1. Higher Quadratic Past Paper Questions

2. Higher Quadratic Past Paper Questions
2001 Paper I Q2
For what values of k does the equation x2 – 5x + (k + 6) = 0 have equal roots?
a = x2 – 5x + (k + 6) = 0
b = – 5
c = (k + 6) Equal roots  b2 – 4ac = 0
(– 5)2 – 4(1)(k + 6) = 0
25 – 4k – 24 = 0
1 – 4k = 0
1 = 4k
k = ¼ 3

3. Higher Quadratic Past Paper Questions
Show that the equation (1 – 2k)x2 – 5kx – 2k = 0 has real roots for all integer
values of k?
a = 1 – 2k (1 – 2k)x2 – 5kx – 2k = 0
b = – 5k
c = - 2k Real roots  b2 – 4ac ≥ 0
(– 5k)2 – 4(1 – 2k)(– 2k) ≥ 0
25k2 + 8k(1 – 2k) ≥ 0
25k2 + 8k – 16k2 ≥ 0
9k2 + 8k ≥ 0
For all integers 9k2 ≥ 0 & 9k2 ≥ 8k
 always real roots for all integer values of k. 5

4. Higher Quadratic Past Paper Questions
Show that the line with equation y = 2x + 1 does not intersect the parabola with
equation y = x2 + 3x + 4
x2 + 3x + 4 = 2x + 1
a = x2 + x + 3 = 0
b = 1
c = Find value of b2 – 4ac
(1)2 – 4(1)(3) = 1 – 12 = -11 < 0
As b2 – 4ac < 0  No intersection occurs. 5

5. Higher Quadratic Past Paper Questions
Show that x = -1 is a solution of the cubic x3 + px2 + px + 1 = 0 1
Hence find the range of values of p for which all the roots are real 7
x3 + px2 + px + 1 = 0
1 p p 1 -1
As Remainder, R = 0  x = -1 is a solution
& x3 + px2 + px + 1 = (x + 1)(1x2 + (p – 1)x + 1) -1
1–p -1 1
P–1 1 1

6. Higher Quadratic Past Paper Questions
(b) Hence find the range of values of p for which all the roots are real 7
From (a)
x3 + px2 + px + 1 = (x + 1)(1x2 + (p – 1)x + 1)
a = If real then b2 – 4ac ≥ 0
b = (p – 1)
c = (p – 1)2 – 4(1)(1) ≥ 0
p2 – 2p + 1 – 4 ≥ 0
p2 – 2p – 3 ≥ 0
(p – 3)(p + 1) ≥ 0
 Real when p ≤ -1 & p ≥ 3 y -1 3 x 7

7. Higher Quadratic Past Paper Questions
Express 2x2 + 4x – 3 in the form f(x) = a(x + b)2 + c 3
Write down the coordinaates of the turning point 1
2x2 + 4x – 3 = 2[x2 + 2x – 3/2]
= 2[(x2 + 2x + 1) – 1 – 3/2]
= 2[(x2 + 2x + 1) – 2/2 – 3/2]
= 2[(x + 1)2 – 5/2]
= 2(x + 1)2 – 5
Happy as + x2  (- 1, - 5) is Min Tpt 4

8. Higher Quadratic Past Paper Questions
Find the range of values of k such that the equation kx2 – x – 1 = 0 has no real roots
a = k kx2 – 1x – 1= 0
b = -1
c = b2 – 4ac < 0  No Real Roots
(-1)2 – 4(k)(-1) < 0
1 + 4k < 0
4k < – 1
k < – ¼ 4

9. Higher Quadratic Past Paper Questions
2008 Paper 1 Q10 Which of the following are true about the equation x2 + x + 1 = 0
(1) The roots are equal (2) The roots are real
A = Neither B = (1) Only C = (2) Only D = Both are True
a = x2 + 1x + 1= 0
b = 1
c = Find nature of b2 – 4ac
(1)2 – 4(1)(1) = 1 – 4 = -3 < 0
As b2 – 4ac < 0  Non real roots are not equal or real
 A – Neither (1) or (2) are true 2

10. Higher Quadratic Past Paper Questions
The graph has an equation of the form y = k(x – a)(x – b).
What is the equation of the graph y 12 1 4 x
Use roots at x = 1 & x = 4 for factors
y = k(x – 1)(x – 4)
12 = k(0 – 1)(0 – 4)
12 = k(-1)(-4)
4k = 12
k = 3
y = 3(x – 1)(x – 4) A 2

11. Higher Quadratic Past Paper Questions
2x2 + 4x + 7 is expressed in the form 2(x + p)2 + q
What is the value of q?
2x2 + 4x + 7 = 2[x2 + 2x + 7/2]
= 2[(x2 + 2x + 1) – 1 + 7/2]
= 2[(x2 + 2x + 1) – 2/2 + 7/2]
= 2[(x + 1)2 + 5/2]
= 2(x + 1)2 + 5
= 2(x + p)2 + q  q = 5 = A 2

12. Higher Quadratic Past Paper Questions
Write x2 – 10x in the form (x + b)2 + c 2
Hence show that g(x) = 1/3 x3 – 5x2 + 27x – 2 is always increasing 4
(a) x2 – 10x + 27 = (x2 – 10x + 25) –
= (x – 5)2 + 2
Increasing  Differentiate g(x) = 1/3 x3 – 5x2 + 27x – 2
(b) g’(x) = x2 – 10x + 27
** Use answer to (a) = (x – 5)2 + 2
(x – 5) > 0 for all x  function is always increasing 6

13. Higher Quadratic Past Paper Questions
Total = 41 Marks