✦ NCF 2023 ALIGNED • CBSE CLASS X ✦
Chapter 11: Areas Related to Circles
Assertion-Reason Question Bank
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Assertion (A)
If perimeter of a circle is \(44\text{ cm}\), its area is \(44\text{ cm}^2\).
Reason (R)
Radius \(r = 44 / (2\pi) = 7\text{ cm}\). Area \(= \pi (7)^2 = 154\text{ cm}^2\).
Correct Option: (D)
Assertion (A) is false (area is \(154\text{ cm}^2\)). Reason (R) is true.
Assertion (A) is false (area is \(154\text{ cm}^2\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The area of a quadrant of a circle with circumference \(22\text{ cm}\) is \(\frac{77}{8}\text{ cm}^2\).
Reason (R)
Circumference \(2\pi r = 22 \implies 2 \times \frac{22}{7} \times r = 22 \implies r = \frac{7}{2}\text{ cm}\). Area of quadrant \(= \frac{1}{4} \pi r^2 = \frac{1}{4} \times \frac{22}{7} \times \frac{49}{4} = \frac{77}{8}\text{ cm}^2\).
Correct Option: (A)
\(r = 7/2\text{ cm}\). Quadrant area \(= (1/4) \pi (7/2)^2 = 77/8\text{ cm}^2\). Reason explains Assertion.
\(r = 7/2\text{ cm}\). Quadrant area \(= (1/4) \pi (7/2)^2 = 77/8\text{ cm}^2\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
Total area of a segment is equal to area of corresponding sector.
Reason (R)
Area of segment = Area of sector - Area of triangle.
Correct Option: (D)
Assertion (A) is false (must subtract triangle area). Reason (R) is true.
Assertion (A) is false (must subtract triangle area). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The length of an arc of a sector of radius \(r\) and angle \(\theta\) is \(\frac{\theta}{360^\circ} 2\pi r\).
Reason (R)
Circumference for \(360^\circ\) is \(2\pi r\), so arc length for central angle \(\theta\) is \(\frac{\theta}{360^\circ} 2\pi r\).
Correct Option: (A)
Unitary method derivation of arc length formula. Reason explains Assertion.
Unitary method derivation of arc length formula. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If radius of circle is halved, its area is halved.
Reason (R)
New area \(A' = \pi (r/2)^2 = \frac{1}{4} \pi r^2\), so area becomes one-fourth.
Correct Option: (D)
Assertion (A) is false (it becomes \(1/4\), not \(1/2\)). Reason (R) is true.
Assertion (A) is false (it becomes \(1/4\), not \(1/2\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
Area of the largest triangle inscribed in a semicircle of radius \(r\) is \(r^2\).
Reason (R)
Base of triangle = diameter \(2r\), maximum height = radius \(r\), so \(\text{Area} = \frac{1}{2} \times 2r \times r = r^2\).
Correct Option: (A)
Base \(= 2r\), height \(= r\), area \(= (1/2)(2r)(r) = r^2\). Reason explains Assertion.
Base \(= 2r\), height \(= r\), area \(= (1/2)(2r)(r) = r^2\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
Perimeter of a sector of radius \(r\) and angle \(\theta\) is \(\frac{\theta}{360^\circ} 2\pi r + 2r\).
Reason (R)
Perimeter of sector includes arc length plus two bounding radii.
Correct Option: (B)
Both A and R are true perimeter definitions.
Both A and R are true perimeter definitions.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
Perimeter of sector of central angle \(90^\circ\) and radius \(r\) is \(\frac{\pi r}{2}\).
Reason (R)
Perimeter includes arc length \(\frac{\pi r}{2}\) plus two radii \(2r\), making total perimeter \(\frac{\pi r}{2} + 2r\).
Correct Option: (D)
Assertion (A) is false (misses \(+2r\)). Reason (R) is true.
Assertion (A) is false (misses \(+2r\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
The minute hand of a clock \(14\text{ cm}\) long sweeps an area of \(\frac{154}{3}\text{ cm}^2\) in \(5\) minutes.
Reason (R)
Angle swept in \(5\) mins \(= \frac{5}{60} \times 360^\circ = 30^\circ\). Area \(= \frac{30}{360} \times \frac{22}{7} \times 14^2 = \frac{1}{12} \times 22 \times 28 = \frac{154}{3}\text{ cm}^2\).
Correct Option: (A)
Angle \(= 30^\circ\). Area \(= (30/360) \pi (14)^2 = 154/3\text{ cm}^2\). Reason explains Assertion.
Angle \(= 30^\circ\). Area \(= (30/360) \pi (14)^2 = 154/3\text{ cm}^2\). Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If perimeter and area of a circle are numerically equal, then radius of circle is \(2\) units.
Reason (R)
\(2\pi r = \pi r^2 \implies r^2 - 2r = 0 \implies r(r-2) = 0 \implies r = 2\) units (since \(r \ne 0\)).
Correct Option: (A)
\(2\pi r = \pi r^2 \implies r = 2\) units. Reason explains Assertion.
\(2\pi r = \pi r^2 \implies r = 2\) units. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If radius of a circle is doubled, its circumference is doubled.
Reason (R)
Circumference \(C = 2\pi r\) is directly proportional to radius \(r\).
Correct Option: (B)
Both A and R are true linear relationships.
Both A and R are true linear relationships.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
Area of a square inscribed in a circle of radius \(r\) is \(2r^2\).
Reason (R)
Diagonal of square \(= 2r\), so \(\text{Area} = \frac{1}{2} d^2 = \frac{1}{2} (2r)^2 = 2r^2\).
Correct Option: (B)
Both A and R are true area derivations.
Both A and R are true area derivations.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
Area of circle circumscribing a square of side \(a\) is \(\frac{\pi a^2}{2}\).
Reason (R)
Radius of circumscribing circle is \(a\).
Correct Option: (C)
Assertion (A) is true (diagonal \(d = a\sqrt{2} \implies r = a/\sqrt{2} \implies \text{Area} = \pi (a/\sqrt{2})^2 = \pi a^2/2\)). Reason (R) is false because radius is \(a/\sqrt{2}\), not \(a\).
Assertion (A) is true (diagonal \(d = a\sqrt{2} \implies r = a/\sqrt{2} \implies \text{Area} = \pi (a/\sqrt{2})^2 = \pi a^2/2\)). Reason (R) is false because radius is \(a/\sqrt{2}\), not \(a\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
If radius of a circle is doubled, its area becomes \(4\) times.
Reason (R)
Area of circle is directly proportional to radius \(r\).
Correct Option: (C)
Assertion (A) is true (\(A' = \pi (2r)^2 = 4\pi r^2\)). Reason (R) is false because area is proportional to \(r^2\) (square of radius), not \(r\).
Assertion (A) is true (\(A' = \pi (2r)^2 = 4\pi r^2\)). Reason (R) is false because area is proportional to \(r^2\) (square of radius), not \(r\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
Area of segment of a circle = Area of corresponding sector - Area of corresponding triangle.
Reason (R)
A segment is the region bounded by a chord and an arc.
Correct Option: (B)
Both A and R are true segment properties.
Both A and R are true segment properties.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
Area of sector of angle \(60^\circ\) of a circle of radius \(6\text{ cm}\) is \(\frac{132}{7}\text{ cm}^2\).
Reason (R)
Formula for sector area is \(\frac{\theta}{180^\circ} \pi r^2\).
Correct Option: (C)
Assertion (A) is true (\((60/360) \times (22/7) \times 36 = 132/7\text{ cm}^2\)). Reason (R) is false because formula denominator is \(360^\circ\), not \(180^\circ\).
Assertion (A) is true (\((60/360) \times (22/7) \times 36 = 132/7\text{ cm}^2\)). Reason (R) is false because formula denominator is \(360^\circ\), not \(180^\circ\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
Angle described by minute hand of a clock in \(1\) minute is \(6^\circ\).
Reason (R)
In \(60\) minutes, minute hand completes \(360^\circ\), so in \(1\) min angle is \(360^\circ/60 = 6^\circ\).
Correct Option: (B)
Both A and R are true clock angle calculations.
Both A and R are true clock angle calculations.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
Area of a ring (annulus) formed by two concentric circles of radii \(R\) and \(r\) (\(R > r\)) is \(\pi(R^2 - r^2)\).
Reason (R)
Area of ring = Area of outer circle - Area of inner circle \(= \pi R^2 - \pi r^2 = \pi(R^2 - r^2)\).
Correct Option: (A)
Outer area minus inner area gives ring area. Reason explains Assertion.
Outer area minus inner area gives ring area. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
If radii of two circles are \(3\text{ cm}\) and \(4\text{ cm}\), radius of circle with area equal to sum of areas is \(7\text{ cm}\).
Reason (R)
\(\pi R^2 = \pi (3^2) + \pi (4^2) = 25\pi \implies R = 5\text{ cm}\).
Correct Option: (D)
Assertion (A) is false (it is \(5\text{ cm}\), not \(7\text{ cm}\)). Reason (R) is true.
Assertion (A) is false (it is \(5\text{ cm}\), not \(7\text{ cm}\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
Area of sector with angle \(\theta\) and radius \(r\) is \(\frac{\theta}{360^\circ} 2\pi r\).
Reason (R)
The correct formula for area of sector is \(\frac{\theta}{360^\circ} \pi r^2\).
Correct Option: (D)
Assertion (A) is false (\(2\pi r\) is arc length, not area). Reason (R) is true.
Assertion (A) is false (\(2\pi r\) is arc length, not area). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
If sum of areas of two circles with radii \(R_1\) and \(R_2\) is equal to area of circle with radius \(R\), then \(R_1^2 + R_2^2 = R^2\).
Reason (R)
The relation between radii is \(R_1 + R_2 = R\).
Correct Option: (C)
Assertion (A) is true (\(\pi R_1^2 + \pi R_2^2 = \pi R^2 \implies R_1^2 + R_2^2 = R^2\)). Reason (R) is false because \(R_1 + R_2 = R\) holds for circumferences, not areas.
Assertion (A) is true (\(\pi R_1^2 + \pi R_2^2 = \pi R^2 \implies R_1^2 + R_2^2 = R^2\)). Reason (R) is false because \(R_1 + R_2 = R\) holds for circumferences, not areas.
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
If wheel radius is \(0.25\text{ m}\), distance covered in \(1000\) revolutions is \(1000\text{ m}\).
Reason (R)
Distance \(= 1000 \times 2\pi r = 1000 \times 2 \times \frac{22}{7} \times 0.25 \approx 1571\text{ m}\).
Correct Option: (D)
Assertion (A) is false (\(1571\text{ m}\), not \(1000\text{ m}\)). Reason (R) is true.
Assertion (A) is false (\(1571\text{ m}\), not \(1000\text{ m}\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
Perimeter of a protractor (semicircle) of diameter \(14\text{ cm}\) is \(36\text{ cm}\).
Reason (R)
Perimeter of protractor is \(\pi r\).
Correct Option: (C)
Assertion (A) is true (\(r=7, P = \pi r + 2r = 22 + 14 = 36\text{ cm}\)). Reason (R) is false because perimeter includes diameter \(2r\) (\(\pi r + 2r\)).
Assertion (A) is true (\(r=7, P = \pi r + 2r = 22 + 14 = 36\text{ cm}\)). Reason (R) is false because perimeter includes diameter \(2r\) (\(\pi r + 2r\)).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
If circumferences of two circles are in ratio \(2:3\), ratio of their areas is \(4:9\).
Reason (R)
Ratio of areas is equal to ratio of circumferences.
Correct Option: (C)
Assertion (A) is true (\((2/3)^2 = 4/9\)). Reason (R) is false because ratio of areas is equal to SQUARE of ratio of circumferences.
Assertion (A) is true (\((2/3)^2 = 4/9\)). Reason (R) is false because ratio of areas is equal to SQUARE of ratio of circumferences.
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.
Assertion (A)
Perimeter of a semicircle of radius \(r\) is \((\pi + 2)r\).
Reason (R)
Perimeter of semicircle consists of semicircular arc \(\pi r\) plus diameter \(2r\).
Correct Option: (B)
Both A and R are true (\(r(\pi+2)\)).
Both A and R are true (\(r(\pi+2)\)).
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
Area of a sector of a circle of radius \(r\) with central angle \(\theta\) is \(\frac{\theta}{360^\circ} \pi r^2\).
Reason (R)
Area of full circle (\(360^\circ\)) is \(\pi r^2\), so by unitary method, area for angle \(\theta\) is \(\frac{\theta}{360^\circ} \pi r^2\).
Correct Option: (A)
Unitary method derivation of sector area formula. Reason explains Assertion.
Unitary method derivation of sector area formula. Reason explains Assertion.
Correct Answer: (A)
Reason explains Assertion.
Reason explains Assertion.
Assertion (A)
Area of sector of angle \(p\) (in degrees) of a circle with radius \(R\) is \(\frac{p}{720^\circ} (2\pi R^2)\).
Reason (R)
\(\frac{p}{720^\circ} (2\pi R^2) = \frac{p}{360^\circ} (\pi R^2)\).
Correct Option: (B)
Both A and R are true (\(2/720 = 1/360\)).
Both A and R are true (\(2/720 = 1/360\)).
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
Angle described by hour hand of clock in \(1\) hour is \(6^\circ\).
Reason (R)
In \(12\) hours, hour hand completes \(360^\circ\), so in \(1\) hour it describes \(360^\circ/12 = 30^\circ\).
Correct Option: (D)
Assertion (A) is false (it is \(30^\circ\)). Reason (R) is true.
Assertion (A) is false (it is \(30^\circ\)). Reason (R) is true.
Correct Answer: (D)
Assertion false; Reason true.
Assertion false; Reason true.
Assertion (A)
Distance covered by a wheel of radius \(r\) in \(1\) revolution is equal to its circumference \(2\pi r\).
Reason (R)
Number of revolutions \(= \frac{\text{Total distance}}{\text{Circumference}}\).
Correct Option: (B)
Both A and R are true wheel motion facts.
Both A and R are true wheel motion facts.
Correct Answer: (B)
Both true.
Both true.
Assertion (A)
Area of major sector = Area of circle - Area of minor sector.
Reason (R)
Sum of central angles of minor and major sectors is \(180^\circ\).
Correct Option: (C)
Assertion (A) is true. Reason (R) is false because sum of central angles around a point is \(360^\circ\), not \(180^\circ\).
Assertion (A) is true. Reason (R) is false because sum of central angles around a point is \(360^\circ\), not \(180^\circ\).
Correct Answer: (C)
Assertion true; Reason false.
Assertion true; Reason false.