M.C.Q

MCQ 511 Mark

In Fig. $A O B$ is a diameter of a circle with centre $O$ such that $A B=34 cm$ and $C D$ is a chord of length 30 cm . The distance between $C D$ and $A B$ is

✓
8 cm
B
15 cm
C
18 cm
D
6 cm

Answer

Correct option: A.

8 cm

(a) 8 cm
Applying Pythagoras theorem in $\triangle O L C$, we obtain
$\begin{array}{ll}& O C^2=O L^2+L C^2 \\
\Rightarrow \quad & O L=\sqrt{O C^2-L C^2}=\sqrt{17^2-15^2}=\sqrt{64} cm=8 cm\end{array}$

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MCQ 521 Mark

In Fig. O is the centre of the circle and diameter AB bisects the chord CD of length 16 cm at point E such that $E B=4 cm$. The radius of the circle is

✓
10 cm
B
12 cm
C
6 cm
D
8 cm

Answer

Correct option: A.

10 cm

(a) 10 cm
In right triangle OEC, we obtain
$\begin{array}{ll}& O C^2=O E^2+E C^2 \\
\Rightarrow & O C^2=(O B-E B)^2+8^2 \\
\Rightarrow & O C^2=(O C-4)^2+64 \\
\Rightarrow & 0=-8 O C+16+64 \Rightarrow O C=10 cm\end{array}$

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MCQ 531 Mark

In Fig. CD is the diameter of a circle with centre $O$ and $C D$ is perpendicular to chord $A B$. If $A B=12 cm$ and $C E=3 cm$, then radius of the circle is

A
6 cm
B
9 cm
✓
7.5 cm
D
8 cm

Answer

Correct option: C.

7.5 cm

(c) 7.5 cm
In right triangle OEA, applying Pythagoras theorem, we obtain
$\begin{array}{ll}& O A^2=O E^2+A E^2 \\
\Rightarrow & O A^2=(O C-C E)^2+6^2 \\
\Rightarrow & O A^2=(O A-3)^2+36 \\
\Rightarrow \quad & 0=-6 \cdot O A+9+36 \Rightarrow O A=\frac{45}{6} cm=7.5 cm\end{array}$

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MCQ 541 Mark

If a chord of a circle is at a distance of 8 cm from the centre of a circle of radius 17 cm , then the length of the chord is

A
25 cm
B
12.5 cm
✓
30 cm
D
9 cm

Answer

Correct option: C.

30 cm

(c) 30cm
Let AB be a chord of a circle of radius 17 cm such that AB is at a distance of 8 cm from the origin. Clearly, L is the mid-point of AB.
In right triangle OLA, applying Pythagoras theorem, we obtain
$\begin{array}{ll}& O A^2=O L^2+A L^2 \Rightarrow A L=\sqrt{O A^2-O L^2}=\sqrt{17^2-8^2}=\sqrt{289-64}=15 cm \\
\therefore \quad & A B=2 A L=2 \times 15 cm=30 cm .\end{array}$

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MCQ 551 Mark

In Fig. if $O A=5 cm, A B=8 cm$ and OD is perpendicular to AB, then CD is equal to

✓
2 cm
B
3 cm
C
4 cm
D
5 cm

Answer

Correct option: A.

2 cm

(a) 2 cm
We have, $A B=8 cm$. Therefore, $A C=B C=4 cm$
Applying Pythagoras theorem in $\triangle O C A$, we obtain
$\begin{aligned}& O A^2=O C^2+A C^2 \Rightarrow O C=\sqrt{O A^2-A C^2}=\sqrt{5^2-4^2}=3 cm \\
\therefore & \quad C D=O D-O C=O A-O C=(5-3)=2 cm\end{aligned}$

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MCQ 561 Mark

In Fig. AB is a diameter of the circle. If $\angle A D C=120^{\circ}$, then $\angle C A B=$

A
$60^{\circ}$
B
$45^{\circ}$
✓
$30^{\circ}$
D
$40^{\circ}$

Answer

Correct option: C.

$30^{\circ}$

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MCQ 571 Mark

In Fig. $\angle B O D=$

✓
$150^{\circ}$
B
$140^{\circ}$
C
$105^{\circ}$
D
$145^{\circ}$

Answer

Correct option: A.

$150^{\circ}$

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MCQ 581 Mark

In Fig. $\angle B C D=$

A
$95^{\circ}$
B
$85^{\circ}$
C
$100^{\circ}$
✓
$105^{\circ}$

Answer

Correct option: D.

$105^{\circ}$

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MCQ 591 Mark

In Fig. $\angle O B D=$

A
$30^{\circ}$
✓
$15^{\circ}$
C
$20^{\circ}$
D
$35^{\circ}$

Answer

Correct option: B.

$15^{\circ}$

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MCQ 601 Mark

In Fig. O is the centre of the circle. If $\angle B A D=75^{\circ}$ and chord BC = chord CD then $\angle B O C=$

A
$80^{\circ}$
B
$65^{\circ}$
✓
$75^{\circ}$
D
$105^{\circ}$

Answer

Correct option: C.

$75^{\circ}$

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MCQ 611 Mark

In Fig. ABCD is a quadrilateral inscribed in a circle. If $\angle D A C=40^{\circ}$ and $\angle B D C=45^{\circ}$, then $\angle B C D=$

A
$85^{\circ}$
✓
$95^{\circ}$
C
$75^{\circ}$
D
$105^{\circ}$

Answer

Correct option: B.

$95^{\circ}$

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MCQ 621 Mark

In Fig. if equilateral triangle ABC is inscribed in a circle and ABCD is a quadrilateral, then $\angle B D C=$

A
$90^{\circ}$
B
$60^{\circ}$
✓
$120^{\circ}$
D
$150^{\circ}$

Answer

Correct option: C.

$120^{\circ}$

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MCQ 631 Mark

In Fig. AOB is a diameter of the circle. If $\angle A D C=120^{\circ}$, then $\angle B A C=$

A
$60^{\circ}$
✓
$30^{\circ}$
C
$45^{\circ}$
D
$20^{\circ}$

Answer

Correct option: B.

$30^{\circ}$

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MCQ 641 Mark

In Fig. ABCD is a cyclic quadrilateral in which AB || DC. If $\angle A=80^{\circ}$, then $\angle B=$

A
$100^{\circ}$
✓
$80^{\circ}$
C
$120^{\circ}$
D
$60^{\circ}$

Answer

Correct option: B.

$80^{\circ}$

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MCQ 651 Mark

In Fig. sides AB and AD of quadrilateral ABCD are produced to E and F respectively. If $\angle C B E=100^{\circ}$, then $\angle C D F=$

A
$100^{\circ}$
✓
$80^{\circ}$
C
$130^{\circ}$
D
$90^{\circ}$

Answer

Correct option: B.

$80^{\circ}$

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MCQ 661 Mark

In Fig. O is the centre of the circle such that $\angle A O C=130^{\circ}$, then $\angle A B C=$

A
$130^{\circ}$
✓
$115^{\circ}$
C
$65^{\circ}$
D
$165^{\circ}$

Answer

Correct option: B.

$115^{\circ}$

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MCQ 671 Mark

In a circle of radius 17 cm, two parallel chords are drawn on opposite side of a diameter. The distance between the chords is 23 cm. If the length of one chord is 16 cm, then the length of the other is

A
34 cm
B
15 cm
C
23 cm
✓
30 cm

Answer

Correct option: D.

30 cm

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MCQ 681 Mark

AB and CD are two parallel chords of a circle with centre O such that AB = 6cm and CD = 12cm The chords are on the same side of the centre and the distance between them is 3 cm. The radius of the circle, is

A
6 cm
B
$5 \sqrt{2} cm$
C
7 cm
✓
$3 \sqrt{5} cm$

Answer

Correct option: D.

$3 \sqrt{5} cm$

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MCQ 691 Mark

If AB is a chord of a circle, P and Q are the two points on the circle different A and B, then

A
$\angle A P B=\angle A Q B$
✓
$\angle A P B+\angle A Q B=180^{\circ}$ or $\angle A P B=\angle A Q B$
C
$\angle A P B+\angle A Q B=90^{\circ}$
D
$\angle A P B+\angle A Q B=180^{\circ}$

Answer

Correct option: B.

$\angle A P B+\angle A Q B=180^{\circ}$ or $\angle A P B=\angle A Q B$

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MCQ 701 Mark

Two equal circles of radius r intersect such that each passes through the of the other. The length of the common chord of the circles, is

A
$\sqrt{r}$
B
$\sqrt{2} r A B$
✓
$\sqrt{3} r$
D
$\frac{\sqrt{3}}{2} r$

Answer

Correct option: C.

$\sqrt{3} r$

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MCQ 711 Mark

In a circle with centre O, AB and CD are two diameters perpendicular each other. The length of chord AC, is

A
2AB
B
$\sqrt{2}$
C
$\frac{1}{2} A B$
✓
$\frac{1}{\sqrt{2}} A B$

Answer

Correct option: D.

$\frac{1}{\sqrt{2}} A B$

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MCQ 721 Mark

In Fig. O is the centre of the circle and $\angle B D C=42^{\circ}$. The measure of $\angle A C B$ is

A
$42^{\circ}$
✓
$48^{\circ}$
C
$58^{\circ}$
D
$52^{\circ}$

Answer

Correct option: B.

$48^{\circ}$

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MCQ 731 Mark

In Fig. chords AD and BC intersect each other at right angles at a point P. If $\angle D A B=35^{\circ}$, then $\angle A D C=$

A
$35^{\circ}$
B
$45^{\circ}$
✓
$55^{\circ}$
D
$65^{\circ}$

Answer

Correct option: C.

$55^{\circ}$

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MCQ 741 Mark

In Fig. if $\angle A B C=45^{\circ}$, then $\angle A O C=$

A
$45^{\circ}$
B
$60^{\circ}$
C
$75^{\circ}$
✓
$90^{\circ}$

Answer

Correct option: D.

$90^{\circ}$

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MCQ 751 Mark

In Fig. if chords AB and CD of the circle intersect each other at right angles, then x + y =

A
$45^{\circ}$
B
$60^{\circ}$
C
$75^{\circ}$
✓
$90^{\circ}$

Answer

Correct option: D.

$90^{\circ}$

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MCQ 761 Mark

Number of circles that can be drawn through three non-collinear points, is

✓
1
B
$0$
C
2
D
3

Answer

Correct option: A.

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MCQ 771 Mark

Angle formed in minor segment of a circle, is

A
acute
✓
obtuse
C
right angle
D
none of these

Answer

Correct option: B.

obtuse

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MCQ 781 Mark

The greatest chord of a circle is called its

A
radius
B
secant
✓
diameter
D
none of these

Answer

Correct option: C.

diameter

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MCQ 791 Mark

If A, B, C are three points on a circle with centre O such that $\angle A O B=90^{\circ}$ and $\angle B O C=120^{\circ}$, then $\angle A B C=$

A
$60^{\circ}$
✓
$75^{\circ}$
C
$90^{\circ}$
D
$135^{\circ}$

Answer

Correct option: B.

$75^{\circ}$

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MCQ 801 Mark

PQRS is a cyclic quadrilateral such that PR is a diameter of the circle. If $\angle Q P R=67^{\circ}$ and $\angle S P R=72^{\circ}$, then $\angle Q R S=$

✓
$41^{\circ}$
B
$23^{\circ}$
C
$67^{\circ}$
D
$18^{\circ}$

Answer

Correct option: A.

$41^{\circ}$

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MCQ 811 Mark

The chord of a circle is equal to its radius. The angle subtended by this chord at the minor arc of the circle, is

A
$60^{\circ}$
B
$75^{\circ}$
C
$120^{\circ}$
✓
$150^{\circ}$

Answer

Correct option: D.

$150^{\circ}$

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MCQ 821 Mark

If ABC is an arc of a circle and $\angle A B C=135^{\circ}$, then the ratio of arc $\widehat{A B C}$ to the circumference, is

✓
1:4
B
3:4
C
3:8
D
1:2

Answer

Correct option: A.

1:4

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MCQ 831 Mark

If two diameters of a circle intersect each other at right angles, then quadrilateral formed by joining their end points is a

A
rhombus
B
rectangle
C
parallelogram
✓
square

Answer

Correct option: D.

square

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MCQ 841 Mark

An equilateral triangle ABC is inscribed in a circle with centre O. The measures of $\angle B O C$ is

A
$30^{\circ}$
B
$60^{\circ}$
C
$90^{\circ}$
✓
$120^{\circ}$

Answer

Correct option: D.

$120^{\circ}$

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MCQ 851 Mark

If A and B are two points on a circle such that $m(\overparen{A B})=260^{\circ}$. A possible value for the angle subtended by arc BA at a point on the circle is

A
$100^{\circ}$
B
$75^{\circ}$
✓
$50^{\circ}$
D
$25^{\circ}$

Answer

Correct option: C.

$50^{\circ}$

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MCQ 861 Mark

In a circle, the major arc is 3 times the minor arc. The corresponding central angles and the degree measures of two arcs are

A
$90^{\circ}$ and $270^{\circ}$
B
$90^{\circ}$ and $90^{\circ}$
✓
$270^{\circ}$ and $90^{\circ}$
D
$60^{\circ}$ and $210^{\circ}$

Answer

Correct option: C.

$270^{\circ}$ and $90^{\circ}$

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MCQ 871 Mark

Let C be the mid-point of an arc AB of a circle such that m$\overparen{A B}=183^{\circ}$. If the region bounded by the arc ACB and line segment AB is denoted by S, then the centre O of the circle lies

✓
in the interior of S
B
in the exterior of S
C
on the segment AB
D
on AB and bisects AB

Answer

Correct option: A.

in the interior of S

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MCQ 881 Mark

If AB, BC and CD are equal chords of a circle with O as centre and AD diameter, than $\angle A O B=$

✓
$60^{\circ}$
B
$90^{\circ}$
C
$120^{\circ}$
D
none of these

Answer

Correct option: A.

$60^{\circ}$

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MCQ 891 Mark

ABC is a triangle with B as right angle, AC = 5 cm and AB = 4 cm. A circle is drawn with A as centre and AC as radius. The length of the chord of this circle passing through C and B is

A
3 cm
B
4 cm
C
5 cm
✓
6 cm

Answer

Correct option: D.

6 cm

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MCQ 901 Mark

One chord of a circle is known to be 10 cm. The radius of this circle must be

A
5 cm
✓
greater than 5 cm
C
greater than or equal to 5 cm
D
less than 5 cm

Answer

Correct option: B.

greater than 5 cm

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MCQ 911 Mark

A chord of length 14 cm is at a distance of 6 cm from the centre of a circle. The length of another chord at a distance of 2 cm from the centre of the circle is

A
12 cm
B
14 cm
C
16 cm
✓
18 cm

Answer

Correct option: D.

18 cm

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MCQ 921 Mark

ABCD is a cyclic quadrilateral such that $\angle A D B=30^{\circ}$ and $\angle D C A=80^{\circ}$, then $\angle D A B=$

✓
$70^{\circ}$
B
$100^{\circ}$
C
$125^{\circ}$
D
$150^{\circ}$

Answer

Correct option: A.

$70^{\circ}$

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MCQ 931 Mark

If O is the centre of a circle of radius rand AB is a chord of the circle at a distance r/2 from O, then $\angle B A O=$

A
$60^{\circ}$
B
$45^{\circ}$
✓
$30^{\circ}$
D
$15^{\circ}$

Answer

Correct option: C.

$30^{\circ}$

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MCQ 941 Mark

The radius of a circle is 6 cm. The perpendicular distance from the centre of the circle to the chord which is 8 cm in length, is

A
$\sqrt{5} cm$
✓
$2 \sqrt{5} cm$
C
$2 \sqrt{7} cm$
D
$\sqrt{7} cm$

Answer

Correct option: B.

$2 \sqrt{5} cm$

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MCQ 951 Mark

If the length of a chord of a circle is 16 cm and is at a distance of 15 cm from the centre of the circle, then the radius of the circle is

A
15 cm
B
16 cm
✓
17 cm
D
34 cm

Answer

Correct option: C.

17 cm

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MATHS M.C.Q