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# 1.    Prove that for each posive integer 'm' the smallest integer which exceeds (Ö3 + 1)2m is divisible by 2m+1.:9.  4x2/{1-Ö(1+2x2)}2 < 2x+9Ans: 10. Show that, if the real numbers a, b, c, A, B, C satisfy: aC-2bB+cA=0 and ac-b2>0 then AC-B2£0.11. Solve for x, y, z: yz = a(y+z) + r zx = a(z+x) + s xy = a(x+y) + t12. Prove that: 2/(x2 - 1) + 4/(x2 - 4) + 6/(x2 - 9) + ... + 20/(x2 - 100) =   11/((x - 1)(x + 10)) + 11/((x - 2)(x + 9)) + ... + 11/((x - 10)(x + 1))32    Evaluate: 0ò11/{ (5+2x-2x2)(1+e(2-4x)) } dx 33    Let P(x)= Õ (x-ai), where i=1 to n. and all ai’s are real. Prove that the derivatives P ‘(x) and P "(x) satisfy the inequality P’(x)2 ³ P(x)P"(x) for all real numbers x. 34    .  Determine the value of 0ò1 xa-1.(ln x)n dx where a Î {2, 3, ...} and n Î N. 32    Let T be an acute angled triangle. Inscribe rectangles R & S as shown. Let A(X) denote the area of any polygon X. Find the maximum value of [A(R)+A(S)]/A(T).1.      Prve that log418 is an irrational number. 