CIE Quadratic Function 2012 to 2020

CIE Quadratic Function 2012 to 2018

**********************************************

CIE Quadratic Function 2012-2018

 1 (CIE 2012, s, paper 11, question 3)

Find the set of values of $k$ for which the line $y=2 x+k$ cuts the curve $y=x^{2}+k x+5$ at two distinct points. $[6]$

2 (CIE 2012, s, paper 21, question 12either)

EITHER

The equation of a curve is $y=2 x^{2}-20 x+37$.

(i) Express $y$ in the form $a(x+b)^{2}+c$, where $a, b$ and $c$ are integers.$[3]$

(ii) Write down the coordinates of the stationary point on the curve.$[1]$

A function $\mathrm{f}$ is defined by $\quad \mathrm{f}: x \mapsto 2 x^{2}-20 x+37$ for $x>k$. Given that the function $\mathrm{f}^{-1}(x)$ exists,

(iii) write down the least possible value of $k$,$[1]$

(iv) sketch the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{f}^{-1}(x)$ on the axes provided,$[2]$

(v) obtain an expression for $f^{-1}$.[3]

3 (CIE 2012, s, paper 22, question 3)

Find the values of $m$ for which the line $y=m x-5$ is a tangent to the curve $y=x^{2}+3 x+4$.

4 (CIE 2012, w, paper 11, question 2)

Find the values of $k$ for which the line $y=k-6 x$ is a tangent to the curve $y=x(2 x+k)$.$[4]$

5 (CIE 2012, w, paper 21, question 1)

Solve the inequality $4 x-9>4 x(5-x)$.

6 (CIE 2012, w, paper 21, question 4) 

It is given that $f(x)=4+8 x-x^{2}$.

(i) Find the value of $a$ and of $b$ for which $f(x)=a-(x+b)^{2}$ and hence write down the coordinates of the stationary point of the curve $y=\mathrm{f}(x)$

(ii) On the axes below, sketch the graph of $y=\mathrm{f}(x)$, showing the coordinates of the point where your graph intersects the $y$-axis. 

7 (CIE 2012, w, paper 22, question 3)

Solve the inequality $4 x(4-x)>7$$[4]$

8 (CIE $2012, \mathrm{w}$, paper 23, question 5$)$

Find the set of values of $m$ for which the line $y=m x+2$ does not meet the curve $y=m x^{2}+7 x+11$[6]

9 (CIE 2013, s, paper 12 , question 4)

Find the set of values of $k$ for which the curve $y=2 x^{2}+k x+2 k-6$ lies above the $x$-axis for all values of $x$

10 (CIE 2013, w, paper 11, question 2)

Find the set of values of $k$ for which the curve $y=(k+1) x^{2}-3 x+(k+1)$ lies below the $x$-axis. $[4]$

11 (CIE 2013, w, paper 21, question 1)

Find the set of values of $x$ for which $x^{2}<6-5 x$.

12 (CIE 2013, w, paper 23, question 3)

Find the set of values of $k$ for which the line $y=3 x-k$ does not meet the curve $y=k x^{2}+11 x-6$.

13 (CIE 2014, s, paper 11, question 4)

Find the set of values of $k$ for which the line $y=k(4 x-3)$ does not intersect the curve $y=4 x^{2}+8 x-8$

14 (CIE 2014, s, paper 13, question 1)

(i) Show that $y=3 x^{2}-6 x+5$ can be written in the form $y=a(x-b)^{2}+c$, where $a, b$ and $c$ are constants to be found.$[3]$

(ii) Hence, or otherwise, find the coordinates of the stationary point of the curve $y=3 x^{2}-6 x+5$.[1]

15 (CIE 2014, s, paper 21, question 1)

Find the set of values of $x$ for which $x(x+2)<x$.

16 (CIE 2014, s, paper 21, question 5)

(i) Express $2 x^{2}-x+6$ in the form $p(x-q)^{2}+r$, where $p, q$ and $r$ are constants to be found.

(ii) Hence state the least value of $2 x^{2}-x+6$ and the value of $x$ at which this occurs.$[2]$

17 (CIE 2014, s, paper 22, question 2)

Find the values of $k$ for which the line $y+k x-2=0 \quad$ is a tangent to the curve $y=2 x^{2}-9 x+4$.

18 (CIE 2014, s, paper 22, question 4)

(i) Express $12 x^{2}-6 x+5$ in the form $p(x-q)^{2}+r$, where $p, q$ and $r$ are constants to be found.$[3]$

(ii) Hence find the greatest value of $\frac{1}{12 x^{2}-6 x+5}$ and state the value of $x$ at which this occurs.

19 (CIE 2014, w, paper 21, question 2) 

Solve the inequality $9 x^{2}+2 x-1<(x+1)^{2}$.

20 (CIE 2015, s, paper 12, question 1)

Given that the graph of $y=(2 k+5) x^{2}+k x+1$ does not meet the $x$-axis, find the possible values of $k .$

21 (CIE 2015, s, paper 21, question 9)

(a) Find the set of values of $x$ for which $4 x^{2}+19 x-5 \leqslant 0$ $[3]$

(b) (i) Express $x^{2}+8 x-9$ in the form $(x+a)^{2}+b$, where $a$ and $b$ are integers.$[2]$

(ii) Use your answer to part (i) to find the greatest value of $9-8 x-x^{2}$ and the value of $x$ at which this occurs. $[2]$

(iii) Sketch the graph of $y=9-8 x-x^{2}$, indicating the coordinates of any points of intersection with the coordinate axes. 

22 (CIE 2015, w, paper 11, question 1)

Find the range of values of $k$ for which the equation $k x^{2}+k=8 x-2 x k$ has 2 real distinct roots. $[4]$

23 (CIE 2015, w, paper 11, question 11)

(a) A function $\mathrm{f}$ is such that $\mathrm{f}(x)=x^{2}+6 x+4$ for $x \geqslant 0$.

(i) Show that $x^{2}+6 x+4$ can be written in the form $(x+a)^{2}+b$, where $a$ and $b$ are integers.$[2]$

(ii) Write down the range of $\mathrm{f}$.$[1]$

(iii) Find $\mathrm{f}^{-1}$ and state its domain.[3]

(b) Functions $g$ and $h$ are such that, for $x \in \mathbb{R}$,$g(x)=\mathrm{e}^{x} \quad$ and $\quad \mathrm{h}(x)=5 x+2 .$ Solve $h^{2} g(x)=37$

24 (CIE 2015, w, paper 13, question 6)

(i) On the axes below, sketch the graph of $y=\left|x^{2}-4 x-12\right|$ showing the coordinates of the points where the graph meets the axes. 

(ii) Find the coordinates of the stationary point on the curve $y=\left|x^{2}-4 x-12\right|$.[2]

(iii) Find the values of $k$ such that the equation $\left|x^{2}-4 x-12\right|=k \quad$ has only 2 solutions. [2]

25 (CIE 2015, w, paper 23, question 2)

Find the values of $k$ for which the line $y=2 x+k+2 \quad$ cuts the curve $\quad y=2 x^{2}+(k+2) x+8$ in two distinct points. $[6]$

26 (CIE 2015, w, paper 23, question 9)

Given that $\mathrm{f}(x)=3 x^{2}+12 x+2$,

(i) find values of $a, b$ and $c$ such that $\mathrm{f}(x)=a(x+b)^{2}+c$,$[3]$

(ii) state the minimum value of $f(x)$ and the value of $x$ at which it occurs,

(iii) solve $\mathrm{f}\left(\frac{1}{y}\right)=0$, giving each answer for $y$ correct to 2 decimal places.$[3]$

27 (CIE 2016, march, paper 12, question 1)

Find the values of $a$ for which the line $y=a x+9$ intersects the curve $y=-2 x^{2}+3 x+1$ at 2 distinct points.[4]

28 (CIE 2016, s, paper 21, question 1)

Find the values of $x$ for which $(x-4)(x+2)>7$

29 (CIE 2016, s, paper 21, question 6)

(i) Express $4 x^{2}+8 x-5$ in the form $p(x+q)^{2}+r$, where $p, q$ and $r$ are constants to be found. [3]

(ii) State the coordinates of the vertex of $y=\left|4 x^{2}+8 x-5\right|$

(iii) On the axes below, sketch the graph of $y=\left|4 x^{2}+8 x-5\right|$, showing the coordinates of the points where the curve meets the axes. 

30 (CIE 2016, s, paper 22, question 1)

(i) Given that $x^{2}+2 k x+4 k-3=0$ has no real roots, show that $k$ satisfies $k^{2}-4 k+3<0$. $\quad[2]$

(ii) Solve the inequality $k^{2}-4 k+3<0 .$

31 (CIE 2016,w, paper 13 , question 3)

(i) Given that $3 x^{2}+p(1-2 x)=-3$, show that, for $x$ to be real, $p^{2}-3 p-9 \geqslant 0$.$[3]$

(ii) Hence find the set of values of $p$ for which $x$ is real, expressing your answer in exact form.

32 (CIE 2016, w, paper 21, question 9)

The line $y=k x-4$, where $k$ is a positive constant, passes through the point $P(0,-4)$ and is a tangent to the curve $-2 y=8$ at the point $T$. Find

(i) the value of $k$,

(ii) the coordinates of $T$,

(iii) the length of $T P$.

33 (CIE 2017, march, paper 22, question 4)

Find the set of values of $k$ for which the line $y=3 x+k$ and the curve $y=2 x^{2}-3 x+4$ do not intersect.[4]

34 (CIE 2017, s, paper 22, question 4)

The point $P$ lies on the curve $y=3 x^{2}-7 x+11$. The normal to the curve at $P$ has equation $5 y+x=k$ Find the coordinates of $P$ and the value of $k$.

35 (CIE 2017, s, paper 22, question 6)

Show that the roots of $p x^{2}+(p-q) x-q=0$ are real for all real values of $p$ and $q$.

36 (CIE 2017, w, paper 13, question 3)

Find the set of values of $k$ for which the equation $k x^{2}+3 x-4+k=0$ has no real roots. $\quad[4]$

37 (CIE 2017, w, paper 21, question 1)

Solve the inequality $(x-1)(x-5)>12$.

38 (CIE 2017, w, paper 21, question 11) 

The line $y=k x+3$, where $k$ is a positive constant, is a tangent to the curve $x^{2}-2 x+y^{2}=8$ at the point $P$.

(i) Find the value of $k$.[4]

(ii) Find the coordinates of $P$.[3]

(iii) Find the equation of the normal to the curve at $P$.$[2]$

39 (CIE 2018, march, paper 22, question 2) 

Determine the set of values of $k$ for which the equation $(3-2 k) x^{2}+(2 k-3) x+1=0$ has no real roots.

40 (CIE 2018, s, paper 12, question 2) 

Find the values of $k$ for which the line $y=1-2 k x$ does not meet the curve $y=9 x^{2}-(3 k+1) x+5$.[5]

***********************************

Answers

1. $k>4$ or $k \leq -4$

2. (i) $y=2(x-5)^{2}-13$

(ii) $(5,-13)$ (iii) 5

(iv) 

(v) $5+\sqrt{(x+13)/2}$

3. $m=9,-3$

4. $k=-2,-18$

5. $x<-.5$ or $x>4.5$

6. (i) $a=20, b=-4,(4,20)$

7. $0.5<x<3.5$

8. $1<m<49$

9. $4<k<12$

10. $k<-5 / 2$

11. $-6<x<1$

12. $k<-2, k>8$

13. $3<k<4$

14. (i) $y=3(x-1)^{2}+2$

(ii) $(1,2)$

15. $-1<x<0$

16. (i) $2(x-1 / 4)^{2}+47 / 8$

(ii) $47 / 8$ is $\min \mathrm{x}=1 / 4$.

17. $k=5,13$

18. (i) $12(x-1 / 4)^{2}+17 / 4$

(ii) $x=1 / 4,4 / 17$

19. $-.5<x<.5$

20. $-2<k<10$

21. (a) $-5 \leq x \leq 1 / 4$

(bi) $a=4, b=-25$, (ii) $-25$

(iii) 

22. $k \leq 2$

23. (a)(i) $(x+3)^{2}-5($ ii $) y \geq 4$

(iii) $y=\sqrt{x+5}-3, \mathrm{D}: x \geq 4$

(b) $x=0$

24 . (i) 

(ii) $(2,16)$, (iii) $k=0, k>16$

25. $k<-12$ or $k>4$

26. (i) $a=3, b=2, c=-10$

(ii) min $-10$ at $x=-2$

(iii) $y=-5.74,-0.26$

27. $a>11 . a<-5$

28. $x<-3$ or $x>5$

29. (i) $4(x+1)^{2}-9$

(ii) $(-1,9)$

(iii) 

30. (ii) $1<k<3$

31. (ii) $p \leq \frac{3-3 \sqrt{5}}{2}, p \geq \frac{3+3 \sqrt{5}}{2}$

32. (i) $k=4 / 3$ (ii) $x=12 / 5, y=-4 / 5$

(iii) 4

33. $k<-1 / 2$

34. $(2,9), \mathrm{k}=47$

35. $(p+q)^{2} \geq 0$

$36 . k<-.5, k>4.5$

37. $x>7$ or $x<-1$

38. (i) $k=3 / 4$ (ii) $(-.8,2.4)$

(iii) $3 y=4-4 x$

39. $-0.5<k<1.5$

40. $-13<k<11$

************************************

CIE Quadratic Function 2018 to 2020

CIE Quadratic Function (2018-2020)

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