Question 2 - A-Level Maths

Edexcel F2 2021 October — Question 2 8 marks

Exam Board	Edexcel
Module	F2 (Further Pure Mathematics 2)
Year	2021
Session	October
Marks	8
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Inequalities
Type	Rational inequality algebraically
Difficulty	Standard +0.3 This is a standard rational inequality requiring systematic algebraic manipulation: moving terms to one side, finding a common denominator, factoring, and analyzing sign changes across critical points (x=0, x=2, roots of numerator). While it requires careful attention to inequality direction when multiplying by squared terms and checking multiple intervals, it follows a well-established procedure taught in Further Pure modules with no novel insight required.
Spec	1.02g Inequalities: linear and quadratic in single variable 1.02h Express solutions: using 'and', 'or', set and interval notation

2. Use algebra to determine the set of values of $x$ for which $$\frac { x } { 2 - x } \leqslant \frac { x + 3 } { x }$$ (Solutions relying entirely on graphical methods are not acceptable.)
(8)

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Question 2:

Way 1:

Answer	Marks	Guidance
Working/Answer	Mark	Notes
$\frac{x}{2-x}, \frac{x+3}{x} \Rightarrow \frac{x}{2-x} - \frac{x+3}{x}\ ``"\ 0$	M1	Collects to one side
$\frac{x^2-(2-x)(x+3)}{x(2-x)}\ ``"\ 0$	M1 A1	M1: Attempt common denominator; A1: Correct fraction
$x = 0,\ 2$	B1	These critical values
$x^2-(2-x)(x+3)=0 \Rightarrow 2x^2+x-6=0 \Rightarrow x=\ldots$	M1	Solves the 3TQ in the numerator
$x = \frac{3}{2},\ -2$	A1	These critical values
$x\ ``"\ {-2},\ 0 < x\ ``"\ \frac{3}{2},\ x > 2$	A1 A1	A1: Any 2 correct with strict inequalities allowed; A1: All correct as shown. Ignore connector between inequalities (allow "or","and","," etc. but not $\cap$)

Alternative 1: $\times x^2(2-x)^2$

Answer	Marks	Guidance
Working/Answer	Mark	Notes
$x^3(2-x)\ ``"\ x(x+3)(2-x)^2$	M1	Multiplies by a positive expression
$x^3(2-x) - x(x+3)(2-x)^2\ ``"\ 0$	M1 A1	Collects to one side; Correct inequality
$x = 0,\ 2$	B1	These critical values
$x(2-x)\left[x^2-(x+3)(2-x)\right]=0 \Rightarrow 2x^2+x-6=0 \Rightarrow x=\ldots$	M1	Attempts to factorise taking out $x(2-x)$ and solves resulting 3TQ. May have quartic and apply factor theorem.
$x = \frac{3}{2},\ -2$	A1	These critical values
$x\ ``"\ {-2},\ 0 < x\ ``"\ \frac{3}{2},\ x > 2$	A1 A1	A1: Any 2 correct with strict inequalities allowed; A1: All correct as shown. Ignore connector, but not $\cap$

## Question 2:

**Way 1:**

| Working/Answer | Mark | Notes |
|---|---|---|
| $\frac{x}{2-x}, \frac{x+3}{x} \Rightarrow \frac{x}{2-x} - \frac{x+3}{x}\ ``"\ 0$ | M1 | Collects to one side |
| $\frac{x^2-(2-x)(x+3)}{x(2-x)}\ ``"\ 0$ | M1 A1 | M1: Attempt common denominator; A1: Correct fraction |
| $x = 0,\ 2$ | B1 | These critical values |
| $x^2-(2-x)(x+3)=0 \Rightarrow 2x^2+x-6=0 \Rightarrow x=\ldots$ | M1 | Solves the 3TQ in the numerator |
| $x = \frac{3}{2},\ -2$ | A1 | These critical values |
| $x\ ``"\ {-2},\ 0 < x\ ``"\ \frac{3}{2},\ x > 2$ | A1 A1 | A1: Any 2 correct with strict inequalities allowed; A1: All correct as shown. Ignore connector between inequalities (allow "or","and","," etc. but not $\cap$) |

**Alternative 1: $\times x^2(2-x)^2$**

| Working/Answer | Mark | Notes |
|---|---|---|
| $x^3(2-x)\ ``"\ x(x+3)(2-x)^2$ | M1 | Multiplies by a positive expression |
| $x^3(2-x) - x(x+3)(2-x)^2\ ``"\ 0$ | M1 A1 | Collects to one side; Correct inequality |
| $x = 0,\ 2$ | B1 | These critical values |
| $x(2-x)\left[x^2-(x+3)(2-x)\right]=0 \Rightarrow 2x^2+x-6=0 \Rightarrow x=\ldots$ | M1 | Attempts to factorise taking out $x(2-x)$ and solves resulting 3TQ. May have quartic and apply factor theorem. |
| $x = \frac{3}{2},\ -2$ | A1 | These critical values |
| $x\ ``"\ {-2},\ 0 < x\ ``"\ \frac{3}{2},\ x > 2$ | A1 A1 | A1: Any 2 correct with strict inequalities allowed; A1: All correct as shown. Ignore connector, but not $\cap$ |

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2. Use algebra to determine the set of values of $x$ for which

$$\frac { x } { 2 - x } \leqslant \frac { x + 3 } { x }$$

(Solutions relying entirely on graphical methods are not acceptable.)\\
(8)\\

\hfill \mbox{\textit{Edexcel F2 2021 Q2 [8]}}

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Q1 4 Q2 8 Q3 6 Q4 9 Q5 8 Q6 9 Q7 11 Q8 11 Q9 9

Edexcel F2 2021 October — Question 2 8 marks

Question 2:

Way 1:

Alternative 1: \(\times x^2(2-x)^2\)

This paper (9 questions)