IGCSE Mathematics 0580 Paper 42, Oct/Nov 2023

Translate triangle $T$ by the vector $\left(\begin{array}{c}-7\\ 1\end{array}\right)$. Label the image $K$.

Describe fully the single transformation that maps triangle $K$ onto triangle $T$.

Reflect triangle $T$ in the line $y=4$.

Rotate triangle $T$ through ${90}^{\circ }$ clockwise about $(0,0)$.

Enlarge triangle $T$ by scale factor $-\frac{1}{2}$, centre $(0,0)$. Label the image $P$.

Describe fully the single transformation that maps triangle $P$ onto triangle $T$.

Daisy records her 50 homework marks. The table shows the results.

Write down the range.

Write down the mode.

Find the median.

Calculate the mean.

21 33 20 25 21 34 22 21 20 30 18 The list shows Ed's scores in 11 tests. Complete the stem-and-leaf diagram to show this information.

Key: $2\mathrm{\mid }5$ represents 25

Find the median.

Find the interquartile range.

The value of Priya's car decreases by 10% every year. The value today is $7695. Calculate the value of the car after one year.

Calculate the value of the car one year ago.

Ali invests $600 at a rate of 2% per year simple interest. Calculate the value of Ali's investment at the end of 5 years.

Sara invests $500 at a rate of $r$ per year compound interest. At the end of 12 years, the value of Sara's investment is $601.35, correct to the nearest cent. Find the value of $r$.

The mass of a radioactive substance decreases exponentially at a rate of 3% each day. Find the overall percentage decrease at the end of 10 days.

Find the number of whole days it takes until the mass of this substance is one half of its original amount.

The diagram shows a sector of a circle that is made into a cone by joining $OA$ to $OB$. The sector angle is $x^{\circ }$ and the radius of the sector is $7.5\mathrm{c}\mathrm{m}$. The base radius of the cone is $1.5\mathrm{c}\mathrm{m}$. Calculate the value of $x$.

The diagram shows a cylinder with radius $8\mathrm{c}\mathrm{m}$ inside a sphere with radius $17\mathrm{c}\mathrm{m}$. Both ends of the cylinder touch the curved surface of the sphere. Show that the height of the cylinder is $30\mathrm{c}\mathrm{m}$.

Calculate the volume of the cylinder as a percentage of the volume of the sphere. [The volume, $V$, of a sphere with radius $r$ is $V=\frac{4}{3}\pi r^3$.]

The diagram shows a solid sphere with radius $6\mathrm{c}\mathrm{m}$ inside a cube with side length $20\mathrm{c}\mathrm{m}$. The cube contains water to a depth of $15\mathrm{c}\mathrm{m}$. The sphere is removed. Calculate the new depth of water in the cube. [The volume, $V$, of a sphere with radius $r$ is $V=\frac{4}{3}\pi r^3$.]

In a shop the cost of a fiction book is $\mathrm{\$}x$ and the cost of a reference book is $\mathrm{\$}(x+2)$. The cost of 11 fiction books is the same as the cost of 10 reference books. Find the value of $x$.

In another shop, the cost of a fiction book is $\mathrm{\$}y$ and the cost of a reference book is $\mathrm{\$}(y+2)$. Maria spends $95 on fiction books and $147 on reference books. She buys a total of 12 books. Show that $6y^2-109y-95=0$.

Factorise $6y^2-109y-95$.

Find the value of $y$.

The diagram shows a right-angled triangle $PQR$ on horizontal ground. $X$ is vertically above $R$ and the angle of elevation of $X$ from $P$ is ${21}^{\circ }$. $XR=2.8\mathrm{m}$ and $RQ=7.1\mathrm{m}$. Calculate the angle of elevation of $X$ from $Q$.

Calculate $PQ$.

Calculate the acute angle $KML$.

The area of triangle $ABC$ is $62.89{\mathrm{c}\mathrm{m}}^2$. Show that angle $BAC={28.4}^{\circ }$, correct to 1 decimal place.

Calculate $BC$.

$AB$ is extended to a point $D$ such that angle $BDC={90}^{\circ }$. Calculate $BD$.

Dice A is rolled once. Write down the probability that it lands on the number 6.

Dice A is rolled 150 times. Find the number of times it is expected to land on the number 6.

Dice A and Dice B are each rolled once. Find the probability that the two numbers they land on have a total of 6.

Find the probability that when the two numbers they land on have a total of 6, both numbers are 3.

Dice B is rolled $n$ times. The probability that on the $n$th roll it first lands on a number 3 is $\frac{32}{729}$. Find the value of $n$.

Differentiate $4x^3-x^4$.

Find the coordinates of $B$.

Find the gradient of the graph at the point $A$.

$ABCDEF$ is a regular hexagon. $DF$, $DA$ and $DB$ are diagonals. Complete the following statements using three different triangles. Triangle DEF is congruent to triangle .................. Triangle .................. is congruent to triangle ..................

$P$ and $Q$ are points on the circle with centre $O$. $TP$ and $TQ$ are tangents to the circle from the point $T$. Complete the following statements and reasons. In triangles $OPT$ and $OQT$ $OP=\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}$ because each is a radius of the circle $OT$ is a common side Angle $OPT=$ angle $\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.}={90}^{\circ }$ because .............................................................................. Triangles $OPT$ and $OQT$ are congruent using the criterion .................. This proves that the tangents $TP$ and $TQ$ are ..................................................

Find $g(3)$.

Find $f(x-2)$, giving your answer in its simplest form.

Find $f^{-1}(x)$.

$gf(x)-g(x)f(x)=3x^3+a{x}^2+bx+c$ Find the value of each of $a$, $b$ and $c$.

Find $h(x)-f(x)$, giving your answer as a single fraction in its simplest form.

$h(x^n)=3x^7$ Find the value of $n$.

Write as column vectors. $\overrightarrow{OB}$

$\overrightarrow{AB}$

Find the equation of the line $AB$. Give your answer in the form $y=mx+c$.

Find the equation of the perpendicular bisector of $AB$. Give your answer in the form $y=mx+c$.

The line $AB$ meets the $y$-axis at $P$. The perpendicular bisector of $AB$ meets the $y$-axis at $Q$. Find the length of $PQ$.