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.
Answer/Explanation
(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).
Answer/Explanation
(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.
Answer/Explanation
\(\left ( 3k \right )^{2}-4\times 2\times k \)
\(9k^{2}-8k> 0\) 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
Answer/Explanation
= 0.
(i) \(-2(x-3) ^{2}\)+15(a=-3,b=15)
(ii) (f(x) ⩽) 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