10 Question MCQ (including Assertion)

Question 1 of 10

The centres of all circles passing through two distinct points \(A\) and \(B\) lie on the:

A Line \(AB\) B Perpendicular bisector of \(AB\) C Circle with diameter \(AB\) D Angle bisector of \(\angle AOB\)

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Question 2 of 10

The theorem states: "If two opposite angles of a quadrilateral add up to \(180^\circ\), then the vertices of the quadrilateral lie on a circle." In an arbitrary quadrilateral \(PQRS\), \(\angle P = 75^\circ\), \(\angle Q = 105^\circ\), and \(\angle R = 105^\circ\). What is the measure of \(\angle S\), and is \(PQRS\) a cyclic quadrilateral?

A \(75^\circ\); Yes, it is cyclic. B \(75^\circ\); No, it is not cyclic. C \(105^\circ\); Yes, it is cyclic. D \(105^\circ\); No, it is not cyclic.

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Question 3 of 10

A rectangle is inscribed in a circular frame. Where is the centre of the circle?

A At any vertex of the rectangle B At the midpoint of any side C At the intersection of the rectangle's diagonals D Outside the rectangle

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Question 4 of 10

A circle is to be constructed with a chord of length \(6\text{ cm}\) at a distance of \(3\text{ cm}\) from the centre. What must the radius be?

A \(3\text{ cm}\) B \(3\sqrt{2}\text{ cm}\) C \(6\text{ cm}\) D \(6\sqrt{2}\text{ cm}\)

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Question 5 of 10

A point \(P\) lies inside a circle with centre \(O\). Which chord through \(P\) is the shortest?

A The chord parallel to \(OP\) B The chord perpendicular to \(OP\) C Any chord through \(P\) D The diameter through \(P\)

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Question 6 of 10

In cyclic quadrilateral \(ABCD\), \(\angle A=68^\circ\). What is \(\angle C\)?

A \(68^\circ\) B \(90^\circ\) C \(112^\circ\) D \(292^\circ\)

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Question 7 of 10

An architect wants to inscribe a parallelogram in a circle. What type of parallelogram must it be?

A Rhombus B Rectangle C Kite D Any parallelogram

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Question 8 of 10

A perfectly circular wheel without spokes or markings is rotated through \(137^\circ\). Why does it appear unchanged?

A A circle has complete rotational symmetry B A circle only has \(180^\circ\) rotational symmetry C Every chord becomes a diameter D Its radius changes during rotation

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Question 9 of 10

At vertex \(D\) of cyclic quadrilateral \(ABCD\), side \(CD\) is extended. The exterior angle at \(D\) is \(118^\circ\). What is \(\angle ABC\)?

A \(62^\circ\) B \(90^\circ\) C \(118^\circ\) D \(242^\circ\)

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Question 10 of 10

Applying the principle: "The line joining the centre of a circle and the midpoint of a chord of the circle is perpendicular to the chord." A steel plate has a circular arc cut. A chord measuring 48 cm is drawn, and a line from the centre to its midpoint measures 7 cm. What is the area of the isosceles triangle formed by the two radii meeting this chord?

A 84 sq cm B 168 sq cm C 336 sq cm D 600 sq cm

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10 Question MCQ (including Assertion)