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


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8 Questions MCQ Test Maths 35 Years JEE Main & Advanced Past year Papers | Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced

Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced for JEE 2022 is part of Maths 35 Years JEE Main & Advanced Past year Papers preparation. The Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced questions and answers have been prepared according to the JEE exam syllabus.The Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced MCQs are made for JEE 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced below.
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*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

Detailed Solution for Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced - 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

Detailed Solution for Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced - 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.

*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

Detailed Solution for Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced - 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

Detailed Solution for Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced - 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

Detailed Solution for Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced - 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 for Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced - 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)

Detailed Solution for Test: MCQs (One or More Correct Option): Properties of Triangle | JEE Advanced - 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 

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






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