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FPM Quadratic Functions (Chapter 2)

1. (2011/june/Paper02/q11a)
f(x)=x2+6x+8 Given that 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 marks)

(b) Hence, or otherwise, find

(i) the value of x for which f(x) has its least value

(ii) the least value of f(x)

The curve C has equation y=x2+6x+8

The line l, with equation y=2x, intersects C at two points. (2 marks)

(c) Find the x-coordinate of each of these two points. (4 marks)

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

2. (2012/jan/paper01/q3)
Solve the inequality 6x219x7<0 (4 marks)

3. (2012/june/paper01/q1)
Find the set of values of x for which (2x+1)(4x)>(x4)(2x3) (4 marks)

4. (2013/jan/Paper01/q3)
f(x)=3x2+6x+7 Given that 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. (3 marks)

(b) Hence, or otherwise, find

(i) the value of x for which 1f(x) is a maximum,

(ii) the maximum value of 1f(x). (2 marks)

5. (2013/june/paper01/q2)
Find the set of values of x for which 3(x+1)2<9x (4 marks)

6. (2014/jan/paper01/q2)
f(x)=2x28x+5 Given that f(x) can be written in the form a(xb)2+c

(a) find the value of a, the value of b and the value of c. (3 marks)

(b) Write down

(i) the minimum value of f(x)

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

7. (2015/jan/paper01/q2)
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+19.6t4.9t2

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

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

(c) Find the acceleration of the stone until it hits the ground. (1 mark)

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

8. ( 2015/ june / paper01/q3)
f(x)=4x28x+7 Given that f(x)=l(xm)2+n, for all values of x,

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

(b) Hence, or otherwise, find

(i) the minimum value of f(x),

(ii) the value of x for which this minimum occurs. (2 marks)

9. (2016/jan/paper01/q2)
Find the set of values of x for which (2x3)2>7x3 (5 marks)

10. (2017/jan/paper01/q3)
Use algebra to find the set of values of x for which (3x1)(x1)<2(3x1) (5 marks)

11. (2018/jan/paper01/q1)
f(x)=6+5x2x2 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 marks)

(b) Hence, or otherwise, find

(i) the maximum value of f(x)

(ii) the value of x for which this maximum occurs. g(x)=6+5x32x6 (2 marks)

(c) Write down

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

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

12. (2019/june/paper01/q5)
f(x)=3x29x+5 Given that f(x) can be written in the form a(xb)2+c, where a,b and c are constants, find

(a) the value of a, the value of b and the value of c. (3 marks)

(b) Hence write down

(i) the minimum value of f(x),

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

13. (2011/june/Paper02/q10)
The roots of the equation x2+6x+2=0 are α and β, where α>β. Without solving the equation

(a) find

(i) the value of α2+β2

(ii) the value of α4+β4 (5 marks)

(b) Show that αβ=27 (3 marks)

(c) Factorise completely α4β4 (2 marks)

(d) Hence find the exact value of α4β4

Given that β4=A+B7 where A and B are positive constants (2 marks)

(e) find the value of A and the value of B. (2 marks)

14. (2012/jan/paper01/q8)
The equation x2+mx+15=0 has roots α and β and the equation x2+hx+k=0 has roots αβ and βα

(a) Write down the value of k (1 mark)

(b) Find an expression for h in terms of m Given that β=2α+1 (6 marks)

(c) find the two possible values of α (3 marks)

(d) Hence find the two possible values of m (3 marks)

15. (2012/june/paper01/q4)
The equation 2x27x+4=0 has roots α and β Without solving this equation, form a quadratic equation with integer coefficients which has roots α+1β and β+1α (8 marks)

16. (2013/jan/Paper01/q10)
f(x)=2x25x+1 The equation f(x)=0 has roots α and β. Without solving the equation

(a) find the value of α2+β2 (3 marks)

(b) show that α4+β4=43316 (2 marks)

(c) form a quadratic equation with integer coefficients which has roots (α2+1α2) and (β2+1β2) (7 marks)

17. (2013/jan/Paper01/q2)
The equation x2+4px+9=0 has unequal real roots. Find the set of possible values of p. (4 marks)

18. (2013/june/paper01/q6)
The equation x2+px+1=0 has roots α and β

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

(i) α+β

(ii) α2+β2

(iii) α3+β3 (6 marks)

(b) Find a quadratic equation, with coefficients expressed in terms of p, which has roots a3 and β3 (2 marks)

19. (2014/jan/paper01/q10)
f(x)=x2+(k3)x+4 The roots of the equation f(x)=0 are α and β

(a) Find, in terms of k, the value of α2+β2 Given that 4(α2+β2)=7α2β2 (3 marks)

(b) without solving the equation f(x)=0, form a quadratic equation, with integer coefficients, which has roots 1α2 and 1β2 (5 marks)

(c) find the possible values of k. (5 marks)

20. (2014/june/paper01/q8)
f(x)=3x2+px7 The equation f(x)=0 has roots α and β.

(a) Without solving the equation

(i) write down the value of α2β2

(ii) find, in terms of p,α2+β2

Given that 3αβ=8 (4 marks)

(b) find the possible values of p

Given also that p is negative, (5 marks)

(c) form an equation with roots 1a2 and 1β2 (3 marks)

21. (2015/jan/paper02/q3)
The equation 2x2+3x+c=0, where c is a constant, has two equal roots.

(a) Find the value of c. (2 marks)

(b) Solve the equation. (2 marks)

22. (2015/jan/paper02/q6)
The equation 2x2+px3=0, where p is a constant, has roots α and β.

(a) Find the value of

(i) αβ

(ii) (α+1β)(β+1α) (4 marks)

(b) Find, in terms of p

(i) α+β

(ii) (α+1β)+(β+1α)

Given that (α+1β)+(β+1α)=2(α+1β)(β+1a) (4 marks)

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

(d) Using the value of p found in part (c), find a quadratic equation, with integer coefficients, which has roots (α+1β) and (β+1α). (2 marks)

23. (2015/ june / paper01 /q5 )

(a) Show that (α+β)(α2αβ+β2)=α3+β3

The roots of the equation 2x2+6x7=0 are α and β where α>β

Without solving the equation, (1 mark)

(b) find the value of α3+β3 (4 marks)

(c) show that αβ=23 (2 marks)

(d) Hence find the exact value of α3β3 (2 marks)

24. (2016/jan/paper02/q5)
Given that α+β=5 and α2+β2=19

(a) show that αβ=3 (2 marks)

(b) Hence form a quadratic equation, with integer coefficients, which has roots α and β (2 marks)

(c) Form a quadratic equation, with integer coefficients, which has roots αβ and βα(5) (5 marks)

25. (2016/june/paper01/q9)
f(x)=3x25x4 The roots of the equation f(x)=0 are α and β

(a) Without solving the equation f(x)=0, form an equation, with integer coefficients, which has

(i) roots αβ and βα (6 marks)

(ii) roots 2α+β and α+2β (5 marks)

(b) Express f(x) in the form A(x+B)2+C, stating the values of the constants A,B and C. (3 marks)

(c) Hence, or otherwise, show that the equation f(x)=8 has no real roots. (2 marks)

26. (2017/jan/paper01/q9)
The equation 3x24x+6=0 has roots α and β.

(a) Without solving the equation, write down

(i) the value of α+β

(ii) the value of αβ (2 marks)

(b) Without solving the equation, show that α3+β3=15227 (3 marks)

(c) Form a quadratic equation, with integer coefficients, that has roots αβ2 and βα2 (5 marks)

27. (2017/june/paper02/q3)

(a) Find the set of possible values of p for which the equation 3x2+px+3=0 has no real roots. (3 marks)

(b) Find the integer values of q for which the equation x2+7x+q2=0 has real roots. (3 marks)

28. (2017/june/paper02/q8)
f(x)=x2+px+7pR The roots of the equation f(x)=0 are α and β

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

(i) α2+β2

(ii) α2β2 Given that 7(α2+β2)=5α2β2 (4 marks)

(b) find the possible values of p Using the positive value of p found in part (b) and without solving the equation f(x)=0 (2 marks)

(c) form a quadratic equation with roots 2pα2 and 2pβ2 (5 marks)

29. (2018/jan/ paper01/q9)
It is given that α and β are such that a+β=52 and αβ=5

(a) Form a quadratic equation with integer coefficients that has roots α and β Without solving the equation found in part (a) (2 marks)

(b) find the value of

(i) α2+β2

(ii) a3+β3 (5 marks)

(c) Hence form a quadratic equation with integer coefficients that has roots (α1α2) and (β1β2) (6 marks)

30. (2018/jan/ paper02/q4)
Here is a quadratic equation 3x2+px+4=0 where p is a constant.

(a) Find the set of values of p for which the equation has two real distinct roots. (5 marks)

(b) List all the possible integer values of p for which the equation has no real roots. (1 mark)

31. (2018/june/paper01/q2)
The equation 3x25x+4=0 has roots α and β. Without solving this equation, form a quadratic equation with integer coefficients that has roots α+12β and β+12α (7 marks)

32. (2019/june/paper02/q10)
The roots of the equation x2+3x5=0 are α and β.

(a) Without solving the equation, find

(i) the value of α2+β2

(ii) the value of α4+β4

Given that α>β and without solving the equation (5 marks)

(b) show that αβ=29 (2 marks)

(c) Factorise α4β4 completely. (3 marks)

(d) Hence find the exact value of α4β4

Given that β4=p+q29 where p and q are positive constants (2 marks)

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

33. (2019/juneR/paper02/q11)
The quadratic equation x2px+q=0 where p>0, has roots α and β

Given that 2αβ=3 and that 4(α2+β2)=k26k3 where k>3

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

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

Given also that 7αβ=3(α+β) (5 marks)

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

(c) Hence form an equation, with integer coefficients, which has roots αα+β and βα+β (5 marks)



 Answer


1. (a) (x+3)21 (b) least at x=3, least value=1 (c) x=6,1 (d) x=2,4 (e) graph (f) Area=2056=20.8 

2. 13<x<312

3. 12<x<4

4. (a) 3(x+1)2+4 (b) (i) x=1 (ii) 14

5. 3<x<23

6. (a) a=2,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,m=1,n=3 (b) (i) 3 (ii) 1

9. x<34 or x>4

10. 13<x<3

11. (a) p=2,q=54,r=738 (b)(i) f(x)=738 (ii) x=54 (c)(i) g(x)=738 (ii) 354

12. (a) a=3,b=32,c=74 (b) x=32,f(x)min=74

13. (a)(i) 32 (ii) 1016 (b) Show (c) (α2+β2)(α+β)(αβ) (d) 3847 (e) A=508,B=192

14. (a) k=1 (b) h=30m215 (c) a=52,3 (d) m=8,812

15. 4x221x+18=0

16. (a) 214 (b) Show (c) 4x2105x+450=0

17. p<32 or p>32

18. (a)(i) p (ii) p22 (iii) 3pp3 (b) x2(3pp)x+1=0

19. (a) (k3)28 (b) 16x228x+1=0 (c) k=9,3

20. (a) (i) 499 (ii) p29+143 (b) p=4,20 (c) 49x258x+9=0

21. (a) c=98 (b) x=34

22. (a)(i) 32( ii )16 (b)(i) p2 (ii) p6(c)p=2 (d) 6x2+2x1=0

23. (a) show (b) 1172 (c) show (d) 25223

24. (a) show (b) x25x+3=0 (c) 3x219x+3=0

25. (a)(i) 12x2+49x+12=0 (ii) 9x245x+38=0 (b) 3(x56)27312 (c) Show

26. (a) (i) α+β=43 (ii) β=2(b) Show(c) 54x2+76x+27=0

27. (a)6<p<6 (b) 72q72

28. (a) (i) p214 (ii) 49 (b) p=±7 (c) x210x+4=0

29. (a) 2x2+5x10=0 (b)(i) 654 (ii) 4258 (c) 200x2+630x567=0

30. (a) p=±43 (b) ±6,±5,±4,±3,±2,±1,0

31. 48x2110x+121=0

32. (a)(i) 19 (ii) 311 (b) Show (c) (αβ)(α+β)(α2+β2) (d) 5729 (e) p=3112,q=572

33. (a) (i) q=32 (ii) p=k32 (b) k=10 (c) 49x249x+6=0


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