JEE Main > Properties of Triangle

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


Q 1.    

Correct4

Incorrect-1

Three distinct points {tex} A , B {/tex} and {tex} C {/tex} are given in the two-dimensional coordinate plane such that the ratio of the distance of any one of them from the point {tex} ( 1,0 ) {/tex} to the distance from the point {tex} ( - 1,0 ) {/tex} is equal to {tex} \frac { 1 } { 3 } {/tex} . Then the circumcentre of the triangle {tex} A B C {/tex} is at the point

A

{tex} \left( \frac { 5 } { 4 } , 0 \right) {/tex}

B

{tex} \left( \frac { 5 } { 2 } , 0 \right) {/tex}

C

{tex} \left( \frac { 5 } { 3 } , 0 \right) {/tex}

{tex} \left( \frac { 5 } { 4} , 0 \right) {/tex}

Explanation

Q 2.    

Correct4

Incorrect-1

For a regular polygon, let' r' and 'R' be the radii of the inscribed and the circumscribed circles. A false statement among the following is

A

There is a regular polygon with {tex} \frac { r } { R } = \frac { 1 } { \sqrt { 2 } } {/tex}

There is a regular polygon with {tex} \frac { r } { R } = \frac { 2 } { 3 } {/tex}

C

There is a regular polygon with {tex} \frac { r } { R } = \frac { \sqrt { 3 } } { 2 } {/tex}

D

There is a regular polygon with {tex} \frac { r } { R } = \frac { 1 } { 2 } {/tex}

Explanation

Q 3.    

Correct4

Incorrect-1

{tex} A B C D {/tex} is a trapezium such that {tex} A B {/tex} and {tex} C D {/tex} are parallel and {tex} B C \perp C D . {/tex} If {tex} \angle A D B = \theta , B C = p {/tex} and {tex} C D = q {/tex} , then {tex} A B {/tex} is equal to

A

{tex} \frac { p ^ { 2 } + q ^ { 2 } \cos \theta } { p \cos \theta + q \sin \theta } {/tex}

B

{tex} \frac { p ^ { 2 } + q ^ { 2 } } { p ^ { 2 } \cos \theta + q ^ { 2 } \sin \theta } {/tex}

C

{tex} \frac { \left( p ^ { 2 } + q ^ { 2 } \right) \sin \theta } { ( p \cos \theta + q \sin \theta ) ^ { 2 } } {/tex}

{tex} \frac { \left( p ^ { 2 } + q ^ { 2 } \right) \sin \theta } { p \cos \theta + q \sin \theta } {/tex}

Explanation

Q 4.    

Correct4

Incorrect-1

The angle of elevation of the top of a vertical tower from a point {tex} P {/tex} on the horizontal ground was observed to be {tex} \alpha . {/tex} After moving a distance 2{tex} \mathrm { m } {/tex} from {tex} P {/tex} towards the foot of the tower, the angle of elevation changes to {tex} \beta . {/tex} Then the height (in metres) of the tower is

{tex} \frac { 2 \sin \alpha \sin \beta } { \sin ( \beta - \alpha ) } {/tex}

B

{tex} \frac { \sin \alpha \sin \beta } { \cos ( \beta - \alpha ) } {/tex}

C

{tex} \frac { 2 \sin ( \beta - \alpha ) } { \sin \alpha \sin \beta } {/tex}

D

{tex} \frac { \cos ( \beta - \alpha ) } { \sin \alpha \sin \beta } {/tex}

Explanation

Q 5.    

Correct4

Incorrect-1

From the top of a 64{tex} \mathrm { m } {/tex} high tower, a stone is thrown upwards vertically with the velocity of 48{tex} \mathrm { m } / \mathrm { s } {/tex} . The greatest height (in meters) attained by the stone, assuming the value of the gravi- tational acceleration, {tex} g = 32 \mathrm { m } / \mathrm { s } ^ { 2 } {/tex} is

100

B

88

C

128

D

112

Explanation

Q 6.    

Correct4

Incorrect-1

{tex} A B C {/tex} is a triangle in a plane with vertices {tex} A ( 2,3,5 ) , B ( - 1,3,2 ) {/tex} and {tex} C ( \lambda , 5 , \mu ) {/tex} . If the median through {tex} A {/tex} is equally inclined to the coordinate axes, then the value of {tex} \left( \lambda ^ { 3 } + \mu ^ { 3 } + 5 \right) {/tex} is

A

1130

1348

C

1077

D

676

Explanation

Q 7.    

Correct4

Incorrect-1

If the lengths of arcs {tex} A B , B C {/tex} and {tex} C A {/tex} of a circle are {tex} 3,4 {/tex} and {tex} 5 , {/tex} respectively, then the area of triangle {tex} A B C {/tex} is

{tex} \frac { 9 \sqrt { 3 } ( \sqrt { 3 } + 1 ) } { \pi ^ { 2 } } {/tex}

B

{tex} \frac { 9 \sqrt { 3 } ( \sqrt { 3 } - 1 ) } { \pi ^ { 2 } } {/tex}

C

{tex} \frac { 9 \sqrt { 3 } ( \sqrt { 3 } - 1 ) } { \pi } {/tex}

D

None of these

Explanation

Q 8.    

Correct4

Incorrect-1

The area of right-angled triangle in terms of {tex} r {/tex} and {tex} r _ { 1 } {/tex} , if {tex} \angle A = 90 ^ { \circ } {/tex} (where {tex} r , r _ { 1 } {/tex} have their usual meanings), is

A

{tex} r + r _ { 1 } {/tex}

{tex} rr_1 {/tex}

C

{tex} r - r _ { 1 } {/tex}

D

{tex} r _ { 1 } - r {/tex}

Explanation

Q 9.    

Correct4

Incorrect-1

If in a {tex} \Delta A B C {/tex} (whose circumcentre is origin), {tex} a \leq \sin A , {/tex} then for any point {tex} ( x , y ) {/tex} inside the circumcircle of {tex} \Delta A B C {/tex}

{tex} | x y | < 1 / 8 {/tex}

B

{tex} | x y | > 1 / 8 {/tex}

C

{tex} 1 / 8 < x y < 1 / 2 {/tex}

D

None of these

Explanation

Q 10.    

Correct4

Incorrect-1

If the sine of the angles of a triangle {tex} A B C {/tex} satisfy the equation {tex} c ^ { 3 } x ^ { 3 } - c ^ { 2 } ( a + b + c ) x ^ { 2 } + \lambda x + \mu = 0 {/tex} (where {tex} a , b , c {/tex} are the sides of {tex} \Delta A B C {/tex} , then triangle {tex} A B C {/tex} is

A

always right-angled for any {tex} \lambda , \mu {/tex}

right-angled only when {tex} \lambda = c ( a b + b c + c a ) , \mu = - a b c {/tex}

C

right-angled only when {tex} \lambda = \frac { c ( a b + b c + c a ) } { 4 } , \mu = \frac { - a b c } { 8 } {/tex}

D

never right-angled

Explanation

Q 11.    

Correct4

Incorrect-1

If {tex} \sin A {/tex} and {tex} \sin B {/tex} of a triangle {tex} A B C {/tex} satisfy {tex} c ^ { 2 } x ^ { 2 } - c ( a + b ) x + a b = 0 {/tex} then the triangle is

A

equilateral

B

isosceles

right-angled

D

acute angled

Explanation

Q 12.    

Correct4

Incorrect-1

{tex} A B C D {/tex} is a quadrilateral circumscribed about a circle of unit radius. Then

A

{tex} A B \sin \frac { C } { 2 } \cdot \sin \frac { A } { 2 } = C D \sin \frac { B } { 2 } \cdot \sin \frac { D } { 2 } {/tex}

{tex} A B \sin \frac { A } { 2 } \cdot \sin = \frac { B } { 2 } = C D \sin \frac { C } { 2 } \cdot \sin \frac { D } { 2 } {/tex}

C

{tex} A B \sin \frac { A } { 2 } \cdot \sin \frac { D } { 2 } = C D \sin \frac { C } { 2 } \cdot \sin \frac { B } { 2 } {/tex}

D

{tex} A B \sin \frac { A } { 2 } \cdot \cos = \frac { B } { 2 } = C D \sin \frac { C } { 2 } \cdot \cos \frac { D } { 2 } {/tex}

Explanation

Q 13.    

Correct4

Incorrect-1

If in triangle {tex} A B C {/tex} , line joining the circumcentre and orthocentre is parallel to side {tex} A C , {/tex} then value of {tex} \tan A\cdot{/tex}tan {tex} C {/tex} is equal to

A

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

3

C

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

D

None of these

Explanation

Q 14.    

Correct4

Incorrect-1

{tex} A A _ { 1 } , B B _ { 1 }, C C _ { 1 } {/tex} are the medians of triangle {tex} A B C {/tex} whose centroid is {tex} G . {/tex} If the points {tex} A , C _ { 1 } {/tex} and {tex} B _ { 1 } {/tex} are concyclic, then

A

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

B

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

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

D

None of these

Explanation

Q 15.    

Correct4

Incorrect-1

The area of a triangle {tex} A B C , {/tex} where {tex} a = 2 ( \sqrt { 3 } + 1 ) , B = 45 ^ { \circ } , C = 60 ^ { \circ } {/tex} is

A

{tex} \sqrt { 3 } ( \sqrt { 3 } + 1 ) {/tex} square unit

B

2{tex} ( \sqrt { 3 } + 1 ) {/tex} square unit

2{tex} \sqrt { 3 } ( \sqrt { 3 } + 1 ) {/tex} square unit

D

{tex} \sqrt { 3 } ( 2 \sqrt { 3 } + 1 ) {/tex} square unit

Explanation

Q 16.    

Correct4

Incorrect-1

In a triangle {tex} A B C {/tex} , the value of \[ \frac { \cos ^ { 2 } B - \cos ^ { 2 } C } { b + c } + \frac { \cos ^ { 2 } C - \cos ^ { 2 } A } { c + a } + \frac { \cos ^ { 2 } A - \cos ^ { 2 } B } { a + b } \text { is } \]

0

B

1

C

2

D

3

Explanation

Q 17.    

Correct4

Incorrect-1

In a triangle {tex} A B C {/tex} if {tex} \cos A + 2 \cos B + \cos C = 2 , {/tex} the sides of the triangle are in

A

{tex} \mathrm { HP } {/tex}

B

{tex} G P {/tex}

{tex} A P {/tex}

D

None of these

Explanation

Q 18.    

Correct4

Incorrect-1

In a triangle, {tex} 1 - \tan \frac { A } { 2 } \tan \frac { B } { 2 } = {/tex}

A

{tex} \frac { 2 } { a + b + c } {/tex}

{tex} \frac { 2 c } { a + b + c } {/tex}

C

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

D

None of these

Explanation

Q 19.    

Correct4

Incorrect-1

In a triangle ABC
{tex} a ^ { 2 } b ^ { 2 } c ^ { 2 } ( \sin 2 A + \sin 2 B + \sin 2 C ) = {/tex}

A

{tex} \Delta ^ { 3 } {/tex}

B

8{tex} \Delta ^ { 3 } {/tex}

C

16{tex} \Delta ^ { 3 } {/tex}

32{tex} \Delta ^ { 3 } {/tex}

Explanation

Q 20.    

Correct4

Incorrect-1

If ex-radii {tex} r _ { 1 } , r _ { 2 } , r _ { 3 } {/tex} of a triangle are in HP then its sides {tex} a , b , c {/tex} are in

{tex} A P {/tex}

B

{tex} G P {/tex}

C

{tex} \mathrm { HP } {/tex}

D

None of these

Explanation

Q 21.    

Correct4

Incorrect-1

In a right-angled {tex} \Delta A B C , \sin ^ { 2 } A + \sin ^ { 2 } B + \sin ^ { 2 } C {/tex} is equal to

A

0

B

1

C

-1

None of these

Explanation

Q 22.    

Correct4

Incorrect-1

In a right-angled {tex} \triangle A B C , \sin ^ { 2 } A + \sin ^ { 2 } B + \sin ^ { 2 } C {/tex} is equal to

A

{tex}\Delta{/tex}

B

2{tex}\Delta{/tex}

C

3{tex} \Delta {/tex}

4{tex} \Delta {/tex}

Explanation

Q 23.    

Correct4

Incorrect-1

If {tex} p _ { 1 } , p _ { 2 } , p _ { 3 } {/tex} are, respectively, the perpendiculars from the vertices of a triangle to the opposite sides, then {tex} p _ { 1 } p _ { 2 } p _ { 3 } {/tex} is equal to

A

{tex} \frac { a ^ { 2 } b ^ { 2 } c ^ { 2 } } { R ^ { 2 } } {/tex}

B

{tex} \frac { a ^ { 2 } b ^ { 2 } c ^ { 2 } } { 4 R ^ { 2 } } {/tex}

C

{tex} \frac { 4 a ^ { 2 } b ^ { 2 } c ^ { 2 } } { R ^ { 2 } } {/tex}

{tex} \frac { a ^ { 2 } b ^ { 2 } c ^ { 2 } } { 8 R ^ { 2 } } {/tex}

Explanation

Q 24.    

Correct4

Incorrect-1

If {tex} p _ { 1 } , p _ { 2 } , p _ { 3 } {/tex} are, respectively, the perpendiculars from the vertices of a triangle to the opposite sides, then {tex} \frac { \cos A } { p _ { 1 } } + \frac { \cos B } { p _ { 2 } } + \frac { \cos C } { p _ { 3 } } {/tex} is equal to

A

1{tex} / r {/tex}

1{tex} / R {/tex}

C

1{tex} / \Delta {/tex}

D

None of these

Explanation

Q 25.    

Correct4

Incorrect-1

If {tex} \Delta = a ^ { 2 } - ( b - c ) ^ { 2 } , {/tex} where {tex} \Delta {/tex} is the area of triangle {tex} A B C , {/tex} then {tex} \tan A {/tex} is equal to

A

{tex} \frac { 15 } { 16 } {/tex}

{tex} \frac { 8 } { 15 } {/tex}

C

{tex} \frac { 8 } { 17 } {/tex}

D

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

Explanation