# JEE Main

Explore popular questions from Trigonometry for JEE Main. This collection covers Trigonometry previous year JEE Main questions hand picked by experienced teachers.

## Mathematics

Trigonometry

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Q 1. A body weighing {tex} 13\mathrm { kg } {/tex} is suspended by two strings {tex} 5 \mathrm { m } {/tex} and {tex} 12 \mathrm { m } {/tex} long, their other ends being fastened to the extremities of a rod {tex} 13 \mathrm { m } {/tex} long. If the rod be so held that the body hangs immediately below the middle point. The tensions in the strings are:

A

12{tex} \mathrm { kg } {/tex} and 13{tex} \mathrm { kg } {/tex}

B

5{tex} \mathrm { kg } {/tex} and 5{tex} \mathrm { kg } {/tex}

5{tex} \mathrm { kg } {/tex} and 12{tex} \mathrm { kg } {/tex}

D

5{tex} \mathrm { kg } {/tex} and 13{tex} \mathrm { kg } {/tex}

##### Explanation

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Q 2. A tower stands at the centre of a circular park. {tex} A {/tex} and {tex} B {/tex} are two points on the boundary of the park such that {tex} A B ( = a ) {/tex} subtends an angle of {tex} 60 ^ { \circ } {/tex} at the foot of the tower, and the angle of elevation of the top of the tower from {tex} A {/tex} or {tex} B {/tex} is {tex} 30 ^ { \circ } {/tex}. The height of the tower is

A

{tex} \frac { 2 a } { \sqrt { 3 } } {/tex}

B

{tex}2 a \sqrt { 3 } {/tex}

{tex} \frac { a } { \sqrt { 3 } } {/tex}

D

{tex} a \sqrt { 3 } {/tex}

##### Explanation

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Q 3. Let {tex} P = ( - 1,0 ) , Q = ( 0,0 ) {/tex} and {tex} R = ( 3,3 \sqrt { 3 } ) {/tex} be three points. The equation of the bisector of the angle {tex} P Q R {/tex} is

{tex} \sqrt { 3 } x + y = 0 {/tex}

B

{tex} x + \frac { \sqrt { 3 } } { 2 } y = 0 {/tex}

C

{tex} \frac { \sqrt { 3 } } { 2 } x + y = 0 {/tex}

D

{tex} x + \sqrt { 3 } y = 0 {/tex}

##### Explanation

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Q 4. A bird is sitting on the top of a vertical pole {tex} 20 \mathrm { m } {/tex} high and its elevation from a point {tex} O {/tex} on the ground is {tex} 45 ^ { \circ } . {/tex} It flies off horizontally straight away from the point {tex} O {/tex} . After one second, the elevation of the bird from {tex} O {/tex} is reduced to {tex} 30 ^ { \circ } . {/tex} Then the speed ( in m/s ) of the bird is

A

20{tex} \sqrt { 2 } {/tex}

20{tex} ( \sqrt { 3 } - 1 ) {/tex}

C

40{tex} ( \sqrt { 2 } - 1 ) {/tex}

D

40{tex} ( \sqrt { 3 } - \sqrt { 2 } ) {/tex}

##### Explanation

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Q 5. If the angles of elevation of the top of a tower from three collinear points {tex} A , B {/tex} and {tex} C , {/tex} on a line leading to the foot of the tower are {tex} 30 ^ { \circ } , 45 ^ { \circ } {/tex} and {tex} 60 ^ { \circ } , {/tex} respectively, then the ratio, {tex} A B : B C , {/tex} is

A

{tex} \sqrt { 3 } : \sqrt { 2 } {/tex}

B

{tex} 1 : \sqrt { 3 } {/tex}

C

{tex} 2 : 3 {/tex}

{tex} \sqrt { 3 } : 1 {/tex}

##### Explanation

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Q 6. In a {tex} \triangle A B C , \frac { a } { b } = 2 + \sqrt { 3 } {/tex} and {tex} \angle C = 60 ^ { \circ } . {/tex} The ordered pair {tex} ( \angle A , \angle B ) {/tex} is equal to

A

{tex} \left( 15 ^ { \circ } , 105 ^ { \circ } \right) {/tex}

{tex} \left( 105 ^ { \circ } , 15 ^ { \circ } \right) {/tex}

C

{tex} \left( 45 ^ { \circ } , 75 ^ { \circ } \right) {/tex}

D

{tex} \left( 75 ^ { \circ } , 45 ^ { \circ } \right) {/tex}

##### Explanation

since {tex}sin75\degree=sin(180\degree-105\degree)=sin105\degree{/tex}

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Q 7. A man is walking towards a vertical pillar in a straight path, at a uniform speed. At a certain point {tex} A {/tex} on the path, he observes that the angle of elevation of the top of the pillar is {tex} 30 ^ { \circ }{/tex}. After walking for {tex}10 \mathrm { min } {/tex} from {tex} A {/tex} in the same direction, at a point {tex} B , {/tex} he observes that the angle of elevation of the top of the pillar is now {tex} 60 ^ { \circ } . {/tex} Then, the time taken (in minutes) by him from {tex} B {/tex} to reach the pillar is

5

B

6

C

10

D

20

##### Explanation

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Q 8. The angle of elevation of the top of a vertical tower from a point {tex} A {/tex} due east of it is {tex} 45 ^ { \circ } . {/tex} The angle of elevation of the top of the same tower from a point {tex} B {/tex} due south of {tex} A {/tex} is {tex} 30 ^ { \circ } {/tex} . If the distance between {tex} A {/tex} and {tex} B {/tex} is {tex} 54\sqrt { 2 } \mathrm { m } {/tex} , then the height of the tower (in metres), is

A

{tex}108{/tex}

B

{tex}36 \sqrt { 3 } {/tex}

C

{tex}54 \sqrt { 3 } {/tex}

{tex}54{/tex}

##### Explanation

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Q 9. If {tex} \tan x = n \cdot \tan y , n \in R ^ { + } , {/tex} then maximum value of {tex} \sec ^ { 2 } ( x - y ) {/tex} is equal to

A

{tex} \frac { ( n + 1 ) ^ { 2 } } { 2 n } {/tex}

B

{tex} \frac { ( n + 1 ) ^ { 2 } } { n } {/tex}

C

{tex} \frac { ( n + 1 ) ^ { 2 } } { 2 } {/tex}

{tex} \frac { ( n + 1 ) ^ { 2 } } { 4 n } {/tex}

##### Explanation

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Q 10. Let {tex} A {/tex} and {tex} B {/tex} denote the statements
{tex} { A : \cos \alpha + \cos \beta + \cos \gamma = 0 }{/tex}
{tex}{ B : \sin \alpha + \sin \beta + \sin \gamma = 0 } {/tex}
If {tex} \cos ( \beta - \gamma ) + \cos ( \gamma - \alpha ) + \cos ( \alpha - \beta ) = - \frac { 3 } { 2 } , {/tex} then

A

{tex} A {/tex} is true and {tex} B {/tex} is false

B

{tex} A {/tex} is false and {tex} B {/tex} is true

Both {tex} A {/tex} and {tex} B {/tex} are true

D

Both {tex} A {/tex} and {tex} B {/tex} are false

##### Explanation

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Q 11. Let {tex} \cos ( \alpha + \beta ) = \frac { 4 } { 5 } {/tex} and {tex} \sin ( \alpha - \beta ) = \frac { 5 } { 13 } , {/tex} where {tex} 0 \leq \alpha , \beta \leq \frac { \pi } { 4 } {/tex} Then {tex} \tan 2 \alpha = {/tex}

{tex} \frac { 56 } { 33 } {/tex}

B

{tex} \frac { 19 } { 12 } {/tex}

C

{tex} \frac { 20 } { 7 } {/tex}

D

{tex} \frac { 25 } { 16 } {/tex}

##### Explanation

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Q 12. If {tex} A = \sin ^ { 2 } x + \cos ^ { 4 } x , {/tex} then for all real {tex} x {/tex}

A

{tex} \frac { 13 } { 16 } \leq A \leq 1 {/tex}

B

{tex} 1 \leq A \leq 2 {/tex}

C

{tex} \frac { 3 } { 4 } \leq A \leq \frac { 13 } { 16 } {/tex}

{tex} \frac { 3 } { 4 } \leq A \leq 1 {/tex}

##### Explanation

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Q 13. In a {tex} \Delta P Q R , {/tex} if {tex} 3 \sin P + 4 \cos Q = 6 {/tex} and {tex} 4 \sin Q + 3 \cos P = 1 , {/tex} then the angle {tex} R {/tex} is equal to

A

{tex} \frac { 5 \pi } { 6 } {/tex}

{tex} \frac { \pi } { 6 } {/tex}

C

{tex} \frac { \pi } { 4 } {/tex}

D

{tex} \frac { 3 \pi } { 4 } {/tex}

##### Explanation

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Q 14. The expression {tex} \frac { \tan A } { 1 - \cot A } + \frac { \cot A } { 1 - \tan A } {/tex} can be written as

{tex} \sec A \cosec A + 1 {/tex}

B

{tex} \tan A + \cot A {/tex}

C

{tex} \sec A + \cosec A {/tex}

D

{tex} \sin A \cos A + 1 {/tex}

##### Explanation

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Q 15. Let {tex} f _ { k } ( x ) = \frac { 1 } { k } \left( \sin ^ { k } x + \cos ^ { k } x \right) , {/tex} where {tex} x \in \mathbb { R } {/tex} and {tex} k \geq 1{/tex}. Then, {tex} f _ { 4 } ( x ) - f _ { 6 } ( x ) {/tex} equals

A

{tex} \frac { 1 } { 4 } {/tex}

{tex} \frac { 1 } { 12 } {/tex}

C

{tex} \frac { 1 } { 6 } {/tex}

D

{tex} \frac { 1 } { 3 } {/tex}

##### Explanation

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Q 16. The number of the values of {tex} \alpha {/tex} in {tex} [ 0,2 \pi ] {/tex} for which {tex} 2 \sin ^ { 3 } \alpha - 7 \sin ^ { 2 } \alpha + 7 \sin \alpha = 2 {/tex} is

A

6

B

4

3

D

1

##### Explanation

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Q 17. If {tex} \cosec \theta = \frac { p + q } { p - q } ( p \neq q \neq 0 ) , {/tex} then {tex} \left| \cot \left( \frac { \pi } { 4 } + \frac { \theta } { 2 } \right) \right| {/tex} is equal to

A

{tex} \sqrt { \frac { p } { q } } {/tex}

{tex} \sqrt { \frac { q } { p } } {/tex}

C

{tex} \sqrt { q p } {/tex}

D

{tex} p q {/tex}

##### Explanation

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Q 18. The most general value of {tex} \theta {/tex} satisfying {tex} 3 - 2 \cos \theta - 4 \sin \theta {/tex} {tex} - \cos 2 \theta + \sin 2 \theta = 0 {/tex} is

{tex}2 n \pi {/tex}

B

{tex} 2 \pi + \pi / 2 {/tex}

C

{tex}4 n \pi {/tex}

D

{tex} 2 n \pi + \pi / 4 {/tex}

##### Explanation

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Q 19. The general solution for {tex} \theta {/tex} if {tex} \sin \left( 2 \theta + \frac { \pi } { 6 } \right) + \cos \left( \theta + \frac { 5 \pi } { 6 } \right) = 2 {/tex} is

{tex} 2 n \pi + \frac { 7 \pi } { 6 } {/tex}

B

{tex} 2 n \pi + \frac { \pi } { 6 } {/tex}

C

{tex} 2 n \pi - \frac { 7 \pi } { 6 } {/tex}

D

None of these

##### Explanation

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Q 20. The number of solutions of the equation {tex} \tan x + \sec x = 2 \cos x {/tex} lying in the interval {tex} [ 0,2 \pi ] {/tex} is

A

0

B

1

2

D

3

##### Explanation

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Q 21. The solution of the equation {tex} \tan \theta \tan 2 \theta = 1 {/tex} is

A

{tex} n \pi + \frac { 5 \pi } { 12 } {/tex}

B

{tex} n \pi - \frac { 5 \pi } { 12 } {/tex}

C

{tex} 2 n \pi \pm \frac { \pi } { 4 } {/tex}

{tex} n \pi \pm \frac { \pi } { 6 } {/tex}

##### Explanation

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Q 22. Find the general solution of the equation {tex}\sin x - 3 \sin 2 x + \sin 3 x = \cos x - 3 \cos 2 x + \cos 3 x {/tex}

A

{tex}\large \frac { n \pi } { 2 } + \frac { 5 \pi } { 12 } {/tex}

B

{tex} n \pi - \frac { 5 \pi } { 12 } {/tex}

{tex}\large \frac { n \pi } { 2 } + \frac { \pi } { 8 } {/tex}

D

{tex} n \pi \pm \frac { \pi } { 8 } {/tex}

##### Explanation

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Q 23. Solve for {tex} x {/tex} the equation {tex} \sin ^ { 3 } x + \sin x \cos x + \cos ^ { 3 } x = 1 : {/tex}

A

2{tex} m \pi {/tex}

B

{tex} ( 4 n + 1 ) \frac { \pi } { 2 } {/tex}

Both A and B

D

None of these

##### Explanation

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Q 24. The equation {tex} e ^ { \sin x } - e ^ { - \sin x } - 4 = 0 {/tex} has

No real solution

B

One real solution

C

Two real solutions

D

Cannot be determined

##### Explanation

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Q 25. If {tex} \tan ( \pi \cos x ) = \cot ( \pi \sin x ) , {/tex} then {tex} \cos \left( x - \frac { \pi } { 4 } \right) {/tex} is

A

{tex} \frac { 1 } { \sqrt { 2 } } {/tex}

{tex} \frac { 1 } { 2 \sqrt { 2 } } {/tex}

C

{tex}\small 0{/tex}

D

None of these