SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
QUADRATIC EQUATIONS (QE) General form :
ax
2
+ bx +c = 0
Solving of QE Factorization Formulae
Completion of Square (COS)
Forming a QE i)
If roots given as values,
ii)
If roots given as unknowns with a QE
Roots = and , ax2 + bx + c = 0
Roots = -3 and 6
New Roots = 2 and 2
Nature of Roots b2 – 4ac > 0
b2 – 4ac = 0
b2 – 4ac < 0
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b2 – 4ac 0
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
2. Given that the QE x2 + (m – 3)x = 2m – 6 has two equal roots, find the values of m.
3. Given that the roots of the QE x2 – hx + 8 = 0 are p and 2p, find the values of h.
4. Given that one of the roots of the QE 2x2 + 18x = 2 – k is twice the other root, find the value of k.
5. Find the value of p for which 2y + x = p is a tangent to the curve y2 + 4x = 20.
6. Solve the equation 2(3x – 1)2 = 18.
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1. One of the roots of quadratic equation 2x2 + kx – 3 = 0 is 3, find the value of k.
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
8. Find the range of values of m such that the equation 2x2 – x = m – 2 has real roots.
9. Find the range of values of x for which (2x + 1)(x + 3) > (x + 3)(x – 3).
10. Find the range of values of k such that the QE x2 + x + 8 = k(2x – k) has two real roots.
11. Solve the QE 2x(x – 5) = (2 – x)(x + 3). Give your answer correct to 4 significant figures.
12. Given the roots of the QE of 4ax2 + bx + 8 = 0 are equal. Express a in terms of b.
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7. Solve the equation (x + 1)(x – 4) = 7. Give your answer correct to 3 significant figures.
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
The QE kx2 + 3 = kx + 5 has two distinct roots. Find the range of values of k.
14. Given m and -3 are the roots of a QE 2 2x – 4x = k – 1, find the value of m and of k.
15.
Given that QE x(3x – p) = 2x – 3 has no roots, find the range of values of p.
16. The QE px2 + px + 2q = 2 + 10x has roots 1/p and q. a) Find the values of p and q. b) Hence, form a QE which has roots p & -3q.
17. Given and are the roots of the QE 2x2 + 7x – 6 = 0, form a QE with roots ( +1) & ( + 1).
18. Find the value of p such that (p – 4)x2 + 2(2 – p)x + p + 1 = 0 has equal roots. Hence, find the root of the equation based on the value of p obtained.
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13.
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
19.
Given that 2 and m – 1 are the roots of the equation x2 + 3x = k, find the values of m and k.
20. Find the range of values of p if the straight line y = px – 5 does not intersect the curve y = x2 – 1.
21.
Given that 3 and m are the roots of the QE 2(x + 1)(x + 2) = k(x – 1). Find the values of m and k.
22. Find the range of values of p where px2 + 2(p + 2)x + p + 7 = 0 has real roots.
24. The roots of QE x2 – 10x + 7 – 3k = 0 are and 4, find a) the value of , b) the value of k.
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23. Given that the roots of the QE x2 + px + q = 0 are and 3, show that 3p2 = 16q
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
25.
Given that and are the roots of the QE 2x2 + 5x – 3 = 0, form a QE which has roots 1 and 1 . 2 2
26. Given 3 and 3 are the roots of a QE. Form a QE if = 6 and + = -5.
27.
Find the range of values of x for which x(x – 2) 35.
28. -3 is one root of the QE 2x2 + px = 3. Find, a) the value of p b) the value of the other root
29. The QE (2x – 5)2 = (p – 10)x has two distinct roots. Find the range of values of p.
x2
30. Prove that the roots of the QE + (2a – 1)x + a2 = 0 is real when a ≤ 1 .
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4
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
QUADRATIC FUNCTIONS (QF) ‘XO’ STEPS – SKETCHING GRAPH ‘XO’ STEPS - COS
Minimum Graph For the case where a is positive (a > 0)
b2 – 4ac > 0
The graph of f(x) touches the x-axis at only one point Two real & equal roots
b2 – 4ac < 0
The graph of f(x) does not meet the x-axis. No real roots
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The graph of f(x) cuts the x-axis at two distinct points Two real & distinct roots
b2 – 4ac = 0
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
Maximum Graph For the case where a is negative (a < 0)
b2 – 4ac > 0
The graph of f(x) cuts the x-axis at two distinct points Two real & distinct roots
b2 – 4ac = 0
The graph of f(x) touches the x-axis at only one point Two real & equal roots
b2 – 4ac < 0
The graph of f(x) does not meet the x-axis. No real roots
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PAPER 1 BREAK BARRIERS TUITION CENTRE – LEARNING MADE PURPOSEFUL & FUN WWW.BREAKBARRIERS.NET
SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
2. Find the range of values of p which satisfies the inequality 2p2 + 7p 4.
3. Find the range of values of m if the equation (2 – 3m)x2 + (4 – m)x + 2 = 0 has no real roots.
4. The QF 4x2 + (12 – 4k)x + 15 – 5k = 0 has two different roots, find the range of values of k.
5. Without using differentiation method, find the minimum value of the function f(x) = 3x2 + x + 2.
6. Given that g(x) = 3x2 – 2x – 8, find the range of values of x so that g(x) is always positive.
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1. Solve the inequality 2(x – 3)2 > 8.
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
8. The QF f(x) = 3 [ (x – 1)2 – 3k ] has a minimum 2 value of 6. Find the value of k.
9. Express y = 1 + 20x – 2x2 in the form y = a(x + p)2 + q. Hence, state i) the minimum value of y, ii) the corresponding value of x.
10. Find the range of values of x for which x(x – 6) ≤ 27.
11. Find the range of the values of x for which 3x(2x – 1) ≤ 2(2x + 5).
12. Solve the quadratic inequality t(15 – 2t) 22.
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7. The expression x2 – x + p, where p is a constant has a minimum value 9/4. Find the value of p.
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
13.
Find the range of values of k if 2x2 + 4x + k is always positive.
14. The minimum value of the QF f(x) = x2 – 10x + k is –10 where k is a constant. Find a) the value of x at which the function f is minimum b) the value of k.
15.
The QF f(x) = x2 + 7x + b has a minimum value of 1/4. Find the value of b.
16. Find the range of values of x for which 5 + 7x – 6x2 0.
18. The QF f(x) = a(x + p)2 = q, where a, p and q are constants, has a maximum value of 5. The equation of axis of symmetry is x = 3. State a) the range of values of a, b) the value of p, c) the value of q. .
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17. The quadratic function f(x) = x2 6x + 5 can be expressed in the form f(x) = (x + m)2 + n where m and n are constants. Find the value of m and of n.
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
19. The following diagram shows the graph of QF y = g(x). The straight line y = -9 is a tangent to the curve y = g(x). y
O
5
x
1
y= 9
a) Write the equation of the axis of symmetry of the curve b) Express g(x) in the form (x + b)2 + c where b and c are constant. 20. The following diagram shows the graph of a quadratic function f ( x) 5 2( x p ) 2 , where p is a constant. The curve y = f(x) has a maximum point at A(1, q), where q is a constant. State y A(1, q)
. 0
x
a) the value of p, b) the value of q, c) the equation of the axis of symmetry.
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21. The following graph shows the curve y = p(x + q)2 = r with the turning point at (4,1). Find the values of p, q and r.
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
22. In the diagram below, (k, 4) is a turning point of a quadratic graph with an equation in the form y = m(x – 1)p + n. Find a) the values of p, n, k and m, b) the equation of the curve formed when the graph shown is reflected on the x-axis.
23. Based on the above information, a) express f(x) in the form of
f(x) = (x + q)2 + r b) Hence, find the maximum point.
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24. Diagram below shows the graph of a QF f(x) = 3(x + p)2 + 2, where p is a constant. The curve y = f(x) has the minimum point (4, q), where q is a constant. State a) the value of p, b) the value of q, c) the equation of the axis of symmetry .
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
25. Diagram below shows the graph of a QF y = 1/2 (x – h)2 + 1, where h is the constant. State
a) the value of h, b) the equation of axis of symmetry, c) the coordinates of minimum point. 26. The QF y = m(x + n)2 + r, where m, n and r are constants, has a minimum value of 2. The equation of the axis of symmetry is x = 1. State a) the range of values of m, b) the value of n, c) the value of r.
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27. In the diagram below, the point A(k, 9) Is the turning point of the curve y = h – (x – 2)2, where h and k are constants. Determine a) the value of h and k, b) the equation of the axis of symmetry .
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
PAPER 2 1. Given the function f(x) = 7 – mx – x2 = 16 – (x + n)2 for all real values of x where m and n are positive, find a) The values of m and n, b) The maximum point of f(x) c) The range of values of x so that f(x) is negative. Hence, sketch the graph of f(x) and state the axis of symmetry.
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2. a) Express QF f(x) = -2x2 + 4x – 3 in the form of a(x + p)2 + q. Hence, state the maximum or minimum value of x. b) Find the range of values of x for which x(x + 4) ≤ 21.
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
3. Given that the QF f(x) = -2x2 – 12x - 23, a) express f(x) in the form m(x + n)2 + p, where m, n and p are constants. b) Determine whether the function f(x) has the minimum or maximum value and state its value.
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4. The QF f(x) = x2 + px +q has a minimum point at (3, -5). a) Without using differentiation, find the values of p and q. b) Hence, find the range of values of x if f(x) – 31 ≤ 0.
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
5. Given the QF f(x) = 6x – 1 – 3x2. a) Express the QF 6x – 1 – 3x2 in the form k + m(x + n)2, where k, m and n are constants. b) Sketch the graph of f(x) c) Find the range of values of p such that 6x – 4 – 3x2 = p has no real roots.
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6. Given that x2 – 3x + 5 = p(x – h)2 + k for all real values of x, where p, h and k are constants. a) State the values of p, h and k, b) Find the minimum or maximum value of x2 – 3x + 5 and the corresponding value of x. c) Sketch a graph of f(x) = x2 – 3x + 5. d) Find the range of values of m such that the equation x2 – 3x + 5 = 2m has two different roots.
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SCHOOL HOLIDAY REVISION COACHING (SHRC) 2013 (PHASE 1)
7. Diagram below shows the curve of a QF f(x) = ½x2 + kx – 6. A is the point of intersection of the quadratic graph and y-axis. The x-intercepts are -6 and 2. y
6
O
2
x
A(0, r)
a) State the value of r and of p.
(p, q)
b) The function can be expressed in the form f(x) = ½(x – p)2 + q, find the value of q and of k.
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c) Determine the range of values of x if f(x) < -6.
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