Practice section 5 of 6

$C:y=(x^2+3x+1)/(2x-1)$.

(i) Sketch C with axial intercepts, asymptotes and stationary points. [5]

(ii) Add a suitable curve to deduce how many real roots $2x^4+6x^3+2x^2+2x-1=0$ has. [3]

$C:y=(-9x^2+6x-1)/(x-1)$.

(i) Without a calculator, find the range of y for which no points lie on C. [3]

(ii) Sketch C with turning points, axial intercepts and asymptotes. [3]

Sketch $y=(2x^2+1)/(x-1)$ with all turning points, axial intercepts and asymptotes. Hence state the values of a for which $2x^2+1=a(x-1)\cos(bx+c)$, x≠1, has no real solution for all b,c. [6]

$C:y=(x^2+2x+3)/(x+1)$.

(i) Find exactly the values of y for which C does not lie. [3]

(ii) Sketch C with axial intercept, asymptotes and stationary points. [3]

(iii) Add a sketch on−π≤x≤πto solve $3(x+1)\cos x-1=x^2+2x+2$. [3]

· Techniques of Differentiation

$G:y=(2x^2+kx-1)/(x+1)$, where k is constant.

(i) Given no stationary point, show that k>1. [2]

(ii) Given y=2 x+1 is an asymptote, show k=3 and sketch G. [4]

· Techniques of Differentiation

$C:y=(x^2+a)/(x-a)$, x≠a, where a is a non-zero constant.

(i) Given the oblique asymptote y=x+2, find a. [1]

(ii) Find the set of values of a if C has two turning points. [3]

(iii) Using a from (i) and an algebraic method, show C cannot exist for y₁<y<y₂, determining y₁,y₂ exactly. [3]

(iv) Hence sketch C, clearly showing the asymptotes and stationary-point coordinates. [3]