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1. Prove that for each posive integer 'm' the smallest integer which exceeds ( Ö 3 + 1) 2m is divisible by 2 m+1 . :9. 4x 2 /{1- Ö (1+2x 2 )} 2 Ans: 10. Show that, if the real numbers a, b, c, A, B, C satisfy: aC-2bB+cA=0 and ac-b 2 >0 then AC-B 2 £ 0. 11. Solve for x, y, z: yz = a(y+z) + r zx = a(z+x) + s xy = a(x+y) + t 12. Prove that: 2/(x 2 - 1) + 4/(x 2 - 4) + 6/(x 2 - 9) + ... + 20/(x 2 - 100) = 11/((x - 1)(x + 10)) + 11/((x - 2)(x + 9)) + ... + 11/((x - 10)(x + 1)) 32 Evaluate: 0 ò 1 1/{ (5+2x-2x 2 )(1+e (2-4x) ) } dx 33 Let P(x)= Õ (x-a i ), where i=1 to n. and all a i ’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 0 ò 1 x a-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 log 4 18 is an irrational number.


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+9


Ans:


 


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) + t


 


12. 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.




Grade:Upto college level

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