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Maths Paper 1, November 2024

49 questions · 150 marks. Answer any question here and see it marked like the exam, mark by mark.

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This is the November 2024 Paper 1 for Mathematics, written as 49 questions that add up to 150 marks. By topic the marks are differential calculus (39), functions and graphs (29), number patterns and sequences (27), algebra: equations and inequalities (24), probability (17) and finance, growth and decay (14). Functions and graphs and differential calculus together carry the largest share, so they repay the most practice. Of the 150 marks, 93 are for the correct final answer, 53 for method and working, 3 for reading data from diagrams, tables and graphs and 1 for explaining.

Questions here are based on the National Senior Certificate paper, in our own words. Each is marked point by point: a method mark for a correct step, an answer mark for the correct result, and your working is read even when the answer is wrong. The Diagnostic Report for this paper says candidates most often lose marks by skipping the check on a solution, misreading "at least", and reading an asymptote off the wrong line. Try the questions in order, or jump to the topic you find hardest.

Where marks are lost: in our words, from Department of Basic Education, NSC 2024 Diagnostic Report, Part 1 (Book 1).

No sign-up to try. Your first 3 answers are marked free. The original paper and memo are on PapersZA.

Every question

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Question 1 · Algebra

[27]

x

x (x + 5) = 0

(2)Mark it 1.1.2

x

3 x^{2} + 2 = 7 x

. Round your answers to TWO decimal places.

(4)Mark it 1.1.3

x

x^{2} - 4 x - 5 > 0

(4)Mark it 1.1.4

x

3^{2 x} - 3^{x + 2} + 8 = 0

. Hint: leave your answers in exact form if needed.

(5)Mark it 1.1.5

x

\sqrt{3 x + 10} - x = 2

x

y

simultaneously:

2 x + y = 5

y^{2} + x y = 3

Look at the product

(1 + \frac{1}{3}) (1 + \frac{1}{4}) (1 + \frac{1}{5}) \dots (1 + \frac{1}{n})

n \geq 3

and the last factor is

1 + \frac{1}{n}

. Find ALL the values of

n

for which this product is a whole number.

Question 2 · Number patterns and sequences

[14]

An arithmetic series has a first term of

4

and a common difference of

3

. The series has

30

terms. Calculate the sum of all

30

(2)Mark it 2.1.2

The first term of an arithmetic series is

6

and the common difference is

4

. The series has

15

terms. It is then extended so that it has

60

terms in total, and the sum of all

60

7440

. Write down an equation, using sigma notation, for the sum of only the new terms that were added to the original series.

(4)Mark it 2.2.1

The first differences of a quadratic number pattern are

4; 6; 8; \dots

T_{60} = 3 500

, calculate the value of

T_{59}

(3)Mark it 2.2.2

The first differences of a quadratic number pattern are:

4; 6; 8; \dots

The third term of the pattern is

22

. Determine the general term,

T_{n}

, of the pattern.

Question 3 · Number patterns and sequences

[10]

A circle with radius 20 cm is drawn. A second, smaller circle is drawn so that it passes through the centre of the first circle and touches the first circle from the inside, as shown in the diagram below. A third circle is drawn in the same way: it passes through the centre of the second circle and touches the second circle from the inside. This carries on, so that every new circle has a radius equal to half of the radius of the circle before it. Write down the radius of the 4th circle.

A circle of radius 8 cm is drawn. A second, smaller circle is drawn so that it passes through the centre of the first circle and touches the first circle from the inside. A third circle is drawn in the same way inside the second circle, passing through the centre of the second circle and touching it from the inside. This pattern of circles carries on, as shown in the diagram below. Calculate the sum of the areas of the first 8 circles. Give your answer correct to two decimal places.

A circle of radius 20 cm is drawn. A second circle is drawn so that it passes through the centre of the first circle and touches the first circle from the inside. A third circle is drawn in the same way inside the second circle: it passes through the centre of the second circle and touches the second circle from the inside. The process carries on forever, as shown in the diagram below. Which circle in this pattern has a diameter of

\frac{5}{64}

Question 4 · Functions and graphs

[9]

g

g (x) = b^{x} + 2

b > 0

P (3; \frac{43}{8})

lies on the graph of

g

. Calculate the value of

b

g (x) = b^{x} + 3

b > 0

, and the point

P (2; \frac{37}{4})

g

. State the set of all possible

y

g

g

g (x) = b^{x} - 2

b > 0

P (2; - \frac{7}{4})

g

. Draw a neat sketch of

g

on your own set of axes, labelling where it cuts each axis and giving the equation of any asymptote.

f (x) = {(\frac{1}{2})}^{x} - 1

is given. C is the point on the graph of

f

y = 7

. Determine the coordinates of

C ’

, the image of C when C is reflected in the line

y = x

Question 5 · Functions and graphs

[10]

The diagram below shows the graph of

f (x) = \frac{a}{x + p} + q

x \in R, x \neq 2

. Write down the value of

p

The diagram below shows the graph of

f (x) = \frac{a}{x + p} + q

(- \infty; 2) \cup (2; \infty)

f

y

(0; 2)

. The straight line

g (x) = x - 5

is a line of symmetry of

f

. Determine the equation of the horizontal asymptote of

f

The diagram below shows the graph of

f (x) = \frac{a}{x + p} + q

x \in R

x \neq 2

. The vertical asymptote is

x = 2

and the horizontal asymptote is

y = - 1

f

y

(0; 2)

. Calculate the value of

a

f (x) = \frac{- 6}{x - 2} - 2

is drawn in the diagram below. Its domain is

x \in (- \infty; 2) \cup (2; \infty)

. The graph cuts the

y

(0; 1)

g (x) = x - 4

is a line of symmetry of

f

. Determine the values of

x

f (x) \geq 0

f

is given in the information above. Graph

f

is transformed to a new graph

h

h

has the same domain and the same range as

f

• the gradient of

h

is negative at every point of its domain, that is

h ’ (x) < 0

. Describe one possible transformation of

f

h

Question 6 · Functions and graphs

[15]

The diagram below shows the graph of

f (x) = - x^{2} + 6 x + 7

, a parabola with turning point B. Calculate the coordinates of B.

The diagram below shows the parabola

f (x) = - x^{2} + 2 x + 8

and a straight line

g

g

cuts the parabola at point A on the left, which lies on the x-axis, and at point C(2;8) on the right. Prove that the straight line

g

has the equation

g (x) = 2 x + 4

The diagram below shows the parabola

f (x) = - x^{2} + 6 x + 7

and the straight line

g (x) = 2 x + 2

. The two graphs cross at A on the x-axis and at C

(5; 12)

. A vertical segment EH joins a point E on

f

to a point H on

g

, somewhere between A and C where the parabola lies above the line. Determine the greatest possible length of EH.

The diagram below shows the parabola

f (x) = - x^{2} + 6 x + 7

and a straight line

g

g

2

and passes through the point

(2; 11)

g (x) = 2 x + 7

k

is obtained by shifting

f

horizontally, so that

k (x) = f (x + m) = - x^{2} - 2 m x - m^{2} + 6 x + 6 m + 7

. Determine the value of

m

g

is a tangent to

k

Question 7 · Finance, growth and decay

[14]

On the day Sipho was born, his aunt invested R10 000 in a savings account. The account earns interest at 4% p.a., compounded quarterly. Calculate how much is in the account on Sipho’s

10^{th}

birthday, if nothing is added or withdrawn.

A laptop loses value over time using the straight-line method of depreciation. After 5 years, its value is exactly one fifth less than its original value, that is, it is worth

\frac{4}{5}

of what it cost. Calculate the annual rate of depreciation, as a percentage.

(2)Mark it 7.3.1

Naledi borrowed R80 000 from a bank on 1 July 2023. She agreed to repay the loan over 4 years in equal monthly instalments of R2 000 each, with the first instalment paid on 1 August 2023. Calculate the total interest that she will pay over the 4 years.

(2)Mark it 7.3.2

Sipho took out a loan of R80 000 on 1 June 2021 at an interest rate of 12% p.a., compounded monthly. He agreed to repay the loan over 5 years by paying monthly instalments of R1 779,55, with the first instalment on 1 July 2021. On 1 June 2023 Sipho paid R16 779,55 into the loan account. This was his usual instalment plus an extra R15 000. He keeps paying the usual instalment every month after that. Calculate how many whole months earlier than planned Sipho will finish repaying the loan.

Question 8 · Differential calculus

[18]

\frac{d}{d x} [7 x^{2} - 4 x]

(2)Mark it 8.1.2

f ’ (x)

f (x) = \frac{3}{x^{4}} - \sqrt[3]{x^{5}}

Find the equation of the tangent to the curve

g (x) = x^{3} - 3 x^{2} + 4 x + 1

at the point where

x = 2

(3)Mark it 8.3.1

g (x) = - 4 x^{2}

. Use first principles to determine

g ’ (x)

(5)Mark it 8.3.2

g

g (x) = - 4 x^{2}

. State one way to restrict the domain of

g

so that its inverse,

g^{- 1}

, is a function.

(1)Mark it 8.3.3

g

g (x) = - 4 x^{2}

. Find the inverse relation of

g

and write it as

y = \dots

, choosing the branch for which

y \leq 0

Question 9 · Differential calculus

[8]

The diagram below shows the graph of a cubic function

g

with turning points P

(1; 8)

(2; 7)

. The graph cuts the

y

(0; 3)

. State the values of

x

g

The diagram below shows the graph of a cubic function

g

with turning points P

(1; 8)

(2; 7)

. The graph cuts the

y

(0; 3)

. Write down the

x

g ’

, the derivative of

g

The diagram below shows the graph of a cubic function

g

with turning points P

(1; 8)

(2; 7)

. The graph cuts the

y

(0; 3)

. Determine the values of

x

g

The diagram below shows the graph of a cubic function

g

with turning points P

(1; 8)

(2; 7)

. The graph cuts the

y

(0; 3)

. Determine all the values of

k

y = g (x) + k

has exactly THREE positive

x

Question 10 · Differential calculus

[8]

A delivery drone flies from a depot and lands at a warehouse. Its speed, in km/h, after

t

hours of flight is given by

v (t) = - 2 t^{2} + 12 t

. Speed is the rate of change of distance with respect to time. Calculate the greatest speed the drone reaches during this flight.

(3)Mark it 10.2

A runner travels from checkpoint A and stops at checkpoint B. Her speed (in km/h) after

t

hours is given by

s ’ (t) = - 6 t^{2} + 24 t

s (t)

is the distance travelled from A. Note: speed is the rate of change of distance with respect to time, and she starts at A when

t = 0

. Calculate the distance between checkpoint A and checkpoint B.

Question 11 · Probability

[9]

Draw a Venn diagram with three overlapping circles to show ALL the learners who write these examinations, and fill in the number in every region.

(3)Mark it 11.2

Calculate the probability that a learner chosen at random writes at least TWO of the three subjects.

(2)Mark it 11.3

Decide whether the events "a learner sits for Mathematics" and "a learner sits for Tourism" are independent. Support your answer with calculations.

Question 12 · Probability

[8]

A school locker uses a 4-character lock code made from the 26 letters of the alphabet and the digits 0 to 9. Every code has the pattern: digit, letter, digit, letter. Letters and digits may be used more than once in a code. Determine how many different lock codes are possible.

(2)Mark it 12.2

Work out how many different passes can be made under these conditions: • No vowel (A, E, I, O, U) may be used • The first letter cannot be M or N • Within the pass, every letter is different and every digit is different • The last digit must be even

(4)Mark it 12.3

A company makes 4-character codes in the form letter | digit | letter | digit. The first digit is chosen from 1 to 9 and the last digit must be an even digit (0, 2, 4, 6 or 8). Letters may be repeated. At present only 20 of the 26 letters of the alphabet may be used, which gives 18 000 different codes. The company now allows 5 of the 6 letters that were left out to be used as well, so 25 letters are available. Calculate the percentage increase in the number of different codes.

Updated 12 November 2026