In $\Delta \;ABC,{\rm{if}}\left( {a + b + c} \right)\left( {a - b + c} \right) = 3ac$, then

$\angle A + \angle C = 90^\circ $

Back to PROPERTIES OF TRIANGLES

Your Progress

0 / 434 Answered

Correct

Incorrect

Accuracy

JEE • MATHEMATICS • PROPERTIES OF TRIANGLES

Properties of triangles practice · Mixed difficulty

Try a question, then open “View answer” to check it. Sign in to submit answers and save progress.

Practice format and chapter quiz

These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

Take a chapter quiz →

Showing 61-90 of 434 questions

61 Easy

61 of 434

Application +4 | -1

If the area of a

\triangle A B C

\lambda

, then

a^2 \sin 2 B+b^2 \sin 2 A

is equal to

View answer and explanation

Correct option: C

4 \lambda

Option C (

4 \lambda

) is correct because using trigonometric identities and the sine rule, the expression simplifies to four times the area of the triangle.

62 Easy

62 of 434

Application +4 | -1

In any

\Delta ABC

under usual notation,

a\left( {b\cos C - c\cos B} \right)

is equal to

View answer and explanation

Correct option: A

{b^2} - {c^2}

Option A (

{b^2} - {c^2}

) is correct because using the cosine rule, the expression simplifies to

{b^2} - {c^2}

63 Easy

63 of 434

Application +4 | -1

If the sides

a,\,\,b

and

c

of a

\Delta \,ABC

are in A.P., then

\left( {\tan \,\,\frac{A}{2} + \tan \,\,\frac{C}{2}} \right):\cot \,\,\frac{B}{2}

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the ratio simplifies to 2/3, which is not among the given options.

64 Medium

64 of 434

Analyzing +4 | -1

In any

\triangle A B C

, the least value of

\left(\frac{\sin ^2 A+\sin A+1}{\sin A}\right)

View answer and explanation

Correct option: A

Option A (3) is correct because the minimum value of the given expression occurs when

\sin A = 1

, and the expression evaluates to 3.

65 Easy

65 of 434

Application +4 | -1

\triangle A B C

is right angled at

B

. Then, the diameter of the incircle of the triangle is

View answer and explanation

Correct option: C

c+a-b

Option C (

c+a-b

) is correct because it represents the diameter (2r) of the incircle of a right-angled triangle.

66 Medium

66 of 434

Analyzing +4 | -1

\triangle A B C, a=5, b=4

and

\cos (A-B)=\frac{31}{32}

, then side

c

View answer and explanation

Correct option: A

Option A (6) is correct because by using the tangent half-angle formula and the cosine rule, the side c can be calculated to be 6.

67 Easy

67 of 434

Application +4 | -1

\Delta \;ABC,\frac{{1 + \cos \left( {A - B} \right)\cos C}}{{1 + \cos \left( {A - C} \right)\cos B}}

is equal to

View answer and explanation

Correct option: D

\frac{{{a^2} + {b^2}}}{{{a^2} + {c^2}}}

Option D ((a²+b²)/(a²+c²)) is correct because the given expression simplifies to (sin²A + sin²B)/(sin²A + sin²C), which is equal to (a²+b²)/(a²+c²) by the sine rule.

68 Easy

68 of 434

Analyzing +4 | -1

\Delta \;ABC,\mathrm{if}{\sin ^2}\frac{A}{2},{\sin ^2}\frac{B}{2},{\sin ^2}\frac{C}{2}

be in HP. Then,

a,b,c

will be in

View answer and explanation

Correct option: C

Option C (HP) is correct because the reciprocals of the given terms are in AP, leading to a, b, c being in HP.

69 Easy

69 of 434

Application +4 | -1

If in a

\Delta ABC,\;a = 6

cm,

b = 8

cm,

c = 10

cm, then the value of

\sin 2A

View answer and explanation

Correct option: D

24/25

Option D (24/25) is correct because using the cosine rule to find cos A, then sin A, and finally sin 2A gives 24/25.

70 Easy

70 of 434

Application +4 | -1

The circumradius of the triangle whose sides are 13, 12 and 5, is

View answer and explanation

Correct option: B

\frac{{13}}{2}

Option B (

\frac{{13}}{2}

) is correct because the triangle is right-angled, and the circumradius is half the length of the hypotenuse.

71 Medium

71 of 434

Application +4 | -1

Two sides of a triangle are

\sqrt{3}-1

and

\sqrt{3}+1

units and their included angle is

60^{\circ}

View answer and explanation

Correct option: A

\sqrt{6}

Option A (

\sqrt{6}

) is correct because using the cosine rule with the given sides and included angle yields a side length of

\sqrt{6}

72 Easy

72 of 434

Application +4 | -1

In a

\Delta \;ABC,\;a\left( {b\cos C - c\cos B} \right)

is equal to

View answer and explanation

Correct option: B

{b^2} - {c^2}

Option B

{b^2} - {c^2}

is correct because it is derived from applying the cosine rule and simplifying the expression.

73 Easy

73 of 434

Application +4 | -1

The greatest angle of a triangle whose sides are

x^2+x+1,2 x+1

and

x^2-1

, is

View answer and explanation

Correct option: C

120^{\circ}

Option A (

120^{\circ}

) is correct because the greatest angle is found using the cosine rule, with

x^2+x+1

being the longest side.

74 Medium

74 of 434

Application +4 | -1

The sides of triangle are in the ratio

1:\sqrt 3 :2

, then the angles of the triangle are in ratio

View answer and explanation

Correct option: D

1:2:3

Option D (

1:2:3

) is correct because the sides are in the ratio

1:\sqrt 3 :2

, indicating a 30-60-90 triangle, and the angles are in the ratio

1:2:3

75 Medium

75 of 434

Analyzing +4 | -1

Consider the following statements :

1.

\frac{{{b^2} - {c^2}}}{{a\sin \left( {B - C} \right)}} = 2R

a\sin \left( {B - C} \right) + b\sin \left( {C - A} \right) + c\sin \left( {A - B} \right) = 0

Which of these is/are correct?

View answer and explanation

Correct option: C

Both (1) and (2)

Option C (Both (1) and (2)) is correct because both statements are trigonometric identities derived from the sine rule and angle relationships in a triangle.

Explore Other Chapters in MATHEMATICS

View All →

SEQUENCE OF SERIES 1764 MCQs DEFINITE INTEGRATION 1443 MCQs MATRICES 1363 MCQs INDEFINITE INTEGRATION 1179 MCQs PROBABILITY 1146 MCQs COMPLEX NUMBERS 980 MCQs

76 Easy

76 of 434

Application +4 | -1

If the angles of a triangle are in the ratio

1\;:2\;:3

, the corresponding sides are in the ratio

View answer and explanation

Correct option: D

1\;:\sqrt 3 \;:2

Option D (

1:\sqrt 3 :2

) is correct because the ratio of sides is the same as the ratio of sines of the angles, and the angles are 30, 60, and 90 degrees.

77 Medium

77 of 434

Application +4 | -1

A tower subtends a right angle at the top of a pole and distance between tower and pole is 10 m,
then minimum height of the tower is

View answer and explanation

Correct option: C

20 m

Option D (20 m) is correct because the minimum height is achieved when the angle is 45 degrees, resulting in a height of 20 m.

78 Medium

78 of 434

Analyzing +4 | -1

If the cotangents of half the angles of a triangle are in AP, then its sides are in

View answer and explanation

Correct option: A

\mathrm{AP}

Option A (AP) is correct because if the cotangents of half the angles of a triangle are in AP, then the sides of the triangle are also in AP.

79 Easy

79 of 434

Application +4 | -1

In a

\Delta ABC,a\cot A + b\cot B + c\cot C

is equal to

View answer and explanation

Correct option: C

2\left( {r + R} \right)

Option C (

2\left( {r + R} \right)

) is correct because

a\cot A + b\cot B + c\cot C

can be simplified to

2R(\cos A + \cos B + \cos C)

, which equals

2(r+R)

80 Easy

80 of 434

Application +4 | -1

\Delta \;ABC,\mathrm{if}\left( {a + b + c} \right)\left( {a - b + c} \right) = 3ac

, then

View answer and explanation

Correct option: A

\angle B = 60^\circ

Option A (

\angle B = 60^\circ

) is correct because by simplifying the given equation and using the cosine rule, we find that angle B is 60 degrees.

81 Easy

81 of 434

Application +4 | -1

In a

\Delta \;ABC

, if

{b^2} + {c^2} = 3{a^2}

, then

\cot B + \cot C - \cot A

is equal to

View answer and explanation

Correct option: C

Option C (0) is correct because substituting the given condition b² + c² = 3a² into the expression and simplifying leads to 0.

82 Easy

82 of 434

Application +4 | -1

If in a

\triangle A B C, a^2+b^2+c^2=8 R^2

, where

R=

circumradius, then the triangle is

View answer and explanation

Correct option: C

right angled

Option C (right angled) is correct because the given condition

a^2 + b^2 + c^2 = 8R^2

leads to one angle being 90 degrees.

83 Medium

83 of 434

Analyzing +4 | -1

A tower subtends angles

\theta, 2 \theta, 3 \theta

at points

A, B

and

C

respectively all lying on a horizontal line through the foot of tower. If

\frac{A B}{B C}=1+\sqrt{2}

, then

\theta

is equal to

View answer and explanation

Correct option: A

\frac{\pi}{8}

Option A (

\frac{\pi}{8}

) is correct because using trigonometric identities and the given ratio, we can derive that

\theta = \frac{\pi}{8}

84 Easy

84 of 434

Application +4 | -1

The vertices of a triangle are

A(1,1), B(4,5)

and

C(6,13)

. Then, the value of

\cos A

View answer and explanation

Correct option: A

\frac{63}{65}

Option A (

\frac{63}{65}

) is correct because it is the cosine of angle A, calculated using the cosine rule with the given vertices of the triangle.

85 Easy

85 of 434

Analyzing +4 | -1

The ex-radii of a triangle

{r_1},\,\,{r_2},\,\,{r_3}

are in harmonic progression, then the sides

a,\,\,b,\,\,c

are

View answer and explanation

Correct option: B

In A.P.

Option B (In A.P.) is correct because the harmonic progression of ex-radii implies an arithmetic progression of the sides.

86 Easy

86 of 434

Application +4 | -1

x,y,z

are perpendicular drawn from the vertices of triangle having sides

a,b

and

c

, then the value of

\frac{{bx}}{c} + \frac{{cy}}{a} + \frac{{az}}{b}

will be

View answer and explanation

Correct option: A

\frac{{{a^2} + {b^2} + {c^2}}}{{2R}}

Option A (

\frac{{{a^2} + {b^2} + {c^2}}}{{2R}}

) is correct because it is derived from the area of the triangle and the circumradius.

87 Hard

87 of 434

Application +4 | -1

In a

\triangle A B C, \frac{r_1}{b c}+\frac{r_2}{c a}+\frac{r_3}{a b}

is equal to

View answer and explanation

Correct option: D

\frac{1}{r}-\frac{1}{2 R}

Option D (

\frac{1}{r}-\frac{1}{2 R}

) is correct because it is derived by using the formulas for exradii and the inradius of a triangle.

88 Easy

88 of 434

Application +4 | -1

In a

\triangle A B C, R=

circumradius and

r=

inradius. The value of

\frac{a \cos A+b \cos B+c \cos C}{a+b+c}

is equal to

View answer and explanation

Correct option: C

\frac{r}{R}

Option C (

r/R

) is correct because by using trigonometric identities and the definitions of inradius and circumradius, the given expression simplifies to

r/R

89 Medium

89 of 434

Application +4 | -1

The angle of elevation of the top

C

of a vertical tower

C D

of height

h

from a point

A

in the horizontal plane is

45^{\circ}

and from a point

B

at a distance

a

from

A

on the line making an angle

30^{\circ}

with

A D

, it is

60^{\circ}

, then

View answer and explanation

Correct option: C

a=h(\sqrt{3}-1)

Option C (

a=h(\sqrt{3}-1)

) is correct because it is derived by using trigonometric relationships and the sine rule in the given geometric setup.

90 Easy

90 of 434

Application +4 | -1

In a

\Delta \,ABC

, if

a,\,\,c,\,\,b

are in A.P., then the value of

\frac{{a\,\,\cos \,\,B - b\,\,\cos \,\,A}}{{a - b}}

, is

View answer and explanation

Correct option: B

\mathrm{2}

Option B (2) is correct because the value of the expression simplifies to 2 using the cosine rule and the given A.P. condition.

Revision notes and key takeaways

Review PROPERTIES OF TRIANGLES in your textbook or class notes, then use this set to check your understanding.

Showing 61-90 of 434 questions

Continue Your Practice

📚 Switch Set (ALL)

Chapter Sets

ALL 434 MCQs EASY 287 MCQs HARD 7 MCQs MEDIUM 140 MCQs

0 /434

0✓ 0✗

0% Accuracy