Most Important Selected Qs for JEE AdvancedMathematicsDefinite Integration
Paragraph: A quadratic polynomial f(x) with positive leading coefficient such that g(x)=f( x) x>0 . Also the curve y=g(x) satisfy the following condition. (a) There is exactly one value for a positive number p such that (p, g(p)) is its extremum point and (p^2, g (p^2 ) ) is its inflection point. (b) Exactly one tangent line can be drawn from the point (0,0) to the curve y=g(x) . Question: The value of f(10) f(2) equ
Options
- A25
- B36
- C41
- D64
Correct answer
C. 41
Step-by-step solution
aligned & f(x)=a x^2+b x+c & g(x)=a ^2 x+b x+c x>0 aligned Given, g^ (p)=0 2 a p+b p =0 p= -b 2 a Also g^ (p^2 )=0 2 a (1- p^2 )-b p^4 =0 1- b 2 a = p^2 1- b 2 a =2 p= -b a b=-2 a Also, aligned & g^ ( )= g( )-0 -0 & 2 a +b = a ^2 +b +c & a ^2 +(b-2 a) +c-b=0 aligned For unique , D=0 aligned & (b-2 a)^2-4 a(c-b)=0 & 4 b^2=4 a(c-b) & 4 a=c+2 a & c=2 a & f(x)=a (x^2-2 x+2 ) aligned Now, proceed.