Further Pure Math (Quadratic Function and Roots)

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 1. (2011/june/Paper02/q11a)

$$\mathrm{f}(x)=x^{2}+6 x+8$$

Given that $\mathrm{f}(x)$ can be expressed in the form $(x+A)^{2}+B$ where $A$ and $B$ are constants,

(a) find the value of $A$ and the value of $B$. $(3)$

(b) Hence, or otherwise, find

(i) the value of $x$ for which $\mathrm{f}(x)$ has its least value

(ii) the least value of $\mathrm{f}(x)$ $(2)$

The curve $C$ has equation $y=x^{2}+6 x+8$

The line $l$, with equation $y=2-x$, intersects $C$ at two points.

(c) Find the $x$-coordinate of each of these two points.

(d) Find the $x$-coordinate of the points where $C$ crosses the $x$-axis. (2)


2. (2012/jan/paper01/q3)

Solve the inequality $6 x^{2}-19 x-7<0$ (4)


3. (2012/june/paper01/q1)

Find the set of values of $x$ for which $(2 x+1)(4-x)>(x-4)(2 x-3)$


4. (2013/jan/Paper01/q3)

$$f(x)=3 x^{2}+6 x+7$$

Given that $\mathrm{f}(x)$ can be written in the form $A(x+B)^{2}+C$, where $A, B$ and $C$ are rational numbers,

(a) find the value of $A$, the value of $B$ and the value of $C$.

(b) Hence, or otherwise, find

(i) the value of $x$ for which $\frac{1}{\mathrm{f}(x)}$ is a maximum,

(ii) the maximum value of $\frac{1}{\mathrm{f}(x)}$.


5. (2013/june/paper01/q2)

Find the set of values of $x$ for which

$$3(x+1)^{2}<9-x$$


6. (2014/jan/paper01/q2)

$$\mathrm{f}(x)=2 x^{2}-8 x+5$$

Given that $\mathrm{f}(x)$ can be written in the form $a(x-b)^{2}+c$

(a) find the value of $a$, the value of $b$ and the value of $c$.

(b) Write down

(i) the minimum value of $\mathrm{f}(x)$

(ii) the value of $x$ at which this minimum occurs. (2)


7. $(2015 / \mathrm{jan} / \mathrm{paper0} 1 / \mathrm{q} 2)$

A small stone is thrown vertically upwards from a point $A$ above the ground. At time $t$ seconds after being thrown from $A$, the height of the stone above the ground is $s$ metres. Until the stone hits the ground, $s=1.4+19.6 t-4.9 t^{2}$

(a) Write down the height of $A$ above the ground. (1)

(b) Find the speed with which the stone was thrown from $A$. (2)

(c) Find the acceleration of the stone until it hits the ground.

(d) Find the greatest height of the stone above the ground. (3)


8. ( $2015 /$ june $/$ paper01/q3)

$$f(x)=4 x^{2}-8 x+7$$

Given that $\mathrm{f}(x)=l(x-m)^{2}+n$, for all values of $x$,

(a) find the value of $l$, the value of $m$ and the value of $n$. (3)

(b) Hence, or otherwise, find

(i) the minimum value of $\mathrm{f}(x)$,

(ii) the value of $x$ for which this minimum occurs.


9. (2016/jan/paper01/q2)

Find the set of values of $x$ for which $(2 x-3)^{2}>7 x-3$ (5)


10. (2017/jan/paper01/q3)

Use algebra to find the set of values of $x$ for which $(3 x-1)(x-1)<2(3 x-1)$ (5)


11. $(2018 / \mathrm{jan} / \mathrm{paper} 01 / \mathrm{q} 1)$

$$f(x)=6+5 x-2 x^{2}$$

Given that $f(x)$ can be written in the form $p(x+q)^{2}+r$, where $p, q$ and $r$ are rational numbers,

(a) find the value of $p$, the value of $q$ and the value of $r .$ (3)

(b) Hence, or otherwise, find

(i) the maximum value of $\mathrm{f}(x)$

(ii) the value of $x$ for which this maximum occurs.

$$ g(x)=6+5 x^{3}-2 x^{6} $$

(c) Write down

(i) the maximum value of $g(x)$,

(ii) the exact value of $x$ for which this maximum occurs. (3)


12. (2019/june/paper01/q5)

$$f(x)=3 x^{2}-9 x+5$$

Given that $\mathrm{f}(x)$ can be written in the form $a(x-b)^{2}+c$, where $a, b$ and $c$ are constants, find

(a) the value of $a$, the value of $b$ and the value of $c$.

(b) Hence write down

(i) the minimum value of $\mathrm{f}(x)$,

(ii) the value of $x$ at which this minimum occurs.


13. (2011/june/Paper02/q10)

The roots of the equation $x^{2}+6 x+2=0$ are $\alpha$ and $\beta$, where $\alpha>\beta$. Without solving the equation

(a) find

(i) the value of $\alpha^{2}+\beta^{2}$

(ii) the value of $\alpha^{4}+\beta^{4}$

(b) Show that $\alpha-\beta=2 \sqrt{7}$

(c) Factorise completely $\alpha^{4}-\beta^{4}$ (2)

(d) Hence find the exact value of $\alpha^{4}-\beta^{4}$ (2)

Given that $\beta^{4}=A+B \sqrt{7}$ where $A$ and $B$ are positive constants

(e) find the value of $A$ and the value of $B$.


14. (2012/jan/paper01/q8)

The equation $x^{2}+m x+15=0$ has roots $\alpha$ and $\beta$ and the equation $x^{2}+h x+k=0$ has roots $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$

(a) Write down the value of $k$

(b) Find an expression for $h$ in terms of $m$

Given that $\beta=2 \alpha+1$

(c) find the two possible values of $\alpha$

(d) Hence find the two possible values of $m$


15. (2012/june/paper01/q4)

The equation $2 x^{2}-7 x+4=0$ has roots $\alpha$ and $\beta$

Without solving this equation, form a quadratic equation with integer coefficients which has roots $\alpha+\frac{1}{\beta}$ and $\beta+\frac{1}{\alpha}$


16. (2013/jan/Paper01/q10)

$$\mathrm{f}(x)=2 x^{2}-5 x+1$$

The equation $\mathrm{f}(x)=0$ has roots $\alpha$ and $\beta$. Without solving the equation

(a) find the value of $\alpha^{2}+\beta^{2}$

(b) show that $\alpha^{4}+\beta^{4}=\frac{433}{16}$

(c) form a quadratic equation with integer coefficients which has roots

$$\left(\alpha^{2}+\frac{1}{\alpha^{2}}\right) \text { and }\left(\beta^{2}+\frac{1}{\beta^{2}}\right)$$


17. (2013/jan/Paper01/q2)

The equation $x^{2}+4 p x+9=0$ has unequal real roots. Find the set of possible values of $p$.


18. (2013/june/paper01/q6)

The equation $x^{2}+p x+1=0$ has roots $\alpha$ and $\beta$

(a) Find, in terms of $p$, an expression for

(i) $\alpha+\beta$

(ii) $\alpha^{2}+\beta^{2}$

(iii) $\alpha^{3}+\beta^{3}$

(b) Find a quadratic equation, with coefficients expressed in terms of $p$, which has roots $a^{3}$ and $\beta^{3}$ (2)


19. (2014/jan/paper01/q10)

$$f(x)=x^{2}+(k-3) x+4$$

The roots of the equation $\mathrm{f}(x)=0$ are $\alpha$ and $\beta$

(a) Find, in terms of $k$, the value of $\alpha^{2}+\beta^{2}$

Given that

$$4\left(\alpha^{2}+\beta^{2}\right)=7 \alpha^{2} \beta^{2}$$

(b) without solving the equation $\mathrm{f}(x)=0$, form a quadratic equation, with integer coefficients, which has roots $\frac{1}{\alpha^{2}}$ and $\frac{1}{\beta^{2}}$

(c) find the possible values of $k$.


20. (2014/june/paper01/q8)

$$\mathrm{f}(x)=3 x^{2}+p x-7$$

The equation $\mathrm{f}(x)=0$ has roots $\alpha$ and $\beta$.

(a) Without solving the equation

(i) write down the value of $\alpha^{2} \beta^{2}$

(ii) find, in terms of $p, \alpha^{2}+\beta^{2}$

Given that $3 \alpha-\beta=8$

(b) find the possible values of $p$

Given also that $p$ is negative,

(c) form an equation with roots $\frac{1}{a^{2}}$ and $\frac{1}{\beta^{2}}$ (3)


21. (2015/jan/paper02/q3)

The equation $2 x^{2}+3 x+c=0$, where $c$ is a constant, has two equal roots.

(a) Find the value of $c$.

(b) Solve the equation. (2)


22. $(2015 / \mathrm{jan} / \mathrm{paper} 02 / \mathrm{q} 6)$

The equation $2 x^{2}+p x-3=0$, where $p$ is a constant, has roots $\alpha$ and $\beta$.

(a) Find the value of

(i) $\alpha \beta$

(ii) $\left(\alpha+\frac{1}{\beta}\right)\left(\beta+\frac{1}{\alpha}\right)$ (4)

(b) Find, in terms of $p$

(i) $\alpha+\beta$

(ii) $\left(\alpha+\frac{1}{\beta}\right)+\left(\beta+\frac{1}{\alpha}\right)$

(4)

Given that $\left(\alpha+\frac{1}{\beta}\right)+\left(\beta+\frac{1}{\alpha}\right)=2\left(\alpha+\frac{1}{\beta}\right)\left(\beta+\frac{1}{a}\right)$

(c) find the value of $p$ (1)

(d) Using the value of $p$ found in part (c), find a quadratic equation, with integer coefficients, which has roots $\left(\alpha+\frac{1}{\beta}\right)$ and $\left(\beta+\frac{1}{\alpha}\right)$.


23. $(2015 /$ june $/$ paper01 $/ \mathrm{q} 5$ )

(a) Show that $(\alpha+\beta)\left(\alpha^{2}-\alpha \beta+\beta^{2}\right)=\alpha^{3}+\beta^{3}$

The roots of the equation $2 x^{2}+6 x-7=0$ are $\alpha$ and $\beta$ where $\alpha>\beta$

Without solving the equation,

(b) find the value of $\alpha^{3}+\beta^{3}$ (4)

(c) show that $\alpha-\beta=\sqrt{23}$ (2)

(d) Hence find the exact value of $\alpha^{3}-\beta^{3}$ (2)


24. (2016/jan/paper02/q5)

Given that $\alpha+\beta=5$ and $\alpha^{2}+\beta^{2}=19$

(a) show that $\alpha \beta=3$

(b) Hence form a quadratic equation, with integer coefficients, which has roots $\alpha$ and $\beta$

(c) Form a quadratic equation, with integer coefficients, which has roots $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$(5)


25. (2016/june/paper01/q9)

$$f(x)=3 x^{2}-5 x-4$$

The roots of the equation $\mathrm{f}(x)=0$ are $\alpha$ and $\beta$

(a) Without solving the equation $\mathrm{f}(x)=0$, form an equation, with integer coefficients, which has

(i) roots $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$

(ii) roots $2 \alpha+\beta$ and $\alpha+2 \beta$

(b) Express $\mathrm{f}(x)$ in the form $A(x+B)^{2}+C$, stating the values of the constants $A, B$ and $C$.

(c) Hence, or otherwise, show that the equation $\mathrm{f}(x)=-8$ has no real roots. (2)


26. (2017/jan/paper01/q9)

The equation $3 x^{2}-4 x+6=0$ has roots $\alpha$ and $\beta$.

(a) Without solving the equation, write down

(i) the value of $\alpha+\beta$

(ii) the value of $\alpha \beta$ (2)

(b) Without solving the equation, show that $\alpha^{3}+\beta^{3}=-\frac{152}{27}$ (3)

(c) Form a quadratic equation, with integer coefficients, that has roots $\frac{\alpha}{\beta^{2}}$ and $\frac{\beta}{\alpha^{2}}$


27. (2017/june/paper02/q3)

(a) Find the set of possible values of $p$ for which the equation $3 x^{2}+p x+3=0$ has no real roots.

(b) Find the integer values of $q$ for which the equation $x^{2}+7 x+q^{2}=0$ has real roots.


28. (2017/june/paper02/q8)

$$\mathrm{f}(x)=x^{2}+p x+7 \quad p \in \mathbb{R}$$

The roots of the equation $\mathrm{f}(x)=0$ are $\alpha$ and $\beta$

(a) Find, in terms of $p$ where necessary,

(i) $\alpha^{2}+\beta^{2}$

(ii) $\alpha^{2} \beta^{2}$

Given that $7\left(\alpha^{2}+\beta^{2}\right)=5 \alpha^{2} \beta^{2}$

(b) find the possible values of $p$ (2)

Using the positive value of $p$ found in part (b) and without solving the equation $\mathrm{f}(x)=0$

(c) form a quadratic equation with roots $\frac{2 p}{\alpha^{2}}$ and $\frac{2 p}{\beta^{2}}$ (5)


29. $(2018 / \mathrm{jan} /$ paper01/q9)

It is given that $\alpha$ and $\beta$ are such that $a+\beta=-\frac{5}{2}$ and $\alpha \beta=-5$

(a) Form a quadratic equation with integer coefficients that has roots $\alpha$ and $\beta$ (2)

Without solving the equation found in part (a)

(b) find the value of

(i) $\alpha^{2}+\beta^{2}$

(ii) $a^{3}+\beta^{3}$

(c) Hence form a quadratic equation with integer coefficients that has roots

$$\left(\alpha-\frac{1}{\alpha^{2}}\right) \text { and }\left(\beta-\frac{1}{\beta^{2}}\right)$$


30. $(2018 / \mathrm{jan} /$ paper02/q4)

Here is a quadratic equation $3 x^{2}+p x+4=0$ where $p$ is a constant.

(a) Find the set of values of $p$ for which the equation has two real distinct roots.

(b) List all the possible integer values of $p$ for which the equation has no real roots.


31. (2018/june/paper01/q2)

The equation $3 x^{2}-5 x+4=0$ has roots $\alpha$ and $\beta$.

Without solving this equation, form a quadratic equation with integer coefficients that has roots $\alpha+\frac{1}{2 \beta}$ and $\beta+\frac{1}{2 \alpha}$


32. (2019/june/paper02/q10)

The roots of the equation $x^{2}+3 x-5=0$ are $\alpha$ and $\beta$.

(a) Without solving the equation, find

(i) the value of $\alpha^{2}+\beta^{2}$

(ii) the value of $\alpha^{4}+\beta^{4}$

Given that $\alpha>\beta$ and without solving the equation

(b) show that $\alpha-\beta=\sqrt{29}$ (2)

(c) Factorise $\alpha^{4}-\beta^{4}$ completely. (3)

(d) Hence find the exact value of $\alpha^{4}-\beta^{4}$ (2)

Given that $\beta^{4}=p+q \sqrt{29}$ where $p$ and $q$ are positive constants

(e) find the value of $p$ and the value of $q$. (3)


33. (2019/juneR/paper02/q11)

The quadratic equation $x^{2}-p x+q=0$ where $p>0$, has roots $\alpha$ and $\beta$

Given that $2 \alpha \beta=3$ and that $4\left(\alpha^{2}+\beta^{2}\right)=k^{2}-6 k-3$ where $k>3$

(a) (i) write down the value of $q$,

(ii) find an expression, in terms of $k$, for $p$.

Given also that $7 \alpha \beta=3(\alpha+\beta)$

(b) find the value of $k$. (2)

(c) Hence form an equation, with integer coefficients, which has roots

$$\frac{\alpha}{\alpha+\beta} \text { and } \frac{\beta}{\alpha+\beta}$$



Answer


1. (a) $(x+3)^{2}-1$ 

2. $-\frac{1}{3}<x<3 \frac{1}{2}$

3. $\frac{1}{2}<x<4$

4. (a) $3(x+1)^{2}+4$ (b) (i) $x=-1$ (ii) $\frac{1}{4}$

5. $-3<x<\frac{2}{3}$

6. (a) $a=2, \quad b=2, c=-3$ (b) (i) $-3$ (ii) 2

7. (a) $1.4$ (b) $v=19.6$ (c) $-9.8$ (d) $21$

8. (a) $l=4, \quad m=1, n=3$ (b) (i) 3 (ii) 1

9. $x<\frac{3}{4}$ or $x>4$

10. $\frac{1}{3}<x<3$

11. (a) $p=-2,q=-\frac 54,r=\frac{73}{8}$ (b)(i) $f(x)= \frac{73}{8}$ (ii) $x=\frac 54$ (c)(i) $g(x)=\frac{73}{8}$ (ii) $\sqrt[3]{\frac 54}$

12. (a) $\quad a=3, b=\frac{3}{2}, c=-\frac{7}{4}$ (b) $x=\frac{3}{2}, f(x)_{\min }=-\frac{7}{4}$

13. (a)(i) 32 (ii) 1016 (b) Show (c) $\left(\alpha^{2}+\beta^{2}\right)(\alpha+\beta)(\alpha-\beta)$ (d) $-384 \sqrt{7}$ (e) $A=508, B=192$

14. (a) $\quad k=1$ (b) $h=\frac{30-m^{2}}{15}$ (c) $a=\frac{5}{2},-3$ (d) $m=8,-8 \frac{1}{2}$

15. $4 x^{2}-21 x+18=0$

16. (a) $\frac{21}{4}$ (b) Show (c) $4 x^{2}-105 x+450=0$

17. $p<-\frac{3}{2}$ or $p>\frac{3}{2}$

18. (a)(i) $-p$ (ii) $p^{2}-2$ (iii) $3 p-p^{3}$ (b) $x^{2}-(3 p-p) x+1=0$

19. (a) $(k-3)^{2}-8$ (b) $16 x^{2}-28 x+1=0$ (c) $k=9,-3$

20. (a) (i) $\frac{49}{9}$ (ii) $\frac{p^{2}}{9}+\frac{14}{3}$ (b) $p=-4,20$ (c) $49 x^{2}-58 x+9=0$

21. (a) $\quad c=\frac{9}{8}$ (b) $x=-\frac{3}{4}$

22. (a)(i) $-\frac{3}{2}($ ii $)-\frac{1}{6}$ (b)(i) $-\frac{p}{2}$ (ii) $-\frac{p}{6}(c) p=2$ (d) $6 x^{2}+2 x-1=0$

23. (a) show (b) $-\frac{117}{2}$ (c) show (d) $\frac{25}{2} \sqrt{23}$

24. (a) show (b) $x^{2}-5 x+3=0$ (c) $3 x^{2}-19 x+3=0$

25. (a)(i) $1 2 x^{2}+49 x+12=0$ (ii) $9 x^{2}-45 x+38=0$ (b) $3\left(x-\frac{5}{6}\right)^{2}-\frac{73}{12}$ (c) Show

26. (a) (i) $\alpha+\beta=\frac{4}{3}$ (ii) $\propto \beta=2$(b) Show(c) $54 x^{2}+76 x+27=0$

27. $(a)-6<p<6 \quad$ (b) $-\frac{7}{2} \leqslant q \leqslant \frac{7}{2}$

28. $(a)$ (i) $p^{2}-14$ (ii) 49 (b) $p=\pm 7$ (c) $x^{2}-10 x+4=0$

29. (a) $2x^2+5x-10=0$ (b)(i) $\frac{65}{4}$ (ii) $-\frac{425}{8}$ (c) $200x^2+630x-567=0$

30. (a) $p=\pm 4\sqrt 3$ (b) $\pm 6,\pm 5, \pm 4, \pm 3,\pm 2,\pm 1,0$

31. $48x^2-110x+121=0$

32. (a)(i) $19$ (ii) $311$ (b) Show (c) $(\alpha-\beta)(\alpha+\beta)\left(\alpha^{2}+\beta^{2}\right)$ (d) $-57 \sqrt{29}$ (e) $p=\frac{311}{2}, q=\frac{57}{2}$

33. (a) (i) $q=\frac 32$ (ii) $p=\frac{k-3}{2}$ (b) $k=10$ (c) $49x^2-49x+6=0$


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