MCQs (One or More Correct Option): Properties of Triangle

*Multiple options can be correct

Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced - Question 1

There exists a triangle ABC satisfying the conditions

A.
b sin A = a, A < π/2
B.
b sin A > a, A > π/2
C.
b sin A > a, A < π/2
D.
b sin A < a, A < π/2,b > a
E.
b sin A < a, A > π/2, b = a

Detailed Solution: Question 1

We have

(a) b sin A = a ⇒ a sin B = a
⇒ sin B = 1 ⇒ B = π/2
Since A < π/2, the ΔABC is possible.
(b) b sin A > a ⇒ a sin B > a ⇒ sin B > 1
Which is impossible. Hence the πossibility (b) is ruled out.
Similarly (c) can be shown to be impossible.
(d) b sin A < a ⇒ a sin B < a ⇒ sin B < 1 so value of ∠B exists.
Now, b > a ⇒ B > A. Since A < π/2
The ΔABC is possible when B > or < π/2.
(e) Since b = a, we have B = A. But A > π/2
∴ B > π/2. But this is not possible for any triangle

*Multiple options can be correct

Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced - Question 2

In a triangle, the lengths of the two larger sides are 10 and 9, respectively. If the angles are in A P. Then the length of the third side can be

A.
5 -√6
B.
3√3
C.
5
D.
5 + √6
E.
none

Detailed Solution: Question 2

Since the angles of triangle are in A.P., Let ∠A = x – d, ∠B = x, ∠C = x + d
Then by ∠ sum property of triangle, we have ∠A +∠ B + ∠C = 180°
⇒ x – d + x + x + d = 180°
⇒ 3x = 180° ⇒ x = 60°
∴ ∠B = 60°

Given that a = 10, b = 9 are the longer sides
(both the values are less than 9 and 10),
both are possible.

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*Multiple options can be correct

Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced - Question 3

If in a triangle PQR, sin P, sin Q, sin R are in A.P., then

A.
the altitudes are in A.P.
B.
the altitudes are in H.P.
C.
the medians are in G.P.
D.
the medians are in A.P.

Detailed Solution: Question 3

In ΔPQR let d₁, d₂, d₃ be the altitudes on QR, RP and PQ respectively.

⇒

⇒ d₁, d₂, d₃ are in H.P. (As given that sin P, sin Q, sin R are in A.P.)

*Multiple options can be correct

Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced - Question 4

Let A₀A₁A₂A₃A₄A₅ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments A₀A₁,A₀A₂ and A₀A₄ is

A.
3/4
B.
3√3
C.
3
D.

Detailed Solution: Question 4

Given that A₀A₁A₂A₃A₄A₅ is a regular hexagon inscribed in a circle of radius 1.

∴ A₀A₁ = 1 = A₁A₂ = A₂A₃ = A₃A₄ = A₄A₅ = A₅A₀
∠A₀A₁A₂ = 60° + 60° = 120°
In ΔA₀A₁A₂, using cosine law we get

∴ (c) is the correct option.

*Multiple options can be correct

Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced - Question 5

In ΔABC, internal angle bisector of ∠A meets side BC in D. DE ⊥ AD meets AC in E and AB in F. Then

A.
AE is HM of b & c
B.
C.
D.
DAEF is isosceles

Detailed Solution: Question 5

By simple geometry in ΔAFE, AF = AE
∴ ΔAFE is an isosceles Δ.
Now Ar (ΔABC) = Ar (ΔABD) + Ar (ΔADC)

*Multiple options can be correct

Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced - Question 6

Let ABC be a triangle such that and let a, b and c denote the lengths of the sides opposite to A, B and C respectively. The value(s) of x for which a = x² + x + 1, b = x² – 1 and c = 2x + 1 is (are)

Detailed Solution: Question 6

Given that a = x² + x + 1, b = x² – 1, c = 2x+ 1 and ∠C = π/6

Now for x = –1and 1, b = 0 which is not possible
Also fo which is not possible

*Multiple options can be correct

Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced - Question 7

In a triangle PQR, P is the largest angle and cos p = 1/3. Further the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

A.
16
B.
18
C.
24
D.
22

Detailed Solution: Question 7

Let PN = x, QL = x + 2, RM = x + 4 where x is an even integer.

Then PM = PN = x, QN = QL = x + 2 and RL = RM = x + 4
So that PQ = 2x + 2, QR = 2x + 6, PR = 2x + 4

*Multiple options can be correct

Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced - Question 8

In a triangle XYZ, let x, y, z be the lengths of sides opposite to the angles X,Y, Z, respectively, and 2s = x + y + z.
and area of incircle of the triangle

A.
area of the triangle
B.
the radius of circumcircle of the triangle
C.
D.

Detailed Solution: Question 8

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MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced

MCQ Practice Test & Solutions: Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced (8 Questions)

There exists a triangle ABC satisfying the conditions

In a triangle, the lengths of the two larger sides are 10 and 9, respectively. If the angles are in A P. Then the length of the third side can be

If in a triangle PQR, sin P, sin Q, sin R are in A.P., then

Let A₀A₁A₂A₃A₄A₅ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments A₀A₁,A₀A₂ and A₀A₄ is

In ΔABC, internal angle bisector of ∠A meets side BC in D. DE ⊥ AD meets AC in E and AB in F. Then

Let ABC be a triangle such that and let a, b and c denote the lengths of the sides opposite to A, B and C respectively. The value(s) of x for which a = x² + x + 1, b = x² – 1 and c = 2x + 1 is (are)

In a triangle PQR, P is the largest angle and cos p = 1/3. Further the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

In a triangle XYZ, let x, y, z be the lengths of sides opposite to the angles X,Y, Z, respectively, and 2s = x + y + z.
and area of incircle of the triangle

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MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced

MCQ Practice Test & Solutions: Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced (8 Questions)

There exists a triangle ABC satisfying the conditions

In a triangle, the lengths of the two larger sides are 10 and 9, respectively. If the angles are in A P. Then the length of the third side can be

If in a triangle PQR, sin P, sin Q, sin R are in A.P., then

Let A0A1A2A3A4A5 be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments A0A1,A0A2 and A0A4 is

In ΔABC, internal angle bisector of ∠A meets side BC in D. DE ⊥ AD meets AC in E and AB in F. Then

Let ABC be a triangle such that and let a, b and c denote the lengths of the sides opposite to A, B and C respectively. The value(s) of x for which a = x2 + x + 1, b = x2 – 1 and c = 2x + 1 is (are)

In a triangle PQR, P is the largest angle and cos p = 1/3. Further the incircle of the triangle touches the sides PQ, QR and RP at N, L and M respectively, such that the lengths of PN, QL and RM are consecutive even integers. Then possible length(s) of the side(s) of the triangle is (are)

In a triangle XYZ, let x, y, z be the lengths of sides opposite to the angles X,Y, Z, respectively, and 2s = x + y + z. and area of incircle of the triangle

Crack JEE with 35 Years of Previous Year Solved Papers

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About this JEE Practice Test

Explore more JEE Study Resources

Let A₀A₁A₂A₃A₄A₅ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments A₀A₁,A₀A₂ and A₀A₄ is

Let ABC be a triangle such that and let a, b and c denote the lengths of the sides opposite to A, B and C respectively. The value(s) of x for which a = x² + x + 1, b = x² – 1 and c = 2x + 1 is (are)

In a triangle XYZ, let x, y, z be the lengths of sides opposite to the angles X,Y, Z, respectively, and 2s = x + y + z.
and area of incircle of the triangle