Practice section 6 of 6

$C:y=f(x)$ where $f(x)=(4x^2+bx+c)/(4x+8)$ and b,c are real constants. C has oblique asymptote y=x−5.

(i) Find b. [2]

(ii) Given c>9, show C does not cut the x-axis. [1]

(iii) For c=13 sketch C, clearly indicating asymptotes, axial intercepts and turning points. [3]

$C:y=(-2x^2-2x+a)/(x+1)$, where a is a non-zero constant. C exists for y≤−2 or y≥b, where b is positive.

(i) Using an analytical method, show a=−2 and hence find b. [5]

(ii) Sketch C, clearly showing turning-point coordinates and asymptotes. [3]

(iii) Show(−3,6) lies on y=px+3 p+6 for every real p. Hence use the sketch in(ii) to find p for which $px+3p+6=(-2x^2-2x-2)/(x+1)$ has two distinct real roots. [2]