Let ABC be an acute-angled triangle; AD be the bisector of angle BAC with D on BC; and BE be the altitude from B on AC. Show that . [weightage 17/100]

Let a, b, c be three natural numbers such that a < b < c and gcd (c – a, c – b) = 1. Suppose there exists an integer d such that a + d, b + d, c + d form the sides of a right-angled triangle. Prove that there exist integers, l,m such that . [Weightage 17/100]

Find all pairs (a, b) of real numbers such that whenever is a root of is also a root of the equation.[Weightage 17/100]

How many 6-digit numbers are there such that-:

The digits of each number are all from the set {1,2,3,4,5}

b)any digit that appears in the number appears at least twice ? (Example: 225252 is valid while 222133 is not) [weightage 17/100]

A trapezium ABCD, in which AB is parallel to CD, is inscribed in a circle with centre O. Suppose the diagonals AC and BD of the trapezium intersect at M, and OM = 2.

If is , find, with proof, the difference between the lengths of the parallel sides.

If is , find, with proof, the difference between the lengths of the parallel sides. [Weightage 17/100]

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