ZIMSEC A Level Pure Mathematics Paper 2 November 2023

Question 101

[3 marks]Indices and logarithms

Simplify

\dfrac{3^{2x-3}\cdot81^{x+\frac12}}{27^x}

, giving the answer as a single power of 3.

Answer this when you sit the paper.

Question 102

[3 marks]Indices and logarithms

Solve

3^{3x-1}=\dfrac{1}{5^{x-1}}

, giving the answer in exact form.

Answer this when you sit the paper.

Question 201

[2 marks]Polynomials: factor theorem and completing the square

Given the polynomial

Q(x)=2x^3+3x^2-4x-1

, evaluate

Q(1)

Answer this when you sit the paper.

Question 202

[3 marks]Polynomials: factor theorem and completing the square

Given

Q(x)=2x^3+3x^2-4x-1

and that

x=1

is a root, find the polynomial

P(x)

such that

Q(x)=(x-1)P(x)

Answer this when you sit the paper.

Question 203

[3 marks]Polynomials: factor theorem and completing the square

Express

P(x)=2x^2+5x+1

in the form

A(x+B)^2+C

Answer this when you sit the paper.

Question 204

[1 marks]Polynomials: factor theorem and completing the square

Given that

P(x)=2\left(x+\dfrac54\right)^2-\dfrac{17}{8}

, state the minimum value of

P(x)

Answer this when you sit the paper.

Question 301

[3 marks]Joint variation

The curved surface area

A

of a cylindrical drum varies jointly as its base diameter

d

and its height

l

, so

A=kdl

. Given

A=2.64\,\text{m}^2

when the base radius is

0.35

m and the height is

1.2

m, find

k

Answer this when you sit the paper.

Question 302

[2 marks]Joint variation

A=kdl

for the curved surface area of a cylindrical drum, the drum is described by its base radius of

0.35

m. Why is

0.70

and not

0.35

substituted for

d

ABecause $d$ in the formula stands for the diameter, which is twice the radius the question quotes.
BBecause the surface area must always be doubled when a cylinder is closed at both of its ends.
CBecause the height of $1.2$ m has to be halved first, and the radius is doubled to compensate for it.
DBecause a curved surface area is measured in square metres, so every length in it is used twice over.

Question 303

[3 marks]Joint variation

Using

A=\dfrac{22}{7}dl

for the curved surface area of a cylindrical drum, calculate the radius, in metres to 2 decimal places, of a cylinder whose curved surface area is

14.52\,\text{m}^2

and whose height is

3

Answer this when you sit the paper.

Question 401

[2 marks]Quadratic equations and the discriminant

Given

f(x)=kx^2+3x+3

and

g(x)=kx+7

, write the equation

f(x)=g(x)

as a quadratic equation equal to zero.

Answer this when you sit the paper.

Question 402

[3 marks]Quadratic equations and the discriminant

Find the range of values of

k

for which

kx^2+(3-k)x-4=0

has two distinct real solutions.

Answer this when you sit the paper.

Question 501

[3 marks]Identities and partial fractions

Given that

2x^3+ax^2+x-3\equiv(bx^2+c)(x-3)

, find the constants

a

b

and

c

Answer this when you sit the paper.

Question 502

[2 marks]Identities and partial fractions

Which is the correct way to set up

\dfrac{1+6x}{(2x^2+1)(x-3)}

in partial fractions?

A $\dfrac{A}{2x^2+1}+\dfrac{Bx+C}{x-3}$ , since the larger of the two denominators always takes the constant numerator.
B $\dfrac{Ax+B}{2x^2+1}+\dfrac{Cx+D}{x-3}$ , since the numerator of the original fraction is itself linear in $x$ .
C $\dfrac{A}{2x^2+1}+\dfrac{B}{x-3}$ , since one unknown constant belongs above each of the two factors of the denominator, whatever the degree of that factor happens to be.
D $\dfrac{Ax+B}{2x^2+1}+\dfrac{C}{x-3}$ , since an irreducible quadratic factor takes a linear numerator and a linear factor takes a constant.

Question 503

[3 marks]Identities and partial fractions

Express

\dfrac{1+6x}{(2x^2+1)(x-3)}

in partial fractions.

Answer this when you sit the paper.

Question 601

[1 marks]Composite functions, area by integration and the trapezium rule

Given

t(x)=e^x

and

h(x)=2x+3

, find

th(x)

Answer this when you sit the paper.

Question 602

[3 marks]Composite functions, area by integration and the trapezium rule

Find, by integration, the exact area between

y=12x+20

y=e^{2x+3}

and the lines

x=-1.5

and

x=0

Answer this when you sit the paper.

Question 603

[2 marks]Composite functions, area by integration and the trapezium rule

The trapezium rule is applied to

\displaystyle\int_{-1.5}^{0}\left(12x+20-e^{2x+3}\right)dx

with 4 ordinates. State the strip width

h

Answer this when you sit the paper.

Question 604

[3 marks]Composite functions, area by integration and the trapezium rule

Estimate

\displaystyle\int_{-1.5}^{0}\left(12x+20-e^{2x+3}\right)dx

using the trapezium rule with 4 ordinates and strip width

0.5

, giving the answer to 2 significant figures.

Answer this when you sit the paper.

Question 605

[3 marks]Composite functions, area by integration and the trapezium rule

The exact area of a region is

6.9572

square units and the trapezium rule estimates it as

6.1749

. Calculate the relative error, to 2 significant figures.

Answer this when you sit the paper.

Question 606

[2 marks]Composite functions, area by integration and the trapezium rule

The trapezium rule applied to

y=12x+20-e^{2x+3}

between

x=-1.5

and

x=0

gives

6.17

, while the exact value is

6.96

. Why does the rule fall short here?

ABecause four ordinates is an even number, and the trapezium rule can only be exact when the count of ordinates used is an odd one instead.
BBecause the curve is concave down over the interval, so each straight chord lies below the curve and every strip is undercounted.
CBecause the interval $-1.5$ to $0$ is negative in width, which subtracts a fixed amount from the total the rule returns.
DBecause the rule always loses accuracy whenever an exponential term appears anywhere in the integrand being used.

Question 701

[3 marks]Trigonometric identities and equations

In proving

\dfrac{2\cot2\theta}{1-\tan^2\theta}\equiv\cot\theta

, the denominator

1-\tan^2\theta

is first written as a single fraction. Which fraction is it?

A $\dfrac{1}{\cos^2\theta}$ , because $1-\tan^2\theta$ is one of the standard Pythagorean identities of trigonometry.
B $\dfrac{\sin^2\theta-\cos^2\theta}{\cos^2\theta}$ , because subtracting a squared tangent reverses the order of the two terms, leaving the sine squared written in front of the cosine squared.
C $\dfrac{\cos^2\theta-\sin^2\theta}{\cos^2\theta}$ , because $\tan^2\theta=\dfrac{\sin^2\theta}{\cos^2\theta}$ and the two terms go over the common denominator $\cos^2\theta$ .
D $\dfrac{\cos^2\theta-\sin^2\theta}{\sin^2\theta}$ , because the tangent is defined as sine over cosine in that particular order.

Question 702

[3 marks]Trigonometric identities and equations

Write

2\csc^2\theta-1

in terms of

\cot\theta

only.

Answer this when you sit the paper.

Question 703

[3 marks]Trigonometric identities and equations

For the quadratic

2\cot^2\theta-\cot\theta+1=0

, treated as a quadratic in

\cot\theta

, calculate the discriminant

b^2-4ac

Answer this when you sit the paper.

Question 704

[3 marks]Trigonometric identities and equations

The equation

\dfrac{2\cot2\theta}{1-\tan^2\theta}=2\csc^2\theta-1

reduces to

2\cot^2\theta-\cot\theta+1=0

, whose discriminant is

-7

. What does that tell you about the solutions for

0^\circ\le\theta\le360^\circ

AThere is one solution, because a negative discriminant leaves only the single repeated root at the vertex.
BThere are no solutions, because a negative discriminant means $\cot\theta$ can take no real value at all.
CThere are exactly two solutions, because every quadratic equation still has two roots when they are counted properly.
DThere are four solutions, because each value of $\cot\theta$ produces two angles within a full turn of $360^\circ$ .

Question 705

[2 marks]Trigonometric identities and equations

State the number of solutions of

\dfrac{2\cot2\theta}{1-\tan^2\theta}=2\csc^2\theta-1

in the range

0^\circ\le\theta\le360^\circ

Answer this when you sit the paper.

Question 801

[3 marks]Determinants and inverse matrices

Expand the determinant of

A=\begin{pmatrix}1&2&1\\x&1&x+1\\2&x-2&0\end{pmatrix}

and give it as a simplified expression in

x

Answer this when you sit the paper.

Question 802

[2 marks]Determinants and inverse matrices

The determinant of

A=\begin{pmatrix}1&2&1\\x&1&x+1\\2&x-2&0\end{pmatrix}

simplifies to

3x+4

. Given that the determinant equals 13 and

x>0

, find

x

Answer this when you sit the paper.

Question 803

[3 marks]Determinants and inverse matrices

For

A=\begin{pmatrix}1&2&1\\3&1&4\\2&1&0\end{pmatrix}

, find the cofactor of the entry in row 1, column 1.

Answer this when you sit the paper.

Question 804

[3 marks]Determinants and inverse matrices

The inverse of

A=\begin{pmatrix}1&2&1\\3&1&4\\2&1&0\end{pmatrix}

can be written as

\dfrac{1}{13}

times a matrix. Give the first row of that matrix.

Answer this when you sit the paper.

Question 805

[3 marks]Determinants and inverse matrices

For two invertible matrices

A

and

B

, which expression equals

(AB)^{-1}

A $\dfrac{1}{AB}$ , because the inverse of any product is simply one divided by the whole of the product itself.
B $(BA)^{-1}$ , because a product of two matrices gives the same result whichever way round the two are written.
C $A^{-1}B^{-1}$ , because taking an inverse acts on each of the two factors separately and leaves the order of the product exactly as it was written.
D $B^{-1}A^{-1}$ , because undoing a product means undoing the last operation first, which reverses the order of the factors.

Question 806

[2 marks]Determinants and inverse matrices

(AB)^{-1}=B^{-1}A^{-1}

works out to

\dfrac{1}{104}

times the matrix

\begin{pmatrix}-11&6&3\\?&?&?\\-1&10&-8\end{pmatrix}

for

B^{-1}=\dfrac18\begin{pmatrix}1&-1&1\\2&1&0\\1&0&3\end{pmatrix}

and

A^{-1}=\dfrac{1}{13}\begin{pmatrix}-4&1&7\\8&-2&-1\\1&3&-5\end{pmatrix}

. Give the missing middle row.

Answer this when you sit the paper.

Question 901

[2 marks]Sequences and series: sums, general term and inequalities

The sum of

n

terms of a sequence is

S_n=\dfrac{n}{4}(5-n)

. Find

S_2

Answer this when you sit the paper.

Question 902

[2 marks]Sequences and series: sums, general term and inequalities

The sum of

n

terms of a sequence is

S_n=\dfrac{n}{4}(5-n)

. Find the third term of the sequence.

Answer this when you sit the paper.

Question 903

[3 marks]Sequences and series: sums, general term and inequalities

A sequence has first three terms

1

0.5

and

0

. Find its general term

T_n

in terms of

n

Answer this when you sit the paper.

Question 904

[3 marks]Sequences and series: sums, general term and inequalities

The sum of

n

terms of a sequence is

S_n=\dfrac{n}{4}(5-n)

. Calculate the number of terms needed for the sum to equal

-9

Answer this when you sit the paper.

Question 905

[3 marks]Sequences and series: sums, general term and inequalities

Solving

\dfrac{n}{4}(5-n)=-9

for the number of terms gives

n=9

and

n=-4

. Why is

n=-4

discarded?

ABecause the formula $S_n=\dfrac{n}{4}(5-n)$ was only stated by the question for values of $n$ larger than 5.
BBecause a negative root of any quadratic equation is always an extraneous solution introduced by the squaring step.
CBecause $n$ counts how many terms have been added, so it can only be a positive whole number.
DBecause the sum $-9$ is itself negative, and two negative quantities cannot appear on opposite sides of one equation.

Question 906

[3 marks]Sequences and series: sums, general term and inequalities

The sum of

n

terms of a sequence is

S_n=\dfrac{n}{4}(5-n)

. Find the range of values of

n

for which

S_n>0

Answer this when you sit the paper.

Question 1001

[2 marks]Coordinate geometry: lines, circles and tangents

A quadrilateral has vertices

A(-1;1)

and

B(3;7)

. Find the gradient of line

AB

Answer this when you sit the paper.

Question 1002

[2 marks]Coordinate geometry: lines, circles and tangents

Find the equation of the line through

A(-1;1)

and

B(3;7)

Answer this when you sit the paper.

Question 1003

[2 marks]Coordinate geometry: lines, circles and tangents

A quadrilateral has vertices

A(-1;1)

and

D(2;-1)

. Find the gradient of line

AD

Answer this when you sit the paper.

Question 1004

[3 marks]Coordinate geometry: lines, circles and tangents

Points

C(4;2)

and

D(2;-1)

are two vertices of a quadrilateral. Find the exact length

CD

Answer this when you sit the paper.

Question 1005

[3 marks]Coordinate geometry: lines, circles and tangents

Find the equation of the circle with centre

D(2;-1)

and radius

\sqrt{13}

in the form

x^2+y^2+ax+by=c

, stating

a

b

and

c

Answer this when you sit the paper.

Question 1006

[3 marks]Coordinate geometry: lines, circles and tangents

The circle

x^2+y^2-4x+2y=8

has centre

D(2;-1)

and passes through

A(-1;1)

, and line

AB

is perpendicular to line

AD

. Why does that make

AB

a tangent to the circle?

ABecause $DA$ is a radius and $AB$ is perpendicular to it at $A$ , the point where that radius meets the circle.
BBecause the gradients of $AB$ and $AD$ multiply to give $-1$ , and a product of $-1$ always signals a tangent.
CBecause $AB$ meets the circle at exactly one point, which is the very definition of a tangent to a curve.
DBecause $B(3;7)$ lies outside the circle, and any line drawn from a point outside a circle must touch it once.

Question 1201

[2 marks]Rational functions, graph transformations and inverse functions

Express

g(x)=\dfrac{4x+5}{2(x+1)}

in the form

A+\dfrac{B}{2(x+1)}

, stating

A

and

B

Answer this when you sit the paper.

Question 1202

[3 marks]Rational functions, graph transformations and inverse functions

Which single transformation maps the graph of

y=\dfrac1x

onto the graph of

y=\dfrac{1}{x+1}

AA translation of 1 unit in the positive $x$ direction, moving the vertical asymptote across to $x=1$ .
BA reflection in the $y$ -axis, which turns the right-hand branch of the curve into the left-hand branch.
CA translation of 1 unit in the negative $x$ direction, moving the vertical asymptote across to $x=-1$ .
DA stretch parallel to the $x$ -axis with factor $1$ , leaving both of the asymptotes exactly where they were.

Question 1203

[3 marks]Rational functions, graph transformations and inverse functions

Which single transformation maps the graph of

y=\dfrac{1}{x+1}

onto the graph of

y=\dfrac{1}{2(x+1)}

AA reflection in the line $y=x$ , which exchanges the roles played by the horizontal and vertical asymptotes.
BA stretch parallel to the $y$ -axis with factor $\dfrac12$ , halving every $y$ -coordinate while the asymptotes stay put.
CA stretch parallel to the $x$ -axis with factor $\dfrac12$ , halving every $x$ -coordinate and carrying the vertical asymptote along with them.
DA translation of $\dfrac12$ unit in the negative $y$ direction, dropping the whole curve down the page a little.

Question 1204

[2 marks]Rational functions, graph transformations and inverse functions

State the equations of the two asymptotes of

y=2+\dfrac{1}{2(x+1)}

Answer this when you sit the paper.

Question 1205

[3 marks]Rational functions, graph transformations and inverse functions

Find the inverse of

g(x)=\dfrac{4x+5}{2x+2}

Answer this when you sit the paper.

Question 1206

[3 marks]Rational functions, graph transformations and inverse functions

State the domain of

g^{-1}(x)=\dfrac{5-2x}{2x-4}

, the inverse of

g(x)=\dfrac{4x+5}{2x+2}

Answer this when you sit the paper.

Pure Mathematics Paper 2 November 2023

Sit this paper online

The questions

Question 101

Question 102

Question 201

Question 202

Question 203

Question 204

Question 301

Question 302

Question 303

Question 401

Question 402

Question 501

Question 502

Question 503

Question 601

Question 602

Question 603

Question 604

Question 605

Question 606

Question 701

Question 702

Question 703

Question 704

Question 705

Question 801

Question 802

Question 803

Question 804

Question 805

Question 806

Question 901

Question 902

Question 903

Question 904

Question 905

Question 906

Question 1001

Question 1002

Question 1003

Question 1004

Question 1005

Question 1006

Question 1201

Question 1202

Question 1203

Question 1204

Question 1205

Question 1206

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