1. (CTE 0606/2018/w/11/q4)
(i) Write $x^{2}-9 x+8$ in the form $(x-p)^{2}-q$, where $p$ and $q$ are constants.
(ii) Hence write down the coordinates of the minimum point on the curve $y=x^{2}-9 x+8$
(iii) On the axes below, sketch the graph of $y=\left|x^{2}-9 x+8\right|$, showing the coordinates of the points where the curve meets the coordinate axes.[3]
(iv) Write down the value of $k$ for which $\left|x^{2}-9 x+8\right|=k$ has exactly 3 solutions.$[1]$
2. $(\mathrm{CIE} 0606 / 2018 / \mathrm{w} / 23 / \mathrm{q} 11)$
A line with equation $y=-5 x+k+5$ is a tangent to a curve with equation $y=7-k x-x^{2}$.
(i) Find the two possible values of $k$.
(ii) Find, for each of your values of $k_{1}$
- the equation of the tangent
- the equation of the curve
- the coordinates of the point of contact of the tangent and the curve.$[5]$
(iii) Find the distance between the two points of contact.$[2]$
3. (CIE 0606/2018/w/23/q3)
(i) Write $8+7 x-x^{2}$ in the form $a-(x-b)^{2}$, where $a$ and $b$ are constants.$[3]$
(ii) Hence state the maximum value of $8+7 x-x^{2}$ and the value of $x$ at which it occurs.
(iii) Using your answer to part (i), or otherwise, solve the equation $8+7 z^{2}-z^{4}=0$.
4. $\left(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 11 / \mathrm{q}^{2}\right)$
Find the values of $k$ for which the line $y=k x-3$ and the curve $y=2 x^{2}+3 x+k$ do not intersect.
5. (CIE $0606 / 2019 / \mathrm{w} / 22 / \mathrm{q} 4)$
Find the values of $k$ for which the line $y=k x+3$ does not meet the curve $y=x^{2}+5 x+12$
6. $(\mathrm{CIE} 0606 / 2019 / \mathrm{w} / 23 / \mathrm{q} 4)$
(i) Given that $y=2 x^{2}-4 x-7$, write $y$ in the form $a(x-b)^{2}+c$, where $a, b$ and $c$ are constants.
(ii) Hence write down the minimum value of $y$ and the value of $x$ at which it occurs.
(iii) Using your answer to part (i), solve the equation $2 p-4 \sqrt{p}-7=0$, giving your answer correct to 2 decimal places.$[3]$
7. $(\mathrm{CIE} 0606 / 2020 / \mathrm{m} / 12 / \mathrm{q} 2)$
Find the values of $k$ for which the line $y=k x+3$ is a tangent to the curve $y=2 x^{2}+4 x+k-1$. [5]
8. (CIE $0606 / 2020 / \mathrm{s} / 13 / \mathrm{q} 4)$
(a) Write $2 x^{2}+3 x-4$ in the form $a(x+b)^{2}+c$, where $a, b$ and $c$ are constants.[3]
(b) Hence write down the coordinates of the stationary point on the curve $y=2 x^{2}+3 x-4$
(c) On the axes below, sketch the graph of $y=\left|2 x^{2}+3 x-4\right|$, showing the exact values of the intercepts of the curve with the coordinate axes.
(d) Find the value of $k$ for which $\left|2 x^{2}+3 x-4\right|=k$ has exactly 3 values of $x$.$[1]$
9. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 21 / \mathrm{q} 2)$
(a) Write $9 x^{2}-12 x+5$ in the form $p(x-q)^{2}+r$, where $p, q$ and $r$ are constants.
(b) Hence write down the coordinates of the minimum point of the curve $y=9 x^{2}-12 x+5$
10. (CIE $\left.0606 / 2020 / \mathrm{w} / 21 / \mathrm{q}^{2}\right)$
Find the coordinates of the points of intersection of the curve $x^{2}+x y=9$ and the line $y=\frac{2}{3} x-2$
11. $(\mathrm{CIE} 0606 / 2020 / \mathrm{s} / 21 / \mathrm{q} 6)$
Find the values of $k$ for which the line $y=k x-7$ and the curve $y=3 x^{2}+8 x+5$ do not intersect.
12. $(\mathrm{CIE} 0606 / 2020 / \mathrm{m} / 22 / \mathrm{q} 1)$
Variables $x$ and $y$ are such that, when $\lg y$ is plotted against $x^{3}$, a straight line graph passing through the points $(6,7)$ and $(10,9)$ is obtained. Find $y$ as a function of $x .$
13. (CTE 0606/2020/s/22/q3)
Find the values of $k$ for which the line $y=x-3$ intersects the curve $y=k^{2} x^{2}+5 k x+1$ at two distinct points.
14. (CIE 0606/2020/s/23/q2)
Find the set of values of $k$ for which $4 x^{2}-4 k x+2 k+3=0$ has no real roots.$[5]$
15. (CIE $0606 / 2020 / \mathrm{w} / 23 / \mathrm{q} 3)$
Find the values of $k$ for which the equation $x^{2}+(k+9) x+9=0$ has two distinct real roots.
16. (CIE $0606 / 2019 / \mathrm{s} / 11 / \mathrm{q} 3)$
The polynomial $\mathrm{p}(x)=(2 x-1)(x+k)-12$, where $k$ is a constant.
(i) Write down the value of $\mathrm{p}(-k)$.
When $p(x)$ is divided by $x+3$ the remainder is 23 .
(ii) Find the value of $k$.
(iii) Using your value of $k$, show that the equation $p(x)=-25$ has no real solutions.
17. (CIE $0606 / 2019 / \mathrm{s} / 13 / \mathrm{q} 3)$
Show that the line $y=m x+4$ will touch or intersect the curve $y=x^{2}+3 x+m$ for all values of $m$
18. $(\mathrm{CIE} 0606 / 2019 / \mathrm{s} / 21 / \mathrm{q} 1)$
Find the values of $x$ for which $x(6 x+7) \geqslant 20$
19. (CIE $\left.0606 / 2019 / \mathrm{s} / 22 / \mathrm{q}^{2}\right)$
Find the values of $k$ for which the equation $(k-1) x^{2}+k x-k=0$ has real and distinct roots.
20. (CIE 0606/2019/s/22/q5)
(i) Express $5 x^{2}-15 x+1$ in the form $p(x+q)^{2}+r$, where $p, q$ and $r$ are constants.
(ii) Hence state the least value of $x^{2}-3 x+0.2$ and the value of $x$ at which this occurs.
21. $(\mathrm{CIE} 0606 / 2020 / \mathrm{w} / 12 / \mathrm{q} 1)$
The curve $y=2 x^{2}+k+4$ intersects the straight line $y=(k+4) x$ at two distinct points. Find the possible values of $k$.$[4]$
22. (CIE $0606 / 2020 / \mathrm{w} / 12 / \mathrm{q} 6)$
$$f(x)=x^{2}+2 x-3 \quad \text { for } x \geqslant-1$$
(a) Given that the minimum value of $x^{2}+2 x-3$ occurs when $x=-1$, explain why $f(x)$ has an inverse.$[1]$
(b) On the axes below, sketch the graph of $y=\mathrm{f}(x)$ and the graph of $y=\mathrm{f}^{-1}(x) .$ Label cach graph and state the intercepts on the coordinate axes.
Answer
1. (i) $\left(x-\frac{9}{2}\right)^{2}-\frac{49}{4}$
(ii) $\left(\frac{9}{2}, \frac{-49}{4}\right)$
(iii)
(iv) $\frac{49}{4}$
2. (i) $k=3,11$
(ii) $k=11, y=-5 x+16, y=7-11 x-x^{2}, k=3, y=-5 x+8, y=$ $7-3 x-x^{2},(-3,31),(1,3)$
(iii) $20 \sqrt{2}$
3. (i) $\frac{81}{4}-\left(x-\frac{7}{2}\right)^{2}$
(ii) $\left(\frac{7}{2}, \frac{81}{4}\right)$
(iii) $z=\pm 2 \sqrt{2}$
4. $\quad-1<k<15$
5. $\quad-1<k<11$
6. (i) $y=2(x-1)^{2}-9$
(ii) $x=1, y=-9$
(iii) $p=4.74$
7. $k=4,12$
8. (a) $2\left(x+\frac{3}{4}\right)^{2}-\frac{41}{8}$
(b) $\left(-\frac{3}{4},-\frac{41}{8}\right)$
(d) $\frac{41}{8}$
9. (a) $9\left(x-\frac{2}{3}\right)^{2}+1$
(b) $\left(\frac{2}{3}, 1\right)$
10. $(3,0),\left(-\frac{9}{5},-\frac{16}{5}\right)$
11. $-4<k<20$
12. $\frac{2}{5}<x<\frac{3}{2}$
13. $k<\frac{1}{9}$ or $k>1$
14. $-1<k<3$
15. $k<-15$ or $k>-3$
16. (i) $-12$ (ii) $k=-2$
(iii) Show
17. Show
18. $\quad x \leqslant-\frac{5}{2}$ or $x \geqslant \frac{4}{3}$
19. $\quad k<0, k>0.8$
20. (i) $5(x-1.5)^{2}-10.25$
(ii) $x=1.5, y=-2.05$
21. $k<-4, k>4$
22 . (a) one - one
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