MCQ Questions for Class 10 Maths Circles with Answers

Free PDF Download of CBSE Class 10 Maths Chapter 10 Circles Multiple Choice Questions with Answers. MCQ Questions for Class 10 Maths with Answers was Prepared Based on Latest Exam Pattern. Students can solve NCERT Class 10 Maths Circles MCQs with Answers to know their preparation level.

Class 10 Maths MCQs Chapter 10 Circles

1. The distance between two parallel tangents of acircle of radius 4 cm is
(a) 2 cm
(b) 4 cm
(c) 6 cm
(d) 8 cm

Answer/ Explanation

Answer: d
Explaination:
MCQ Questions for Class 10 Maths Circles with Answers 3
Reason: Here radius, r = 4 cm
Required distance,
AB = OA + OB
= r + r = 2r = 2×4 = 8 cm


2. In the given figure, if ZRPS = 25°, the value of ZROS is
MCQ Questions for Class 10 Maths Circles with Answers 1
(a) 135°
(b) 145°
(c) 165°
(d) 155°

Answer/ Explanation

Answer: d
Explaination: Reason: Since OR ⊥ PR and OS ⊥ PS
∴ ∠ORP = ∠OSP = 90°
In □ ORPS, ∠ROS + ∠ORP + ∠RPS + ∠OSP = 360°
∠ROS + 90° + 25° + 90° = 360°
∠ROS = 360° – 205° = 155°


3. A tangent is drawn from a point at a distance of 17 cm of circle C(0, r) of radius 8 cm. The length of its tangent is
(a) 5 cm
(b) 9 cm
(c) 15 cm
(d) 23 cm

Answer/ Explanation

Answer: c
Explaination:
MCQ Questions for Class 10 Maths Circles with Answers 4
Reason: In rt ∆OAP, AP² + OA² = OP²
⇒ AP² + (8)² = (17)² => AP² + 64 = 289
⇒ AP² = 289 – 64 = 225
∴ AP = √225 = 15 cm


4. The length of tangents drawn from an external point to the circle
(a) are equal
(b) are not equal
(c) sometimes are equal
(d) are not defined

Answer/ Explanation

Answer: a
Explaination: Reason: Since the length of tangents drawn from an external point to a circle are equal.


5. Number of tangents drawn at a point of the , circle is/are
(a) one
(b) two
(c) none
(d) infinite

Answer/ Explanation

Answer: a
Explaination: Reason: There is only one tangent at a point of the circle.


6. The tangents drawn at the extremities of the diameter of a circle are
(a) perpendicular
(b) parallel
(c) equal
(d) none of these

Answer/ Explanation

Answer: b
Explaination:
MCQ Questions for Class 10 Maths Circles with Answers 5
Reason: Since OP ⊥ AB and OQ ⊥ CD
∴ Z1 = 90° and Z2 = 90°
⇒ ∠1 = Z2, which are alternate angles.
∴ AB || CD


7. Tangents from an external point to a circle are
(a) equal
(b) not equal
(c) parallel
(d) perpendicular

Answer/ Explanation

Answer: a
Explaination: Reason: Tangents from external points to a circle are equal.


8. The length of a tangent drawn from a point at a distance of 10 cm of circle is 8 cm. The radius of the circle is
(a) 4 cm
(b) 5 cm
(c) 6 cm
(d) 7 cm

Answer/ Explanation

Answer: c
Explaination:
MCQ Questions for Class 10 Maths Circles with Answers 6
Reason: In rt. AOAP, we have
OA² + AP² = OP²
⇒ OA² + (8)² = (10)2
⇒ OA² + 64 = 100
⇒ OA² = 100 – 64 = 36
∴ OA = √36 = 6 cm


9. In given figure, CP and CQ are tangents to a circle with centre O. ARB is another tangent touching the circle at R. If CP = 11 cm and BC = 6 cm then the length of BR is
MCQ Questions for Class 10 Maths Circles with Answers 2
(a) 6 cm
(b) 5 cm
(c) 4 cm
(d) 3 cm

Answer/ Explanation

Answer: b
Explaination: Reason: Since
BQ = BR …(i) [∵ Tangents drawn from external points are equal]
CQ = CP …[Using (i)]
BC + BQ = 11
⇒ 6 + BR = 11
⇒ BR = 11 – 6 = 5 cm


10. From a point P which is at a distance of 13 cm from the centre O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle are drawn. Then the area of the quadrilateral PQOR is
(a) 60 cm²
(b) 65 cm²
(c) 30 cm²
(d) 32.5 cm²

Answer/ Explanation

Answer: a
Explaination:
MCQ Questions for Class 10 Maths Circles with Answers 10
Reason: OP² = OQ² + PQ²
169 = 25 + PQ²
PQ² = 144
PQ = 12
Area PQOR = ar (AOPQ) + ar (AOPR)
= \(\frac{1}{2}\) × 12 × 5 + \(\frac{1}{2}\) × 12 × 5 = 60 cm²


11. In the given figure, AB and AC are tangents to the circle with centre O such that ∠BAC = 40°, then ∠BOC is equal to [AI2011]
MCQ Questions for Class 10 Maths Circles with Answers 8
(a) 40°
(b) 50°
(c) 140°
(d) 150°

Answer/ Explanation

Answer: c
Explaination:
(c) In quadrilateral ABOC
∠ABO + ∠BOC + ∠OCA + ∠BAC = 360°
⇒ 90° + ∠BOC + 90° + 40° = 360°
⇒ ∠BOC = 360° – 220° = 140°


12. In the given figure, point P is 26 cm away from the centre O of a circle and the length PT of the tangent drawn from P to the circle is 24 cm. Then the radius of the circle is [Foreign 2011]
MCQ Questions for Class 10 Maths Circles with Answers 9
(a) 25 cm
(b) 26 cm
(c) 24 cm
(d) 10 cm

Answer/ Explanation

Answer: d
Explaination: (d) v OT is radius and PT is tangent
MCQ Questions for Class 10 Maths Circles with Answers 21
∴ OT ⊥ PT
Now, in AOTP,
⇒ OP² = PT² + OT²
⇒ 26² = 24² + OT²
⇒ 676 – 576 = OT²
⇒ 100 = OT²
⇒ 10 cm = OT


13. A line through point of contact and passing through centre of circle is known as
(a) tangent
(b) chord
(c) normal
(d) segment

Answer/ Explanation

Answer: c
Explaination: (c) normal


14. C (O, r1) and C(O, r2) are two concentric circles with r1 > r2 AB is a chord of C(O, r1) touching C(O, r,2) at C then
(a) AB = r1
(b) AB = r2
(c) AC = BC
(d) AB = r1 + r2

Answer/ Explanation

Answer: c
Explaination:
(c) ∵ AB touches
C(0, r<sub>2</sub>)
∴ OC ⊥ AB
MCQ Questions for Class 10 Maths Circles with Answers 22
Also, perpendicular from the centre to a chord bisects the chord.
∴ AC = BC


15. Two parallel lines touch the circle at
points A and B respectively. If area of the circle is 25 n cm2, then AB is equal to
(a) 5 cm
(b) 8 cm
(c) 10 cm
(d) 25 cm

Answer/ Explanation

Answer: c
Explaination:
(c) Let radius of circle = R
∴ πR² = 25π
⇒ R = 5 cm
∴ Distance between two parallel tangents = diameter = 2 × 5 = 10 cm.


16. From a point P which is at a distance of 13 cm from the centre O of a circle of radius 5 cm, the pair of tangents PQ and PR to the circle are drawn. Then the area of the quadrilateral PQOR is [NCERT Exemplar Problems]
(a) 60 cm²
(b) 65 cm²
(c) 30 cm²
(d) 32.5 cm²

Answer/ Explanation

Answer: a
Explaination:
MCQ Questions for Class 10 Maths Circles with Answers 23


17. In figure AT is a tangent to the circle with centre O such that OT = 4 cm and ∠OTA = 30°. Then AT is equal to [NCERT Exemplar Problems]
(a) 4 cm
(b) 2 cm
(c) 2√3 cm
(d) 4√3 cm
MCQ Questions for Class 10 Maths Circles with Answers 10

Answer/ Explanation

Answer: c
Explaination:
MCQ Questions for Class 10 Maths Circles with Answers 24


18. In figure if O is centre of a circle, PQ is a chord and the tangent PR at P makes an angle of 50° with PQ, then ∠POQ is equal to [NCERT Exemplar Problems]
MCQ Questions for Class 10 Maths Circles with Answers 11
(a) 100°
(b) 80°
(c) 90°
(d) 75°

Answer/ Explanation

Answer: a
Explaination:
(a) OP ⊥ PR [Y Tangent and radius are ⊥ to each other at the point of contact]
∠OPQ = 90° – 50° = 40°
OP = OQ [Radii]
∴ ∠OPQ = ∠OQP = 40°
In ∆OPQ,
⇒ ∠POQ + ∠OPQ + ∠OQP = 180°
⇒ ∠POQ + 40° + 40° = 180°
∠POQ = 180° – 80° = 100°.


19. Match the column:

(1) The tangent at any point of a circle is … (A) known as tangent to the circle
(2) The line containing the radius through the point of contact is … (B) to the radius through the point of
(3) The lengths of tangents drawn from an external point to a circle are… (C) called the ‘normal’ to circle
(4) When two end points of the corresponding chord of a  secant coincide, it is … (D) equal

(a) 1 → A, 2 → B, 3 → C, 4 → D
(b) 1 → B, 2 → A, 3 → D, 4 → C
(c) 1 → D, 2 → A, 3 → C, 4 → B
(d) 1 → B, 2 → C, 3 → D, 4 → A

Answer/ Explanation

Answer:
Explaination: Properties of circle.


20. In figure, O is the centre of a circle, AB is a chord and AT is the tangent at A. If ∠AOB = 100°, then ∠BAT is equal to [Delhi 2011]
MCQ Questions for Class 10 Maths Circles with Answers 12
(a) 100°
(b) 40°
(c) 50°
(d) 90°

Answer/ Explanation

Answer: c
Explaination:
(c) ∠AOB = 100°
∠OAB = ∠OBA (∵ OA and OB are radii)
Now, in ∆AOB,
∠AOB + ∠OAB + ∠OBA = 180°
(Angle sum property of A)
⇒ 100° + x + x = 180° [Let ∠OAB = ∠OBA = x]
⇒ 2x = 180° – 100°
⇒ 2x = 80°
⇒ x = 40°
Also, ∠OAB + ∠BAT = 90°
[∵ OA is radius and TA is tangent at A]
⇒ 40° + ZBAT = 90°
⇒ ∠BAT = 50°


21. In the figure PA and PB are tangents to the circle with centre O. If ∠APB = 60°, then ∠OAB is [Delhi 2011]
MCQ Questions for Class 10 Maths Circles with Answers 13
(a) 30°
(b) 60°
(c) 90°
(d) 15°

Answer/ Explanation

Answer: a
Explaination:
(a) Given ∠APB = 60°
∵ ∠APB + ∠PAB + ∠PBA = 180°
⇒ APB + x + x = 180°
[∵ PA = PB ∴ ∠PAB = ∠PBA = x (say)]
⇒ 60° + 2x = 180°
⇒ 2x = 180° – 60°
⇒ 2x = 120°
⇒ x = \(\frac{120°}{2}\) = 60°
Also, ∠OAP = 90°
⇒ ∠OAB + ∠PAB = 90°
⇒ ∠OAB + 60°= 90°
⇒ ∠OAB = 30°


22. In the given figure, TP and TQ are two tangents to a circle with centre O, such that ∠POQ = 110°. Then ∠PTQ is equal to [Foreign 2011]
MCQ Questions for Class 10 Maths Circles with Answers 14
(a) 55°
(b) 70°
(c) 110°
(d) 90°

Answer/ Explanation

Answer: b
Explaination:
(b) In quadrilateral POQT,
∠PTQ + ∠TPO + ∠TQO + ∠POQ
= 360°
⇒ ∠PTQ + 90° + 90° + 110° = 360°
⇒ ∠PTQ + 290° = 360°
⇒ ∠PTQ = 360° – 290° = 70°


23. Infigure,PQandPRaretangentstoacirclewith centreA. If ∠QPA=27°, then ∠QAR equals to [Foreign 2012]
MCQ Questions for Class 10 Maths Circles with Answers 15
(a) 63°
(b) 153°
(c) 110°
(d) 90°

Answer/ Explanation

Answer: c
Explaination:
(c) ∠QPA = ∠RPA
[∵ ∆AQP ≅ ∆ARP (RHS congruence rule)]
⇒ ∠RPA = 27°
∴ ∠QPR = ∠QPA + ∠RPA
= 27° + 27° = 54°
MCQ Questions for Class 10 Maths Circles with Answers 27
Now,
∠QAR + ∠AQP + ∠ARP + ∠QPR = 360°
⇒ ∠QAR = 90° + 90° + 54° = 360°
⇒ ∠QAR = 360° – 234° = 126°


24. In figure if PQR is the tangent to a circle at Q whose centre is O, AB is a chord parallel to PR and ∠BQR = 70°, then ∠AQB is equal to [NCERT Exemplar Problems]
MCQ Questions for Class 10 Maths Circles with Answers 16
(a) 20°
(b) 40°
(c) 35°
(d) 45°

Answer/ Explanation

Answer:
Explaination:
(b) AB || PR
∴ ∠ABQ =∠BQR
[Alternate interior angles]
⇒ ∠ABQ = 70°
Also, ∠BQR = ∠BAQ [Angles in alternate segment]
⇒ ∠BAQ = 70°
In ∆AQB,
∠BAQ + ∠ABQ + ∠AQB = 180°
⇒ 70° + 70° + ∠AQB = 180°
⇒ ∠AQB = 180° – 140° = 40°.


25. The common point of the tangent and the circle is called _____ .

Answer/ Explanation

Answer:
Explaination: point of contact


26. Two concentric circles are of radii 13 cm and 5 cm. The length of the chord of larger circle which touches the smaller circle is _____ .

Answer/ Explanation

Answer:
Explaination: 24 cm. Hint: ∵ AB touches the smaller circle
MCQ Questions for Class 10 Maths Circles with Answers 33
∴ OC ⊥ AB and hence AC = BC
In right ∆OCA,
OA² = OC² + AC²
⇒ AC²= 13² – 5²
⇒ AC = 12
∴ AB = 2 × 12 = 24 cm.


27. A quadrilateral ABCD is drawn to circumscribe a circle. If AB =12 cm, BC = 15 cm and CD = 14 cm, then AD is equal to _____ .

Answer/ Explanation

Answer:
Explaination: 11 cm. Hint: AB + CD = BC + AD
MCQ Questions for Class 10 Maths Circles with Answers 26
⇒ 12+ 14= 15+ AD
⇒ AD =11 cm.


28. Number of tangents to a circle which are parallel to a secant is ____ .

Answer/ Explanation

Answer:
Explaination: 2


29. A tangent PQ at a point P of a circle of radius 7 cm meets a line through centre O at a point Q so that OQ = 25 cm length PQ is ____ .

Answer/ Explanation

Answer: 24 cm
Explaination:
MCQ Questions for Class 10 Maths Circles with Answers 28
24 cm.
Hint: PQ² = OQ² – OP² = 252²- 7²
⇒ PQ = 24 cm.


30. If the angle between two tangents drawn from an external point P to a circle of radius a and centre O, is 60°, then find the length of OP. [Delhi 2017]

Answer/ Explanation

Answer:
Explaination:
MCQ Questions for Class 10 Maths Circles with Answers 29


31. In the given fig., there are two concentric circles with centre O. PRT and PQS are tangents to the inner circle from a point P lying on the outer circle. If PR = 5 cm, find the length of PS. [Delhi 2017 (C)]
MCQ Questions for Class 10 Maths Circles with Answers 17

Answer/ Explanation

Answer:
Explaination:
PR and PQ are tangents to the circle from common external point P.
MCQ Questions for Class 10 Maths Circles with Answers 30
∴ PR = PQ
∴ PQ = 5 cm (∵ PR = 5 cm)
PS = 2PQ (∵ OQ ⊥ PS)
= 2 × 5 = 10 cm


32. In the given Fig., O is the centre of the circle, PQ is a chord and PT is tangent to the circle at P. If ∠POQ = 70°, find ∠TPQ. [AI 2017 (C)]
MCQ Questions for Class 10 Maths Circles with Answers 18

Answer/ Explanation

Answer:
Explaination:
Given: O is centre of the circle, PQ is chord and PT is tangent at P.
To find: ∠TPQ
Solution: In ∆OPQ
∠POQ + ∠OPQ + ∠OQP = 180°
⇒ 70° + ∠OPQ + ∠OPQ = 180°
(∵ OP = OQ, radii of the circle)
⇒ 70° + 2∠OPQ = 180°
⇒ 2∠OPQ = 110°
⇒ ∠OPQ = 55°
OP is perpendicular to the tangent at P.
∴ ∠OPT = 90°
⇒ ∠OPQ + ∠TPQ = 90°
⇒ 55° + ∠TPQ = 90°
⇒ ∠TPQ = 90° – 55°
∠TPQ = 35°


33. What is the distance between two parallel tangents of a circle of radius 7 cm?

Answer/ Explanation

Answer:
Explaination:
Two parallel tangents of a circle can be drawn only at the end points of the diameter
MCQ Questions for Class 10 Maths Circles with Answers 31
⇒ l<sub>1</sub> || l<sub>2</sub>
⇒ Distance between l<sub>1</sub> and l<sub>2</sub> = AB
= Diameter of the circle
= 2 × r = 2 × 7 cm = 14 cm


34. In the given figure, find ∠QSR.
MCQ Questions for Class 10 Maths Circles with Answers 19

Answer/ Explanation

Answer:
Explaination:
Given: PQ and PR are tangents to a circle with centre O and ∠QPR = 50°.
To find: ∠QSR
∠QOR + ∠QPR =180°
⇒ ∠QOR +50° = 180°
⇒ ∠QOR = 130°
⇒ ∠QSR = \(\frac{1}{2}\) ∠QOR [Degree measure theorem]
⇒ ∠QSR =\(\frac{1}{2}\) × 130° = 65°


35. In the given figure, AB, AC and AD are tangents. If AB = 5 cm, find AD.
MCQ Questions for Class 10 Maths Circles with Answers 20

Answer/ Explanation

Answer:
Explaination:
Given: AB, AC and AD are tangents. AB = 5 cm.
MCQ Questions for Class 10 Maths Circles with Answers 32
To find: AD
Sol. AB and AC are tangents from the same point to the circle with centre O.
⇒ AB = AC …(i)
(Length of the tangents from the same external point are equal).
AC and AD are tangents from the same point to the circle with centre O.
⇒ AC = AD …(ii)
(Length of the tangents from the same external point are equal)
From (i) and (ii)
∵ AB = AC = AD = 5 cm


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