CIE Quadratic Function (-2020)

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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