CBSE Class 9 Maths Ganita Manjari Part 1 Ch 05 I'm Up and Down and Round and Round MCQs with Answers Set 01

Answer: (b) 75°

Explanation:
1. Equal chords are equidistant from the centre, so OP = OQ and triangle OPQ is isosceles.
2. \( \angle OPQ = \angle OQP = \frac{180^\circ - 150^\circ}{2} = 15^\circ \).
3. OP ⊥ AB, so \( \angle APO = 90^\circ \).
4. \( \angle APQ = \angle APO - \angle OPQ = 90^\circ - 15^\circ = 75^\circ \).

Answer: (d) 8 cm

Explanation:
1. Radius = 34 ÷ 2 = 17 cm. The perpendicular from the centre bisects AB, so half of AB = 15 cm.
2. Distance = \( \sqrt{17^2 - 15^2} = \sqrt{289 - 225} = \sqrt{64} = 8 \) cm.

Answer: (a) 2 cm

Explanation:
1. The perpendicular from the centre bisects the chord, so AC = 4 cm.
2. In right triangle OCA: \( OC = \sqrt{5^2 - 4^2} = 3 \) cm.
3. OD is a radius, so OD = 5 cm and CD = OD − OC = 5 − 3 = 2 cm.

Answer: (c) 10 cm

Explanation:
1. ∠ABC = 90°, so AC is a diameter of the circle through A, B and C (angle in a semicircle).
2. \( AC = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = 20 \) cm.
3. Radius = 20 ÷ 2 = 10 cm.

Answer: (b) radii

Explanation:
1. The size of a circle is fixed entirely by its radius.
2. So two circles are congruent exactly when their radii are equal.

Answer: (c) rectangle

Explanation:
1. Both diagonals are diameters, so they are equal in length.
2. Both pass through the centre, so they bisect each other.
3. A quadrilateral whose diagonals are equal and bisect each other is a rectangle. (It is a square only if the diagonals are also perpendicular.)

Answer: (c) in the exterior of the circle

Explanation:
1. Points at distance less than r from O are inside, equal to r are on the circle, and more than r are outside.
2. Since OP > r, P is in the exterior of the circle.

Answer: (b) 1

Explanation:
1. Three points on a circle can never be in one straight line, so they are non-collinear.
2. Through three non-collinear points, exactly one circle can be drawn.

Answer: (a) First

Explanation:
1. In a circle, a chord closer to the centre is longer.
2. Wire 1 is closer (3 cm < 4 cm), so Wire 1 is longer.

Answer: (a) 2 times

Explanation:
1. Circumference = \( 2\pi r \). With radius 2r, it becomes \( 2\pi(2r) = 2 \times 2\pi r \).
2. So the circumference becomes 2 times. (The area would become 4 times.)

Answer: (a) is the perpendicular bisector of the common chord

Explanation:
1. Each centre is equidistant from the two ends of the common chord.
2. So both centres lie on the perpendicular bisector of the common chord, and the line joining them is that bisector.

Answer: (b) 70°

Explanation:
1. Equal chords of a circle subtend equal angles at the centre.
2. So ∠COD = ∠AOB = 70°.

Answer: (c) 2r = a√2

Explanation:
1. The diagonal of the square is a diameter of the circle, so diagonal = 2r.
2. Diagonal of a square of side a = \( a\sqrt{2} \).
3. So \( 2r = a\sqrt{2} \).

Answer: (a) 48°

Explanation:
1. AB is a diameter, so ∠ACB = 90° (angle in a semicircle).
2. In triangle ABC: ∠BAC = 180° − 90° − 42° = 48°.
3. ∠BDC and ∠BAC stand on the same arc BC, so they are equal. ∠BDC = 48°.

Answer: (c) AB = CD = EF

Explanation:
1. OL, OM and ON are the distances of the chords from the centre.
2. Chords that are equidistant from the centre are equal, so AB = CD = EF.

Answer: (c) A is true, but R is false.

Explanation:
1. Every diameter equals 2r, so all diameters have the same length. The Assertion is true.
2. The diameter is twice the radius, not half of it. The Reason is false.

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. A quadrilateral is cyclic when each pair of opposite angles adds up to 180°.
2. In a square, each angle is 90°, so opposite angles add up to 90° + 90° = 180°. The Assertion is true.
3. The Reason states this property, which is true for a square and is exactly why it is cyclic.