CBSE Class 10 Mathematics Areas related to circles MCQs Set 06

Question. A race track is in the form of a ring whose inner and outer circumference are 437 m and 503 m respectively. The area of the track is
(a) 66 sq. cm.
(b) 4935 sq. cm.
(c) 9870 sq. cm
(d) None of these
Answer: B
\(2\pi r_1 = 503\) and \(2\pi r_2 = 437\)
\(r_1 = \frac{503}{2\pi}\) and \(r_2 = \frac{437}{2\pi}\)
Area of ring \(= \pi(r_1 + r_2)(r_1 - r_2)\)
\(= \pi (\frac{503 + 437}{2\pi}) (\frac{503 - 437}{2\pi})\)
\(= \pi (\frac{940}{2\pi}) (\frac{66}{2\pi}) = 235 \times \frac{66}{2\pi} \times 7 / \text{corrected notation}\)
\(= 235 \times 21 = 4935 \text{ sq. cm.}\)

Question. If the sum of the circumferences of two circles with diameters \(d_1\) and \(d_2\) is equal to the circumference of a circle of diameter \(d\), then
(a) \(d_1^2 + d_2^2 = d^2\)
(b) \(d_1 + d_2 = d\)
(c) \(d_1 + d_2 > d\)
(d) \(d_1 + d_2 < d\)
Answer: B
\(\pi d_1 + \pi d_2 = \pi d\)
\(d_1 + d_2 = d\)

Question. If the circumference of a circle increases from \(4\pi\) to \(8\pi\), then its area is
(a) halved
(b) doubled
(c) tripled
(d) quadrupled
Answer: D
\(2\pi r = 4\pi \Rightarrow r = 2\). Area \(= \pi(2)^2 = 4\pi\)
When, \(2\pi r = 8\pi \Rightarrow r = 4\). Area \(= 16\pi\)

Question. If the radius of a circle is diminished by 10%, then its area is diminished by
(a) 10%
(b) 19%
(c) 36%
(d) 20%
Answer: B
Let \(r\) be the radius, area \(= \pi r^2\)
When \(r\) is diminished by 10%, area \(= \pi(r - r/10)^2 = \pi r^2 (\frac{81}{100})\)
Thus area is diminished by \((1 - \frac{81}{100})\% = 19\%\)

Question. If the perimeter of a semi-circular protractor is 36 cm, then its diameter is
(a) 10 cm
(b) 14 cm
(c) 12 cm
(d) 16 cm
Answer: B
Perimeter \(= \frac{2\pi r}{2} + 2r = \pi r + 2r\)
\((\pi + 2)r = 36 \Rightarrow (\frac{36}{7})r = 36 \Rightarrow r = 7 \text{ cm}\)
Hence, diameter \(= 7 \times 2 = 14 \text{ cm}\)

Question. The area of a circular path of uniform width 'd' surrounding a circular region of radius 'r' is
(a) \(\pi d(2r + d)\)
(b) \(\pi(2r + d)r\)
(c) \(\pi(d + r)d\)
(d) \(\pi(d + r)r\)
Answer: A

Question. In a circle of radius 14 cm, an arc subtends an angle of \(45^\circ\) at the centre, then the area of the sector is:
(a) 71 cm\(^2\)
(b) 76 cm\(^2\)
(c) 77 cm\(^2\)
(d) 154 cm\(^2\)
Answer: C

Question. Area of the largest triangle that can be inscribed in a semi-circle of radius \(r\) units is:
(a) \(r^2\) sq units
(b) \(\frac{1}{2}r^2\) sq units
(c) \(2r^2\) sq units
(d) \(\sqrt{2}r^2\) sq units
Answer: A

FILL IN THE BLANK

Question. A sector of a circle is called a .......... sector if the minor arc of the circle is a part of its boundary.
Answer: minor

Question. Angle formed by two radii at the centre is known as ..........
Answer: central angle

Question. Concentric circles are circles having same...........
Answer: centre

Question. The area of a circle is the measurement of the region ..........
Answer: interior of the boundary

Question. enclosed by its ..........
Answer: boundary

Question. Segment is the region enclosed between chord and ..........
Answer: arc

Question. If the area of a circle is \(154 \text{ cm}^2\), then its circumference is .........
Answer: \(44 \text{ cm}\)

Question. Pie (\(\pi\)) is the ratio between circumference and .......... of the circle.
Answer: diameter

Question. Area of a sector of a circle with radius \(6 \text{ cm}\) if angle of the sector is \(60^\circ\), is ........
Answer: \(\frac{132}{7} \text{ cm}^2\)

Question. \(2\pi r\) is .......... of a circle.
Answer: circumference

TRUE/FALSE

Question. Distance travelled by a circular wheel of diameter \(d \text{ cm}\) in one revolution in \(2\pi d \text{ cm}\).
Answer: False

Question. The numerical value of the area of a circle is greater than the numerical value of its circumference.
Answer: False

Question. The length of a rope by which a cow must be tethered in order that it may be able to graze of an area of \(616 \text{ cm}^2\) is \(18 \text{ m}\).
Answer: False

Question. The areas of two sectors of two different circles with equal corresponding arc lengths are equal.
Answer: False

Question. The perimeter of a square circumscribing a circle of radius \(a \text{ cm}\), is \(8a \text{ cm}\).
Answer: True

Question. If the boundary of a segment is a minor arc of a circle then the corresponding segment is called a minor segment.
Answer: True

Question. The length of an arc of a sector of a circle of radius \(r\) units and of centre angle \(\theta\) is \(\frac{\theta}{360^\circ} \times \pi r^2\).
Answer: False

Question. The area of the largest circle that can be drawn inside a rectangle of length \(a \text{ cm}\) and breadth \(b \text{ cm} (a > b)\) is \(\frac{\pi b^2}{4} \text{ cm}^2\).
Answer: True

Question. If the circumference of a circle is \(88 \text{ cm}\), then its radius is \(14 \text{ cm}\).
Answer: True

Question. If diameter of a circle is \(p \text{ cm}\), then area of square inscribed in it is \(p^2 \text{ cm}^2\).
Answer: False

ASSERTION AND REASON

DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.

Question. Assertion : If the outer and inner diameter of a circular path is \(10 \text{ m}\) and \(6 \text{ m}\) then area of the path is \(16\pi \text{ m}^2\).
Reason : If \(R\) and \(r\) be the radius of outer and inner circular path \( = \pi(R^2 - r^2)\).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: A
Area \( = \pi [(\frac{10}{2})^2 - (\frac{6}{2})^2] = \pi (25 - 9) = 16\pi\).

Question. Assertion : A bicycle wheel makes \(5000\) revolutions in covering \(11 \text{ km}\). Then diameter of the wheel is \(35 \text{ cm}\).
Reason : Area of segment of a circle is \(\frac{\theta}{360} \times \pi r^2 - \frac{1}{2} r^2 \sin \theta\).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: D
\(2\pi r = \frac{11000}{5000} \text{ m} = \frac{11}{5} \times 100 \text{ cm}\). \(2r = \frac{11 \times 20}{22} \times 7 = 70\). Diameter \( = 70 \text{ cm}\).

Question. Assertion : If a wire of length \(22 \text{ cm}\) is bent in the shape of a circle, then area of the circle so formed is \(40 \text{ cm}^2\).
Reason : Circumference of the circle = length of the wire.
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: D
(2\pi r = 22 \Rightarrow r = 3.5 \text{ cm}\). Area \( = \frac{22}{7} \times 3.5 \times 3.5 = 38.5 \text{ cm}^2\).

Question. Assertion : If the circumference of two circles are in the ratio \(2 : 3\) then ratio of their areas is \(4 : 9\).
Reason : The circumference of a circle of radius \(r\) is \(2\pi r\) and its area is \(\pi r^2\).
(a) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
(b) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
(c) Assertion (A) is true but reason (R) is false.
(d) Assertion (A) is false but reason (R) is true.
Answer: A
\(\frac{2\pi r_1}{2\pi r_2} = \frac{2}{3} \Rightarrow \frac{r_1}{r_2} = \frac{2}{3}\). Ratio of areas \( = \frac{\pi r_1^2}{\pi r_2^2} = (\frac{r_1}{r_2})^2 = (\frac{2}{3})^2 = \frac{4}{9}\).