| DR | |
$e^x = 3 + 2e^y$ | M1 | Attempt to eliminate one variable
$(3 + 2e^y)^2 - 4e^{2y} = 33$ | A1 | Obtain correct equation in one variable – allow unsimplified
| | or $e^{2x} - 4(0.5e^x - 1.5)^2 = 33$
$9 + 12e^y + 4e^{2y} - 4e^{2y} = 33$ | M1 | Simplify and attempt to solve
$12e^y = 24$ | | or $6e^y = 42$ etc
$e^y = 2$ | A1 | Obtain $y = \ln 2$
$y = \ln 2$ | |
$e^x - 4 = 3$ | A1 | Obtain $x = \ln 7$, using either equation
$e^x = 7$ | [5] |
$x = \ln 7$ | |

## Question 5(a):

$f(x + h) - f(x) = [(x + h)^2 - 4(x + h)] - [x^2 - 4x]$ | M1 | Attempt to simplify $f(x+h) - f(x)$
$= x^2 + 2xh + h^2 - 4x - 4h - x^2 + 4x$ | |
$= 2xh + h^2 - 4h$ | A1 | Correct expression for $f(x+h) - f(x)$
$\frac{f(x+h) - f(x)}{h} = \frac{2xh + h^2 - 4h}{h}$ | M1 | Attempt $\frac{f(x+h) - f(x)}{h}$
$= 2x + h - 4$ | A1 | Obtain correct expression
$f'(x) = \lim_{h \to 0}(2x + h - 4) = 2x - 4$ | A1 | Complete proof by considering limit as $h \to 0$
| [5] |

## Question 5(b):

$y = x^2 - 4x + c$ | B1 | Correct equation, including $c$
$7 = 4 - 8 + c$ | M1 | Attempt to find $c$
$c = 11$ | A1 | Obtain correct equation
$y = x^2 - 4x + 11$ | [3] |

## Question 6(a):

Geometric sequence, as multiplying by a common ratio each time | B1 | Identify geometric with reasoning
| [1] | Allow GP or similar

## Question 6(b):

$u_{20} = 500 \times 0.8^{19}$ | M1 | Attempt $u_{20}$ using $ar^{n-1}$, with $a = 500$ and $r = 0.8$
| | DR so method must be seen
$= 7.21$ | A1 | Obtain 7.21 or better (7.205759)
| [2] |

## Question 6(c):

$S_{20} = \frac{500(1 - 0.8^{20})}{1 - 0.8}$ | M1 | Attempt $u_{20}$ using correct formula, with $a = 500$ and $r = \pm 0.8$
| | DR so method must be seen
$= 2471$ | A1 | Obtain 2471 or better (2471.17696)
| [2] |

## Question 6(d):

$\frac{u_k}{1 - 0.8} = 1024$ | M1 | Attempt to use correct $S_\infty$ formula, equate to 1024 and attempt $u_k$
| | OR attempt $S_\infty - S_{k-1} = 1024$
$u_k = \frac{1024}{5}$ | A1 | Obtain correct value for first term in this sequence
| | Could use other notation, OR obtain correct unsimplified equation
$500 \times 0.8^{k-1} = \frac{1024}{5}$ | M1 | Equate $500 \times 0.8^{k-1}$ to their value for $u_k$ and rearrange to $0.8^{k-1} = c$
$0.8^{k-1} = \frac{256}{625}$ | |
$k - 1 = \log_{0.8}\left(\frac{256}{625}\right) = 4$ | M1 | Correct use of logs to attempt $k - 1$
| | Allow M1 if using logs to solve $0.8^k = \frac{256}{625}$
$k = 5$ | A1 | Obtain $k = 5$
| [5] | DR so method must be seen

## Question 7(a):

$\frac{dV}{dt} = -kV$ | B1 | Set up correct differential equation
| | Allow $k$ for $-k$
$-20 = -k \times 500$ so $k = 0.04$ | B1 | Correct value for $k$ – may be seen later
| | Or $k = -0.04$
$\int -0.04dt = \int\frac{1}{V}dV$ | M1 | Separate variables and attempt integration
$-0.04t = \ln V + c$ | A1 | Correct integral – could still be in terms of $k$
| | Accept no + c here
$c = -\ln 500$ | M1* | Use $t = 0, V = 500$ to find $c$
$-0.04t = \ln 250 - \ln 500$ | M1dep* | Attempt to find $t$ when $V = 250$
$t = 17.3$ hours | A1 | Obtain 17.3 hours, or better (17 hours and 20 minutes)
| [7] | Units needed

### Question 7(a) - Alternate method:

$\frac{dV}{dt} = -kV$ | B1 | Set up correct differential equation
$-20 = -k \times 500$ so $k = 0.04$ | B1 | Correct value for $k$ – may be seen later
| | Or $k = -0.04$
$\int_0^T -0.04dt = \int_{500}^{250} \frac{1}{V}dV$ | M1 | Separate variables and attempt integration of LHS
| | OR Use of $t = 0, V = 500$
$-0.04T = -0.693...$ | M1 | Use of $t = 0, V = 500$
$T = 17$ hours | M1 | Use of $t = T, V = 250$ (accept $t = t$)
| | Use of V limits 500 and 250 (either way round)
| [7] | Units needed
| | 17.3286…

## Question 7(b):

E.g. Assumes that temperature remains constant | B1 | Any valid assumption made
E.g. Assume that the snowball remains a sphere throughout | [1] |

## Question 7(c):

Not very realistic as volume never equals 0, so snowball never melts completely | B1 | Consider long term prediction
| [1] |

## Question 8(a):

$(1 + 2x)^3 = 1 + x$ | B1 | Obtain correct first two terms
| | Must be simplified
$+ \frac{(\frac{1}{2})(-\frac{1}{2})(2x)^2}{2} + \frac{(\frac{1}{2})(-\frac{1}{2})(-\frac{3}{2})(2x)^3}{6}$ | M1 | Attempt at least one more term
$= 1 + x - \frac{1}{2}x^2 + \frac{1}{2}x^3$ | A1 | Obtain correct third term
| | Must be simplified
| A1 | Obtain correct fourth term
| [4] | Must be simplified

## Question 8(b):

$(1 + 9x^2)^{-1} = 1 - 9x^2$ | B1 | Correct expansion soi
$(1 - 9x^2)(1 + x - \frac{1}{2}x^2 + \frac{1}{2}x^3)$ | M1 | Attempt expansion
$= 1 + x - \frac{19}{2}x^2 - \frac{17}{2}x^3$ | A1FT | Obtain correct expansion
| [3] | FT their (i) – must be 4 terms

## Question 8(c):

$(1 + 2x)^{\frac{1}{2}} \Rightarrow |x| < \frac{1}{4}$ | M1 | At least one correct condition seen
$(1 + 9x^2)^{-1} \Rightarrow |x| < \frac{1}{3}$ | | oe
hence $|x| < \frac{1}{3}$ | A1 | Correct conclusion, from both correct conditions
| [2] | oe

## Question 9(a):

$f'(x) = -12x(x^2 + a)^{-2}$ | M1 | Attempt differentiation to obtain $kx(x^2 + a)^{-2}$
| A1 | Obtain fully correct derivative
For $x > 0, -12x < 0$ and $(x^2 + a)^2 > 0$ | M1 | Attempt to show that $f'(x) < 0$
negative divided by positive is always negative, hence function is decreasing | E1 | Fully convincing argument
| [4] |

## Question 9(b):

$f''(x) = -12(x^2 + a)^{-2} + 48x^2(x^2 + a)^{-3}$ | M1 | Attempt use of product, or quotient, rule
$f''(x) = 0$ | A1 | Obtain correct expression
(and $f'(x) \neq 0$ since $f'(x) = 0$ only when $x = 0$) | B1 | Identify condition for a point of inflection
| | Seen or implied
$-12(x^2 + a) + 48x^2 = 0$ | M1 | Attempt correct process to solve for $x$
$36x^2 = 12a$ | |
$x^2 = \frac{a}{3}$ | |
$x = \sqrt{\frac{a}{3}}, y = \frac{9}{2a}$ | A1 | Obtain correct coordinates
| [5] | A0 if $x = \pm\sqrt{\frac{a}{3}}$

## Question 10(a):

At $A, y = 0$ so $-\tan\frac{h}{2}\theta = -\frac{1}{4}$ | M1 | Attempt $x$-coordinate at $A$
$\frac{2}{3}x - \frac{1}{4}\pi = 0$ | |
$x = \frac{3}{8}\pi$ so $A$ is $(\frac{3}{8}\pi, 0)$ | A1 | Obtain $(\frac{3}{8}\pi, 0)$
| | Allow decimal equiv 2.094...
At $B, x = 0$ so $y = 0.808$ | B1 | Obtain (0, 0.808), or better
| [3] |

## Question 10(b):

reflection in the $x$-axis | B1 | Stated at any point
Translation by $\pm\frac{1}{4}\pi$ and stretch by sf 2 or $\frac{1}{2}$, both in the $x$ direction | M1 | Translation in the $x$ direction by $\frac{1}{4}\pi$ (or $\pm\frac{1}{4}\pi$) and stretch by sf 2 or $\frac{1}{2}$, both in the $x$ direction
translation in the $x$ direction by $\frac{1}{4}\pi$ | A1 | Must use 'factor' or 'scale factor'
then stretch in $x$ direction by sf 2 | | Allow stretch then translation, as long as details are commensurate with order
| [3] |

## Question 10(c):

$0 < 0.808$ | M1 | Substitute $x = 0$ and $x = 1$ into both sides of the equation
$1 > 0.501$ | |
change in inequality sign hence $0 <$ root $< 1$ | E1 | Conclude appropriately
| [2] | Refer to change in sign

## Question 10(d):

eg $x_1 = 0.5, x_2 = 0.6730, 0.6179, 0.6360, 0.6301, 0.6320, 0.6314, 0.6316, 0.6315, 0.6315...$ | B1 | Correct first iterate for $0 < x < 1$
hence root is 0.632 | M1 | Attempt correct iterative process
| | At least 2 more values
| A1 | Obtain root as 0.632
| [3] | Must be 3sf

## Question 10(e):

add $y = x$ to diagram in P.A.B. and show first iteration | M1 | Vertical line from $x_1$ and horizontal line to $y = x$
at least 4 more lines to show cobweb | A1 | ie 2 vertical and 2 horizontal lines
| [2] |

## Question 11(a):

$\frac{(x - 4)(x + 3) + (3x + 1)(x + 2)}{(x + 2)(x - 1)(x + 3)}$ | M1 | Attempt to use a common denominator
| | Could be quartic denominator if repeated factor not spotted
$= \frac{x^2 - x - 12 + 3x^2 + 7x + 2}{(x + 2)(x - 1)(x + 3)}$ | A1 | Obtain correct unsimplified fraction
$= \frac{4x^2 + 6x - 10}{(x + 2)(x - 1)(x + 3)}$ | M1 | Expand and simplify numerator
$= \frac{2(2x + 5)(x - 1)}{(x + 2)(x - 1)(x + 3)}$ | M1 | Attempt to factorise numerator
$= \frac{2(2x + 5)}{(x + 2)(x + 3)}$ | A1 | Obtain given answer, with sufficient detail shown
| [5] | AG

## Question 11(b):

$\int f(x)dx = 2\ln(x^2 + 5x + 6)$ | M1 | Obtain integral of $k\ln(x^2 + 5x + 6)$
| | no need for modulus signs or using partial fractions or $2(\ln(x+3) + \ln(x+2))$
$2\ln[(a + 4)^2 + 5(a + 4) + 6)] - 2\ln(a^2 + 5a + 6)]$ | A1 | Obtain correct $2\ln(x^2 + 5x + 6)$
$2\ln\frac{a^2 + 13a + 42}{a^2 + 5a + 6} = 2\ln 3$ | M1 | Attempt use of limits
$a^2 + 13a + 42 = 3$ | M1 | Equate to 2ln3 and remove logs
$a^2 + 5a + 6$ | | Using valid method
$2a^2 + 2a - 24 = 0$ | A1 | Obtain correct three term quadratic
$2(a + 4)(a - 3) = 0$ | M1 | Attempt to solve quadratic
$a = 3$ | A1 | Obtain $a = 3$ only
| [7] | DR so method must be seen
| | A0 if $a = -4$ also given

## Question 12(a):

$\frac{\tan 2\theta + \tan\theta}{1 - \tan 2\theta\tan\theta}$ | B1 | Correct expression
$\frac{\frac{2\tan\theta}{1 - \tan^2\theta} + \tan\theta}{1 - \frac{2\tan\theta}{1 - \tan^2\theta}\tan\theta}$ | B1 | Correct expression in terms of $\tan\theta$
$= \frac{2\tan\theta + \tan\theta(1 - \tan^2\theta)}{(1 - \tan^2\theta) - 2\tan^2\theta}$ | M1 | Attempt to simplify
$= \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}$ | A1 | Complete proof to show given identity convincingly
| [4] | AG

## Question 12(b):

$3 \times \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta} = \tan\theta + k$ | M1 | Equate and attempt rearrange
$9\tan\theta - 3\tan^3\theta = (\tan\theta + k)(1 - 3\tan^2\theta)$ | |
$9\tan\theta - 3\tan^3\theta = \tan\theta - 3\tan^3\theta + k - 3k\tan^2\theta$ | |
$3k\tan^2\theta + 8\tan\theta - k = 0$ | A1 | Correct 3 term quadratic
$b^2 - 4ac = 64 + 12k^2$ | A1FT | Correct discriminant
| | Could be within quadratic formula
$k^2 > 0$, so $64 + 12k^2 > 0$ so equation will always have two distinct roots | M1 | Consider sign of correct discriminant and hence number of roots
$\tan\theta = c$ will always give one value for $\theta$, which will be between $0°$ and $90°$ for $c > 0$ and between $90°$ and $180°$ if $c < 0$ | A1 | Conclude by justifying two values for $\theta$
so two distinct roots for $\tan\theta$ will always give two values for $\theta$ between $0°$ and $180°$ | [5] |