A hexagon is inscribed in a circle of radius r. Two of its sides have length 1, two have length 2 and the last two have length 3. Prove that r is a root of the equation `2r^(3) -7r-3=0.`

Two sides of a triangle have lengths of 5 and 19. Can the third side have a length of 13?

If the sum of two roots of the equation x^3-px^2 + qx-r =0 is zero, then:

Two wires A and B are of lengths 40 cm and 30 cm. A is bent into a circle of radius r and B into an arc of radius r. A current i_(1) is passed through A and i_(2) through B. To have the same magnetic induction at the centre, the ratio of i_(1) : i_(2) is

(c) 103 (d) 10 18. A circle is inscribed in an equilateral triangle with side lengths 6 unit. Another circle is drawn inside the triangle (but outside the first circle), tangent to the first circle and two of the sides of the triangle. The radius of the smaller circle is (b) 2/3 (a) 1/ root3 (c) 2 (d) 1

Two equal circles of radius r intersect such that each passes through the centre of the other. The length of the common chord of the circles is sqrt(r) (b) sqrt(2)\ r\ A B (c) sqrt(3)\ r\ (d) (sqrt(3))/2\ r

The circles having radii r_1 and r_2 intersect orthogonally. Find the length of their common chord.

The length of the common chord of two circles of radii r_1 and r_2 which intersect at right angles is

A force F is needed to break a copper wire having radius R. The force needed to break a copper wire of same length and radius 2R will be

Prove that r_1+r_2+r_3-r=4R