# JEE Advanced > Properties of Triangle

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

Physics
Chemistry
Mathematics
Q 1.

Correct4

Incorrect-1

If the bisector of the angle {tex} P {/tex} of a triangle {tex} P Q R {/tex} meets {tex} Q R {/tex} in {tex} S , {/tex} then

A

{tex} Q S = S R {/tex}

B

{tex} Q S: S R = P R: P Q {/tex}

{tex} Q S: S R = P Q: P R {/tex}

D

None of these

##### Explanation

Q 2.

Correct4

Incorrect-1

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

Q 3.

Correct4

Incorrect-1

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

Q 4.

Correct4

Incorrect-1

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

Q 5.

Correct4

Incorrect-1

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

Q 6.

Correct4

Incorrect-1

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

Q 7.

Correct4

Incorrect-1

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

Q 8.

Correct4

Incorrect-1

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

Q 9.

Correct4

Incorrect-1

A man from the top of a 100 metres high tower sees a car moving towards the tower at an angle of depression of {tex} 30 ^ { \circ } . {/tex} After some time, the angle of depression becomes {tex} 60 ^ { \circ } . {/tex} The distance (in metres) travelled by the car during this time is

A

{tex} 100 \sqrt { 3 } {/tex}

{tex} 200 \sqrt { 3 } / 3 {/tex}

C

{tex} 100 \sqrt { 3 / 3 } {/tex}

D

{tex} 200 \sqrt { 3 } {/tex}

##### Explanation

Q 10.

Correct4

Incorrect-1

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

Q 11.

Correct4

Incorrect-1

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

Q 12.

Correct4

Incorrect-1

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

Q 13.

Correct4

Incorrect-1

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

Q 14.

Correct4

Incorrect-1

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

Q 15.

Correct4

Incorrect-1

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

Q 16.

Correct4

Incorrect-1

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

Q 17.

Correct4

Incorrect-1

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

Q 18.

Correct4

Incorrect-1

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

Q 19.

Correct4

Incorrect-1

In a triangle the sum of two sides is {tex} x {/tex} and the product of the same sides is {tex} y . {/tex} If {tex} x ^ { 2 } - c ^ { 2 } = y , {/tex} where {tex} c {/tex} is the third side of the triangle, then the ratio of the in radius to the circum-radius of the triangle is

A

{tex} \frac { 3 y } { 2 x ( x + c ) } {/tex}

{tex} \frac { 3 y } { 2 c ( x + c ) } {/tex}

C

{tex} \frac { 3 y } { 4 x ( x + c ) } {/tex}

D

{tex} \frac { 3 y } { 4 c ( x + c ) } {/tex}