Question

A curve has equation $$y=x^{2}-x+3$$  and a line has equation y = 3x + a, where a is a constant.
(i) Show that the x-coordinates of the points of intersection of the line and the curve are given by the equation $$x^{2}-4x+(3+a)=0$$.
(ii) For the case where the line intersects the curve at two points, it is given that the x-coordinate of one of the points of intersection is −1. Find the x-coordinate of the other point of intersection.
(iii) For the case where the line is a tangent to the curve at a point P, find the value of a and the coordinates of P.

(i)$$x^{2}-x+3=3x+a\rightarrow x^{2}-4x+(3-a)=0$$

(ii)$$5+(3-a)=0\rightarrow a=8$$

$$x^{2}-4x-5=0\rightarrow x=5$$

(iii) $$16-4\left ( 3-a \right )=0$$  (applying $$b^{2}-4ac=0$$)

a=-1

$$(x-2)^{2}=0\rightarrow x=2,y=5$$

#### Question.

(a) Find the values of the constant m for which the line y = mx is a tangent to the curve $$y = 2x^2 – 4x + 8$$.
(b) The function f is defined for x \in R by $$f(x) = x^2 + ax + b$$, where a and b are constants. The
solutions of the equation f(x) = 0 are x = 1 and x = 9.
Find
(i) the values of a and b,
(ii) the coordinates of the vertex of the curve y = f(x).

(i)$$y=2x^{2}-4x+8$$

Equates with y=mx and selects a,b,c

Uses $$b^{2}=4ac$$

$$\rightarrow a=-10,b=9$$

(iii) Calculus or $$x=\frac{1}{2}\left ( 1+9 \right )$$ by symmetry

$$\rightarrow \left ( 5,-16 \right )$$

#### Question.

Find the set of values of k for which the equation $$2x^2 + 3kx + k = 0$$ has distinct real roots.

$$\left ( 3k \right )^{2}-4\times 2\times k$$

$$9k^{2}-8k> 0$$ soi Allow   $$9k^{2}-8k\geq 0$$

0,8/9 soi

k<0,k>8/9 (or 0.889)

Question

The function f is defined by $$f(x)=-2x^{2}+12x-3 for \varepsilon R$$

(i) Express $$-2x^{2}+12x-3 in the from 2(x+a)^{a}+b,$$where a and b are constants.

(ii) State the greatest value of fx.

The function g is defined by g(x) = 2x + 5 for x ∈ >.

(iii) Find the values of x for which gf(x )+ 1

(i) $$-2(x-3) ^{2}$$+15(a=-3,b=15)
(iii) gf(x) =$$(-2x^{2}+12x-3)+5=-4x^{2}+24x-6+5 gf(x)+1=0\rightarrow -4x^{2}+24x$$=0 x=0 or 6