In a triangle P Q R , let ∠ P Q R = 30 ° and the sides P Q and Q R have lengths 10 3 a n d 10 units, respectively. Then, which of the following statement(s) is (are) TRUE?

The area of the circumcircle of the triangle P Q R is 100 π sq. units

EASY

Earn 100

If in a $△ A B C$ , the altitudes from the vertices $A, B, C$ on opposite sides are in H.P., then $\sin A, \sin B, \sin C$ are in

50% studentsanswered this correctly

Important Questions on Sequence and Series

MEDIUM

Mathematics>Algebra>Sequence and Series>Harmonic Progression

Let

A B C D

be a convex cyclic quadrilateral. Suppose

P

is a point in the plane of the quadrilateral such that sum of its distances from the vertices of

A B C D

is the least. If

{P A, P B, P C, P D} = \{3, 4, 6, 8\}

, what is the maximum possible area of

A B C D ?

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

Let

A B C

be a triangle such that

A B = 4, B C = 5

and

C A = 6

. Choose points

D, E, F

A B, B C, C A

respectively, such that

A D = 2, B E = 3, C F = 4

. Then

\frac{area Δ D E F}{area Δ A B C}

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

15

and

9

are lengths of two medians of a triangle, what is the maximum possible area of the triangle to the nearest integer?

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

Let

A B C D

be a square and

E

be a point outside

A B C D

such that

E, A, C

are collinear in that order. Suppose

E B = E D = \sqrt{130}

and the areas of triangle

E A B

and square

A B C D

are equal. Then the area of square

A B C D

is :

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

Suppose we have two circles of radius $2$ each in the plane such that the distance between their centres is $2 \sqrt{3}$ . The area of the region common to both circles lies between

MEDIUM

Mathematics>Algebra>Sequence and Series>Harmonic Progression

A triangle

A B C

has area of

P

square units and circumference

2 S

units. If

h_{1}, h_{2}

and

h_{3}

are respectively the length of the altitudes of the triangle drawn from the vertices

A, B

and

C,

then

P^{2} [\frac{{(h_{1} h_{2} + h_{2} h_{3} + h_{3} h_{1})}^{2}}{h_{1}^{2} h_{2}^{2} h_{3}^{2}} - 2] =

MEDIUM

Mathematics>Algebra>Sequence and Series>Harmonic Progression

ΔABC,

the lengths of sides

A C

and

A B

are

12 cm

and

5 cm,

respectively. If the area of

∆ ABC

30 {cm}^{2}

and

R

and

r

are respectively the radii of circumcircle and incircle of

ΔABC

, then the value of

2 R + r (in cm)

is equal to ______ .

MEDIUM

Mathematics>Algebra>Sequence and Series>Harmonic Progression

With the usual notation in

∆ A B C

, if

∠ A + ∠ B = 120 °, a = \sqrt{3} + 1

units and

b = \sqrt{3} - 1

units, then the ratio

∠ A : ∠ B

MEDIUM

Mathematics>Algebra>Sequence and Series>Harmonic Progression

Let

X, Y, Z

be respectively the areas of a regular pentagon, regular hexagon and regular heptagon which are inscribed in a circle of radius

1

. Then

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

In the figure given below, if the areas of the two regions are equal then which of the following is true?

Question Image

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

Denote Area

(∆ X Y Z), P (∆ X Y Z)

and

|X Y|

by area of the triangle

∆ X Y Z

, perimeter of the triangle

∆ X Y Z

and length of the line segment

X Y

respectively.
Let

A B C D

be a convex quadrangle and the diagonals

A C

and

B D

intersect at

O

. Then

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

In a rectangle

A B C D,

points

X

and

Y

are the mid-points of

A D

and

D C

respectively. Lines

B X

and

C D

when extended intersect at

E

and lines

B Y

and

A D

when extended intersect at

F

. If the area of rectangle

A B C D

60

square units, then the area of

△ B E F

(in square units) is

MEDIUM

Mathematics>Algebra>Sequence and Series>Harmonic Progression

p_{1}, p_{2}, p_{3}

are altitudes of a triangle

A B C

drawn from the vertices

A, B, C

respectively. If

∆

is the area of the triangle and

2 s

is the sum of the sides, then

\frac{1}{p_{1}} + \frac{1}{p_{2}} - \frac{1}{p_{3}} =

MEDIUM

Mathematics>Algebra>Sequence and Series>Harmonic Progression

In a

∆ A B C,

a = 2 x, b = 2 y

and

∠ C = 120 °,

then the area of the triangle is

MEDIUM

Mathematics>Algebra>Sequence and Series>Harmonic Progression

Let

C

be the centre of the circle

x^{2} + y^{2} - x + 2 y = \frac{11}{4}

and

P

be a point on the circle. A line passes through the point

C

, makes an angle of

\frac{π}{4}

with the line

C P

and intersects the circle at the points

Q

and

R

. Then the area of the triangle

P Q R

(in

{unit}^{2}

) is

MEDIUM

Mathematics>Algebra>Sequence and Series>Harmonic Progression

In a

∆ A B C

, if

\tan \frac{A}{2} = \frac{5}{6}, \tan \frac{C}{2} = \frac{2}{5}

, then

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

In a triangle

P Q R,

let

∠ P Q R = 30 °

and the sides

P Q

and

Q R

have lengths

10 \sqrt{3} a n d 10

units, respectively. Then, which of the following statement(s) is (are) TRUE?

MEDIUM

Mathematics>Algebra>Sequence and Series>Harmonic Progression

In a $Δ A B C$ , points $X$ and $Y$ are on $A B$ and $A C$ , respectively, such that $X Y$ is parallel to $B C$ . Which of the two following equalities always hold? (Here, $[P Q R]$ denotes the area of $Δ P Q R$ ).

$I$ . $[B C X] = [B C Y]$

$II$ . $[A C X] \cdot [A B Y] = [A X Y] \cdot [A B C]$

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

Let

A B C D

be a square and let

P

be point on segment

C D

such that

D P : P C = 1 : 2

. Let

Q

be a point on segment

A P

such that

∠ B Q P = 90^{o}

. Then the ratio of the area of quadrilateral

P Q B C

to the area of the square

A B C D

HARD

Mathematics>Algebra>Sequence and Series>Harmonic Progression

In the figure given below, $A B C D E F$ is a regular hexagon of side length $1$ unit, $A F P S$ and $A B Q R$ are squares. Then the ratio $\frac{area of △ A P Q}{area of △ S R P}$ equals

Question Image

If in a △ABC, the altitudes from the vertices A, B, C on opposite sides are in H.P., then sin A, sinB, sinC are in

Important Questions on Sequence and Series

Learn and Prepare for any exam you want !

If in a $△ A B C$ , the altitudes from the vertices $A, B, C$ on opposite sides are in H.P., then $\sin A, \sin B, \sin C$ are in