## Question 6:

| Answer | Marks | Guidance |
|--------|-------|----------|
| State, at some stage, $a(4+b)^{\frac{1}{2}}=18$ | B1 | |
| Obtain derivative $\frac{4}{4x-7}$ for $C_1$ | B1 | |
| Obtain derivative $kx(x^2+b)^{-\frac{1}{2}}$ for $C_2$ | M1 | Any non-zero constant $k$ |
| Obtain correct $ax(x^2+b)^{-\frac{1}{2}}$ | A1 | |
| Equate derivatives with $x=2$; Attempt values of $a$ and $b$ from two equations involving $a$ and $(4+b)^{\frac{1}{2}}$ | M1, M1 | Using correct process; Correct equations are $a(4+b)^{\frac{1}{2}}=18$ and $2a(4+b)^{-\frac{1}{2}}=4$ |
| Obtain $a=6$ | A1 | |
| Obtain $b=5$ | A1 [8] | |

## Question 7i:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Draw more or less correct sketch of $y=\cos^{-1}x$ existing in first and second quadrants | *B1 | Ignore any curve outside $0 \leq y \leq \pi$; condone no or wrong intercepts on axes |
| Draw U-shaped parabola passing through origin and showing minimum point | *B1 | Curve must exist in first and third quadrants |
| Indicate one intersection in first quadrant by blob or reference in words or $\ldots$ | B1 [3] | Dep *B *B |

## Question 7ii:

| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain correct first iterate showing at least 4 s.f. rounded or truncated | B1 | |
| Show iterative process to produce at least three iterates in all showing at least 3 s.f. | M1 | Implied by incorrect values apparently converging | 0.25, 0.23965…, 0.24250…, 0.24172…, 0.24193… |
| Obtain at least four correct iterates in all showing at least 4 s.f. | A1 | Allowing recovery after error |
| Conclude with value 0.242 | A1 [4] | Answer to be clearly indicated by underlining final value in sequence or by separate statement; answer required to precisely 3 s.f.; allow final A1 even if iterates have been shown to only 3 s.f.; answer only earns 0/4 |

## Question 7iii:

| Answer | Marks | Guidance |
|--------|-------|----------|
| State $y=-\cos^{-1}(-x)$ or $y=\cos^{-1}x - \pi$ | B1 | |
| State $y=x(-2x+5)$ or equiv | B1 | Allow $y=--x(2(-x)+5)$ or similar; condone missing $y=$ in each case |
| State $-0.242$ for $x$-coordinate | B1 FT | Following their answer to (ii); allow greater accuracy here |
| State $-1.33$ for $y$-coordinate | B1 [4] | Allow value rounding to $-1.33$ |

## Question 8i:

| Answer | Marks | Guidance |
|--------|-------|----------|
| State range of f is $f(x) \geq 3a$ or $y \geq 3a$ | B1 | Allow $f \geq 3a$ or equiv expression in words but $3a$ to be included |
| State range of g is all real numbers or equiv such as $y \in \mathbb{R}$ (real numbers) | B1 [2] | |

## Question [Part ii]:

| Answer | Mark | Guidance |
|--------|------|----------|
| State function is not $1-1$ or different $x$-values give same $y$-value or equiv | B1 | No credit for 'no inverse due to modulus' nor for 'cannot be reflected across $y=x$' |
| Obtain form $k(y+4a)$ or $k(x+4a)$ | M1 | For non-zero constant $k$ |
| Obtain $\frac{1}{5}(x+4a)$ or $\frac{1}{5}x+\frac{4}{5}a$ | A1 **[3]** | Must finally be in terms of $x$ |

## Question [Part iii]:

| Answer | Mark | Guidance |
|--------|------|----------|
| **Either** Attempt composition of functions the right way round | M1 | Earned for $5$(what they think $f(x)$ is) $-4a$ |
| Obtain $5\|2x+a\|+11a=31a$ or equiv | A1 | |
| **Or** Apply their $g^{-1}$ to $31a$ | M1 | |
| Obtain $\|2x+a\|+3a=7a$ or equiv | A1 | |
| **Either** Solve $2x+a=4a$ and obtain $\frac{3}{2}a$ | B1 FT | Following their $\|2x+a\|=ka$ |
| Solve linear equation in which signs of (their) $2x$ and (their) $4a$ are different | M1 | Condone other sign slips |
| Obtain $-\frac{5}{2}a$ | A1 | And no others; obtaining $-\frac{5}{2}a$ and then concluding $\frac{5}{2}a$ is A0 |
| **Or** Square both sides and obtain $4x^2+4ax-15a^2=0$ | B1 FT | Following their $\|2x+a\|=ka$ |
| Solve 3-term quadratic equation to obtain two values | M1 | Allow M1 if factorisation wrong but expansion gives correct first and third terms; allow M1 if incorrect use of formula involves only one error |
| Obtain $-\frac{5}{2}a,\ \frac{3}{2}a$ | A1 **[5]** | And no others; continuing from two correct answers to conclude $\frac{5}{2}a,\ \frac{3}{2}a$ is A0 |

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## Question 9i:

| Answer | Mark | Guidance |
|--------|------|----------|
| Use $\sin 2\theta = 2\sin\theta\cos\theta$ | B1 | |
| State $\frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta}$ or $\tan\theta+\frac{1}{\tan\theta}$ | B1 | Perhaps as part of expression |
| Simplify using correct identities | M1 | Note that going directly from $2\sin^2\theta+2\cos^2\theta$ to 2 is M0 but $2(\sin^2\theta+\cos^2\theta)$ to 2 is M1A1 |
| Obtain 2 correctly | A1 **[4]** | AG; necessary detail needed |

## Question 9ii(a):

| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain expression involving at least one of $\sin\frac{1}{6}\pi$ and $\sin\frac{1}{4}\pi$ | M1 | |
| Obtain $\frac{2}{\sin\frac{1}{6}\pi}+\frac{2}{\sin\frac{1}{4}\pi}$ | A1 | Or equiv involving cosecant |
| Obtain $4+2\sqrt{2}$ or exact equiv | A1 **[3]** | Answer only is 0/3 |

## Question 9ii(b):

| Answer | Mark | Guidance |
|--------|------|----------|
| Use $\sin 4\theta = 2\sin 2\theta\cos 2\theta$ | B1 | |
| Obtain $\cos 2\theta = \frac{1}{4}$ or $\cos^2\theta=\frac{5}{8}$ or $\sin^2\theta=\frac{3}{8}$ | B1 | |
| Obtain $0.659$ or $0.66$ | B1 **[3]** | Or greater accuracy; and no others between 0 and $\frac{1}{2}\pi$; allow $0.21\pi$ but not $0.659\pi$; answer only earns 0/3 |

## Question 9ii(c):

| Answer | Mark | Guidance |
|--------|------|----------|
| Express in form $k_1\sin^4\theta\times\frac{k_2}{\sin^3\theta}$ | M1 | |
| Obtain $4\sin^4\theta\times\frac{8}{\sin^3\theta}$ and hence $32\sin\theta$ | A1 **[2]** | A0 if $(-2\sin^2\theta)^2$ involved in simplification |