Cambridge IGCSE™

* 7 0 2 9 1 0 6 8 3 8 *

ADDITIONAL MATHEMATICS 0606/12

Paper 1 February/March 2023

2 hours

You must answer on the question paper.

No additional materials are needed.

INSTRUCTIONS
● Answer all questions.
● Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs.
● Write your name, centre number and candidate number in the boxes at the top of the page.
● Write your answer to each question in the space provided.
● Do not use an erasable pen or correction fluid.
● Do not write on any bar codes.
● You should use a calculator where appropriate.
● You must show all necessary working clearly; no marks will be given for unsupported answers from a
calculator.
● Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in
degrees, unless a different level of accuracy is specified in the question.

INFORMATION
● The total mark for this paper is 80.
● The number of marks for each question or part question is shown in brackets [ ].

This document has 16 pages.

DC (PQ/CT) 312456/2
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Mathematical
MathematicalFormulae
Formulae

1. ALGEBRA

Quadratic Equation

For the equation ax 2 + bx + c = 0 ,

- b ! b 2 - 4ac
x=
2a

Binomial Theorem
n n n
(a + b) n = a n + e o a n - 1 b + e o a n - 2 b 2 + f + e o a n - r b r + f + b n
1 2 r

n
where n is a positive integer and e o =
n!
r (n - r) !r!

Arithmetic series un = a + (n - 1) d
1 1
Sn = n (a + l ) = n {2a + (n - 1) d}
2 2

Geometric series un = ar n - 1
a (1 - r n )
Sn = ( r ! 1)
1-r
a
S3 = ( r 1 1)
1-r

2. TRIGONOMETRY

Identities

sin 2 A + cos 2 A = 1
sec 2 A = 1 + tan 2 A
cosec 2 A = 1 + cot 2 A

Formulae for ∆ABC

a b c
= =
sin A sin B sin C
a 2 = b 2 + c 2 - 2bc cos A
1
T = bc sin A
2

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1 Find the exact values of k such that the straight line y = 1-k-x is a tangent to the curve
y = kx 2 + x + 2k . [4]

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2 A curve has equation y = (5 - x) (x + 2) 2 .

(a) Find the x-coordinates of the stationary points on the curve. [4]

(b) On the axes below, sketch the graph of y = (5 - x) (x + 2) 2 , stating the coordinates of the points
where the curve meets the axes. [3]

O x

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(c) Find the values of k for which the equation k = (5 - x) (x + 2) 2 has one distinct root only. [3]

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Find the coefficient of x 8 in the expansion of `1 - x 2jb2x - l .

10
1
3 [5]
x

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4 (a) Write 3 lg x - 2 lg y 2 - 3 as a single logarithm to base 10. [3]

5
(b) Solve the equation log3 x + log x 3 = . [5]
2

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The table shows values of the variables x and y, which are related by an equation of the form y = Ab x ,
2
5
where A and b are constants.

x 1 1.5 2 2.5
y 2.0 11.3 128 2896

(a) Use the data to draw a straight line graph of ln y against x 2 . [2]

ln y
8

2
0 1 2 3 4 5 6 7 x

–1
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(b) Use your graph to estimate the values of A and b. Give your answers correct to 1 significant figure.
[5]

(c) Estimate the value of y when x = 1.75. [2]

(d) Estimate the positive value of x when y = 20 . [2]

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- 25 17 26
6 Given that f m (x) = (5x + 2) , f l (6) = and f (6) = , find an expression for f (x) . [8]
3 3

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7 (a) A 5-character password is to be formed from the following 13 characters.

Letters A B C D E

Numbers 9 8 7 6 5

Symbols * # !

No character may be used more than once in any password.

(i) Find the number of possible passwords that can be formed. [1]

(ii) Find the number of possible passwords that contain at least one symbol. [2]

(b) Given that 16 # n C12 = (n - 10) # n + 1 C11 , find the value of n. [3]

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8
y

3
y = 2-
6y = 9 - 2x x-1

O A C x

3
The diagram shows part of the curve y = 2 - and the straight line 6y = 9 - 2x . The curve
x-1
intersects the x-axis at point A and the line at point B. The line intersects the x-axis at point C. Find
the area of the shaded region ABC, giving your answer in the form p + ln q , where p and q are rational
numbers. [11]

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Additional working space for Question 8.

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9 In this question, all lengths are in metres.

J2 + 12tN
(a) A particle P has position vector KK OO at a time t seconds, t H 0 .
L 5 - 5t P

(i) Write down the initial position vector of P. [1]

(ii) Find the speed of P. [2]

J 158N
(iii) Determine whether P passes through the point with position vector KK OO . [2]
L- 48 P

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(b)
O

a b c

A B C

The diagram shows the triangle OAC. The point B lies on AC such that AB : AC = 1: 4 . Given that
OA = a , OB = b and OC = c , find c in terms of a and b. [3]

Question 10 is printed on the next page.

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10 (a) It is given that 2 + cos i = x for 1 1 x 1 3 and 2 cosec i = y for y 2 2 . Find y in terms of x.
[4]

z z
(b) Solve the equation 3 cos = 3 sin for - 4r 1 z 1 4r . [5]
2 2

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