Advanced Higher Maths 2021 Paper 1 — Questions & Marking Instructions

Product, quotient, chain rules

2 Marks

2021 P1 Q1(a)

Differentiate $\displaystyle y = x^3e^{5x}.$

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\displaystyle 3x^2e^{5x} + 5x^3e^{5x}

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Mark	Awarded for	Illustrative answer
(a) •¹	evidence use of product rule	$\displaystyle 3x^2(\ldots) + \ldots$ OR $\displaystyle \ldots + 5e^{5x}(\ldots)$
(a) •²	complete differentiation	$\displaystyle 3x^2e^{5x} + 5x^3e^{5x}$

Product, quotient, chain rules

2 Marks

2021 P1 Q1(b)

Given $\displaystyle y = \frac{\tan x}{x^6 + 1}$
find $\displaystyle \frac{dy}{dx}.$

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\displaystyle \frac{(x^6 + 1)\sec^2 x - 6x^5\tan x}{(x^6 + 1)^2}

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Mark	Awarded for	Illustrative answer
(b) •³	evidence use of quotient rule with denominator and one term of the numerator correct	$\displaystyle \dfrac{(x^6 + 1)\sec^2 x - \ldots}{(x^6 + 1)^2}$ OR $\displaystyle \dfrac{\ldots - 6x^5\tan x}{(x^6 + 1)^2}$
(b) •⁴	complete differentiation	$\displaystyle \dfrac{(x^6 + 1)\sec^2 x - 6x^5\tan x}{(x^6 + 1)^2}$

Matrix operationsDeterminant and inverse

(2, 2)4 Marks

2021 P1 Q2

Matrices $\displaystyle A$ and $\displaystyle B$ are defined as follows

$\displaystyle A = \begin{pmatrix} -2 & 4 \\ -3 & 7 \end{pmatrix}$ ,
$\displaystyle B = \begin{pmatrix} 4 & 0 \\ 2 & 3 \\ -2 & 1 \end{pmatrix}.$

(a)Find $\displaystyle AB'$ , where $\displaystyle B'$ is the transpose of $\displaystyle B.$

(b)Find $\displaystyle A^{-1}.$

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(a)

\displaystyle \begin{pmatrix} -8 & 8 & 8 \\ -12 & 15 & 13 \end{pmatrix}

(b)

\displaystyle \frac{1}{2}\begin{pmatrix} -7 & 4 \\ -3 & 2 \end{pmatrix}

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Mark	Awarded for	Illustrative answer
(a) •¹	state the transpose of $\displaystyle B$	$\displaystyle B' = \begin{pmatrix}4 & 2 & -2\\0 & 3 & 1\end{pmatrix}$
(a) •²	calculate $\displaystyle AB'$	$\displaystyle \begin{pmatrix}-8 & 8 & 8\\-12 & 15 & 13\end{pmatrix}$
(b) •³	calculate the determinant of $\displaystyle A$	$\displaystyle \det A = -2$
(b) •⁴	find the inverse of $\displaystyle A$	$\displaystyle \dfrac{1}{2}\begin{pmatrix}-7 & 4\\-3 & 2\end{pmatrix}$

Integration by substitution

2 Marks

2021 P1 Q3

Use the substitution $\displaystyle u = \sin \theta$ to find $\displaystyle \int \cos \theta \sin^3 \theta \, d\theta.$
Write your answer in terms of $\displaystyle \theta.$

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\displaystyle \frac{1}{4}\sin^4 \theta + c

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Mark	Awarded for	Illustrative answer
•¹	rewrite integral	$\displaystyle \displaystyle\int u^3\,du$
•²	integrate and rewrite in terms of $\displaystyle \theta$ , including constant of integration	$\displaystyle \dfrac{1}{4}\sin^4\theta + c$

Systems of Equations

4 Marks

2021 P1 Q4

A system of equations is given by

$\displaystyle x + 2y + z = 5$
$\displaystyle 3x - y + 2z = 4$
$\displaystyle -2x + 3y + \lambda z = -8$

where $\displaystyle \lambda \in \mathbb{R}.$

Use Gaussian elimination to determine the value of $\displaystyle \lambda$ for which this system of equations has no solution.

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\displaystyle -1

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Mark	Awarded for	Illustrative answer
•¹	set up augmented matrix	$\displaystyle \left(\begin{array}{ccc\|c}1 & 2 & 1 & 5\\3 & -1 & 2 & 4\\-2 & 3 & \lambda & -8\end{array}\right)$
•²	obtain two zeros	eg $\displaystyle \left(\begin{array}{ccc\|c}1 & 2 & 1 & 5\\0 & -7 & -1 & -11\\0 & 7 & 2 + \lambda & 2\end{array}\right)$
•³	complete row operations	$\displaystyle \left(\begin{array}{ccc\|c}1 & 2 & 1 & 5\\0 & -7 & -1 & -11\\0 & 0 & 1 + \lambda & -9\end{array}\right)$
•⁴	write down value of $\displaystyle \lambda$	$\displaystyle -1$

Volume of revolution

2 Marks

2021 P1 Q5

A solid is formed by rotating the curve with equation $\displaystyle y = 2\sqrt{x}$ between $\displaystyle x = 3$ and $\displaystyle x = 5$ through $\displaystyle 2\pi$ radians about the $\displaystyle x$ -axis.
Calculate the exact value of the volume of this solid.

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\displaystyle 32\pi

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Mark	Awarded for	Illustrative answer
•¹	correct form of integral	$\displaystyle \pi\displaystyle\int_3^5 y^2\,dx$
•²	evaluate	$\displaystyle 32\pi$ (cubic units)

Rectilinear motion

(2, 2)4 Marks

2021 P1 Q6

The velocity, $\displaystyle v$ ms $\displaystyle ^{-1}$ , of a particle after $\displaystyle t$ seconds is given by $\displaystyle v = 3t^2 - e^{-2t}.$ At time $\displaystyle t = 0$ the displacement of the particle is zero.

(a)Find an expression for the displacement of the particle.

(b)Calculate the acceleration of the particle when $\displaystyle t = 0.$

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(a)

\displaystyle t^3 + \frac{1}{2}e^{-2t} - \frac{1}{2}

(b)

\displaystyle 2

ms

\displaystyle ^{-2}

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Mark	Awarded for	Illustrative answer
(a) •¹	start to integrate	$\displaystyle t^3\ldots$ or $\displaystyle \ldots - \dfrac{1}{-2}e^{-2t}$
(a) •²	find expression	$\displaystyle t^3 + \dfrac{1}{2}e^{-2t} - \dfrac{1}{2}$
(b) •³	differentiate	$\displaystyle 6t + 2e^{-2t}$
(b) •⁴	calculate acceleration	$\displaystyle 2\ \text{ms}^{-2}$

AsymptotesSketching related functions

(1, 2, 1, 1, 1)6 Marks

2021 P1 Q7

A function is defined on a suitable domain by $\displaystyle f(x) = \frac{x^2}{x - 2}.$

(a)For the graph of $\displaystyle y = f(x)$

(i) state the equation of the vertical asymptote
(ii) find the equation of the non-vertical asymptote. Justify your answer.

The turning points on the graph are (0, 0) and (4, 8). There are no other stationary points.

(b)On the diagram provided, sketch the graph of $\displaystyle y = f(x).$

(c)(i) On the diagram provided, sketch the graph of $\displaystyle y = |f(x)|.$ Show all asymptotes.

(c)(ii) State the values of $\displaystyle k$ for which $\displaystyle |f(x)| = k$ has exactly two distinct solutions.

Blank axes for (b) and (c)(i), labelled x and y

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(a)(i)

\displaystyle x = 2

(a)(ii)

\displaystyle y = x + 2

(Justification: As

\displaystyle x \rightarrow \pm \infty

,

\displaystyle \frac{4}{x - 2} \rightarrow 0

)
(b) A sketch showing the shape of the curve approaching the asymptotes (

\displaystyle x = 2

and

\displaystyle y = x + 2

), with turning points at (0, 0) and (4, 8).
(c)(i) A sketch with the negative section reflected above the

\displaystyle x

-axis: the left branch approaches

\displaystyle y = -x - 2

and rises to

\displaystyle +\infty

as

\displaystyle x \rightarrow 2^-

, and the right branch is unchanged, with its minimum at (4, 8). Asymptotes:

\displaystyle x = 2

,

\displaystyle y = x + 2

and

\displaystyle y = -x - 2.

(c)(ii)

\displaystyle 0 < k < 8

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Mark	Awarded for	Illustrative answer
(a)(i) •¹	state vertical asymptote	$\displaystyle x = 2$
(a)(ii) •²	complete algebraic division and restate function	$\displaystyle x + 2 + \dfrac{4}{x - 2}$
(a)(ii) •³	state non-vertical asymptote with justification	eg As $\displaystyle x \to \pm\infty$ , $\displaystyle \dfrac{4}{x - 2} \to 0$ $\displaystyle \therefore y = x + 2$
(b) •⁴	sketch showing shape of curve with approach to asymptotes	a sketch: the dashed asymptotes $\displaystyle x = 2$ and $\displaystyle y = x + 2$ ; left branch rising along $\displaystyle y = x + 2$ to the maximum at $\displaystyle (0, 0)$ , then falling to $\displaystyle -\infty$ as $\displaystyle x \to 2^-$ ; right branch coming down from $\displaystyle +\infty$ as $\displaystyle x \to 2^+$ to the minimum at $\displaystyle (4, 8)$ , then rising along $\displaystyle y = x + 2$
(c)(i) •⁵	sketch showing shape of curve with approach to asymptotes	a sketch of $\displaystyle y = \|f(x)\|$ : the part below the $\displaystyle x$ -axis reflected, so the left branch runs down along $\displaystyle y = -x - 2$ (the reflection of $\displaystyle y = x + 2$ , drawn dashed) to $\displaystyle (0, 0)$ and then rises to $\displaystyle +\infty$ as $\displaystyle x \to 2^-$ ; the right branch is unchanged, with its minimum at $\displaystyle (4, 8)$ ; asymptotes $\displaystyle x = 2$ , $\displaystyle y = x + 2$ and $\displaystyle y = -x - 2$ drawn
(c)(ii) •⁶	state values of $\displaystyle k$	$\displaystyle 0 < k < 8$

Second order non-homogeneous DE

9 Marks

2021 P1 Q8

Find the particular solution of the differential equation

$\displaystyle \frac{d^2y}{dx^2} + \frac{dy}{dx} - 6y = 35e^{2x}$

given $\displaystyle y = 5$ and $\displaystyle \frac{dy}{dx} = 12$ when $\displaystyle x = 0.$

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\displaystyle y = 4e^{2x} + e^{-3x} + 7xe^{2x}

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Mark	Awarded for	Illustrative answer
•¹	solve auxiliary equation	$\displaystyle m = 2,\ m = -3$
•²	state complementary function	$\displaystyle y = Ae^{2x} + Be^{-3x}$
•³	state particular integral	$\displaystyle y = Cxe^{2x}$
•⁴	differentiate complementary function	$\displaystyle \dfrac{dy}{dx} = Ce^{2x} + 2Cxe^{2x}$ , $\displaystyle \dfrac{d^2y}{dx^2} = 4Ce^{2x} + 4Cxe^{2x}$
•⁵	evaluate $\displaystyle C$	$\displaystyle C = 7$
•⁶	general solution stated or implied by •⁹	$\displaystyle y = Ae^{2x} + Be^{-3x} + 7xe^{2x}$
•⁷	differentiate	$\displaystyle \dfrac{dy}{dx} = 2Ae^{2x} - 3Be^{-3x} + 7e^{2x} + 14xe^{2x}$
•⁸	form equations and solve for one constant	$\displaystyle A = 4$ or $\displaystyle B = 1$
•⁹	give particular solution	$\displaystyle y = 4e^{2x} + e^{-3x} + 7xe^{2x}$

2021 Paper 1

Question 1(a)

Question 1(b)

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8