If $P = \frac{1}{2}\,\,{\sin ^2}\theta \,\, + \,\,\frac{1}{3}\,\,{\cos ^2}\theta $, then

$ - \frac{{\sqrt {13} }}{6} \le P \le \frac{{\sqrt {13} }}{6}$

The set of values of $x$ in $\left( { - \pi ,\pi } \right)$ satisfying the inequation $\left| {4\sin x - 1} \right| < \sqrt 5 $ is

$\left( { - \,\pi /10,\,\,\pi } \right)$

$\left( { - \,\pi ,\;\,3\pi /10} \right)$

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JEE • MATHEMATICS • TRIGONOMETRY RATIOS

Trigonometry ratios practice · Hard

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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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1 Hard

1 of 19

Application +4 | -1

\tan \alpha=\frac{m}{m+1}

and

\tan \beta=\frac{1}{2 m+1}

, then

\alpha+\beta=

View answer and explanation

Correct option: B

\frac{\pi}{4}

Option B (

\frac{{\pi}}{{4}}

) is correct because by using the formula for

\tan(\alpha + \beta)

, we can show that

\tan(\alpha + \beta) = 1

, which implies

\alpha + \beta = \frac{{\pi}}{{4}}

2 Hard

2 of 19

Analyzing +4 | -1

y = \frac{{{{\sec }^2}\theta \,\, - \,\,\tan \theta }}{{{{\sec }^2}\theta \,\, + \,\,\tan \theta }}

, then

View answer and explanation

Correct option: A

\frac{1}{3} < y < 3

Option A (

\frac{1}{3} < y < 3

) is correct because by substituting

x = \tan \theta

and analyzing the resulting quadratic equation, we find that the range of y is between

\frac{1}{3}

and 3.

3 Hard

3 of 19

Application +4 | -1

The extreme values of

4\cos \left( {{x^2}} \right)\cos \left( {\frac{\mathrm{\pi }}{3} + {x^2}} \right)\cos \left( {\frac{\mathrm{\pi }}{3} - {x^2}} \right)

over

R

, are

View answer and explanation

Correct option: A

- 1,1

Option A (-1, 1) is correct because the extreme values are found by simplifying the trigonometric expression and determining the range of the resulting cosine function.

4 Hard

4 of 19

Analyzing +4 | -1

The number of all possible triplets

\left( {{a_1},\,\,{a_2},\,{a_3}} \right)

such that

{a_1} + {a_2}\,\,\cos \,\,2x + {a_3}{\sin ^2}x = 0

for all

x

View answer and explanation

Correct option: D

None of these

The correct answer is None of these because there are infinitely many triplets that satisfy the given condition.

5 Hard

5 of 19

Application +4 | -1

If the roots of

x^3-7 x^2+5=0

are

\tan \alpha, \tan \beta

and

\tan \gamma

then find the value of

\sec ^2 \alpha \sec ^2 \beta \sec ^2 \gamma

View answer and explanation

Correct option: A

145

Option A (145) is correct because by transforming the given cubic equation into one with roots

\sec^2 \alpha

\sec^2 \beta

, and

\sec^2 \gamma

, the product of the roots of the new equation is 145.

6 Hard

6 of 19

Application +4 | -1

The minimum value of the function

f(x)=3 \tan x+4 \cot x

where

x \in\left(0, \frac{\pi}{2}\right)

View answer and explanation

Correct option: B

2 \sqrt{12}

Option B (2 sqrt(12)) is correct because the minimum value of the function can be found using AM >= GM inequality, which gives 3tan(x) + 4cot(x) >= 2*sqrt(3tan(x)*4cot(x)) = 2*sqrt(12).

7 Hard

7 of 19

Analyzing +4 | -1

The range of values of

a

for which the function

f\left( x \right) = \left( {{a^2} - 7a + 12} \right)\cos x + 2\left( {a - 4} \right)x + 3{e^5}

does not possess critical points, is

View answer and explanation

Correct option: B

\left( {1,\;4} \right) \cup \left( {4,\;5} \right)

Option B (

\left( {1,\;4} \right) \cup \left( {4,\;5} \right)

) is correct because it represents the values of 'a' for which the function has no critical points, derived from the condition that the derivative has no real solutions.

8 Hard

8 of 19

Application +4 | -1

How many times do the minute and hour hands of a clock coincide in a period of

12 \mathrm{~h}

View answer and explanation

Correct option: D

Option D (11) is correct because the minute and hour hands coincide 11 times in a 12-hour period.

9 Hard

9 of 19

Analyzing +4 | -1

\left| {\,\cos \,\,\theta \,\,\left\{ {\,\sin \,\,\theta + \sqrt {{{\sin }^2}\,\,\theta + {{\sin }^2}\,\,\alpha } } \right\}\,} \right| \le k

, then the value of

k

View answer and explanation

Correct option: B

\sqrt {1 + {{\sin }^2}\,\,\alpha }

Option B (

\sqrt {1 + {{\sin }^2}\,\alpha }

) is correct because it represents the maximum value of the given expression, derived using trigonometric manipulations and inequalities.

10 Hard

10 of 19

Application +4 | -1

P = \frac{1}{2}\,\,{\sin ^2}\theta \,\, + \,\,\frac{1}{3}\,\,{\cos ^2}\theta

, then

View answer and explanation

Correct option: A

\frac{1}{3} \le P \le \frac{1}{2}

Option A (

\frac{1}{3} \le P \le \frac{1}{2}

) is correct because by rewriting P in terms of cos²θ and using the range of cos²θ, we can find the range of P.

11 Hard

11 of 19

Application +4 | -1

The minimum value of

{27^{\cos 2x}}{.81^{\sin 2x}}

, is

View answer and explanation

Correct option: A

\frac{1}{{243}}

Option A (

\frac{1}{{243}}

) is correct because the minimum value is found by minimizing the exponent of 3 using trigonometric identities.

12 Hard

12 of 19

Application +4 | -1

x=\sin \theta|\sin \theta|, y=\cos \theta|\cos \theta|

, where

\frac{99 \pi}{2} \leq \theta \leq 50 \pi

, then

View answer and explanation

Correct option: D

y-x=1

Option D (

y-x=1

) is correct because within the specified range of

\theta

, x becomes

-\sin^2 \theta

and y becomes

\cos^2 \theta

, leading to

y-x = \cos^2 \theta + \sin^2 \theta = 1

13 Hard

13 of 19

Application +4 | -1

ABCD

is a convex quadrilateral such that

4\sec A + 5 = 0

then the quadratic equation whose roots are

\tan A

and

\mathrm{cosec}\;A

View answer and explanation

Correct option: B

12{x^2} - 11x - 15 = 0

The correct answer is

12{x^2} - 11x - 15 = 0

because it is the quadratic equation formed with roots

\tan A

and

\csc A

derived from the given condition.

14 Hard

14 of 19

Application +4 | -1

A watch which gains 5 seconds in 3 minutes was set right at

7 \mathrm{am}

. In the afternoon of the same day, when the watch indicated quarter past 4 o'clock, the true time was

View answer and explanation

Correct option: B

14 \cdot \frac{22}{37} \min

to 4

Option B (

14 \frac{22}{37}

min to 4) is correct because it calculates the true time by accounting for the watch's gain over the period from 7 am to 4:15 pm.

15 Hard

15 of 19

Application +4 | -1

How many times do the hands of the watch form a right angle during a complete day?

View answer and explanation

Correct option: D

Option D (44) is correct because the hands of a watch form a right angle 44 times during a complete day.

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

16 Hard

16 of 19

Application +4 | -1

The set of values of

x

\left( { - \pi ,\pi } \right)

satisfying the inequation

\left| {4\sin x - 1} \right| < \sqrt 5

View answer and explanation

Correct option: A

\left( { - \,\pi /10,\;\,3\pi /10} \right)

Option A

\left( { - \,\pi /10,\;\,\,3\pi /10} \right)

is correct because it represents the interval of

x

values that satisfy the inequality

\left| {4\sin x - 1} \right| < \sqrt 5

within the given range of

\left( { - \pi ,\pi } \right)

17 Hard

17 of 19

Analyzing +4 | -1

Let

f\left( x \right) = \cos x\sin 2x

Then,

\min \left| {f\left( x \right): - \pi \le x \le \pi } \right|

is greater than

View answer and explanation

Correct option: A

\frac{{ - 9}}{7}

Option A (

\frac{{ - 9}}{7}

) is correct because the minimum value of the function is greater than -9/7.

18 Hard

18 of 19

Application +4 | -1

\cos \alpha=3 / 5, \cos \beta=4 / 5, \alpha, \beta>0

and are acute angles, then

\cos \frac{\alpha-\beta}{2}

View answer and explanation

Correct option: D

None of these

The correct answer is None of these, as the calculated value of

\cos \frac{\alpha-\beta}{2}

\frac{7}{5 \sqrt{2}}

, which is not among the given options.

19 Hard

19 of 19

Application +4 | -1

\alpha

and

\beta

are acute such that

\tan (\alpha+\beta)

and

\tan (\alpha-\beta)

satisfy the equation

x^2-4 x+1=0

, then

(\alpha, \beta)=

View answer and explanation

Correct option: C

\left(45^{\circ}, 30^{\circ}\right)

Option C ((45°, 30°)) is correct because solving the quadratic equation and using the tangent addition and subtraction formulas, we find that α = 45° and β = 30°.

Revision notes and key takeaways

Review TRIGONOMETRY RATIOS in your textbook or class notes, then use this set to check your understanding.

Key Takeaways

✓ This set contains 19 single-correct-answer MCQs on TRIGONOMETRY RATIOS.
✓ Try each question before opening its answer. Read the explanation where one is available.
✓ Sign in to save submitted answers. Revisit incorrect answers alongside your textbook or class notes.

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