

and

\tan B = \frac{1}{3}

11 Easy

11 of 112

Application +4 | -1

The period of the function

\sin \left( {\frac{{2x}}{3}} \right) + \sin \left( {\frac{{3x}}{2}} \right)

View answer and explanation

Correct option: D

12\pi

Option D (

12\pi

) is correct because it is the least common multiple of the individual periods of the sine functions.

12 Easy

12 of 112

Application +4 | -1

The value of

\cos \,\,\frac{\pi }{7}\,\,\cos \,\,\frac{{2\pi }}{7}\,\,\cos \,\,\frac{{3\pi }}{7}

, is

View answer and explanation

Correct option: A

\frac{1}{8}

Option A

\frac{1}{8}

is correct because by using the identity

\cos x \cos 2x \cos 4x = \frac{{\sin 8x}}{{8\sin x}}

and simplifying, the value of the expression is

\frac{1}{8}

13 Easy

13 of 112

Analyzing +4 | -1

The value of

\;1 + \cos \,\,56^\circ + \cos \,\,58^\circ - \cos \,\,66^\circ \;

is equal to

View answer and explanation

Correct option: B

4\,\cos \,\,28^\circ \,\,\cos \,\,29^\circ \,\,\sin \,\,33^\circ

Option B (

4\,\cos \,28^\circ \,\cos \,29^\circ \,\sin \,33^\circ

) is correct because by using trigonometric identities such as

1 + \cos 2x = 2{\cos ^2}x

and the sum-to-product formulas, the expression can be simplified to this form.

14 Easy

14 of 112

Application +4 | -1

The minimum value of f

f\left( x \right) = {\sin ^4}\,\,x + {\cos ^4}\,\,x,\,\;0 \le \,\,x\,\, \le \,\,\frac{\pi }{2}

View answer and explanation

Correct option: D

\frac{1}{2}

Option D (

\frac{1}{2}

) is correct because the minimum value of the function

f(x) = \sin^4 x + \cos^4 x

is found by rewriting it in terms of

\sin^2 2x

and minimizing that expression.

15 Easy

15 of 112

Application +4 | -1

The value of

\cos \,\,\frac{\pi }{9}\,\,\cos \,\,\frac{{2\,\pi }}{9}\,\,\cos \,\,\frac{{3\,\pi }}{9}\,\,\cos \,\,\frac{{4\,\pi }}{9}

View answer and explanation

Correct option: B

\frac{1}{16}

Option B (

\frac{1}{16}

) is correct because the value of

\cos \,\frac{\pi }{9}\,\,\cos \,\frac{{2\,\pi }}{9}\,\,\cos \,\frac{{3\,\pi }}{9}\,\,\cos \,\frac{{4\,\pi }}{9}

\frac{1}{16}

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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

16 Easy

16 of 112

Application +4 | -1

The period of the function

f\left( x \right) = {\sin ^4}x + {\cos ^4}x

View answer and explanation

Correct option: B

\pi /2

Option B (

\pi /2

) is correct because the function can be simplified to a cosine function with a period of

\pi/2

17 Easy

17 of 112

Application +4 | -1

The value of

\sin \,\,12^\circ \,\,\sin \,\,48^\circ \,\,\sin \,\,54^\circ \mathrm{\;}

is equal to

View answer and explanation

Correct option: C

\frac{1}{{8}}

Option C

\frac{1}{{8}}

is correct because using the product-to-sum identities and the value of

\cos 36^\circ

, the expression simplifies to

\frac{1}{8}

18 Easy

18 of 112

Analyzing +4 | -1

\frac{{\sin \,\,7\theta + 6\,\,\sin \,\,5\theta + 17\,\,\sin \,\,3\theta + 12\,\,\sin \,\,\theta }}{{\sin \,\,6\theta + 5\,\,\sin \,\,4\theta + 12\,\,\sin \,\,2\theta }}

is equal to

View answer and explanation

Correct option: A

2\,\,\cos \,\,\theta

Option A (

2 \cos \theta

) is correct because by applying trigonometric identities and simplifying, the given expression reduces to

2 \cos \theta

19 Easy

19 of 112

Analyzing +4 | -1

Let

A

and

B

denote the statements

\begin{array}{l} A:\cos \,\,\alpha + \cos \,\,\beta + \cos \,\,\gamma = 0\\ B:\sin \,\,\alpha + \sin \,\,\beta + \sin \,\,\gamma = 0 \end{array}

\cos \,\,\left( {\beta - \gamma } \right) + \cos \left( {\gamma - \alpha } \right) + \cos \left( {\alpha - \beta } \right) = - \frac{3}{2}

, then

View answer and explanation

Correct option: C

Both

A

and

B

are true

Option C (Both

A

and

B

are true) is correct because the given condition implies that both

(\cos \,\alpha + \cos \,\beta + \cos \,\gamma)^2

and

(\sin \,\alpha + \sin \,\beta + \sin \,\gamma)^2

must be zero, hence both statements are true.

20 Easy

20 of 112

Application +4 | -1

In a

\Delta \,ABC,\,\,\cos \,\,A = \cos \,\,B\,\,\cos \,\,C

then

\cot \,\,B\,\,\cot \,\,C

is equal to

View answer and explanation

Correct option: C

1/2

Option C (8 : 7 : 6) is correct because it is the ratio of

s-a : s-b : s-c

where

r_1, r_2, r_3

are the exradii of the triangle.

21 Easy

21 of 112

Application +4 | -1

n \in N

, and the period of

\frac{{\cos nx}}{{\sin \left( {\frac{x}{n}} \right)}}\;is\;4\pi

, then n is equal to

View answer and explanation

Correct option: C

Option C (2) is correct because the period of the combined function is found by equating the period of the sine function to

4\pi

and solving for n.

22 Easy

22 of 112

Application +4 | -1

\sin \,\,B = \frac{1}{5}\,\,\sin \,\,\left( {2A + B} \right)

, then

\frac{{\tan \,\,\left( {A + B} \right)}}{{\tan \,\,A}}

is equal to

View answer and explanation

Correct option: C

3/2

Option A (2) is correct because if

\sin x + \csc x = 2

, then

\sin x = 1

, and therefore

\sin^n x + \csc^n x = 1 + 1 = 2

23 Easy

23 of 112

Analyzing +4 | -1

A = \tan \,\,6^\circ \,\,\tan \,\,42^\circ

and

B = \cot \,\,66^\circ \cot \,\,78^\circ

, then

View answer and explanation

Correct option: C

A = B

Option C (

A = B

) is correct because simplifying the expression

\frac{A}{B}

using trigonometric identities results in 1, thus A = B.

24 Easy

24 of 112

Analyzing +4 | -1

Consider the following statements:

1.If

\sin A = \sin B

then we have

\sin \,\,2A = \sin \,\,2B\;

always

2.The value of

\mathrm{\;cos}\,\,\frac{\pi }{7}\,\,\cos \,\,\frac{{4\,\pi }}{7}\,\,\cos \,\,\frac{{5\,\pi }}{7}\mathrm{\;is}\,\;\frac{1}{4}

Which of the statements given above is/are correct?

View answer and explanation

Correct option: D

Neither

\left( \mathrm{1} \right)

nor

\left( \mathrm{2} \right)

Option D (Neither (1) nor (2)) is correct because statement 1 is not always true, and the value in statement 2 is

\frac{1}{8}

, not

\frac{1}{4}

25 Easy

25 of 112

Application +4 | -1

The maximum value of

\frac{1}{{3\,\sin \,\,\theta - 4\,\,\cos \,\,\theta + 7}}

, is

View answer and explanation

Correct option: A

\frac{1}{{12}}

Option A (

\frac{1}{{12}}

) is correct because the maximum value of the given expression is obtained when the denominator is minimized, which occurs when

3\sin \theta - 4\cos \theta

is at its minimum value of -5.

26 Easy

26 of 112

Application +4 | -1

The maximum value of

5\cos \,\,\theta + 3\cos \left( {\theta + \frac{\pi }{3}} \right) + 3

View answer and explanation

Correct option: C

Option C (10) is correct because it is derived by simplifying the expression and finding the maximum value of the resulting trigonometric function.

27 Easy

27 of 112

Analyzing +4 | -1

The value of

\tan 9^\circ - \tan 27^\circ - \tan 63^\circ + \tan 81^\circ

View answer and explanation

Correct option: C

Option C (4) is correct because the expression simplifies to 4 using trigonometric identities.

28 Easy

28 of 112

Application +4 | -1

\sin \,\,47^\circ - \sin \,\,25^\circ + \sin \,\,61^\circ - \sin \,\,11^\circ =

View answer and explanation

Correct option: A

\cos \,\,7^\circ

Option A (

\cos \,7^\circ

) is correct because it is the simplified form of the given trigonometric expression after applying sum-to-product and product-to-sum identities.

29 Easy

29 of 112

Application +4 | -1

The value of

\frac{{\sin \,\,85^\circ - \sin \,35^\circ }}{{\cos \,\,65^\circ }}

View answer and explanation

Correct option: C

\mathrm{1}

Option C (1) is correct because it is obtained by applying the sine subtraction formula and simplifying the expression.

30 Easy

30 of 112

Application +4 | -1

The maximum of the function

3\,\,\cos \,\,x - 4\,\,\sin \,\,x

View answer and explanation

Correct option: D

Option D (5) is correct because it represents the maximum value of the function, calculated using the formula

\sqrt{a^2 + b^2}

for functions of the form

a \cos x - b \sin x

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Key Takeaways

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