Advanced Higher Maths 2021 Paper 1 — Questions & Marking Instructions

1(a)

Question 1(a)

Differentiation

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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Question 1(a)

•¹ evidence use of product rule
$\displaystyle 3x^2(...) + ...$ OR $\displaystyle ... + 5e^{5x}(...)$

•² complete differentiation
$\displaystyle 3x^2e^{5x} + 5x^3e^{5x}$

1(b)

Question 1(b)

Differentiation

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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Question 1(b)

•³ evidence use of quotient rule with denominator and one term of the numerator correct
$\displaystyle \frac{(x^6+1)\sec^2 x - ...}{(x^6+1)^2}$ OR $\displaystyle \frac{... - 6x^5\tan x}{(x^6+1)^2}$

•⁴ complete differentiation
$\displaystyle \frac{(x^6+1)\sec^2 x - 6x^5\tan x}{(x^6+1)^2}$

2

Question 2

Matrices

(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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Question 2(a)

•¹ state the transpose of B
$\displaystyle B' = \begin{pmatrix} 4 & 2 & -2 \\ 0 & 3 & 1 \end{pmatrix}$

•² calculate AB'
$\displaystyle \begin{pmatrix} -8 & 8 & 8 \\ -12 & 15 & 13 \end{pmatrix}$

Question 2(b)

•³ calculate the determinant of A
$\displaystyle \det A = -2$

•⁴ find the inverse of A
$\displaystyle \frac{1}{2}\begin{pmatrix} -7 & 4 \\ -3 & 2 \end{pmatrix}$

3

Question 3

Integration

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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Question 3

•¹ rewrite integral
$\displaystyle \int u^3 \, du$

•² integrate and rewrite in terms of $\displaystyle \theta$ , including constant of integration
$\displaystyle \frac{1}{4}\sin^4 \theta + c$

4

Question 4

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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Question 4

•¹ set up augmented matrix
$\displaystyle \begin{pmatrix} 1 & 2 & 1 & 5 \\ 3 & -1 & 2 & 4 \\ -2 & 3 & \lambda & -8 \end{pmatrix}$

•² obtain two zeros
e.g. $\displaystyle \begin{pmatrix} 1 & 2 & 1 & 5 \\ 0 & -7 & -1 & -11 \\ 0 & 7 & 2+\lambda & 2 \end{pmatrix}$

•³ complete row operations
e.g. $\displaystyle \begin{pmatrix} 1 & 2 & 1 & 5 \\ 0 & -7 & -1 & -11 \\ 0 & 0 & 1+\lambda & -9 \end{pmatrix}$

•⁴ write down value of $\displaystyle \lambda$
$\displaystyle -1$

5

Question 5

Integration

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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Question 5

•¹ correct form of integral
$\displaystyle \pi\int_{3}^{5} y^2 \, dx$

•² evaluate
$\displaystyle 32\pi$ (cubic units)

6

Question 6

DifferentiationIntegration

(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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Question 6(a)

•¹ start to integrate
$\displaystyle t^3 ...$ or $\displaystyle ... - \frac{1}{-2}e^{-2t}$

•² find expression
$\displaystyle t^3 + \frac{1}{2}e^{-2t} - \frac{1}{2}$

Question 6(b)

•³ differentiate
$\displaystyle 6t + 2e^{-2t}$

•⁴ calculate acceleration
$\displaystyle 2 \text{ ms}^{-2}$

7

Question 7

Functions & Graphs

(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.

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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 showing the shape of the curve approaching the asymptotes, with the negative section reflected above the

\displaystyle x

-axis.
(c)(ii)

\displaystyle 0 < k < 8

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Question 7(a)(i)

•¹ state vertical asymptote
$\displaystyle x = 2$

Question 7(a)(ii)

•² complete algebraic division and restate function
$\displaystyle x + 2 + \frac{4}{x - 2}$

•³ state non-vertical asymptote with justification
e.g. As $\displaystyle x \rightarrow \pm \infty$ , $\displaystyle \frac{4}{x - 2} \rightarrow 0$
$\displaystyle y = x + 2$

Question 7(b)

•⁴ sketch showing shape of curve with approach to asymptotes
Valid sketch showing asymptotes $\displaystyle x = 2$ and $\displaystyle y = x + 2$ with turning points at $\displaystyle (0, 0)$ and $\displaystyle (4, 8).$

Question 7(c)(i)

•⁵ sketch showing shape of curve with approach to asymptotes
Valid sketch showing the negative section of the curve reflected above the $\displaystyle x$ -axis.

Question 7(c)(ii)

•⁶ state values of k
$\displaystyle 0 < k < 8$

8

Question 8

Differential Equations

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 $\displaystyle x = 0.$

Show answer

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

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Question 8

•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
$\displaystyle \frac{dy}{dx} = Ce^{2x} + 2Cxe^{2x}$
$\displaystyle \frac{d^2y}{dx^2} = 4Ce^{2x} + 4Cxe^{2x}$

•⁵ evaluate C
$\displaystyle C = 7$

•⁶ general solution stated or implied
$\displaystyle y = Ae^{2x} + Be^{-3x} + 7xe^{2x}$

•⁷ differentiate
$\displaystyle \frac{dy}{dx} = 2Ae^{2x} - 3Be^{-3x} + 7e^{2x} + 14xe^{2x}$

•⁸ form equations and solve for one constant
$\displaystyle A = 4$ or $\displaystyle B = 1$

•⁹ give particular solution
$\displaystyle y = 4e^{2x} + e^{-3x} + 7xe^{2x}$