AOD

b

d

P(x) is sort of like a derivative

so let f(x)= integral p(x)

Then get roots of f(x) and use rolle’s theorem to get 2 roots of p(x)

put f’’(x)=0

Then find value of x in terms of a and put x<0

ans: a

c

c

Solve each integral and then diff wrt t

c

on solving via derivative, we also need to check if we calculated the minima or maxima

in this case, we calculated the minima

a

direct AM GM

a

c

on taking derivative, we get x^2 (…) > 1-cos3x

1-cos3x ranges from [0,1]

So x^2 > 1/…

So a^2+2/a^2-4 < 0

a^2 -4 <0

c

derivative must be >1

So minimum value of derivative >1

So f’’(x) = 0 to find relation between t and (1-t)

Put that value in f’(x) to get t> cos^2/3

so t> 1/3

a

justg keep making relations and eventually you’ll get ans

b

convert fractional to x1 - box x

sigma x1 = 0 and you cna find box x

use AM GM on mx + 1/x and make it equal to 1

Ans B

c

10 d

11 c

12 b

d

a

b

a

Write the expressions in det as a,b,c

on solving, youll get det = 3abc - a^3-b^3-c^3

open the identity to get f(x) and g(x)