SECTION F — Assertion and Reasoning Type Questions [1 Marks Each]
Directions: Each of the following questions consists of two statements, namely, Assertion (A) and Reason (R). Select the correct option from the choices given below:
- (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.
-
Q1. Assertion (A): The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Reason (R): The tangent line touches the circle at only one common point, and the shortest distance from the center to a line on the tangent is the radius. -
Q2. Assertion (A): The lengths of tangents drawn from an external point to a circle are equal.
Reason (R): Triangles formed by joining the external point, center, and points of contact are congruent by the RHS congruence criterion. -
Q3. Assertion (A): No tangent can be drawn to a circle from a point lying inside the circle.
Reason (R): Any line passing through an interior point of a circle intersects the circle at two distinct points, making it a secant rather than a tangent. -
Q4. Assertion (A): Exactly two tangents can be drawn to a circle from an external point.
Reason (R): A quadratic equation derived from the distance formula between an external point and a circle always yields two distinct real solutions for the points of contact. -
Q5. Assertion (A): A circle can have at most two parallel tangents at the same time for any given direction of a secant line.
Reason (R): The tangents drawn at the opposite ends of any diameter of a circle are always parallel to each other. -
Q6. Assertion (A): If a parallelogram circumscribes a circle, it is a rhombus.
Reason (R): The opposite sides of a parallelogram are equal, and the sum of opposite sides of any circumscribed quadrilateral are equal ($AB + CD = AD + BC$). -
Q7. Assertion (A): The angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the center.
Reason (R): The opposite angles of the quadrilateral formed by the two radii and two tangents at the points of contact sum up to $180^\circ$ because the angles between the radius and tangent are $90^\circ$ each. -
Q8. Assertion (A): The perpendicular drawn from the center of a circle to a chord bisects the chord.
Reason (R): The radius to the midpoint of a chord forms congruent right-angled triangles with the endpoints of the chord via RHS criterion. -
Q9. Assertion (A): Equal chords of a circle are equidistant from the center.
Reason (R): Equal chords subtend equal angles at the center of the circle, making their perpendicular drop lengths equal. -
Q10. Assertion (A): The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
Reason (R): The exterior angle of a triangle is equal to the sum of the two interior opposite angles, which is applied to the isosceles triangles formed by the radii. -
Q11. Assertion (A): The angle in a semicircle is a right angle ($90^\circ$).
Reason (R): The angle subtended by a diameter at any point on the circle is half of the straight angle ($180^\circ$) subtended at the center. -
Q12. Assertion (A): Angles in the same segment of a circle are equal.
Reason (R): Every angle in the same segment subtends the same arc at the circumference, and angles subtended by the same arc at the circumference are equal. -
Q13. Assertion (A): The sum of either pair of opposite angles of a cyclic quadrilateral is $180^\circ$.
Reason (R): The opposite angles of a cyclic quadrilateral are complementary because their respective subtended arcs combine to form the full circle ($360^\circ$). *(Note: Opposite angles are supplementary, summing to $180^\circ$, not complementary, making Reason false).* -
Q14. Assertion (A): If a side of a cyclic quadrilateral is produced, the exterior angle so formed is equal to the interior opposite angle.
Reason (R): The exterior angle and the adjacent interior angle form a linear pair summing to $180^\circ$, which equals the supplementary sum of opposite cyclic angles. -
Q15. Assertion (A): If a pair of opposite angles of a quadrilateral is supplementary, then the quadrilateral is cyclic.
Reason (R): A circle can always be uniquely circumscribed through any four random non-collinear points in a plane. *(Note: Only three non-collinear points uniquely determine a circle; four random points are not always concyclic, making Reason false).* -
Q16. Assertion (A): The angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment.
Reason (R): This theorem is a direct consequence of the properties of cyclic quadrilaterals and the right-angled relationship between the radius and the tangent. -
Q17. Assertion (A): If two chords $AB$ and $CD$ intersect inside a circle at point $P$, then $AP \cdot PB = CP \cdot PD$.
Reason (R): The triangles formed by joining the chord endpoints are similar ($\triangle APC \sim \triangle DPB$) due to vertically opposite angles and angles in the same segment. -
Q18. Assertion (A): If a secant and a tangent are drawn from an external point to a circle, the product of the external segment and the full secant equals the square of the tangent length.
Reason (R): The relation is derived from the similarity of the triangle formed by the tangent and the secant sharing a common angle and using angle-angle (AA) similarity. -
Q19. Assertion (A): A chord of a larger concentric circle that touches a smaller concentric circle is bisected at the point of contact.
Reason (R): The radius of the inner circle acts as a perpendicular dropped from the center to the chord of the outer circle, which bisects the chord. -
Q20. Assertion (A): The distance between two parallel tangents of a circle of radius $r$ is equal to $2r$.
Reason (R): The line segment joining the points of contact of two parallel tangents passes through the center and forms a diameter of the circle.
Academic Repository Disclaimer & எச்சரிக்கை அறிவிப்பு :
English: These model solutions are structured to global academic standards for self-assessment and practice purposes. Students are advised to verify with official textbooks for board examinations.
தமிழ்: இந்த மாதிரி கணிதத் தீர்வுகள் மாணவர்களின் சுய கற்றல் மற்றும் பயிற்சித் திறனுக்காக உலகளாவிய கல்வித் தரத்தில் உருவாக்கப்பட்டுள்ளன. தேர்வுகளுக்குத் தயாராகும் போது உங்கள் அதிகாரப்பூர்வ பாடப்புத்தகத்துடன் சரிபார்க்கவும்.
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