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1. 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 $\angle CED > 45^\circ$ . [weightage 17/100]
2. 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 $c + d = l^{2} + m^{2}$. [Weightage 17/100]
3. Find all pairs (a, b) of real numbers such that whenever $\alpha$ is a root of $x^{2} + ax + b = 0, \alpha^{2} - 2$ is also a root of the equation.[Weightage 17/100]
4. How many 6-digit numbers are there such that-:
1. The digits of each number are all from the set {1,2,3,4,5}
2. b)any digit that appears in the number appears at least twice ?
(Example: 225252 is valid while 222133 is not) [weightage 17/100]
5. 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.
1. If $\angle AMB$ is $60^\circ$ , find, with proof, the difference between the lengths of the parallel sides.
2. If $\angle AMD$ is $60^\circ$ , find, with proof, the difference between the lengths of the parallel sides.
[Weightage 17/100]
6. Prove that:
1. $5<\sqrt {5}+\sqrt [3]{5}+\sqrt [4]{5}$
2. $8>\sqrt {8}+\sqrt [3]{8}+\sqrt [4]{8}$
3. $n>\sqrt {n}+\sqrt [3]{n}+\sqrt [4]{n}$ for all integers $\ngeq 9$.[Weightage 16/100]