Moderate -0.5 This is a straightforward substitution question requiring students to recognize the quartic as a quadratic in u, solve u² - 5u + 4 = 0 (which factors easily), then back-substitute to find x. The substitution is explicitly given, making this easier than average but still requiring multiple algebraic steps and careful manipulation.
Use the given substitution to obtain a quadratic or factorise into 2 brackets each containing \((3x-2)^2\); No marks if evidence of "square rooting" e.g. "\((3x-2)^2 - 5(3x-2) + 2\) (or 4) \(= 0\)"; No marks if straight to quadratic formula to get \(x =\) "1" \(x =\) "4" and no further working
\((u-1)(u-4) = 0\)
DM1
Correct method to solve a quadratic
\(u = 1\) or \(u = 4\)
A1
Correct values for \(u\); SR 1) If M0 Spotted solutions www B1 each; Justifies 4 solutions exactly B2
\(3x - 2 = \pm 1\) or \(3x - 2 = \pm 2\)
M1
Attempt to square root and rearrange to obtain \(x\) OR expand, rearrange and solve quadratic (at least one); SR 2) If first 3 marks awarded, spotted solutions 2 correct B1; Other 2 correct B1; Justifies 4 solutions exactly B1
\(x = 1\) or \(\frac{1}{3}\) or \(\frac{4}{3}\) or \(0\)
A1
2 correct values
A1
6 marks (total 6); All 4 correct values (\(\frac{0}{3} =\) A0)
*Alternative scheme for candidates who multiply out:*
Answer
Marks
Guidance
Answer
Mark
Guidance
Attempt to expand \((3x-2)^4\) and \((3x-2)^2\)
M1
\(81x^4 - 216x^3 + 171x^2 - 36x = 0\)
A1
\(x = 0\) a solution or \(x\) a factor of the quartic
A1
Attempt to use factor theorem to factorise their cubic
M1*
Correct method to solve quadratic
DM1
All 4 solutions correct
A1
# Question 4:
| Answer | Mark | Guidance |
|--------|------|----------|
| $u^2 - 5u + 4 = 0$ | M1* | Use the given substitution to obtain a quadratic or factorise into 2 brackets each containing $(3x-2)^2$; **No marks** if evidence of "square rooting" e.g. "$(3x-2)^2 - 5(3x-2) + 2$ (or 4) $= 0$"; **No marks** if straight to quadratic formula to get $x =$ "1" $x =$ "4" and no further working |
| $(u-1)(u-4) = 0$ | DM1 | Correct method to solve a quadratic |
| $u = 1$ or $u = 4$ | A1 | Correct values for $u$; **SR 1)** If **M0** Spotted solutions **www B1** each; Justifies 4 solutions exactly **B2** |
| $3x - 2 = \pm 1$ or $3x - 2 = \pm 2$ | M1 | Attempt to square root and rearrange to obtain $x$ **OR** expand, rearrange and solve quadratic (at least one); **SR 2)** If first 3 marks awarded, spotted solutions 2 correct **B1**; Other 2 correct **B1**; Justifies 4 solutions exactly **B1** |
| $x = 1$ or $\frac{1}{3}$ or $\frac{4}{3}$ or $0$ | A1 | 2 correct values |
| | A1 | **6 marks** (total **6**); All 4 correct values ($\frac{0}{3} =$ **A0**) |
*Alternative scheme for candidates who multiply out:*
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt to expand $(3x-2)^4$ and $(3x-2)^2$ | M1 | |
| $81x^4 - 216x^3 + 171x^2 - 36x = 0$ | A1 | |
| $x = 0$ a solution or $x$ a factor of the quartic | A1 | |
| Attempt to use factor theorem to factorise their cubic | M1* | |
| Correct method to solve quadratic | DM1 | |
| All 4 solutions correct | A1 | |
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