1. If f(x) is a monotonically increasing function " x Î R, f "(x) > 0 and f -1 (x) exists, then prove that å {f -1 (x i )/3} -1 ({x 1 +x 2 +x 3 }/3), i=1,2,3 2. Discuss the applicability of Rolle’s theorem to f(x)=log[x 2 +ab/{(a+b)x}] in the interval [a,b]. 3. y = f(x) be a curve passing through (1,1) such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area 2. Form the differential equation and determine all such possible curves. 4. Let f be a real-valued function defined for all real nos. x such that, for some positive constant a, the equation f(x+a) = 1/2 + Ö (f(x)-(f(x)) 2 ) holds for all x. (a) Prove that the function f is periodic. (b) For a=1, give an example of a non-constant function with the required properties. Sum the series: 1 + 4x + 9x 2 + ... If w is a root of x 4 =1 then Show that a + b w + c w 2 + d w 3 is a factor of a b c d b c d a c d a b d a b c Hence Show that the det is equal to -(a+b+c+d)(a-b+c-d){(a-c) 2 +(b-d) 2 }. Prove that if a > 0, b > 0 then for any x and y the following inequality holds true: a.2 x +b.3 y +1 £ Ö (4 x +9 y +1) Ö (a 2 +b 2 +1) Given 6 numbers which satisfy the relations: y 2 + yz + z 2 = a 2 z 2 + zx + x 2 = b 2 x 2 + xy + y 2 = c 2 Determine the sum x+y+z in terms of a, b, c. Give geometrical interpretation if the numbers are all positive. When 0 3 ) + x 2 /(1-x 3 )(1-x 5 ) + x 4 /(1-x 5 )(1-x 7 ) + .... 10. Find all real p, q, a, b such that we have (2x-1) 20 - (ax+b) 20 = (x 2 +px+q) 10 for allx. 11. f(x+y) = f(x) + f(y) + 2xy - 1 " x,y. f is differentiable and f ‘(0) = cos a . Prove that f(x) > 0 " x Î R.

1.      If f(x) is a monotonically increasing function " x Î R, f "(x) > 0 and f ^-1(x) exists, then prove that å{f ^-1(x_i)/3} < f ^-1({x₁+x₂+x₃}/3), i=1,2,3

 

2.      Discuss the applicability of Rolle’s theorem to f(x)=log[x²+ab/{(a+b)x}] in the interval [a,b].

 

3.       y = f(x) be a curve passing through (1,1) such that the triangle formed by the coordinate axes and the tangent at any point of the curve lies in the first quadrant and has area 2. Form the differential equation and determine all such possible curves.

 

4.        Let f be a real-valued function defined for all real nos. x such that, for some positive constant a, the equation f(x+a) = 1/2 + Ö(f(x)-(f(x))²) holds for all x. (a) Prove that the function f is periodic. (b) For a=1, give an example of a non-constant function with the required properties.

1. Sum the series: 1 + 4x + 9x² + ...


2. If w is a root of x⁴=1 then Show that a + bw + cw² + dw³ is a factor of

a	b	c	d
b	c	d	a
c	d	a	b
d	a	b	c

     

                                                                                                                                                       

 

 

Hence Show that the det is equal to -(a+b+c+d)(a-b+c-d){(a-c)²+(b-d)²}.

1. Prove that if a > 0, b > 0 then for any x and y the following inequality holds true: a.2^x+b.3^y+1 £ Ö(4^x+9^y+1)Ö(a²+b²+1)


2. Given 6 numbers which satisfy the relations:
   y² + yz + z² = a²
   z² + zx + x² = b²
   x² + xy + y² = c²
   Determine the sum x+y+z in terms of a, b, c. Give geometrical interpretation if the numbers are all positive.


3. When 03) + x²/(1-x³)(1-x⁵) + x⁴/(1-x⁵)(1-x⁷) + ....

10.   Find all real p, q, a, b such that we have (2x-1)²⁰ - (ax+b)²⁰ = (x²+px+q)¹⁰ for allx.

 

11.  f(x+y) = f(x) + f(y) + 2xy - 1 " x,y. f is differentiable and f ‘(0) = cos a. Prove that f(x) > 0 " x Î R.