IOQM mock test 3
Q1. Define n a ! n_a! for n n and a a positive to be n a ! = n ( n − a ) ( n − 2 a ) ( n − 3 a ) … ( n − k a ) n_a! = n(n - a)(n - 2a)(n - 3a) \ldots (n - ka) where k k is the greatest integer for which n > k a n > ka . If the quotient 72 8 ! / 18 2 ! 72_8! / 18_2! is equal to 2 x 2^x , find x x . Solution: Given number equals: 72.(72-8).(72-16)...(72-64)/18.(18-2).(18-4)...(18-16) = 4^9 = 2^18 Q2. If p , q p, q are two real numbers satisfying the relationships 2 p 2 − 3 p − 1 = 0 2p^2 - 3p - 1 = 0 and q 2 + 3 q − 2 = 0 q^2 + 3q - 2 = 0 and p q ≠ 1 pq \ne 1 , find the value of p q + p + 1 q \frac{pq + p + 1}{q} . Solution: Clearly p, 1/q are roots of 2x^2 - 3x - 1 = 0. p + 1/q = 3/2 p/q = -1/2 (pq + p + 1)/q = p + p/q + 1/q = 1 = Answer Q3. Convex quadrilateral ABCD has A B = 9 AB = 9 and C D = 12 CD = 12 . Diagonals AC and BD intersect at E, A C = 14 AC = 14 , and △ A E D \triangle AED and △ B E C \triangle BEC have equal areas. What is AE? Solutio...