#### If -5 is a root of the quadratic equation 2x^{2} + Px – 15 = 0 and the quadratic equation P(x^{2} + x) + K = 0 has equal roots, find the value of K.

#### Let P and Q be the points of trisection of the line segment joining the points A(2, –2) and B(–7, 4) such that P is nearer to A. Find the coordinates of P and Q.

#### A quadrilateral ABCD is drawn to circumscribe a circle, with centre O, in such a way that the sides AB, BC, CD and DA touch the circle at the points P, Q, R and S respectively. Prove that AB +CD = BC + DA.

#### Prove that the points (3, 0), (6, 4) and (–1, 3) are the vertices of a right angled isosceles triangle.

#### The 4th term of an A.P. is zero. Prove that the 25th term of the A.P. is three times its 11th term.

#### From an external point P, two tangents PT and PS are drawn to a circle with centre O and radius r. If OP = 2r, show that ∠OTS = ∠OST = 30°.