If $a,b,c$ are sides of a triangle, then

$\left| {\sqrt a - \sqrt b } \right| > \sqrt c $ (if $c$ is smallest)

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JEE • MATHEMATICS • PROPERTIES OF TRIANGLES

Properties of triangles practice · Medium

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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.

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61 Medium

61 of 140

Application +4 | -1

If in a triangle

ABC

, right angled at

B,\,\,s - a = 3,\,\,s - c = 2

, then the values of

a

and

c

are respectively

View answer and explanation

Correct option: C

4,\,\;3

Option C (

4, 3

) is correct because by using the given equations and the Pythagorean theorem, we can solve for the values of

a

and

c

62 Medium

62 of 140

Application +4 | -1

a,b,c

are sides of a triangle, then

View answer and explanation

Correct option: A

\sqrt a + \sqrt b > \sqrt c

Option A

\sqrt a + \sqrt b > \sqrt c

is correct because it follows from the triangle inequality theorem.

63 Medium

63 of 140

Application +4 | -1

r_1=r_2+r_3+r

, then triangle is

View answer and explanation

Correct option: A

right angled

Option A (right angled) is correct because the given condition leads to the Pythagorean theorem, indicating a right-angled triangle.

64 Medium

64 of 140

Understanding +4 | -1

In a

\triangle A B C

, the inradius and three exradii are

r, r_1, r_2

and

r_3

, respectively. In usual notations, the value of

r \cdot r_1 \cdot r_2 \cdot r_3

is equal to

View answer and explanation

Correct option: B

\Delta^2

Option B (

\Delta^2

) is correct because the product of the inradius and exradii is equal to the square of the area of the triangle.

65 Medium

65 of 140

Application +4 | -1

A pole indined from vertical at

15^{\circ}

towards the sun. Length of the shadow of the pole is

\sqrt{3}+1

meter at the time when angle of elevation of the sun is

30^{\circ}

, then length of pole is

View answer and explanation

Correct option: B

\frac{\sqrt{2}}{\sqrt{3}-1} m

Option B (

\frac{\sqrt{2}}{\sqrt{3}-1} m

) is correct because it applies the sine rule to the triangle formed by the pole, its shadow, and the sun's rays, accounting for the pole's inclination.

66 Medium

66 of 140

Application +4 | -1

The abscissa of the point on the curve

9 y^2

=x^3

, the normal at which cuts off equal intercepts on the coordinate axes is

View answer and explanation

Correct option: B

Option B (4) is correct because the abscissa of the point is 4, which is found by setting the intercepts of the normal equal and solving for x.

67 Medium

67 of 140

Application +4 | -1

If in a

\Delta ABC,a = 15,b = 36,c = 39

, then

\sin \frac{C}{2}

is equal to

View answer and explanation

Correct option: C

\frac{1}{{\sqrt 2 }}

Option C

\frac{1}{{\sqrt 2 }}

is correct because it is the value of

\sin \frac{C}{2}

calculated using the given sides of the triangle.

68 Medium

68 of 140

Analyzing +4 | -1

A B C

is a triangle whose internal angles are

A, B, C

and

\triangle D E F

is pedal triangle of

\triangle A B C

, then the ratio of the area of

\triangle D E F

to area of

\triangle A B C

View answer and explanation

Correct option: A

2 \cos A \cos B \cos C

The correct answer is A (

2 \cos A \cos B \cos C

). The ratio of the area of the pedal triangle to the area of the original triangle is

2 \cos A \cos B \cos C

69 Medium

69 of 140

Application +4 | -1

A stone is projected vertically upward moves such that its distance feet in time t sec is given by

s=112 t-16 t^2

then the maximum height reached by the stone in (feet) is

View answer and explanation

Correct option: A

196

Option A (196) is correct because it is the maximum height reached by the stone, calculated by finding the time when the velocity is zero and substituting it into the distance equation.

70 Medium

70 of 140

Application +4 | -1

\triangle A B C, a: b: c=(1+x): 1:(1-x)

, where

x \in(0,1)

. If

\angle A=\frac{\pi}{2}+\angle C

, then

x

is equal to

View answer and explanation

Correct option: C

\frac{1}{\sqrt{7}}

Option C (

\frac{1}{\sqrt{7}}

) is correct because it is the value of x that satisfies the given conditions and trigonometric relationships in the triangle.

71 Medium

71 of 140

Application +4 | -1

\Delta \;ABC

, if

\tan \frac{A}{2}\tan \frac{C}{2} = \frac{1}{2}

, then

a,\;b,\;c

are in

View answer and explanation

Correct option: D

None of these

Option D (None of these) is correct because the given condition

\tan \frac{A}{2}\tan \frac{C}{2} = \frac{1}{2}

leads to the relation

a + c = 3b

, which does not satisfy the conditions for AP, GP, or HP.

72 Medium

72 of 140

Application +4 | -1

In a

\triangle A B C, A=\frac{2 \pi}{3}, b-c=3 \sqrt{3}

and ar

(\triangle A B C)=\frac{9 \sqrt{3}}{2} \mathrm{~cm}^2

. Then a is

View answer and explanation

Correct option: B

9 \mathrm{~cm}

Option B (

9 \mathrm{~cm}

.) is correct because by using the area and the given angle, we can find bc, and then using the cosine rule, we can find a.

73 Medium

73 of 140

Analyzing +4 | -1

0 < x < \frac{\pi }{2}

, then the largest angle of a triangle whose sides are

1,\sin x,\cos x

View answer and explanation

Correct option: B

\frac{\pi }{2}

Option B (

\frac{\pi }{2}

) is correct because since

0 < \sin x < 1

and

0 < \cos x < 1

, the side with length 1 is the largest, and the angle opposite to it is a right angle.

74 Medium

74 of 140

Analyzing +4 | -1

In a

\Delta \,ABC

\begin{array}{l} \sin A + \sin B + \sin C = 1 + \sqrt 2 \\ \mathrm{and},\mathrm{\;}\cos A + \cos B + \cos C = \sqrt 2 \end{array}

if, the triangle is

View answer and explanation

Correct option: D

Right angled isosceles

Option D (Right angled isosceles) is correct because it is the only type of triangle where the given conditions

\sin A + \sin B + \sin C = 1 + \sqrt 2

and

\cos A + \cos B + \cos C = \sqrt 2

hold true.

75 Medium

75 of 140

Application +4 | -1

In a triangle

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

is equal to

View answer and explanation

Correct option: A

\frac{r}{R}

Option A (

\frac{r}{R}

) is correct because, using trigonometric identities and the sine rule, the given expression simplifies to

\frac{r}{R}

, where

r

is the inradius and

R

is the circumradius of the triangle.

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76 Medium

76 of 140

Analyzing +4 | -1

If in a

\Delta \;ABC,\;a\tan A + b\tan B = \left( {a + b} \right)\tan \frac{{A + B}}{2}

, then

View answer and explanation

Correct option: A

A = B

Option A (

A = B

) is correct because the given trigonometric equation simplifies to A = B.

77 Medium

77 of 140

Application +4 | -1

The radii

r_1, r_2, r_3

of escribed circles of the triangle

A B C

are in H.P. If its area is

24 \mathrm{sq}

. cm and its perimeter is

24 \mathrm{~cm}

, then the lengths of its sides are -

View answer and explanation

Correct option: C

6,8,10

Option C (

6,8,10

) is correct because the sides are in A.P. and satisfy the area and perimeter conditions.

78 Medium

78 of 140

Analyzing +4 | -1

\triangle A B C, \sin A \sin B+\sin B \sin C+\sin C \sin A=\frac{9}{4}

and

a=2

, then the value of

\sqrt{3} \Delta

, where

\Delta

is area of triangle, is

View answer and explanation

Correct option: A

Option A (3) is correct because the given condition implies that the triangle is equilateral with side 2, and its area multiplied by

\sqrt{3}

is 3.

79 Medium

79 of 140

Analyzing +4 | -1

If in a

\triangle A B C

\cos A \cos B+\sin A \sin B \sin C=1

, then

a: b: c

is equal to

View answer and explanation

Correct option: A

1: 1: \sqrt{2}

Option A (

1: 1: \sqrt{2}

) is correct because the given condition implies that the triangle is a right-angled isosceles triangle, leading to the ratio of sides

1: 1: \sqrt{2}

80 Medium

80 of 140

Application +4 | -1

If in a

\Delta \,ABC,\,\,\angle A = \pi /3

and

AD

is a median, then

View answer and explanation

Correct option: B

4\;A{D^2} = {b^2} + {c^2} + bc

Option B (

4\;A{D^2} = {b^2} + {c^2} + bc

) is correct because it follows from applying the cosine rule to find the length of the median AD.

81 Medium

81 of 140

Application +4 | -1

If the angles of a triangle are in the ratio

1: 1: 4

, then the ratio of the perimeter of the triangle to its largest side is

View answer and explanation

Correct option: D

\sqrt{3}+2: \sqrt{3}

Option D (

\sqrt{3}+2:\sqrt{3}

) is correct because it represents the ratio of the perimeter to the largest side of the triangle with angles in the ratio 1:1:4.

82 Medium

82 of 140

Application +4 | -1

In any

\triangle A B C

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

is equal to

View answer and explanation

Correct option: D

4 R \sin A \sin B \sin C

Option D (

4 R \sin A \sin B \sin C

) is correct because by using the sine rule and trigonometric identities, the expression simplifies to

4 R \sin A \sin B \sin C

83 Medium

83 of 140

Application +4 | -1

If the angles of a

\Delta \;ABC

be in AP, then

View answer and explanation

Correct option: B

{b^2} = {a^2} + {c^2} - ac

Option B (

{b^2} = {a^2} + {c^2} - ac

) is correct because when the angles are in AP, one angle is 60 degrees, and the cosine rule simplifies to this form.

84 Medium

84 of 140

Application +4 | -1

If in a triangle,

R

and

r

are the circumradius and inradius respectively, then the HM of the exradii of the triangle is

View answer and explanation

Correct option: A

3 r

Option A (

3r

) is correct because the harmonic mean of the exradii is equal to three times the inradius.

85 Medium

85 of 140

Application +4 | -1

If the angle

\mathrm{A}

of a triangle

\mathrm{ABC}

is given by the equation

5 \cos A+3=0

, then

\sin A

and

\tan A

are the roots of the equation

View answer and explanation

Correct option: D

15 x^2+8 x-16=0

Option D (

15 x^2+8 x-16=0

) is correct because it is the quadratic equation with roots sin A and tan A, derived from the given cosine value.

86 Medium

86 of 140

Application +4 | -1

In a

\triangle \mathrm{ABC}

, if angle

\mathrm{C}

is obtuse, then

View answer and explanation

Correct option: A

\tan A \tan B < 1

Option A (

\tan A \tan B < 1

) is correct because when angle C is obtuse, the product of tan A and tan B must be less than 1.

87 Medium

87 of 140

Application +4 | -1

The inradius of the triangle whose sides are 3, 5, 6 is

View answer and explanation

Correct option: A

\sqrt {\frac{8}{7}}

Option A (

\sqrt{\frac{8}{7}}

) is correct because it is the inradius calculated using the formula

r = \frac{\Delta}{s}

, where

\Delta

is the area and

s

is the semi-perimeter.

88 Medium

88 of 140

Analyzing +4 | -1

A, B

and

C

are angles of triangle such that

\angle A

is obtuse, then

\tan B \tan C

will be less than

View answer and explanation

Correct option: C

Option C (1) is correct because, given that angle A is obtuse, the product of tan B and tan C must be less than 1.

89 Medium

89 of 140

Application +4 | -1

R

and

R^{\prime}

are the resultant of two forces

\frac{P}{Q}

and

\frac{{Q}}{{P}}({P}>{Q})

according as they are like or unlike such that

{R}: {R}^{\prime}=25: 7

, then

{P}: {Q}=

View answer and explanation

Correct option: C

4: 3

Option C (

4: 3

) is correct because it is derived by setting up a ratio of the resultants and solving for the ratio of the forces P and Q.

90 Medium

90 of 140

Application +4 | -1

In any

\triangle A B C, 2 a \cos \frac{B}{2} \cos \frac{C}{2}

is equal to

View answer and explanation

Correct option: B

(a+b+c) \sin \frac{A}{2}

Option B (

(a+b+c) \sin \frac{A}{2}

) is correct because it is derived using trigonometric identities and the properties of a triangle.

Revision notes and key takeaways

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

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Properties of triangles practice · Medium

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