2018 Paper 1

•¹ start differentiation
$\displaystyle \frac{1}{\sqrt{1-(3x)^2}} \times \dots$

•² apply chain rule and complete differentiation
$\displaystyle \dots \frac{3}{\sqrt{1-9x^2}}$

•³ evidence use of quotient rule with denominator and one term of numerator correct
$\displaystyle \frac{35(7x+1)e^{5x}-\dots}{(7x+1)^2}$ OR $\displaystyle \frac{\dots-7e^{5x}}{(7x+1)^2}$

•⁴ complete differentiation
$\displaystyle \frac{5(7x+1)e^{5x}-7e^{5x}}{(7x+1)^2}$

•⁵ start to differentiate product with one term correct
$\displaystyle \frac{dy}{dx}\cos x + \dots$ OR $\displaystyle -y\sin x + \dots$

•⁶ complete differentiation of product
$\displaystyle \frac{dy}{dx}\cos x - y\sin x$

•⁷ differentiate remaining terms
$\displaystyle + 2y\frac{dy}{dx} = 6$

•⁸ express derivative explicitly in terms of x and y
$\displaystyle \frac{dy}{dx} = \frac{6+y\sin x}{\cos x+2y}$

•¹ state expression
$\displaystyle \frac{3x-7}{x^2-2x-15} = \frac{A}{x+3} + \frac{B}{x-5}$

•² form equation and find one unknown
$\displaystyle 3x-7 = A(x-5) + B(x+3)$ AND eg $\displaystyle A=2$

•³ find second unknown and write integral expression
$\displaystyle B=1$ AND $\displaystyle \int (\frac{2}{x+3} + \frac{1}{x-5})dx$

•⁴ integrate
$\displaystyle 2\ln|x+3| + \ln|x-5| + c$

•¹ state general term
$\displaystyle \binom{9}{r}(2x)^{9-r}(\frac{5}{x^2})^r$

•² simplify powers of x OR coefficients
$\displaystyle 2^{9-r}5^r$ OR $\displaystyle x^{9-3r}$

•³ state simplified general term
$\displaystyle \binom{9}{r}2^{9-r}5^rx^{9-3r}$

•⁴ determine value of r
$\displaystyle r=3$

•⁵ evaluate term
$\displaystyle 672000$

•¹ state conjugate
$\displaystyle \overline{z}_2 = p+6i$ stated or implied

•² substitute for $\displaystyle z_1$, $\displaystyle \overline{z}_2$, expand and apply $\displaystyle i^2 = -1$
$\displaystyle (2p-18) + (3p+12)i$

•³ find value of p
Value of p found from equating real or imaginary part

•¹ start process
$\displaystyle 306 = 2 \times 119 + 68$

•² obtain remainder of 17
$\displaystyle 68 = 1 \times 51 + 17$

•³ express gcd in terms of 306 and 119
$\displaystyle 17 = -1 \times 119 + 2(306 - 2 \times 119)$

•⁴ obtain a and b
$\displaystyle a=2$, $\displaystyle b=-5$

•¹ find $\displaystyle \frac{dy}{dt}$
$\displaystyle \frac{dy}{dt} = \frac{3}{3t+2}$

•² complete differentiation and relate derivatives
$\displaystyle \frac{dx}{dt} = 2t$ and $\displaystyle \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ stated or implied

•³ evaluate gradient
$\displaystyle \frac{dy}{dx} = -\frac{9}{2}$

•⁴ find coordinates
$\displaystyle x = \frac{10}{9}$, $\displaystyle y = 0$

•⁵ state equation of tangent
$\displaystyle y = -\frac{9}{2}x + 5$

•¹ state transpose of C
$\displaystyle C' = \begin{pmatrix} -2 & 1 & 1 \\ 1 & -1 & 0 \\ 2 & 0 & -1 \end{pmatrix}$ stated or implied

•² obtain matrix
$\displaystyle 2C' - D = \begin{pmatrix} -5 & 1 & 0 \\ -k-1 & -2 & -2 \\ 3 & -1 & -3 \end{pmatrix}$

•³ begin to find determinant
$\displaystyle 1\begin{vmatrix} 1 & 2 \\ 1 & 1 \end{vmatrix} - (k+3)\begin{vmatrix} 1 & 2 \\ 1 & 1 \end{vmatrix} + \dots$

•⁴ simplify expression
$\displaystyle k+3$

•⁵ state value of k
$\displaystyle -3$

•¹ differentiate
$\displaystyle \frac{du}{d\theta} = \cos\theta$

•² find limits for u
$\displaystyle u = \frac{1}{2}$, $\displaystyle u = 1$

•³ rewrite integral
$\displaystyle 2\int_{1/2}^1 u^4 du$

•⁴ integrate and evaluate
$\displaystyle \frac{2}{5}[u^5]_{1/2}^1 = \frac{31}{80}$

•¹ form the sum of three consecutive integers
$\displaystyle (n-1) + n + (n+1)$

•² communication
$\displaystyle 3n$ which is divisible by 3

•³ appropriate form for odd number, decomposed into two consecutive integers
$\displaystyle 2k+1 = k + (k+1)$, $\displaystyle k \in \mathbb{Z}$

•¹ substitute, collect real and imaginary parts and equate moduli
$\displaystyle |x+iy| = |(x-2)+(y+2)i|$

•² process to obtain a linear equation in x and y
eg $\displaystyle y=x-2$

•³ sketch consistent with equation / complete sketch / interpret conditions
A straight line exhibiting bisection, perpendicularity passing through $\displaystyle (2,0)$ and $\displaystyle (0,-2)$

•¹ obtain A
$\displaystyle \begin{pmatrix} \cos\dfrac{\pi}{3} & -\sin\dfrac{\pi}{3} \\[7pt] \sin\dfrac{\pi}{3} & \cos\dfrac{\pi}{3} \end{pmatrix}$

•² obtain B
$\displaystyle \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$

•³ correct order for multiplication
$\displaystyle P = BA$ = $\displaystyle \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \frac{1}{2}\begin{pmatrix} 1 & -\sqrt{3} \\ \sqrt{3} & 1 \end{pmatrix}$

•⁴ multiplication completed and appearance of exact values
$\displaystyle \frac{1}{2}\begin{pmatrix} 1 & -\sqrt{3} \\ -\sqrt{3} & -1 \end{pmatrix}$

•⁵ valid explanation
eg compare the elements of P with the general form of a rotation matrix

•¹ show true for $\displaystyle n=1$
LHS: $\displaystyle 3^0=1$ RHS: $\displaystyle \frac{1}{2}(3^1-1)=1$ So true for $\displaystyle n=1$

•² assume (statement) true for $\displaystyle n=k$ AND consider whether (statement) true for $\displaystyle n=k+1$
Assume $\displaystyle \sum_{r=1}^k 3^{r-1} = \frac{1}{2}(3^k-1)$ AND $\displaystyle \sum_{r=1}^{k+1} 3^{r-1} = \dots$

•³ correct statement for sum to $\displaystyle (k+1)$ terms using inductive hypothesis
$\displaystyle \dots = \frac{1}{2}(3^k-1) + 3^{(k+1)-1}$

•⁴ combine terms in 3
$\displaystyle \frac{3}{2} \times 3^k - \frac{1}{2}$

•⁵ express sum explicitly in terms of $\displaystyle (k+1)$ or achieve stated aim/goal AND communicate
$\displaystyle \frac{1}{2}(3^{(k+1)}-1)$ AND If true for $\displaystyle n=k$ then true for $\displaystyle n=k+1.$ Also shown true for $\displaystyle n=1$ therefore, by induction, true for all positive integers n.

•¹ Determine the relationship between x and h
$\displaystyle x^2 + h^2 = 2500$ or $\displaystyle h = \sqrt{2500-x^2}$

•² interpret rate of change of x
$\displaystyle \frac{dx}{dt} = -0.3$

•³ find $\displaystyle \frac{dh}{dx}$
$\displaystyle \frac{dh}{dx} = -2x \cdot \frac{1}{2}(2500-x^2)^{-1/2}$

•⁴ form relationship
$\displaystyle \frac{dh}{dt} = \frac{dh}{dx} \cdot \frac{dx}{dt}$ stated or implied

•⁵ multiply by $\displaystyle \frac{dx}{dt}$
$\displaystyle \frac{dh}{dt} = \frac{0.3x}{\sqrt{2500-x^2}}$

•⁶ evaluate
$\displaystyle \frac{dh}{dt} = \frac{9}{40} \text{ cm s}^{-1}$

•¹ multiply first term by a power of the common ratio
$\displaystyle 80(\frac{1}{3})^6$

•² find term
$\displaystyle \frac{80}{729}$

•³ substitute
$\displaystyle \frac{80}{1-1/3}$

•⁴ find sum to infinity
$\displaystyle 120$

•⁵ substitute
$\displaystyle \frac{5}{2}(2 \times 80 + (5-1)d) = 240$

•⁶ find common difference
$\displaystyle -16$

•⁷ find simplified expression
$\displaystyle 96 - 16n$

•⁸ set up equation
$\displaystyle \frac{n}{2}[160+(n-1)(-16)] = 144$

•⁹ obtain quadratic equation in general form
$\displaystyle 16n^2 - 176n + 288 = 0$

•¹⁰ find values of n
$\displaystyle n=2$, $\displaystyle n=9$

•¹ start integration by parts
$\displaystyle -\frac{x}{3}\cos 3x - \dots$

•² complete integration by parts
$\displaystyle \dots - \int -\frac{1}{3}\cos 3x dx$

•³ complete integration
$\displaystyle -\frac{x}{3}\cos 3x + \frac{1}{9}\sin 3x + c$

•⁴ identify integral form of integrating factor
$\displaystyle e^{\int -\frac{2}{x} dx}$

•⁵ determine integrating factor
$\displaystyle \frac{1}{x^2}$

•⁶ rewrite as integral equation
$\displaystyle \frac{1}{x^2}y = \int x\sin 3x dx$

•⁷ integrate
$\displaystyle \frac{1}{x^2}y = -\frac{x}{3}\cos 3x + \frac{1}{9}\sin 3x + c$

•⁸ evaluate constant
$\displaystyle c = -\frac{\pi}{3}$

•⁹ form particular solution
$\displaystyle y = -\frac{x^3}{3}\cos 3x + \frac{x^2}{9}\sin 3x - \frac{\pi x^2}{3}$

•¹ set up augmented matrix
$\displaystyle \begin{bmatrix} 1 & -2 & 1 & -4 \\ 3 & -5 & -2 & 1 \\ -7 & 11 & a & -11 \end{bmatrix}$

•² obtain two zeros
$\displaystyle \begin{bmatrix} 1 & -2 & 1 & -4 \\ 0 & 1 & -5 & 13 \\ 0 & -3 & a+7 & -39 \end{bmatrix}$

•³ complete row operations
$\displaystyle \begin{bmatrix} 1 & -2 & 1 & -4 \\ 0 & 1 & -5 & 13 \\ 0 & 0 & a-8 & 0 \end{bmatrix}$

•⁴ obtain value for a
$\displaystyle a=8$

•⁵ introduce parameter and substitute
$\displaystyle z = t$, $\displaystyle y - 5t = 13$

•⁶ equation of line
$\displaystyle x = 22+9t$, $\displaystyle y = 13+5t$, $\displaystyle z = t$

•⁷ write down normals
$\displaystyle \begin{pmatrix} 1 \\ -2 \\ 1 \end{pmatrix}$, $\displaystyle \begin{pmatrix} -3 \\ 5 \\ 2 \end{pmatrix}$ stated or implied

•⁸ start to find angle
$\displaystyle \cos \theta = \frac{-11}{\sqrt{38}\sqrt{6}}$ OR $\displaystyle \frac{11}{\sqrt{38}\sqrt{6}}$

•⁹ find acute angle
Accept answer in degrees which rounds to $\displaystyle 43^\circ$

•¹⁰ explanation
Planes $\displaystyle \pi_2$ and $\displaystyle \pi_4$ are parallel because the normal of $\displaystyle \pi_4$ is a multiple of the normal of $\displaystyle \pi_2.$

•¹ first derivative and two evaluations OR all three derivatives OR all four evaluations OR write down Maclaurin series for $\displaystyle e^x$
$\displaystyle f(x) = e^{2x}$ , $\displaystyle f(0)=1$
$\displaystyle f'(x) = 2e^{2x}$ , $\displaystyle f'(0)=2$
$\displaystyle f''(x) = 4e^{2x}$ , $\displaystyle f''(0)=4$
$\displaystyle f'''(x) = 8e^{2x}$ , $\displaystyle f'''(0)=8$

•² obtain expression OR substitute
$\displaystyle f(x) = 1 + 2x + 2x^2 + \frac{4}{3}x^3\dots$

•³ find $\displaystyle g''(x)$
$\displaystyle g''(x) = 2\sec x\sec x\tan x$

•⁴ evidence of product rule
$\displaystyle g'''(x) = 2\sec^2 x(\dots) + (\dots)\tan x$

•⁵ complete proof
$\displaystyle g'''(x) = 2\sec^2 x(\sec^2 x) + (4\sec^2 x\tan x)\tan x$

•⁶ completes ALL evaluations
$\displaystyle g(0)=0$, $\displaystyle g'(0)=1$, $\displaystyle g''(0)=0$, $\displaystyle g'''(0)=2$

•⁷ substitute
$\displaystyle g(x) = x + \frac{1}{3}x^3\dots$

•⁸ multiply expressions
$\displaystyle (1 + 2x + 2x^2 + \dots)(x + \frac{1}{3}x^3\dots)$

•⁹ multiply out and simplify
$\displaystyle x + 2x^2 + \frac{7}{3}x^3\dots$

•¹⁰ write down terms
$\displaystyle 1 + 4x + 7x^2$