CIE Quadratic Function (Additional Mathematics -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. $-13<k<11$




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