Advanced Higher Maths 2021 Paper 2 — Questions & Marking Instructions

1

Question 1

Differentiation

2 Marks

2021 P2 Q1

Given $\displaystyle f(x) = 3\sec 2x$ find the exact value of $\displaystyle f'\left(\frac{\pi}{8}\right).$

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\displaystyle 6\sqrt{2}

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Question 1

•¹ differentiate
$\displaystyle f^{\prime}(x)=3\sec 2x \tan 2x \times 2$

•² evaluate
$\displaystyle 6\sqrt{2}$

2

Question 2

Number Theory

(3, 1)4 Marks

2021 P2 Q2

(a)Use the Euclidean algorithm to find integers $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle 105a + 72b = 3.$

(b)Hence find integers $\displaystyle x$ and $\displaystyle y$ such that $\displaystyle 105x + 72y = 360.$

Show answer

(a)

\displaystyle a = 11

,

\displaystyle b = -16

(b)

\displaystyle x = 1320

,

\displaystyle y = -1920

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Question 2(a)

•¹ complete algorithm
$\displaystyle 105 = 72 + 33$
$\displaystyle 72 = 2 \times 33 + 6$
$\displaystyle 33 = 5 \times 6 + 3$
$\displaystyle 6 = 2 \times 3$

•² equates gcd and evidence of substitution
$\displaystyle 3 = 33 - 5 \times (72 - 2 \times 33)$

•³ a and b obtained
$\displaystyle a = 11$ , $\displaystyle b = -16$

Question 2(b)

•⁴ find x and y
$\displaystyle x = 1320$ , $\displaystyle y = -1920$

3

Question 3

Integration

3 Marks

2021 P2 Q3

Use integration by parts to find $\displaystyle \int (2x + 3)\cos 4x \,dx.$

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\displaystyle \frac{1}{4}(2x + 3)\sin 4x + \frac{1}{16}\cos 4x + c

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Question 3

•¹ use integration by parts and start process
$\displaystyle \frac{1}{4}(2x+3)\sin 4x - ...$

•² complete application
$\displaystyle ... - \int \frac{2}{4}\sin 4x \, dx$

•³ complete integration
$\displaystyle \frac{1}{4}(2x+3)\sin 4x + \frac{1}{8}\cos 4x + c$

4

Question 4

Differentiation

(3, 2)5 Marks

2021 P2 Q4

A curve is defined parametrically by $\displaystyle x = \sin^{-1} 2t$ and $\displaystyle y = \tan^{-1} t.$

(a)Find $\displaystyle \frac{dx}{dt}$ and $\displaystyle \frac{dy}{dt}.$

(b)When $\displaystyle t = 0$ find the equation of the tangent to the curve.

Show answer

(a)

\displaystyle \frac{dx}{dt} = \frac{2}{\sqrt{1 - 4t^2}}

\displaystyle \frac{dy}{dt} = \frac{1}{1 + t^2}

(b)

\displaystyle y = \frac{1}{2}x

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Question 4(a)

•¹ start to find dx/dt
$\displaystyle \frac{1}{\sqrt{1-(2t)^2}}$

•² complete dx/dt
$\displaystyle ... \times 2$

•³ state dy/dt
$\displaystyle \frac{1}{1+t^2}$

Question 4(b)

•⁴ begin process
$\displaystyle (0,0)$ OR $\displaystyle \frac{dy}{dx} = \frac{1}{2}$

•⁵ find the equation of the tangent
$\displaystyle y = \frac{1}{2}x$

5

Question 5

Matrices

(2, 2)4 Marks

2021 P2 Q5

A non-singular matrix $\displaystyle A$ satisfies the equation $\displaystyle A^2 = 2A + 5I$ , where $\displaystyle I$ is the identity matrix.

(a)Express $\displaystyle A^4$ in the form $\displaystyle pA + qI$ , where $\displaystyle p, q \in \mathbb{Z}.$

(b)Express $\displaystyle A^{-1}$ in the form $\displaystyle rA + sI$ , where $\displaystyle r, s \in \mathbb{Q}.$

Show answer

(a)

\displaystyle 28A + 45I

(b)

\displaystyle \frac{1}{5}A - \frac{2}{5}I

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Question 5(a)

•¹ find expression for A⁴ in powers of two and less
$\displaystyle 4A^2 + 20A + 25I$ OR $\displaystyle 9A^2 + 10A$

•² substitute for A² and simplify
$\displaystyle 28A + 45I$

Question 5(b)

•³ evidence of strategy
e.g. $\displaystyle A^{-1}A^2 = 2A^{-1}A + 5A^{-1}I$ OR $\displaystyle A(A-2I) = 5I$

•⁴ state expression
$\displaystyle \frac{1}{5}A - \frac{2}{5}I$

6

Question 6

Differential Equations

4 Marks

2021 P2 Q6

Solve the differential equation

$\displaystyle \frac{dy}{dx} + 2xy = 14xe^{-x^2}$

given that when $\displaystyle x = 0$ , $\displaystyle y = 3.$ Express $\displaystyle y$ in terms of $\displaystyle x.$

Show answer

\displaystyle y = \frac{7x^2 + 3}{e^{x^2}}

(or

\displaystyle y = (7x^2 + 3)e^{-x^2}

)

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Question 6

•¹ find integrating factor
$\displaystyle e^{x^2}$

•² write in correct form
$\displaystyle e^{x^2}y = \int 14x \, dx$

•³ integrate right hand side including constant of integration
$\displaystyle e^{x^2}y = 7x^2 + c$

•⁴ find particular solution
$\displaystyle y = \frac{7x^2+3}{e^{x^2}}$

7

Question 7

Binomial TheoremComplex Numbers

(3, 3)6 Marks

2021 P2 Q7

A complex number is defined by $\displaystyle z = a + 2i$ where $\displaystyle a$ is a positive real number.

(a)State and simplify the binomial expansion of $\displaystyle z^3.$

(b)Given that $\displaystyle z^3 + 3z = b + 148i$ where $\displaystyle b$ is a real number, find the values of $\displaystyle a$ and $\displaystyle b.$

Show answer

(a)

\displaystyle a^3 - 12a + (6a^2 - 8)i

(b)

\displaystyle a = 5

,

\displaystyle b = 80

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Question 7(a)

•¹ binomial expansion
$\displaystyle \binom{3}{0}a^3 + \binom{3}{1}a^2(2i) + \binom{3}{2}a(2i)^2 + \binom{3}{3}(2i)^3$

•² binomial coefficients and powers of 2
Stated or implied by $\displaystyle a^3 + 6a^2i + 12ai^2 + 8i^3$

•³ simplify
$\displaystyle a^3 - 12a + (6a^2 - 8)i$

Question 7(b)

•⁴ substitute
$\displaystyle a^3 - 12a + (6a^2 - 8)i + 3a + 6i = b + 148i$

•⁵ equate imaginary and real parts
$\displaystyle 6a^2 - 2 = 148$ and $\displaystyle a^3 - 9a = b$

•⁶ find a and b
$\displaystyle a = 5$ , $\displaystyle b = 80$

8

Question 8

Differentiation

(4, 3)7 Marks

2021 P2 Q8

A curve is defined by $\displaystyle x^2y^3 + e^{2y} = 5.$

(a)Find $\displaystyle \frac{dy}{dx}$ in terms of $\displaystyle x$ and $\displaystyle y.$

(b)Show that there is only one stationary point on the curve.

Show answer

(a)

\displaystyle \frac{-2xy^3}{3x^2y^2 + 2e^{2y}}

(b) Setting

\displaystyle \frac{dy}{dx} = 0

gives

\displaystyle -2xy^3 = 0

, so

\displaystyle x = 0

or

\displaystyle y = 0.

When

\displaystyle x = 0

,

\displaystyle e^{2y} = 5 \Rightarrow y = \frac{1}{2}\ln 5.

When

\displaystyle y = 0

,

\displaystyle 0 + 1 = 5

which has no solution.
Hence there is only one stationary point at

\displaystyle (0, \frac{1}{2}\ln 5).

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Question 8(a)

•¹ start to differentiate product with one term correct
$\displaystyle 2xy^3$ or $\displaystyle 3x^2y^2\frac{dy}{dx}$

•² complete differentiation of product
$\displaystyle 3x^2y^2\frac{dy}{dx}$ or $\displaystyle 2xy^3$

•³ differentiate remaining terms
$\displaystyle ... + 2e^{2y}\frac{dy}{dx} = 0$

•⁴ write derivative explicitly in terms of x and y
$\displaystyle \frac{dy}{dx} = \frac{-2xy^3}{3x^2y^2 + 2e^{2y}}$

Question 8(b)

•⁵ express condition for stationary point
$\displaystyle \frac{-2xy^3}{3x^2y^2 + 2e^{2y}} = 0$

•⁶ state corresponding values of both x and y
$\displaystyle x = 0$ AND $\displaystyle y = 0$

•⁷ show that there is one value of y when x = 0 but no value of x when y = 0
$\displaystyle (0, \frac{1}{2}\ln 5)$ AND e.g. $\displaystyle 1 = 5$ ... no solution

9

Question 9

Partial FractionsDifferential Equations

(2, 8)10 Marks

2021 P2 Q9

(a)Express $\displaystyle \frac{1}{x(5 - x)}$ in partial fractions.

A small island is being populated by seals. The size of the seal population can be modelled by the differential equation

$\displaystyle \frac{dP}{dt} = \frac{1}{100}P(5 - P), \quad 0 < P < 5$

where $\displaystyle P$ (in hundreds) is the number of seals on the island $\displaystyle t$ years after the seals arrive.

(b)Given that there are 250 seals after 10 years, find an expression for $\displaystyle P$ in terms of $\displaystyle t.$

Show answer

(a)

\displaystyle \frac{1}{5x} + \frac{1}{5(5 - x)}

(b)

\displaystyle P = \frac{5e^{0.05t - 0.5}}{1 + e^{0.05t - 0.5}}

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Question 9(a)

•¹ write template
$\displaystyle \frac{1}{x(5-x)} = \frac{A}{x} + \frac{B}{5-x}$

•² find constants and express in partial fractions
$\displaystyle \frac{1}{5x} + \frac{1}{5(5-x)}$

Question 9(b)

•³ write as integral equation
$\displaystyle \int \frac{1}{P(5-P)} \, dP = \int \frac{1}{100} \, dt$

•⁴ write LHS integral in partial fractions
$\displaystyle \frac{1}{5} \int \left( \frac{1}{P} + \frac{1}{5-P} \right) \, dP$

•⁵ complete integration
$\displaystyle \frac{1}{5}(\ln P - \ln(5-P))$

•⁶ substitute values
$\displaystyle \frac{1}{5}(\ln P - \ln(5-P)) = \frac{1}{100}t + c$

•⁷ constant of integration
$\displaystyle \frac{1}{5}(\ln 2.5 - \ln(5-2.5)) = \frac{1}{10} + c$

•⁸ evaluate
$\displaystyle c = -\frac{1}{10}$

•⁹ take exponentials
$\displaystyle \frac{P}{5-P} = e^{0.05t-0.5}$

•¹⁰ write expression in terms of t
$\displaystyle P = \frac{5e^{0.05t-0.5}}{1+e^{0.05t-0.5}}$

10

Question 10

Methods of Proof

5 Marks

2021 P2 Q10

Prove by induction that

$\displaystyle \sum_{r=2}^{n} \frac{1}{r(r - 1)} = \frac{n - 1}{n}$

for all positive integers $\displaystyle n \ge 2.$

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Proof by induction showing true for

\displaystyle n=2

(LHS = RHS =

\displaystyle \frac{1}{2}

), assuming true for

\displaystyle n=k

, and showing the sum to

\displaystyle k+1

simplifies to

\displaystyle \frac{k}{k+1}

, concluding the proof for all integers

\displaystyle n \ge 2.

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Question 10

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

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

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

•⁴ express as a single fraction
$\displaystyle \frac{(k-1)(k+1)+1}{k(k+1)}$

•⁵ express explicitly in terms of (k + 1) AND communicate
$\displaystyle \frac{k^2}{k(k+1)}$ leading to $\displaystyle \frac{k+1-1}{k+1}$
If true for $\displaystyle n=k$ then true for $\displaystyle n=k+1.$ Also shown true for $\displaystyle n=2$ therefore, by induction, true for all $\displaystyle n \ge 2.$

11(a)(b)

Question 11(a)(b)

Sequences & Series

(1, 1, 1, 1)4 Marks

2021 P2 Q11(a)(b)

Three consecutive terms of an arithmetic sequence are given by $\displaystyle x - 1, x - 7, 2x - 9.$

(a)(i) Find the common difference.
(ii) Hence find the value of $\displaystyle x.$

(b)Given that $\displaystyle x - 1$ is the $\displaystyle 21^{st}$ term, find

(i) the value of the first term
(ii) a simplified expression for the $\displaystyle n^{th}$ term of the sequence.

Show answer

(a)(i)

\displaystyle -6

(a)(ii)

\displaystyle x = -4

(b)(i)

\displaystyle 115

(b)(ii)

\displaystyle 121 - 6n

11(c)(d)

Question 11(c)(d)

Sequences & Series

(3, 1, 2)6 Marks

2021 P2 Q11(c)(d)

Three consecutive terms of a geometric sequence are given by $\displaystyle y - 1, y - 7, 2y - 9.$

(c)Find the two possible values of $\displaystyle y$ and the corresponding common ratios.

One of the values of $\displaystyle y$ gives an associated geometric series which has a sum to infinity.

(d)(i) Identify the value of $\displaystyle y$ and justify your answer.
(ii) Determine whether $\displaystyle \frac{64}{3}$ is a possible value for this sum to infinity. Give a reason for your answer.

Show answer

(c)

\displaystyle y = 5

with common ratio

\displaystyle -\frac{1}{2}

\displaystyle y = -8

with common ratio

\displaystyle \frac{5}{3}

(d)(i)

\displaystyle y = 5

because

\displaystyle \left|-\frac{1}{2}\right| < 1.

(d)(ii) No. If the sum is

\displaystyle \frac{64}{3}

, then

\displaystyle a = 32

, which would mean the term

\displaystyle y-1

would be -4 (not 4 as required by

\displaystyle y=5

).

12

Question 12

Vectors

(4, 1, 1, 2)8 Marks

2021 P2 Q12

The points $\displaystyle A(4, 0, 8)$ , $\displaystyle B(6, -5, 4)$ and $\displaystyle C(3, 4, 11)$ all lie on the plane $\displaystyle \pi_1.$

(a)Find the Cartesian equation of $\displaystyle \pi_1.$

The plane $\displaystyle \pi_2$ is parallel to $\displaystyle \pi_1$ and passes through the origin.

(b)State the equation of $\displaystyle \pi_2.$

A sphere touches $\displaystyle \pi_1$ , where $\displaystyle A$ is the point of contact. The sphere also has a single point of contact, $\displaystyle Q$ , with $\displaystyle \pi_2.$

(c)(i) Find parametric equations for the line $\displaystyle AQ.$
(ii) Hence find the coordinates for $\displaystyle Q.$

Show answer

(a)

\displaystyle x - 2y + 3z = 28

(b)

\displaystyle x - 2y + 3z = 0

(c)(i)

\displaystyle x = 4 + t

,

\displaystyle y = -2t

,

\displaystyle z = 8 + 3t

(c)(ii)

\displaystyle Q(2, 4, 2)

Show marking instructions

Question 12(a)

•¹ find two directed line segments
$\displaystyle \vec{AB} = \begin{pmatrix} 2 \\ -5 \\ -4 \end{pmatrix}$ , $\displaystyle \vec{AC} = \begin{pmatrix} -1 \\ 4 \\ 3 \end{pmatrix}$

•² begin to find vector product
e.g. $\displaystyle \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & -5 & -4 \\ -1 & 4 & 3 \end{vmatrix}$

•³ calculate a normal vector
e.g. $\displaystyle \mathbf{n} = \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}$

•⁴ obtain equation
$\displaystyle x - 2y + 3z = 28$

Question 12(b)

•⁵ state equation of π₂
$\displaystyle x - 2y + 3z = 0$

Question 12(c)(i)

•⁶ find parametric equations
$\displaystyle x = 4 + t$
$\displaystyle y = -2t$
$\displaystyle z = 8 + 3t$

Question 12(c)(ii)

•⁷ substitute into equation of π₂
$\displaystyle 4 + t - 2(-2t) + 3(8 + 3t) = 0$

•⁸ find coordinates of Q
$\displaystyle (2, 4, 2)$

13

Question 13

Complex Numbers

(1, 1, 1, 2, 2)7 Marks

2021 P2 Q13

(a)Express $\displaystyle -1$ in the form $\displaystyle \cos\theta + i\sin\theta.$

The complex number $\displaystyle z_1$ is defined by $\displaystyle z_1 = \cos\frac{\pi}{5} + i\sin\frac{\pi}{5}.$

(b)Use de Moivre's theorem to show that $\displaystyle z_1$ is a root of the equation $\displaystyle z^5 + 1 = 0.$

The complex number $\displaystyle z_2$ is also a root of the equation $\displaystyle z^5 + 1 = 0.$ Roots $\displaystyle z_1$ and $\displaystyle z_2$ have been plotted on an Argand diagram, as shown.

(c)Express $\displaystyle z_2$ in the form $\displaystyle \cos\theta + i\sin\theta.$

The remaining roots of the equation $\displaystyle z^5 + 1 = 0$ are $\displaystyle z_3$ , $\displaystyle z_4$ and $\displaystyle z_5.$

(d)Express $\displaystyle z_3$ , $\displaystyle z_4$ and $\displaystyle z_5$ in the form $\displaystyle \cos\theta + i\sin\theta$ , where $\displaystyle -\pi < \theta \le \pi.$

(e) Given $\displaystyle z_1 + z_2 + z_3 + z_4 + z_5 = 0$ , show algebraically that

$\displaystyle \cos\frac{\pi}{5} + \cos\frac{3\pi}{5} = \frac{1}{2}.$

Show answer

(a)

\displaystyle \cos\pi + i\sin\pi

(b)

\displaystyle z_1^5 = \cos\left(5 \times \frac{\pi}{5}\right) + i\sin\left(5 \times \frac{\pi}{5}\right) = \cos\pi + i\sin\pi = -1.

Therefore,

\displaystyle z_1^5 + 1 = -1 + 1 = 0.

(c)

\displaystyle \cos\frac{3\pi}{5} + i\sin\frac{3\pi}{5}

(d)

\displaystyle \cos\left(-\frac{\pi}{5}\right) + i\sin\left(-\frac{\pi}{5}\right)

,

\displaystyle \cos\left(-\frac{3\pi}{5}\right) + i\sin\left(-\frac{3\pi}{5}\right)

and

\displaystyle \cos\pi + i\sin\pi

(e) Equating the real parts to zero gives:

\displaystyle \cos\frac{\pi}{5} + \cos\frac{3\pi}{5} + \cos\left(-\frac{\pi}{5}\right) + \cos\left(-\frac{3\pi}{5}\right) + \cos\pi = 0

Since

\displaystyle \cos(-\theta) = \cos(\theta)

and

\displaystyle \cos\pi = -1

:

\displaystyle 2\cos\frac{\pi}{5} + 2\cos\frac{3\pi}{5} - 1 = 0

\displaystyle 2\left(\cos\frac{\pi}{5} + \cos\frac{3\pi}{5}\right) = 1

\displaystyle \cos\frac{\pi}{5} + \cos\frac{3\pi}{5} = \frac{1}{2}

Show marking instructions

Question 13(a)

•¹ express in appropriate form
$\displaystyle -1 = \cos \pi + i \sin \pi$

Question 13(b)

•² verify root
$\displaystyle \cos \pi + i \sin \pi = -1$

Question 13(c)

•³ polar form
$\displaystyle z_2 = \cos \frac{3\pi}{5} + i \sin \frac{3\pi}{5}$

Question 13(d)

•⁴,⁵ remaining roots
$\displaystyle \cos(-\frac{\pi}{5}) + i \sin(-\frac{\pi}{5})$ or $\displaystyle \cos(-\frac{3\pi}{5}) + i \sin(-\frac{3\pi}{5})$

Question 13(e)

•⁶ equate real part to zero
$\displaystyle \cos \frac{\pi}{5} + \cos \frac{3\pi}{5} + \cos(-\frac{\pi}{5}) + \cos(-\frac{3\pi}{5}) - 1 = 0$
leading to
$\displaystyle 2 \cos \frac{\pi}{5} + 2 \cos \frac{3\pi}{5} = 1$

•⁷ complete proof
$\displaystyle \cos \frac{\pi}{5} + \cos \frac{3\pi}{5} = \frac{1}{2}$