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JEE Advanced

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

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Q 1. The general solution of {tex} \sin x - 3 \sin 2 x + \sin 3 x = \cos x - 3 \cos 2 x + \cos 3 x {/tex} is

A

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

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

C

{tex} ( - 1 ) ^ { n } \frac { n \pi } { 2 } + \frac { \pi } { 8 } {/tex}

D

{tex} 2 n \pi + \cos ^ { - 1 } \frac { 3 } { 2 } {/tex}

Explanation

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Q 2. Let {tex} \mathrm { S } = \left\{ \mathrm { x } \in ( - \pi , \pi ): \mathrm { x } \neq 0 , \pm \frac { \pi } { 2 } \right\} . {/tex} The sum of all distinct
solutions of the equation {tex} \sqrt { 3 } \sec x + \mathrm {cosec} x + 2 ( \tan x -\cot x ) {/tex} {tex} = 0 {/tex} in the set {tex} S {/tex} is equal to

A

{tex} - \frac { 7 \pi } { 9 } {/tex}

B

{tex} - \frac { 2 \pi } { 9 } {/tex}

0

D

{tex} \frac { 5 \pi } { 9 } {/tex}

Explanation



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Q 3. The value of {tex} \displaystyle \sum _ { k = 1 } ^ { 13 } \frac { 1 } { \sin \left( \frac { \pi } { 4 } + \frac { ( k - 1 ) \pi } { 6 } \right) \sin \left( \frac { \pi } { 4 } + \frac { k \pi } { 6 } \right) } {/tex} is equal to

A

{tex} 3 - \sqrt { 3 } {/tex}

B

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

{tex} 2 ( \sqrt { 3 } - 1 ) {/tex}

D

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

Explanation



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Q 4. The value of {tex} \tan \left[ \cos ^ { - 1 } \left( \frac { 4 } { 5 } \right) + \tan ^ { - 1 } \left( \frac { 2 } { 3 } \right) \right] {/tex} is

A

{tex} \frac { 6 } { 17 } {/tex}

B

{tex} \frac { 7 } { 16 } {/tex}

C

{tex} \frac { 16 } { 7 } {/tex}

none

Explanation

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Q 5. The number of real solutions of

{tex} \tan ^ { - 1 } \sqrt { x ( x + 1 ) } + \sin ^ { - 1 } \sqrt { x ^ { 2 } + x + 1 } = \pi / 2 {/tex} is

A

zero

B

one

two

D

infinite

Explanation

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Q 6.
for {tex} 0 < | x | < \sqrt { 2 } , {/tex} then {tex} x {/tex} equals

A

{tex} 1 / 2 {/tex}

1

C

{tex} - 1 / 2 {/tex}

D

{tex} -1 {/tex}

Explanation

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Q 7. The value of {tex} x {/tex} for which {tex} \sin \left( \cot ^ { - 1 } ( 1 + x ) \right) = \cos \left( \tan ^ { - 1 } x \right) {/tex} is

A

{tex} 1 / 2 {/tex}

B

{tex}1{/tex}

C

{tex}0{/tex}

{tex} - 1 / 2 {/tex}

Explanation

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Q 8. If {tex} 0 < x < 1 , {/tex} then
{tex} \sqrt { 1 + x ^ { 2 } } \left[ \left\{ x \cos \left( \cot ^ { - 1 } x \right) + \sin \left( \cot ^ { - 1 } x \right) \right\} ^ { 2 } - 1 \right] ^ { 1 / 2 } = {/tex}

A

{tex} \frac { x } { \sqrt { 1 + x ^ { 2 } } } {/tex}

B

{tex} x {/tex}

{tex} x \sqrt { 1 + x ^ { 2 } } {/tex}

D

{tex} \sqrt { 1 + x ^ { 2 } } {/tex}

Explanation


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Q 9. The value of {tex} \cot \left( \sum _ { n = 1 } ^ { 23 } \cot ^ { - 1 } \left( 1 + \sum _ { k = 1 } ^ { n } 2 k \right) \right) {/tex} is

A

{tex} \frac { 23 } { 25 } {/tex}

{tex} \frac { 25 } { 23 } {/tex}

C

{tex} \frac { 23 } { 24 } {/tex}

D

{tex} \frac { 24 } { 23 } {/tex}

Explanation

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Q 10. From the top of a light-house {tex}60{/tex} metres high with its base at the sea-level, the angle of depression of a boat is {tex} 15 ^ { \circ } . {/tex} The distance of the boat from the foot of the light house is

A

{tex} \left( \frac { \sqrt { 3 } - 1 } { \sqrt { 3 } + 1 } \right) 60 {/tex} metres

{tex} \left( \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 } \right) 60 {/tex} metres

C

{tex} \left( \frac { \sqrt { 3 } + 1 } { \sqrt { 3 } - 1 } \right) ^ { 2 } {/tex} metres

D

none of these

Explanation

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Q 11. In a triangle {tex} A B C , {/tex} angle {tex} A {/tex} is greater than angle {tex} B . {/tex} If the measures of angles {tex} A {/tex} and {tex} B {/tex} satisfy the equation {tex} 3 \sin x - 4 \sin ^ { 3 } x - k = 0,0 < k < 1 , {/tex} then the measure of angle {tex} \mathrm { C } {/tex} is

A

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

B

{tex} \frac { \pi } { 2 } {/tex}

{tex} \frac { 2 \pi } { 3 } {/tex}

D

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

Explanation

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Q 12. If the lengths of the sides of triangle are {tex} 3,5,7 {/tex} then the largest angle of the triangle is

A

{tex} \frac { \pi } { 2 } {/tex}

B

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

{tex} \frac { 2 \pi } { 3 } {/tex}

D

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

Explanation

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Q 13. In a triangle {tex} A B C , \angle B = \frac { \pi } { 3 } {/tex} and {tex} \angle C = \frac { \pi } { 4 } {/tex}. Let {tex} D {/tex} divide {tex} B C {/tex} internally in the ratio 1 : 3 then {tex} \frac { \sin \angle B A D } { \sin \angle C A D } {/tex} is equal to

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

B

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

C

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

D

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

Explanation


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Q 14. In a triangle {tex} A B C , 2 a c \sin \frac { 1 } { 2 } ( A - B + C ) = {/tex}

A

{tex} a ^ { 2 } + b ^ { 2 } - c ^ { 2 } {/tex}

{tex} c ^ { 2 } + a ^ { 2 } - b ^ { 2 } {/tex}

C

{tex} b ^ { 2 } - c ^ { 2 } - a ^ { 2 } {/tex}

D

{tex} c ^ { 2 } - a ^ { 2 } - b ^ { 2 } {/tex}

Explanation

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Q 15. In a triangle {tex} A B C , {/tex} let {tex} \angle C = \frac { \pi } { 2 } . {/tex} If {tex} r {/tex} is the inradius and {tex} R {/tex} is the circumradius of the triangle, then {tex} 2 ( r + R ) {/tex} is equal to

{tex} a + b {/tex}

B

{tex} b + c {/tex}

C

{tex} c + a {/tex}

D

{tex} a + b + c {/tex}

Explanation

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Q 16. A pole stands vertically inside a triangular park {tex} \Delta A B C . {/tex} If the angle of elevation of the top of the pole from each corner of the park is same, then in {tex} \triangle A B C {/tex} the foot of the pole is at the

A

centroid

circumcentre

C

incentre

D

orthocentre

Explanation

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Q 17. Which of the following pieces of data does NOT uniquely determine an acute-angled triangle {tex} A B C ( R {/tex} being the radius of the circumcircle)?

A

{tex} a , \sin A , \sin B {/tex}

B

{tex} a , b , c {/tex}

C

{tex} a , \sin B , R {/tex}

{tex} a , \sin A , R {/tex}

Explanation

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Q 18. If the angles of a triangle are in the ratio {tex} 4: 1: 1 , {/tex} then the ratio of the longest side to the perimeter is

{tex} \sqrt { 3 }: ( 2 + \sqrt { 3 } ) {/tex}

B

{tex}1:6 {/tex}

C

{tex} 1: 2 + \sqrt { 3 } {/tex}

D

{tex}2:3 {/tex}

Explanation

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Q 19. The sides of a triangle are in the ratio {tex} 1: \sqrt { 3 }: 2 , {/tex} then the angles of the triangle are in the ratio

A

1 : 3 : 5

B

2 : 3 : 4

C

3 : 2 : 1

1 : 2 : 3

Explanation

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Q 20. In an equilateral triangle, 3 coins of radii 1 unit each are kept so that they touch each other and also the sides of {tex} f {/tex} the triangle. Area of the triangle is

A

{tex} 4 + 2 \sqrt { 3 } {/tex}

{tex} 6 + 4 \sqrt { 3 } {/tex}

C

{tex} 12 + \frac { 7 \sqrt { 3 } } { 4 } {/tex}

D

{tex} 3 + \frac { 7 \sqrt { 3 } } { 4 } {/tex}

Explanation



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Q 21. In a triangle {tex} A B C , a , b , c {/tex} are the lengths of its sides and {tex} A , B , {/tex} {tex} C {/tex} are the angles of triangle {tex} A B C . {/tex} The correct relation is given by

A

{tex} ( b - c ) \sin \left( \frac { B - C } { 2 } \right) = a \cos \frac { A } { 2 } {/tex}

{tex} ( b - c ) \cos \left( \frac { A } { 2 } \right) = a \sin \frac { B - C } { 2 } {/tex}

C

{tex} ( b + c ) \sin \left( \frac { B + C } { 2 } \right) = a \cos \frac { A } { 2 } {/tex}

D

{tex} ( b - c ) \cos \left( \frac { A } { 2 } \right) = 2 a \sin \frac { B + C } { 2 } {/tex}

Explanation

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Q 22. One angle of an isosceles {tex} \Delta {/tex} is {tex} 120 ^ { \circ } {/tex} and radius of its incircle {tex} = \sqrt { 3 } . {/tex} Then the area of the triangle in sq. units is

A

{tex} 7 + 12 \sqrt { 3 } {/tex}

B

{tex} 12 - 7 \sqrt { 3 } {/tex}

{tex} 12 + 7 \sqrt { 3 } {/tex}

D

{tex} 4 \pi {/tex}

Explanation

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Q 23. Let {tex} A B C D {/tex} be a quadrilateral with area {tex} 18 , {/tex} with side {tex} A B {/tex} parallel to the side {tex} C D {/tex} and {tex} 2 A B = C D {/tex}. Let {tex} A D {/tex} be perpendicular to {tex} A B {/tex} and {tex} C D {/tex}. If a circle is drawn inside the quadrilateral {tex} A B C D {/tex} touching all the sides, then its radius is

A

{tex}3{/tex}

{tex}2{/tex}

C

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

D

{tex}1{/tex}

Explanation



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Q 24. If the angles {tex} A , B {/tex} and {tex} C {/tex} of a triangle are in an arithmetic progression and if {tex} a , b {/tex} and {tex} c {/tex} denote the lengths of the sides opposite to {tex} A , B {/tex} and {tex} C {/tex} respectively, then the value of the
expression {tex} \frac { a } { c } \sin 2 C + \frac { c } { a } \sin 2 A {/tex} is

A

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

B

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

C

{tex}1{/tex}

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

Explanation

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Q 25. Let {tex} P Q R {/tex} be a triangle of area {tex} \Delta {/tex} with {tex} a = 2 , b = \frac { 7 } { 2 } {/tex} and {tex} c = \frac { 5 } { 2 } {/tex} where {tex} a , b , {/tex} and {tex} c {/tex} are the lengths of the sides of the triangle opposite to the angles at {tex} P , Q {/tex} and {tex} R {/tex} respectively. Then {tex} \frac { 2 \sin P - \sin 2 P } { 2 \sin P + \sin 2 P } {/tex} equals.

A

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

B

{tex} \frac { 45 } { 4 \Delta } {/tex}

{tex} \left( \frac { 3 } { 4 \Delta } \right) ^ { 2 } {/tex}

D

{tex} \left( \frac { 45 } { 4 \Delta } \right) ^ { 2 } {/tex}

Explanation