CBSE Class 10 Mathematics Chapter 10 Circles — 20 MCQs with Answers

The cornerstone theorem of this chapter states that the angle a chord subtends at the centre of a circle is exactly twice the angle it subtends at any point on the major arc. This theorem appears in nearly every CBSE paper, often disguised in diagrams where you must identify the central and inscribed angles first. Mastering this concept unlocks most circle geometry problems. The four MCQs below test your ability to apply the 2:1 ratio, recognize when a diameter is involved (leading to 180° at centre), and solve reverse problems where the circumference angle is unknown. Remember: always check which arc the point lies on—major or minor—because angles on opposite arcs are supplementary.

When a chord divides a circle into two arcs (or segments), every angle subtended by that chord from any point on the same arc is equal. This is a direct consequence of the angle-at-centre theorem: since all points on the same arc subtend the same central angle, they all yield the same inscribed angle (half that central angle). CBSE loves to test this by drawing multiple points on one arc and asking you to identify equal angles or to prove congruence. The four MCQs below require you to spot equal angles, apply the property in cyclic figures, and distinguish between angles on the major versus minor arc. Always mark the arc first—this visual step prevents mix-ups.

One of the most elegant results in circle geometry: any angle subtended by a diameter from a point on the circumference is a right angle (90°). This follows from the angle-at-centre theorem—since a diameter subtends 180° at the centre, the inscribed angle is 180° ÷ 2 = 90°. CBSE frequently uses this in proof-based questions and diagram-based MCQs where you must identify the diameter first. The four MCQs below test recognition of diameters, application of the 90° property, and reverse reasoning (if an angle is 90°, the chord must be a diameter). This property is also a quick way to verify right-angled triangles inscribed in circles.

A cyclic quadrilateral has all four vertices on the circumference of a circle, and its defining property is that opposite angles are supplementary—they add to 180°. This property appears in nearly every CBSE paper, either as a direct MCQ or embedded in a diagram-based problem. The converse is equally testable: if opposite angles sum to 180°, the quadrilateral is cyclic. The four MCQs below test forward application (finding unknown angles), converse reasoning (proving a quadrilateral is cyclic), and multi-step problems involving exterior angles. Always check both pairs of opposite angles to confirm the cyclic property, and remember that the sum of all four angles in any quadrilateral is 360°—a useful cross-check.

CBSE Class 10 Mathematics papers now include higher-order thinking questions that combine multiple theorems or require you to reason across steps. Assertion-reason MCQs test both your knowledge of a statement and your ability to justify it. The four MCQs below are at HOTS level: they involve combining the angle-at-centre theorem with cyclic quadrilateral properties, proving quadrilaterals are cyclic, and solving for unknowns in complex diagrams. Approach these by sketching a diagram if one is not provided, labeling all known angles, and applying theorems one step at a time. These questions separate average scorers from high achievers in board exams—practice them under timed conditions.

Multiple-choice questions in the CBSE Class 10 Mathematics paper carry 1 mark each and are designed to test quick recall, formula application, and logical reasoning under strict time limits. The 2025 paper typically includes 20 MCQs across all chapters, and you have roughly 1 minute per question. Success depends on three habits: (i) read the question twice before looking at options to avoid distraction, (ii) eliminate obviously wrong answers first to improve your odds if you must guess, and (iii) always verify your answer by a quick reverse calculation or substitution—many MCQs are designed to trap students who rush. For assertion-reason questions, evaluate the assertion independently first, then check if the reason logically supports it. If you are stuck, mark the question, move on, and return with fresh eyes in the last 10 minutes. Never leave an MCQ blank—there is no negative marking, so educated guessing is better than omission. Practice 15-20 MCQs daily from each chapter in the final month before boards to build pattern recognition and reduce silly errors. Time yourself strictly: 20 MCQs in 20 minutes is the target pace.

Chapter 10 Circles has only a handful of core theorems, but CBSE exam questions combine them in dozens of creative ways—cyclic quadrilaterals nested inside triangles, chords that are also diameters, angles on opposite arcs. Textbook examples cover the basics, but board-level MCQs demand pattern recognition that comes only from solving 50+ varied problems. CBSETUTOR.ai gives your child exactly that: a 24×7 AI tutor trained on NCERT and past CBSE papers, accessible on any device. Your child can snap a photo of any circle geometry problem—whether from school worksheets, sample papers, or even a tricky board paper from 2023—and get a step-by-step solution in seconds, written in clear, exam-friendly language. The AI explains why the angle-at-centre theorem applies, how to spot a cyclic quadrilateral, and which property to use when. Beyond Circles, CBSETUTOR.ai covers every Class 10 chapter in Mathematics, Science, and Social Science at one flat price: ₹999 per month for Classes 6 to 12, no hidden fees, no per-question charges. Start with a 3-day free trial—no credit card required. Most Delhi NCR and Mumbai families see a jump in confidence within the first week because the child gets instant help exactly when they are stuck, not three days later when the tutor visits. Visit CBSETUTOR.ai today and turn Circles from a confusion into a scoring chapter.