$A: C = 2m + 10$ | M1 | Either correct equation
$B: C = 2.4m + 3$ | A1 | Both equations correct and variables defined
$2.4m + 3 = 2m + 10$ | M1 | Attempt to solve simultaneously
$0.4m = 7$ | | Attempt to solve their two equations
$m = 17.5$ | A1 | Obtain 17.5 miles
Same cost for a journey of 17.5 miles | [4] | Units needed

## Question 2(a)(i):

$2 + 5 + x + x + (x + 2) + (x + 5) < 44$ or equivalent | B1 | Correct inequality
| [1] | Must be < only

## Question 2(a)(ii):

$x(x + 2) + 10 \geq 45$ or equivalent | B1 | Correct inequality relating to area
| [1] | Must be $\geq$ only

## Question 2(b):

$x < 7.5$ | B1FT | Obtain $x < 7.5$ from linear inequality
| | FT their linear inequality in (a)
$x^2 + 2x - 35 \geq 0$ | M1 | Attempt to solve three term quadratic
Critical values are $-7$ and $5$ | | BC: Both values of $x$ needed
$x \leq -7, x \geq 5$ | A1FT | Choose 'outside' region for inequality
but $x$ is a length so $x \geq 5$ | | FT their quadratic inequality in (b), as long as one positive root and one negative root
$\{x: 5 \leq x < 7.5\}$ or $[5, 7.5)$ | B1 | Single correct interval – any correct notation
| [4] | Condone $5 \leq x < 7.5$
| | B1M1A0B1 possible if no reason for rejecting $-7$

## Question 3(a):

| DR | |
$\cos\alpha = \pm\frac{1}{3}\sqrt{5}$ | M1 | Attempt Pythagoras on correct right-angled triangle
| | or $\frac{9}{4} + \cos^2\alpha = 1$ or $\cos^2\alpha = \frac{5}{9}$
$\pm\frac{1}{3}\sqrt{5}$ or eg $\pm\sqrt{\frac{5}{9}}$ | A1 | Obtain $\pm\frac{1}{3}\sqrt{5}$ or eg $\pm\frac{\sqrt{5}}{3}$
| [2] | A0 for answer only as DR

## Question 3(b):

$2\sec^2\beta - 2 - 7\sec\beta + 5 = 0$ | M1 | Attempt to use $\tan^2\beta = \sec^2\beta - 1$
| | Allow equiv methods involving $\cos\beta$
$2\sec^2\beta - 7\sec\beta + 3 = 0$ | A1 | Obtain correct equation
$(\sec\beta - 3)(2\sec\beta - 1) = 0$ | M1 | Attempt to solve quadratic
$\sec\beta = 3, \sec\beta = 0.5$ | |
$\sec\beta \geq 1$ for $0° \leq \beta < 90°$ | A1 | Obtain $\sec\beta = 3$ only
hence $\sec\beta = 3$ | | $\sec\beta = 0.5$ must be seen and discarded with a reason
| [4] | Allow more general reason such as $|\sec\beta| \geq 1$