Mathematics Circles Exam Prep
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🎯 Exam Preparation Kit
Mathematics Circles Exam Prep
English 11 leveled MCQs Free
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Sample questions with model answers
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5 marks Long answer
If a circle touches the side BC of a triangle ABC at P and extended sides AB and AC at Q and R, respectively, prove that $AQ = \frac{1}{2}$ (perimeter of the $\triangle ABC$).
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Use tangent segments and triangle properties.
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5 marks Long answer
A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively. Find the sides AB and AC.
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Apply the properties of tangents and segments.
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3 marks Medium answer
Prove that the angle between the 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 centre.
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Use properties of tangents and angles in circles.
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3 marks Medium answer
XY and X'Y' are two parallel tangents to a circle with centre O and another tangent AB with point of contact C intersecting XY at A and X'Y' at B. Prove that $\angle AOB = 90^\circ$.
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Apply properties of tangents and angles.
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3 marks Medium answer
Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
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Use properties of cyclic quadrilaterals.
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3 marks Medium answer
If a, b, c are the sides of a right triangle where c is the hypotenuse, prove that the radius r of the circle which touches the sides of the triangle is given by $r = \frac{(a+b-c)}{2}$.
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Derive using triangle properties.
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1 marks Short answer
If PQ and PR are tangents to the circle with centre O and radius 4 cm such that $\angle QPR = 90^\circ$, then the length OP is
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8 cm
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1 marks Short answer
If tangents PA and PB from a point P to a circle with centre O are inclined to each other at an angle of $80^\circ$, then $\angle POA$ is equal to:
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$60^\circ$
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1 marks Short answer
Two concentric circles are of radii 5 cm and 3 cm. The length of the chord of the larger circle which touches the smaller circle is:
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6 cm
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1 marks Short answer
If $d_1$ and $d_2$ (where $d_2 > d_1$) are the diameters of two concentric circles and c is the length of a chord of the larger circle which is completely tangent to the smaller circle, then which relation is true?
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$d_2^2 = c^2 + d_1^2$
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1 marks Short answer
Assertion (A): In the below figure, AB and CD are common tangents to circles which touch each other at D. If AB = 8 cm, then the length of CD is 4 cm. Reason (R): A tangent to a circle is perpendicular to the radius through the point of contact.
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Assertion is false; reason is true.
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