Pre-U Pre-U 9794/2 2018 June — Question 4 12 marks

Exam BoardPre-U
ModulePre-U 9794/2 (Pre-U Mathematics Paper 2)
Year2018
SessionJune
Marks12
TopicIndices and Surds
TypeSolve power equations
DifficultyModerate -0.3 This is a straightforward surd equation requiring a standard substitution (u = √x) to convert to a quadratic, then solving and back-substituting. The algebraic manipulation to express the answer in the required form is routine. Slightly easier than average as it follows a well-practiced technique with no conceptual surprises.
Spec1.02b Surds: manipulation and rationalising denominators1.02f Solve quadratic equations: including in a function of unknown
4 Solve the equation \(x + 2 \sqrt { x } - 6 = 0\), giving your answer in the form \(x = c + d \sqrt { 7 }\) where \(c\) and \(d\) are integers.
Question 4
- \(u^2 + 2u - 6 = 0\) M1 Attempt to rewrite equation as a quadratic e.g. using \(u = \sqrt{x}\)
- \((u+1)^2 - 7 = 0\); \(u+1 = \sqrt{7}\) M1 Dep on first M mark. Attempt to solve resulting quadratic
- \(u = -1 + \sqrt{7}\) A1 Allow unsimplified equiv. Could still be \(-1 \pm \sqrt{7}\)
- \(x = (-1+\sqrt{7})^2\) M1 Recognise that root(s) need to be squared to obtain \(x\)
- \(x = (1 - 2\sqrt{7} + 7)\) M1 Attempt correct method to square two term surd
- \(x = 8 - 2\sqrt{7}\) A1 Obtain \(8 - 2\sqrt{7}\) only
OR
- \((x-6)^2 = (-2\sqrt{x})^2\); \(x^2 - 12x + 36 = 4x\) M1 Rearrange to appropriate form and attempt to square both sides. M0 for squaring term by term
- \(x^2 - 16x + 36 = 0\) M1 Gather like terms
- Obtain correct quadratic A1
- \(x = \frac{16 - \sqrt{256-144}}{2}\) M1 Dep on first M mark. Attempt to solve quadratic – any valid method. Condone \(\pm\) in quadratic formula, but not \(+\)
- Attempt to simplify to required form M1
- \(x = 8 - 2\sqrt{7}\) A1 Obtain \(8 - 2\sqrt{7}\) only
**Question 4**

- $u^2 + 2u - 6 = 0$ **M1** Attempt to rewrite equation as a quadratic e.g. using $u = \sqrt{x}$
- $(u+1)^2 - 7 = 0$; $u+1 = \sqrt{7}$ **M1** Dep on first M mark. Attempt to solve resulting quadratic
- $u = -1 + \sqrt{7}$ **A1** Allow unsimplified equiv. Could still be $-1 \pm \sqrt{7}$
- $x = (-1+\sqrt{7})^2$ **M1** Recognise that root(s) need to be squared to obtain $x$
- $x = (1 - 2\sqrt{7} + 7)$ **M1** Attempt correct method to square two term surd
- $x = 8 - 2\sqrt{7}$ **A1** Obtain $8 - 2\sqrt{7}$ only

OR

- $(x-6)^2 = (-2\sqrt{x})^2$; $x^2 - 12x + 36 = 4x$ **M1** Rearrange to appropriate form and attempt to square both sides. M0 for squaring term by term
- $x^2 - 16x + 36 = 0$ **M1** Gather like terms
- Obtain correct quadratic **A1**
- $x = \frac{16 - \sqrt{256-144}}{2}$ **M1** Dep on first M mark. Attempt to solve quadratic – any valid method. Condone $\pm$ in quadratic formula, but not $+$
- Attempt to simplify to required form **M1**
- $x = 8 - 2\sqrt{7}$ **A1** Obtain $8 - 2\sqrt{7}$ only
4 Solve the equation $x + 2 \sqrt { x } - 6 = 0$, giving your answer in the form $x = c + d \sqrt { 7 }$ where $c$ and $d$ are integers.

\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2018 Q4 [12]}}
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