Challenging +1.2 This is a Further Maths modulus inequality requiring systematic case analysis (splitting at critical points where x²-2=0 and considering sign changes), solving two quadratic inequalities, and careful combination of solution sets. More demanding than standard A-level but routine for FP2 students who know the technique.
Form 3TQ and attempt to solve - may be implied by correct value(s) (allow decimals 4.449, -0.449)
\(x = 2 \pm \sqrt{6}\) or \(2 + \sqrt{6}\)
A1
Correct exact values or value (NB: Corresponding 3TQ must have been seen)
\(x^2 - 2 = -4x\) ②
M1
Form 3TQ and attempt to solve - may be implied by correct value(s) (allow decimals -4.449, 0.449)
\(x = -2 \pm \sqrt{6}\) or \(x = -2 + \sqrt{6}\)
A1
Correct exact values or value (NB: Corresponding 3TQ must have been seen)
\(x >\) larger root of ① or \(x <\) larger root of ②
dM1
Forms at least one of the required inequalities using their exact values. Must be a strict inequality. Depends on either previous M mark
One of \(x < -2+\sqrt{6}\) or \(x > 2+\sqrt{6}\)
A1
Or exact equivalent
Both of \(x < -2+\sqrt{6}\) or \(x > 2+\sqrt{6}\)
A1
No others seen. Exact equivalents allowed. Allow "or" or "and" but not \(\cap\) if set notation used
ALT method:
Answer
Marks
Guidance
Answer/Working
Mark
Guidance
\((x^2-2)^2 = 16x^2\)
M1
Square both sides and attempt to solve quadratic in \(x^2\) - may be implied by correct value(s) (allow decimals 19.79, -0.202)
\(x^2 = 10 \pm \sqrt{96}\)
A1
\(x^2 = 10 \pm 4\sqrt{6}\) oe
\(x = 2\pm\sqrt{6}\) and \(x = -2\pm\sqrt{6}\) (\(x = 2+\sqrt{6}\) and \(x = -2+\sqrt{6}\) sufficient)
M1A1
Valid attempt required to find exact form for \(x\) e.g. \((a+\sqrt{b})^2 = 10\pm\sqrt{96}\)
\(x >\) largest root or \(x <\) 2nd largest root
dM1
As main scheme
As main scheme
A1, A1
As main scheme
Total: 7
## Question 4:
$|x^2 - 2| > 4x$
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x^2 - 2 = 4x$ ① | M1 | Form 3TQ **and** attempt to solve - may be implied by correct value(s) (allow decimals 4.449, -0.449) |
| $x = 2 \pm \sqrt{6}$ or $2 + \sqrt{6}$ | A1 | Correct exact values or value (NB: Corresponding 3TQ must have been seen) |
| $x^2 - 2 = -4x$ ② | M1 | Form 3TQ **and** attempt to solve - may be implied by correct value(s) (allow decimals -4.449, 0.449) |
| $x = -2 \pm \sqrt{6}$ or $x = -2 + \sqrt{6}$ | A1 | Correct exact values or value (NB: Corresponding 3TQ must have been seen) |
| $x >$ larger root of ① or $x <$ larger root of ② | dM1 | Forms at least one of the required inequalities using their exact values. Must be a strict inequality. Depends on **either** previous M mark |
| One of $x < -2+\sqrt{6}$ or $x > 2+\sqrt{6}$ | A1 | Or exact equivalent |
| Both of $x < -2+\sqrt{6}$ or $x > 2+\sqrt{6}$ | A1 | No others seen. Exact equivalents allowed. Allow "or" or "and" but not $\cap$ if set notation used |
**ALT method:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x^2-2)^2 = 16x^2$ | M1 | Square both sides **and** attempt to solve quadratic in $x^2$ - may be implied by correct value(s) (allow decimals 19.79, -0.202) |
| $x^2 = 10 \pm \sqrt{96}$ | A1 | $x^2 = 10 \pm 4\sqrt{6}$ oe |
| $x = 2\pm\sqrt{6}$ and $x = -2\pm\sqrt{6}$ ($x = 2+\sqrt{6}$ and $x = -2+\sqrt{6}$ sufficient) | M1A1 | Valid attempt required to find exact form for $x$ e.g. $(a+\sqrt{b})^2 = 10\pm\sqrt{96}$ |
| $x >$ largest root or $x <$ 2nd largest root | dM1 | As main scheme |
| As main scheme | A1, A1 | As main scheme |
**Total: 7**
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