Cambridge IGCSE Mathematics - International - 0607 - Advanced Paper 4 2021 Winter Zone 3

The table shows the marks scored by 180 students in an examination.

[Table_1]

(a) (i) Write down the mode. ................................................ [1]

(ii) Write down the range. ................................................ [1]

(iii) Find the median. ................................................ [1]

(iv) Find the interquartile range. ................................................ [1]

(v) Calculate the mean. ................................................ [2]

(b) A different group of 140 students take the same examination. The marks of the two groups are combined and the mean mark of the 320 students is 6.5. Find the mean mark of the 140 students.

................................................ [2]

You may use this grid to help you answer this question.

Transformation P is a rotation of 180° about the origin.
Transformation Q is a reflection in the line $y = x$.
(a) Find the coordinates of the image of the point (5, 2) under transformation P.
( ....................... , ....................... ) [1]
(b) Find the coordinates of the image of the point (5, 2) under transformation Q.
( ....................... , ....................... ) [1]
(c) Find the coordinates of the image of the point $(x, y)$ under transformation P followed by transformation Q.
( ....................... , ....................... ) [2]
(d) Describe fully the single transformation that is equivalent to transformation Q followed by transformation P.
...................................................................................................................
...................................................................................................................[2]

Anna flies by plane from Manchester (UK) to Goa (India). The plane flies a distance of 7650 km.
(a) The flight takes 8.5 hours.
(i) Calculate the average speed of the plane.
....................................... km/h [1]
(ii) The plane leaves Manchester at 2045. The local time in Goa is 5 hours 30 minutes ahead of the local time in Manchester. Find the local time in Goa when the plane lands.
....................................... [2]

(b) The exchange rate is 1 pound (£) = 90 Indian rupees (INR).
(i) The cost of the flight is £299. Calculate the cost of the flight in Indian rupees.
INR ....................................... [1]
(ii) Anna returns to Manchester with 4014 Indian rupees. She changes this money into pounds.
Calculate this amount in pounds.
£ ....................................... [1]

The points $A (2, 5)$ and $B (10, 1)$ are shown on the diagram.
(a) Find the gradient of the line $AB$. [2]
(b) Find the equation of the line $AB$. Give your answer in the form $y = mx + c$. [2]
(c) The point $C$ has coordinates $(6, k)$ where $k > 0$. The line $CA$ is perpendicular to the line $AB$ and $AC = AB$. Find $k$. [3]
(d) The point $D$ is such that $ABDC$ is a square. Find the coordinates of $D$. [2]
(e) Find the area of triangle $BCD$. [3]

(a) Alana and Beau share $200 in the ratio $x : y$.
An expression for the amount of money Alana receives is $\frac{200x}{x+y}$.
(i) Write down an expression for the amount of money Beau receives.
........................................................... [1]
(ii) Alana and Beau are each given an extra $50. The ratio of the total amount of money that each person now has is $3:1$. Find the value of $\frac{x}{y}$.
$\frac{x}{y} = ...........................................................$ [5]

(b) (i) On 1 January each year Bruno invests $1000 in Bank A. Bank A pays simple interest at a rate of 4% per year.
Show that the total value of Bruno’s investment in Bank A at the end of 4 years is $4400.
[3]
(ii) On 1 January each year Bruno also invests $1000 in Bank B. Bank B pays compound interest at a rate of 3.5% per year.
Find the total value of Bruno’s investment in Bank B at the end of 4 years.
$...........................................................$ [3]

The Venn diagram shows the sets $P$, $F$ and $M$.
[Image_1: Venn Diagram with sets P, F, M]

$U = \{ \text{integer values of } x \mid 2 \leq x \leq 12 \}$
$P = \{ \text{prime numbers} \}$
$F = \{ \text{factors of } 12 \}$
$M = \{ \text{multiples of } 3 \}$

(a) List the elements of set $P$ and the elements of set $F$.

$P = \text{.................................}$
$F = \text{.................................}$ [2]

(b) Write each element of $U$ in the correct region of the Venn diagram. [2]

(c) List the elements of:

(i) $F \cup M$,
\text{.................................} [1]

(ii) $P' \cap M$,
\text{.................................} [1]

(iii) $(P \cup F \cup M)'$.
\text{.................................} [1]

(d) Find $n((P \cap F)' \cap M)$.
\text{.................................} [1]

y\text{ varies inversely as the square of }x.\ y = 5 \text{ when } x = 3.\
(a)\ (i)\ \text{Find } y \text{ in terms of } x.\
\ \\ y = \text{..................................................} [2]\
(ii)\ \text{Find the value of } x \text{ when } y = 20.\ \\ x = \text{...............................................} [2]\
(b)\ z \text{ varies directly as the square root of } y.\ z = 12 \text{ when } y = 9.\
\textbf{Use your answer to part (a)(i) to find } z \text{ in terms of } x.\
\\ z = \text{.....................................................} [3]

Given $f(x) = 3x - x^3$ for $-2 \leq x \leq 2$.
(a) On the diagram, sketch the graph of $y = f(x)$. [2]
(b) Find the coordinates of the local maximum.
\( (\text{....................} , \text{....................}) \) [1]
(c) Write down the $x$-coordinates of the points where the curve meets the $x$-axis.
\( x = \text{................} , \ x = \text{................} , \ x = \text{................} \) [2]
(d) (i) Describe fully the \textit{single} transformation that maps $y = f(x)$ onto $y = f(x+1)$. [2]
......................................................................................................................
......................................................................................................................
(ii) Solve $f(x) = f(x+1)$ for $-2 \leq x \leq 2$. [2]
...........................................................
(iii) Solve $f(x) \geq f(x+1)$ for $-2 \leq x \leq 2$. [2]
...........................................................

!
A, B and C lie on a circle, centre O.
AP and BP are tangents to the circle.
AB intersects OP at D and angle OAB = x^\circ.
(a) Write down the size of angle OBP.

Angle \( OBP = \text{........................} \) \[1\]
(b) Find, in terms of \( x \),
(i) angle AOD,

Angle \( AOD = \text{........................} \) \[1\]
(ii) angle ACB,

Angle \( ACB = \text{........................} \) \[1\]
(iii) angle APB.

Angle \( APB = \text{........................} \) \[1\]
(c) Write down the mathematical name of quadrilateral AOBP.

\text{........................} \[1\]
(d) Write down
(i) two triangles that are congruent,

\text{........................} \[1\]
(ii) two triangles that are similar but not congruent.

\text{........................} \[1\]

The diagram shows a solid made from a cylinder, a hemisphere and a cone, each with radius 4 cm. The cylinder has length 16 cm. The slant height of the cone is 12 cm.
(a) Find the volume of the solid.
............................................ $cm^{3}$ [5]

(b) Show that the total surface area of the solid is $208\pi \ cm^{2}$.
[4]

(c) A mathematically similar solid has a total surface area of $468\pi \ cm^{2}$.

Find the radius of the cylinder in this solid.
............................................ $cm$ [3]

Angles $ACB$ and $ACD$ are obtuse.
(a) Show that $AC = 95.9$ m correct to the nearest 0.1 metre. [3]
(b) Find angle $ACD$.
Angle $ACD = .............................................$ [4]
(c) The area of triangle $ABD$ is $5137$ m$^2$.
Calculate the area of triangle $BCD$.
...................................... m$^2$ [4]

(a) Solve.
(i) $9 = 5 - \frac{2}{x}$
$x = \text{................................................}$ [3]

(ii) $\frac{6}{x-4} > 3$
$\text{....................................................}$ [3]

(b) (i) Solve the equation, giving your answers correct to 3 significant figures.
$2x^2 - 5x + 1 = 0$
$x = \text{............... or } x = \text{...............}$ [3]

(ii) Use your answers to part (b)(i) to solve
$2(\tan y)^2 - 5(\tan y) + 1 = 0$ for $0^\circ \leq y \leq 180^\circ$.
$y = \text{............ or } y = \text{............}$ [2]

Two bags each contain only blue balls and red balls.
Bag 1 contains 7 blue balls and 3 red balls.
Bag 2 contains 3 blue balls and 7 red balls.
Maria chooses a ball at random from Bag 1 and puts it into Bag 2.
(a) Find the probability that the ball chosen is blue. ........................................... [1]
(b) Maria now chooses a ball at random from Bag 2 and puts it into Bag 1.
(i) Find the probability that both balls chosen are red. ........................................... [2]
(ii) Find the probability that one of the balls chosen is red and the other is blue. ........................................... [3]
(iii) Find the probability that there are now exactly 7 blue balls in Bag 1. ........................................... [3]