Standard +0.3 This is a standard Further Pure modulus equation question requiring systematic case analysis (positive/negative cases), solving resulting quadratics, and checking solutions against constraints. While it involves multiple steps and a sketch, the techniques are routine for FP2 students with no novel insight required. Slightly easier than average A-level due to straightforward algebraic manipulation.
3. (a) Use algebra to find the exact solutions of the equation
$$\left| 2 x ^ { 2 } + x - 6 \right| = 6 - 3 x$$
(b) On the same diagram, sketch the curve with equation \(y = \left| 2 x ^ { 2 } + x - 6 \right|\) and the line with equation \(y = 6 - 3 x\).
(c) Find the set of values of \(x\) for which
$$\left| 2 x ^ { 2 } + x - 6 \right| > 6 - 3 x$$
(3)(Total 12 marks)
(b) Accept if parts (a) and (b) done in reverse order
Curved shape
B1
Line
B1
At least 3 intersections
B1
3 marks total
(c) Using all 4 CVs and getting all into inequalities
M1
\(x > \sqrt{7} - 1, x < -\sqrt{7} - 1\) both
A1 ft
fit their greatest positive and their least negative CVs
\(0 < x < 1\)
A1
3 marks total [12]
**(a)** $2x^2 + x - 6 = 6 - 3x$ | M1 | Rearranging to standard form
Leading to $x^2 + 2x - 6 = 0$ | M1 A1 | Using surds required
$-2x^2 - x + 6 = 6 - 3x$ | M1 |
Leading to $2x^2 - 2x = 0 \Rightarrow x = 0, 1$ | A1 A1 | 6 marks total
**(b)** Accept if parts (a) and (b) done in reverse order | |
Curved shape | B1 |
Line | B1 |
At least 3 intersections | B1 | 3 marks total
**(c)** Using all 4 CVs and getting all into inequalities | M1 |
$x > \sqrt{7} - 1, x < -\sqrt{7} - 1$ both | A1 ft |
fit their greatest positive and their least negative CVs | |
$0 < x < 1$ | A1 | 3 marks total [12]
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3. (a) Use algebra to find the exact solutions of the equation
$$\left| 2 x ^ { 2 } + x - 6 \right| = 6 - 3 x$$
(b) On the same diagram, sketch the curve with equation $y = \left| 2 x ^ { 2 } + x - 6 \right|$ and the line with equation $y = 6 - 3 x$.\\
(c) Find the set of values of $x$ for which
$$\left| 2 x ^ { 2 } + x - 6 \right| > 6 - 3 x$$
(3)(Total 12 marks)\\
\hfill \mbox{\textit{Edexcel FP2 2006 Q3 [12]}}