Consider a circle with centre at C. Let OP, OQ denote respectively the tangents to the circle drawn from a point O outside the circle. Let R be a point on OP and S be a point on OQ such that OR×SQ = OS×RP. Which of the following statements is/are correct

1. If X is the circle with centre at O and radius OR, and Y is the circle with centre at O and radius OS, then X = Y.
2. ∠POC+∠QCO = 90⁰

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Let ABC be a triangle in which AB = AC. Let L be the locus of points X inside or on the triangle such that BX = CX. Which of the following statements are correct?

1. L is a straight line passing through A and in-centre of triangle ABC is on L.
2. L is a straight line passing through A and orthocentre of triangle ABC is a point on L.
3. L is a straight line passing through A and centroid of triangle ABC is a point on L.

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Consider a circle with centre at O and radius 7 cm. Let QR be a chord of length 2 cm and let P be the midpoint of QR. Let CD be another chord of this circle passing through P such that ∠CPQ is acute. If M is the midpoint of CD and MP = √24 cm, then which of the following statements are correct

1. ∠QPD = 135⁰
2. If CP = m cm and PD = n cm, then m and n are the roots of the quadratic equation x²-10x+1 = 0.
3. The ratio of the area of triangle OPR to the area of triangle OMP is 1:2√2.

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There are five lines in a plane, no two of which are parallel. The maximum number of points in which they can intersect is

Which of the following is correct in respect of the equation 3-tan²θ = ∝ (1-3 tan²θ)? (Given that ∝ is a real number.)

In a triangle ABC if A-B = ∏/2, then C+2B is equal to