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Test: Single Correct MCQs: Properties of Triangle | JEE Advanced


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19 Questions MCQ Test Maths 35 Years JEE Main & Advanced Past year Papers | Test: Single Correct MCQs: Properties of Triangle | JEE Advanced

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Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 1

If the bisector of the angle P of a triangle PQR meets QR in S, then

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 1

QS : SR = PQ : PR (as bisector of an angle, in a triangle, divides the opposite side in the same ratio as the sides containing the angle.)

Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 2

From the top of a light-house 60 metres high with its base at the sea-level, the angle of depression of a boat is 15°. The distance of the boat from the foot of the light house is

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 2


Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 3

In a triangle ABC, angle A is greater than angle B. If the measures of angles A and B satisfy the equation 3 sin x – 4 sin3x – k = 0, 0 < k < 1, then the measure of angle C is

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 3

Given that A > B and 3 sin x – 4 sin 3 x – k = 0, 0 <  k < 1
⇒ sin 3 x = k
As A and B satisfy above eq. (given)
∴ sin 3A = k,  sin 3B = k
⇒ sin 3A – sin 3B = 0


⇒ A + B = 60°  or  A = B
But given that A  >  B,   ∴  A ≠ B
Thus, A + B = 60°
But A + B + C = 180°  ⇒ C = 180° –  60° =  120°
∴ C = 2π3

Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 4

If the lengths of the sides of triangle are 3, 5, 7 then the largest angle of the triangle is

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 4

Let  a = 3, b = 5, c = 7 then the largest ∠ is  opp. to the longest side, i.e., ∠C

Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 5

In a triangle ABC,   Let D divide BC internally in the ratio 

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 5

In ΔABD, applying Sine law we get


⇒  

In ΔACD, applying Sine law, we get

⇒      ....(2)

Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 6

In a triangle ABC, 

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 6

We know that A + B + C = 180°
⇒ A + C – B = 180° – 2B

Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 7

In a triangle ABC, let ∠C = π/2. If r is the inradius and R is the circumradius of the triangle, then 2(r + R) is equal to

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 7

We know by Sine rule


⇒ r = s - c (∴ ∠C = 90º)
⇒ a + b - c= 2r
or   2r + c = a+b
or   2r +2R = a+b (Using C = 2R)

Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 8

A pole stands vertically inside a triangular park ΔABC. If the angle of elevation of the top of the pole from each corner of the park is same, then in ΔABC the foot of the pole is at the

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 8


Let OP = Pole,    
∠PAO = ∠PBO = ∠PCQ = α


Similarly OA = OP  cot a .... (2)
Similarly OC = OP  cot a .... (3)
From  (1),  (2)  and (3), OA = OB = OC ⇒ O  is the point of circumcentre of the triangle ABC.

Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 9

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

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 9


Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 10

Which of the following pieces of data does NOT uniquely determine an acute-angled triangle ABC (R being the radius of the circumcircle)?

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 10

We know by Sine law in ΔABC as


(a) If we know a  , sin A  , sin B we can find b , c ; values of ∠’s  A , B and C all.
(b) Using a , b , c we can find ∠A  , ÐB, ∠C using cosine law.
(c) a , sin B , R are given then sin A , b and hence sin (A + B) and then C can be found.
(d) If we know a , sin A , R then we know  only the ratio  we can  not determine the values of b , c , sin B , sin C seperately.
∴  D can not be determined in  this case.

Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 11

If the angles of a triangle are in the ratio 4 : 1 : 1, then the ratio of the longest side to the perimeter is

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 11

Given that  A : B : C = 4 : 1 : 1
Let A = 4x,  B =  x , C = x
But A + B + C = 180°
∴ 4x + x + x = 180°  ⇒ x = 30°
∴ A = 120° ,  B = 30° , C  = 30°


∴ Ratio of longest side to the perimeter

Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 12

The sides of a triangle are in th e rat io 1 : √3 : 2, then the angles of the triangle are in the ratio

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 12

Sides are in the Ratio 1 : √3 :2

Let a = k , b = √3 k and c = 2k

Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 13

In an equil ateral triangle, 3 coins of radii 1 unit each are kept so that they touch each other and also the sides of the triangle. Area of the triangle is

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 13

The situation is as shown in the figure. For circle with centre C2, BP and BP' are two tangents from B to circle, therefore BC2 must be the ∠bisector of ∠B. But ∠B = 60° (∴ ΔABC is an equilateral Δ)



Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 14

In a triangle ABC, a, b, c are the lengths of its sides and A, B, C are the angles of triangle ABC. The correct  relation is given by

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 14

Let us consider   which is involved in each of the these options.

Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 15

One angle of an isosceles Δ is 120º and radius of its incircle = √3 . Then the area of the triangle in sq. units is

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 15

By Sine law in ΔABC


Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 16

Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and 2AB = CD. Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 16

Given AB || CD,  CD = 2AB Let AB = a then CD = 2a Let radius of circle be r.
Let circle touches AB at P, BC at Q, AD at R and CD at S.
Then AR = AP = r, BP = BQ = a – r
DR = DS = r  and CQ = CS = 2a – r
In ΔBEC BC2 = BE2 + EC2 ⇒ (a – r + 2a – r)2 = (2r)2 + (a)2
⇒  9a2 + 4r2 – 12ar = 4r2 + a2

⇒      ....(1)

Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 17

If the angles A, B and C of a triangle are in an arithmetic progression and if a, b and c denote the lengths of the sides opposite to A, B and C respectively, then the value of the expression 

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 17

∵ A, B, C are in A.P..

= 2(sin A cos C + cos A sin C ) =2 sin( A + C )=2 sin B

Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 18

Let PQR be a triangle of area  where a, b, and c are the lengths of the sides of the triangle 

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 18

We have,


Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 19

In a triangle the sum of two sides is x and the product of the same sides is y. If x2 – c2 = y, where c is the third side of the triangle, then the ratio of the in radius to the circum-radius of the triangle is

Detailed Solution for Test: Single Correct MCQs: Properties of Triangle | JEE Advanced - Question 19

Let two sides of Δ be a and b.
Then a + b = x and ab = y
Also given x2 – c2 = y, where c is the third side of Δ.



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