9)

let g(x) = x^3 + ax^2 + bx + c

==> f(x) = g(|x|)

==> the graph of f(x) = graph of g(x) when x>=0

& the graph of f(x)= reflection of f(x) in y axis when x<=0. (since f(x) is even)

we know the graph of h(x)=x^3 + 5 (an example). now we can twist g(x) in a way that it becomes h(x) as then it will never touch/cut x axis (when x>=0) ie (for understanding) put a=b=0 and c=5.

hence we can construct such g(x) which will never cut x axis when x>=0. for x<=0, it i.e. f(x) is nothing but reflection of g(x) when x>=0. hence minimum no roots possible.

**(a)**11)

f(x)={x-[x]}

which is nothing but a fractional func.

can be rewritten as follows:

f(x) = x+1 ; -1 <=x<0

= x ; 0 <=x<1

=x-1 ; 1 <=x<2

and so on....

on seeing it closely (and/or seeing the plot), it is clear that its range is from [0,1) and it is periodic with periodicity 1. (just plot it for clarity)

So I= Integral (2 to 343) of [{x-[x]}^2] = 341* integral (from 0 to 1) of [{x-[x]}^2] = 341* integral (from 0 to 1) of [x^2] = 341/3

** (c)**15)

if we change the order of integration without changing the limits of integration, it becomes fairly simple and ans comes out to be

**D**. however I wasn't able to integrate after changing the order of integration in a proper way. I might be making some mistake. ans comes out to be

**D ** but I am not satisfied with my solution.

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"You don't have to believe in God, but you should believe in The Book." -Paul Erdős