id stringlengths 1 4 | question stringlengths 31 709 | options listlengths 6 6 | image_path stringlengths 12 15 | image imagewidth (px) 56 2.52k | answer stringclasses 5
values | solution stringclasses 259
values | level int64 1 5 | subject stringclasses 16
values | Answer(Option) stringclasses 5
values |
|---|---|---|---|---|---|---|---|---|---|
1585 | The diagram on the right shows a square with side $3 \mathrm{~cm}$ inside a square with side $7 \mathrm{~cm}$ and another square with side $5 \mathrm{~cm}$ which intersects the first two squares. What is the difference between the area of the black region and the total area of the grey regions? <image1> | [
"$0 \\mathrm{~cm}^{2}$",
"$10 \\mathrm{~cm}^{2}$",
"$11 \\mathrm{~cm}^{2}$",
"$15 \\mathrm{~cm}^{2}$",
"more information needed",
"I don't know"
] | images/1585.jpg | D | Let the area of the white hexagon be $x \mathrm{~cm}^{2}$, as indicated in the diagram. Then the black area is $49-(9+x)=(40-x) \mathrm{cm}^{2}$. The total of the grey areas is $(25-x) \mathrm{cm}^{2}$. Thus the difference between the areas of the black and grey ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2... | 3 | metric geometry - area | D | |
1587 | Each of the nine paths in a park is $100 \mathrm{~m}$ long. Ann wants to go from $X$ to $Y$ without going along any path more than once. What is the length of the longest route she can choose? <image1> | [
"$900 \\mathrm{~m}$",
"$800 \\mathrm{~m}$",
"$700 \\mathrm{~m}$",
"$600 \\mathrm{~m}$",
"$500 \\mathrm{~m}$",
"I don't know"
] | images/1587.jpg | C | Altogether there are nine paths, making $900 \mathrm{~m}$ of path in total. The route is to start at $\mathrm{X}$ and not repeat any path. This means that only one of the paths from $\mathrm{X}$ can be used in the route. [Otherwise the other path would be used to bring Ann back to $\mathrm{X}$ and there would be no pat... | 3 | graph theory | C | |
1589 | Werner folds a sheet of paper as shown in the diagram and makes two straight cuts with a pair of scissors. He then opens up the paper again. Which of the following shapes cannot be the result? <image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1589.jpg | D | The shapes given in options $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and $\mathrm{E}$ can be obtained by cutting the paper as shown.  ![](https://cdn.mathpix.com/cropped/2023_12_27_0f4ed2787981... | 3 | transformation geometry | D | |
1590 | Mrs Gardner has beds for peas and strawberries in her rectangular garden. This year, by moving the boundary between them, she changed her rectangular pea bed to a square by lengthening one of its sides by 3 metres. As a result of this change, the area of the strawberry bed reduced by $15 \mathrm{~m}^{2}$. What was the ... | [
"$5 \\mathrm{~m}^{2}$",
"$9 \\mathrm{~m}^{2}$",
"$10 \\mathrm{~m}^{2}$",
"$15 \\mathrm{~m}^{2}$",
"$18 \\mathrm{~m}^{2}$",
"I don't know"
] | images/1590.jpg | C | If the length of the pea bed is increased by $3 \mathrm{~m}$, the length of the strawberry bed is decreased by $3 \mathrm{~m}$. As the area of the strawberry bed is reduced by $15 \mathrm{~m}^{2}$, this means that the width of the strawberry bed (and hence of the pea bed) was $(15 \div 3) \mathrm{m}=5 \mathrm{~m}$. The... | 3 | metric geometry - area | C | |
1593 | Three small equilateral triangles of the same size are cut from the corners of a larger equilateral triangle with sides $6 \mathrm{~cm}$ as shown. The sum of the perimeters of the three small triangles is equal to the perimeter of the remaining hexagon. What is the side-length of one of the small triangles? <image1> | [
"$1 \\mathrm{~cm}$",
"$1.2 \\mathrm{~cm}$",
"$1.25 \\mathrm{~cm}$",
"$1.5 \\mathrm{~cm}$",
"$2 \\mathrm{~cm}$",
"I don't know"
] | images/1593.jpg | D | If we let the length of the side of one of the removed triangles be $x \mathrm{~cm}$, the perimeter of the remaining hexagon will be $3 x+3(6-2 x) \mathrm{cm}$. Hence we have $3(3 x)=3 x+3(6-2 x)$ which has solution $x=18 / 12=1.5$. | 2 | metric geometry - length | D | |
1594 | A cube is being rolled on a plane so it turns around its edges. Its bottom face passes through the positions $1,2,3,4,5,6$ and 7 in that order, as shown. Which of these two positions were occupied by the same face of the cube? <image1> | [
"1 and 7",
"1 and 6",
"1 and 5",
"2 and 7",
"2 and 6",
"I don't know"
] | images/1594.jpg | B | Imagine the grid is sticky so that when the cube rolls over it, each cell of the grid fastens to the face of the cube touching it. The result would be equivalent to taking the arrangement of cells as shown, cutting it out and folding it into a cube. The latter is possible (for example) with 5 on ![](https://cdn.mathpi... | 4 | solid geometry | B | |
1595 | In the diagram, $W X Y Z$ is a square, $M$ is the midpoint of $W Z$ and $M N$ is perpendicular to $W Y$. What is the ratio of the area of the shaded triangle $M N Y$ to the area of the square? <image1> | [
"1:6",
"1:5",
"7:36",
"3:16",
"7:40",
"I don't know"
] | images/1595.jpg | D | Introduce point $T$, the mid-point of $W Y$. $M T$ is parallel to $Z Y$ and half the length. The area of triangle $W M T$ is $\frac{1}{4}$ of the area of triangle $W Z Y=\frac{1}{8}$ of the area of the square. Also, the area of triangle $W M N$ is $\frac{1}{2}$ of the area of triangle $W M T=\frac{1}{16}$ of the area o... | 3 | metric geometry - area | D | |
1596 | A large triangle is divided into four smaller triangles and three quadrilaterals by three straight line segments. The sum of the perimeters of the three quadrilaterals is $25 \mathrm{~cm}$. The sum of the perimeters of the four triangles is $20 \mathrm{~cm}$. The perimeter of the original triangle is $19 \mathrm{~cm}$.... | [
"$11 \\mathrm{~cm}$",
"$12 \\mathrm{~cm}$",
"$13 \\mathrm{~cm}$",
"$15 \\mathrm{~cm}$",
"$16 \\mathrm{~cm}$",
"I don't know"
] | images/1596.jpg | C | If we add together the sum of the perimeters of the quadrilaterals and the four smaller triangles we get $45 \mathrm{~cm}$. This distance equals twice the sum of the lengths of the three line segments plus the length of the perimeter of the triangle. Hence the sum of the lengths of the line segments is $(45-19) / 2=13 ... | 2 | metric geometry - length | C | |
1600 | In the diagram, $\alpha=55^{\circ}, \beta=40^{\circ}$ and $\gamma=35^{\circ}$. What is the value of $\delta$ ? <image1> | [
"$100^{\\circ}$",
"$105^{\\circ}$",
"$120^{\\circ}$",
"$125^{\\circ}$",
"$130^{\\circ}$",
"I don't know"
] | images/1600.jpg | E | Let $\theta$ be the angle as shown in the diagram. As the exterior angle of a triangle is equal to the sum of the two interior opposite angles, we have $\theta=\alpha+\beta$ and $\delta=\gamma+\theta$. This gives $\delta=\alpha+\beta+\gamma=55^{\circ}+40^{\circ}+35^{\circ}=130^{\circ}$.  \div(x$-coordinate $)$. Which of the four poi... | [
"P",
"Q",
"R",
"S",
"It depends on the rectangle.",
"I don't know"
] | images/1601.jpg | A | The value calculated for all four points will be negative. The least value will be obtained by calculating the most negative $y$-coordinate $\div$ least positive $x$ coordinate. The most negative $y$-coordinates are at $P$ and $Q$ while the least positive $x$-coordinates are at $P$ and $S$. Hence the point that will gi... | 3 | analytic geometry | A | |
1603 | John has made a building of unit cubes standing on a $4 \times 4$ grid. The diagram shows the number of cubes standing on each cell. When John looks horizontally at the building from behind, what does he see? <image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1603.jpg | C | Looking horizontally from behind, John will see the largest number of cubes in each column in the table. This means that, from his left, he will see 2, 3, 3 and 4 cubes. Therefore, the shape he will see is $\mathrm{C}$. | 4 | descriptive geometry | C | |
1604 | The diagram shows a shaded quadrilateral $P Q R S$ drawn on a grid. Each cell of the grid has sides of length $2 \mathrm{~cm}$. What is the area of quadrilateral $P Q R S$ ? <image1> | [
"$96 \\mathrm{~cm}^{2}$",
"$84 \\mathrm{~cm}^{2}$",
"$76 \\mathrm{~cm}^{2}$",
"$88 \\mathrm{~cm}^{2}$",
"$104 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1604.jpg | B | Surround the quadrilateral $P Q R S$ by a rectangle with sides parallel to the grid lines as shown. The area of the rectangle is $14 \times 10=140 \mathrm{~cm}^{2}$. The area of quadrilateral $P Q R S$ can be calculated by subtracting from this the sum of the areas of the four triangles and one square that lie outside ... | 4 | combinatorial geometry | B | |
1605 | One of the following nets cannot be folded along the dashed lines shown to form a cube. Which one?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1605.jpg | C | In each net, the central $4 \times 1$ rectangle can be folded round to form the front, the sides and the back of a cube. The remaining triangles, if correctly positioned, will then fold to form the top and the bottom of the cube. To complete the cube, the triangles must fold down so that the shorter sides of each trian... | 4 | solid geometry | C | |
1611 | In the diagram, the area of each circle is $1 \mathrm{~cm}^{2}$. The area common to any two overlapping circles is $\frac{1}{8} \mathrm{~cm}^{2}$. What is the area of the region covered by the five circles? <image1> | [
"$4 \\mathrm{~cm}^{2}$",
"$\\frac{9}{2} \\mathrm{~cm}^{2}$",
"$\\frac{35}{8} \\mathrm{~cm}^{2}$",
"$\\frac{39}{8} \\mathrm{~cm}^{2}$",
"$\\frac{19}{4} \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1611.jpg | B | There are four regions where two circles overlap. Therefore the area covered by the five circles in $\mathrm{cm}^{2}$ is $5 \times 1-4 \times \frac{1}{8}=\frac{9}{2}$. | 3 | metric geometry - area | B | |
1612 | The heart and the arrow are in the positions shown in the figure. At the same time the heart and the arrow start moving. The arrow moves three places clockwise and then stops and the heart moves four places anticlockwise and then stops. They repeat the same routine over and over again. After how many routines will the ... | [
"7",
"8",
"9",
"10",
"It will never happen",
"I don't know"
] | images/1612.jpg | E | The figure contains seven regions. An anticlockwise rotation of four regions on such a figure is equivalent to a clockwise rotation of three regions. Hence, each routine involves the two symbols moving three regions clockwise and so they will never land in the same region. | 3 | transformation geometry | E | |
1614 | The diagram shows the triangle $P Q R$ in which $R H$ is a perpendicular height and $P S$ is the angle bisector at $P$. The obtuse angle between $R H$ and $P S$ is four times angle $S P Q$. What is angle $R P Q$ ? <image1> | [
"$30^{\\circ}$",
"$45^{\\circ}$",
"$60^{\\circ}$",
"$75^{\\circ}$",
"$90^{\\circ}$",
"I don't know"
] | images/1614.jpg | C | Let $X$ be the point where $R H$ meets $P S$. In $\triangle H X P$, $\alpha+90^{\circ}+\angle H X P=180^{\circ}$. This gives $\angle H X P=90^{\circ}-\alpha$. Angles on a straight line add to $180^{\circ}$ so $4 \alpha+90^{\circ}-\alpha=180^{\circ}$ with solution $\alpha=30^{\circ}$. Hence the size of $\angle R P Q$ is... | 5 | metric geometry - angle | C | |
1618 | My umbrella has KANGAROO written on top as shown in the diagram. Which one of the following pictures also shows my umbrella?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1618.jpg | E | In diagrams $A, C$ and $D$, the letters ' $N$ ', ' $R$ ' and ' $G$ ' respectively have been reversed. In diagram $\mathrm{B}$, the letters are not in the order they appear on the original umbrella. Hence only option E shows part of the original umbrella. (This is immediately clear if you turn the question paper round ... | 4 | descriptive geometry | E | |
1619 | Four identical small rectangles are put together to form a large rectangle as shown. The length of a shorter side of each small rectangle is $10 \mathrm{~cm}$. What is the length of a longer side of the large rectangle? <image1> | [
"$50 \\mathrm{~cm}$",
"$40 \\mathrm{~cm}$",
"$30 \\mathrm{~cm}$",
"$20 \\mathrm{~cm}$",
"$10 \\mathrm{~cm}$",
"I don't know"
] | images/1619.jpg | B | From the diagram, the length of a small rectangle is twice the width. Hence the length of a small rectangle is $20 \mathrm{~cm}$. Therefore the length of the large rectangle, in $\mathrm{cm}$, is $20+2 \times 10=40$. | 2 | metric geometry - length | B | |
1620 | In the diagram, the centre of the top square is directly above the common edge of the lower two squares. Each square has sides of length $1 \mathrm{~cm}$. What is the area of the shaded region? <image1> | [
"$\\frac{3}{4} \\mathrm{~cm}^{2}$",
"$\\frac{7}{8} \\mathrm{~cm}^{2}$",
"$1 \\mathrm{~cm}^{2}$",
"$1 \\frac{1}{4} \\mathrm{~cm}^{2}$",
"$1 \\frac{1}{2} \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1620.jpg | C | The centre of the top square is directly above the common edge of the lower two squares. Hence a rectangle half the size of the square, and so of area $\frac{1}{2} \mathrm{~cm}^{2}$, can be added to the diagram to form a right-angled triangle as shown. The area of the shaded region and the added rectangle is equal to $... | 3 | metric geometry - area | C | |
1621 | A bush has 10 branches. Each branch has either 5 leaves only or 2 leaves and 1 flower. Which of the following could be the total number of leaves the bush has? <image1> | [
"45",
"39",
"37",
"31",
"None of A to D",
"I don't know"
] | images/1621.jpg | E | The maximum number of leaves the bush could have is $10 \times 5=50$. Each branch that has two leaves and a flower instead of five leaves reduces the number of leaves the bush has by three. Therefore the total number of leaves the bush has is of the form $50-3 n$ where $n$ is the number of branches with two leaves and ... | 4 | algebra | E | |
1623 | Luis wants to make a pattern by colouring the sides of the triangles shown in the diagram. He wants each triangle to have one red side, one green side and one blue side. Luis has already coloured some of the sides as shown. What colour can he use for the side marked $x$ ? <image1> | [
"only green",
"only blue",
"only red",
"either blue or red",
"The task is impossible",
"I don't know"
] | images/1623.jpg | A |  Label the internal sides of the diagram $a, b, c, d$ and $e$ as shown. The side labelled $a$ is in a triangle with a green side and in a triangle with a blue side and so is to be coloured re... | 4 | logic | A | |
1625 | A square with area $30 \mathrm{~cm}^{2}$ is divided in two by a diagonal and then into triangles as shown. The areas of some of these triangles are given in the diagram (which is not drawn to scale). Which part of the diagonal is the longest? <image1> | [
"$a$",
"$b$",
"$C$",
"$d$",
"$e$",
"I don't know"
] | images/1625.jpg | D | Label the corners of the square $A, B, C$ and $D$ going anticlockwise from the top left corner. Draw in the lines from each marked point on the diagonal to $B$ and to $D$. All the triangles with a base on the diagonal and a vertex at $B$ or $D$ have the same perpendicular height. Hence their areas are directly proporti... | 2 | metric geometry - length | D | |
1627 | The triangle in the diagram contains a right angle. What is the sum of the other two marked angles on the diagram? <image1> | [
"$150^{\\circ}$",
"$180^{\\circ}$",
"$270^{\\circ}$",
"$320^{\\circ}$",
"$360^{\\circ}$",
"I don't know"
] | images/1627.jpg | C | Since angles in a triangle add to $180^{\circ}$ and one angle is given as $90^{\circ}$, the two blank angles in the triangle add to $90^{\circ}$. Since angles on a straight line add to $180^{\circ}$, the sum of the two marked angles and the two blank angles in the triangle is $2 \times 180^{\circ}=360^{\circ}$. Therefo... | 5 | metric geometry - angle | C | |
1628 | Joanna turns over the card shown about its lower edge and then about its right-hand edge, as indicated in the diagram.
<image1>
What does she see?
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1628.jpg | B | When the card is turned about its lower edge, the light grey triangle will be at the top and the dark grey triangle will be on the left. When this is turned about its right-hand edge, the light grey triangle will be at the top and the dark grey triangle will be on the right. Therefore Joanna will see option B. | 3 | transformation geometry | B | |
1631 | Three equilateral triangles are cut from the corners of a large equilateral triangle to form an irregular hexagon, as shown in the diagram.
The perimeter of the large equilateral triangle is $60 \mathrm{~cm}$. The perimeter of the irregular hexagon is $40 \mathrm{~cm}$. What is the sum of the perimeters of the triangle... | [
"$60 \\mathrm{~cm}$",
"$66 \\mathrm{~cm}$",
"$72 \\mathrm{~cm}$",
"$75 \\mathrm{~cm}$",
"$81 \\mathrm{~cm}$",
"I don't know"
] | images/1631.jpg | A | Let the lengths of the sides of the equilateral triangles that are cut off be $x \mathrm{~cm}, y \mathrm{~cm}$ and $z \mathrm{~cm}$, as shown in the diagram. The length of a side of the large equilateral triangle is $\frac{1}{3} \times 60 \mathrm{~cm}=20 \mathrm{~cm}$. The perimeter of the irregular hexagon is $40 \mat... | 2 | metric geometry - length | A | |
1632 | A $3 \mathrm{~cm}$ wide strip is grey on one side and white on the other. Maria folds the strip, so that it fits inside a rectangle of length $27 \mathrm{~cm}$, as shown. The grey trapeziums are identical. What is the length of the original strip?
<image1> | [
"$36 \\mathrm{~cm}$",
"$48 \\mathrm{~cm}$",
"$54 \\mathrm{~cm}$",
"$57 \\mathrm{~cm}$",
"$81 \\mathrm{~cm}$",
"I don't know"
] | images/1632.jpg | D | Let the length of the shorter of the two parallel sides of the grey trapeziums be $x$ $\mathrm{cm}$. Since the folded shape is $27 \mathrm{~cm}$ long and the strip is $3 \mathrm{~cm}$ wide, we have $3+x+3+x+3+x+3+x+3=27$ which has solution $x=3$. Hence the length of the longer of the two parallel sides of the grey trap... | 3 | transformation geometry | D | |
1634 | Inside a square of area $36 \mathrm{~cm}^{2}$, there are shaded regions as shown. The total shaded area is $27 \mathrm{~cm}^{2}$. What is the value of $p+q+r+s$ ? <image1> | [
"$4 \\mathrm{~cm}$",
"$6 \\mathrm{~cm}$",
"$8 \\mathrm{~cm}$",
"$9 \\mathrm{~cm}$",
"$10 \\mathrm{~cm}$",
"I don't know"
] | images/1634.jpg | D | Since the area of the square is $36 \mathrm{~cm}^{2}$, the length of a side of the square is $6 \mathrm{~cm}$. Since the shaded area is $27 \mathrm{~cm}^{2}$, the area not shaded is $(36-27) \mathrm{cm}^{2}=9 \mathrm{~cm}^{2}$. Let $a \mathrm{~cm}, b \mathrm{~cm}$ and $c \mathrm{~cm}$ be the lengths of the parts of the... | 2 | metric geometry - length | D | |
1636 | The diagram shows a pentagon. The lengths of the sides of the pentagon are given in the diagram.
Sepideh draws five circles with centres $A, B, C, D$ and $E$ such that the two circles with centres at the ends of a side of the pentagon touch on that side. Which point is the centre of the largest circle that she draws? <... | [
"$A$",
"$B$",
"$C$",
"$D$",
"$E$",
"I don't know"
] | images/1636.jpg | A | Let the radius of the circle with centre $A$ be $x \mathrm{~cm}$. Therefore, since the circles drawn on each side of the pentagon touch, the radius of the circle with centre $B$ is $(16-x) \mathrm{cm}$. Similarly, the radius of the circle with centre $C$ is $(14-(16-x)) \mathrm{cm}=(x-2) \mathrm{cm}$, the radius of the... | 2 | metric geometry - length | A | |
1638 | A $3 \times 3 \times 3$ cube is built from 15 black cubes and 12 white cubes. Five faces of the larger cube are shown.
<image1>
Which of the following is the sixth face of the larger cube?
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1638.jpg | A | Note first that a small cube in the centre of a face of the large cube will only appear on one face while a cube appearing on the edge of a face of the large cube will appear on two faces and a cube appearing at a corner of the face of the large cube will appear on three faces. Hence, the total number of white faces on... | 4 | descriptive geometry | A | |
1639 | The diagram shows two rectangles whose corresponding sides are parallel as shown. What is the difference between the lengths of the perimeters of the two rectangles? <image1> | [
"$12 \\mathrm{~m}$",
"$16 \\mathrm{~m}$",
"$20 \\mathrm{~m}$",
"$22 \\mathrm{~m}$",
"$24 \\mathrm{~m}$",
"I don't know"
] | images/1639.jpg | E | The length of the outer rectangle is $(3+4) \mathrm{m}=7 \mathrm{~m}$ longer than the length of the inner rectangle. The height of the outer rectangle is $(2+3) \mathrm{m}=5 \mathrm{~m}$ longer than the height of the inner rectangle. Hence the length of the perimeter of the outer rectangle is $(2 \times 7+2 \times 5) \... | 2 | metric geometry - length | E | |
1640 | The diagram shows four overlapping hearts. The areas of the hearts are $1 \mathrm{~cm}^{2}, 4 \mathrm{~cm}^{2}, 9 \mathrm{~cm}^{2}$ and $16 \mathrm{~cm}^{2}$. What is the total shaded area? <image1> | [
"$9 \\mathrm{~cm}^{2}$",
"$10 \\mathrm{~cm}^{2}$",
"$11 \\mathrm{~cm}^{2}$",
"$12 \\mathrm{~cm}^{2}$",
"$13 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1640.jpg | B | Since the areas of the four hearts are $1 \mathrm{~cm}^{2}, 4 \mathrm{~cm}^{2}, 9 \mathrm{~cm}^{2}$ and $16 \mathrm{~cm}^{2}$, the outer and inner shaded regions have areas $16 \mathrm{~cm}^{2}-9 \mathrm{~cm}^{2}=7 \mathrm{~cm}^{2}$ and $4 \mathrm{~cm}^{2}-1 \mathrm{~cm}^{2}=3 \mathrm{~cm}^{2}$ respectively. Therefore ... | 3 | metric geometry - area | B | |
1641 | Adam the Ant started at the left-hand end of a pole and crawled $\frac{2}{3}$ of its length. Benny the Beetle started at the right-hand end of the same pole and crawled $\frac{3}{4}$ of its length. What fraction of the length of the pole are Adam and Benny now apart?
<image1> | [
"$\\frac{3}{8}$",
"$\\frac{1}{12}$",
"$\\frac{5}{7}$",
"$\\frac{1}{2}$",
"$\\frac{5}{12}$",
"I don't know"
] | images/1641.jpg | E | Adam the Ant has crawled $\frac{2}{3}$ of the length of the pole and so is $\frac{1}{3}$ of the length of the pole from the right-hand end. Benny the Beetle has crawled $\frac{3}{4}$ of the length of the pole and so is $\frac{1}{4}$ of the length of the pole from the left-hand end. Hence the fraction of the length of t... | 5 | arithmetic | E | |
1643 | Two segments, each $1 \mathrm{~cm}$ long, are marked on opposite sides of a square of side $8 \mathrm{~cm}$. The ends of the segments are joined as shown in the diagram. What is the total shaded area? <image1> | [
"$2 \\mathrm{~cm}^{2}$",
"$4 \\mathrm{~cm}^{2}$",
"$6.4 \\mathrm{~cm}^{2}$",
"$8 \\mathrm{~cm}^{2}$",
"$10 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1643.jpg | B | Let the height of the lower triangle be $h \mathrm{~cm}$. Therefore the height of the upper triangle is $(8-h) \mathrm{cm}$. Hence the shaded area in $\mathrm{cm}^{2}$ is $\frac{1}{2} \times 1 \times h+\frac{1}{2} \times 1 \times(8-h)=$ $\frac{1}{2} \times(h+8-h)=4$. | 3 | metric geometry - area | B | |
1649 | The diagram shows a parallelogram $W X Y Z$ with area $S$. The diagonals of the parallelogram meet at the point $O$. The point $M$ is on the edge $Z Y$. The lines $W M$ and $Z X$ meet at $N$. The lines $M X$ and $W Y$ meet at $P$. The sum of the areas of triangles $W N Z$ and $X Y P$ is $\frac{1}{3} S$. What is the are... | [
"$\\frac{1}{6} S$",
"$\\frac{1}{8} S$",
"$\\frac{1}{10} S$",
"$\\frac{1}{12} S$",
"$\\frac{1}{14} S$",
"I don't know"
] | images/1649.jpg | D | The area of parallelogram $W X Y Z$ is $S$. Therefore the area of triangle $W X M$, which has the same base and height, is $\frac{1}{2} S$. Hence the sum of the areas of triangle $W M Z$ and triangle $X Y M$ is also $\frac{1}{2} S$. The sum of the areas of triangle $W N Z$ and triangle $X Y P$ is given as $\frac{1}{3} ... | 3 | metric geometry - area | D | |
1650 | The faces of a cube are painted black, white or grey. Each face is only painted one colour and opposite faces are painted the same colour. Which of the following is a possible net for the cube?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1650.jpg | B | Each of the nets shown has two faces of each colour. The question tells us that the two faces are opposite each other, so they cannot have an edge in common. This eliminates all the nets except net $\mathrm{B}$. | 4 | solid geometry | B | |
1651 | The large rectangle shown is made up of nine identical rectangles whose longest sides are $10 \mathrm{~cm}$ long. What is the perimeter of the large rectangle? <image1> | [
"$40 \\mathrm{~cm}$",
"$48 \\mathrm{~cm}$",
"$76 \\mathrm{~cm}$",
"$81 \\mathrm{~cm}$",
"$90 \\mathrm{~cm}$",
"I don't know"
] | images/1651.jpg | C | Since the horizontal lengths of the small rectangles are $10 \mathrm{~cm}$, the length of the large rectangle is $2 \times 10 \mathrm{~cm}=20 \mathrm{~cm}$. Also, since the sum of the heights of five small rectangles is equal to the length of the large rectangle, the height of a small rectangle is $20 \mathrm{~cm} \div... | 4 | combinatorial geometry | C | |
1652 | The diagram shows a rectangle of size $7 \mathrm{~cm} \times 11 \mathrm{~cm}$ containing two circles that each touch three of the sides of the rectangle. What is the distance between the centres of the two circles?
<image1> | [
"$2 \\mathrm{~cm}$",
"$2.5 \\mathrm{~cm}$",
"$3 \\mathrm{~cm}$",
"$3.5 \\mathrm{~cm}$",
"$4 \\mathrm{~cm}$",
"I don't know"
] | images/1652.jpg | E | From the diagram, it can be seen that the distance between the centres of the two circles is $11 \mathrm{~cm}-2 \times$ the radius of a circle or equivalently $11 \mathrm{~cm}$ - the diameter of each circle. Since this diameter is $7 \mathrm{~cm}$, the distance between the centres is $11 \mathrm{~cm}-7 \mathrm{~cm}=4 \... | 2 | metric geometry - length | E | |
1653 | Square $A B C D$ has sides of length $3 \mathrm{~cm}$. The points $M$ and $N$ lie on $A D$ and $A B$ so that $C M$ and $C N$ split the square into three pieces of the same area. What is the length of $D M$ ? <image1> | [
"$0.5 \\mathrm{~cm}$",
"$1 \\mathrm{~cm}$",
"$1.5 \\mathrm{~cm}$",
"$2 \\mathrm{~cm}$",
"$2.5 \\mathrm{~cm}$",
"I don't know"
] | images/1653.jpg | D | The area of square $A B C D$ is $(3 \times 3) \mathrm{cm}^{2}=9 \mathrm{~cm}^{2}$. Hence the area of each piece is $\frac{1}{3} \times 9 \mathrm{~cm}^{2}=3 \mathrm{~cm}^{2}$. Since the area of a triangle is equal to half its base multiplied by its perpendicular height, we have $\frac{1}{2} \times D M \times D C=3 \math... | 2 | metric geometry - length | D | |
1655 | Valeriu draws a zig-zag line inside a rectangle, creating angles of $10^{\circ}, 14^{\circ}, 33^{\circ}$ and $26^{\circ}$ as shown. What is the size of the angle marked $\theta$ ? <image1> | [
"$11^{\\circ}$",
"$12^{\\circ}$",
"$16^{\\circ}$",
"$17^{\\circ}$",
"$33^{\\circ}$",
"I don't know"
] | images/1655.jpg | A | Add to the diagram three lines parallel to two of the sides of the rectangle, creating angles $a, b, c$, $d, e$ and $f$ as shown. Since alternate angles formed by parallel lines are equal, we have $a=26^{\circ}$ and $f=10^{\circ}$. Since $a+b=33^{\circ}$ and $e+f=14^{\circ}$, we have $b=7^{\circ}$ and $e=4^{\circ}$. S... | 5 | metric geometry - angle | A | |
1658 | Ruth and Sarah decide to have a race. Ruth runs around the perimeter of the pool shown in the diagram while Sarah swims lengths of the pool.
Ruth runs three times as fast as Sarah swims. Sarah swims six lengths of the pool in the same time Ruth runs around the pool five times. How wide is the pool?
<image1> | [
"$25 \\mathrm{~m}$",
"$40 \\mathrm{~m}$",
"$50 \\mathrm{~m}$",
"$80 \\mathrm{~m}$",
"$180 \\mathrm{~m}$",
"I don't know"
] | images/1658.jpg | B | Let the width of the pool be $x \mathrm{~m}$. Therefore the total distance Ruth runs is $5(2 \times 50+2 x) \mathrm{m}=(500+10 x) \mathrm{m}$. The total distance Sarah swims is $6 \times 50 \mathrm{~m}=300 \mathrm{~m}$. Since Ruth runs three times as fast as Sarah swims, $500+10 x=3 \times 300$. Therefore $10 x=400$ an... | 2 | metric geometry - length | B | |
1659 | Freda's flying club designed a flag of a flying dove on a square grid as shown.
The area of the dove is $192 \mathrm{~cm}^{2}$. All parts of the perimeter of the dove are either quarter-circles or straight lines. What are the dimensions of the flag?
<image1> | [
"$6 \\mathrm{~cm} \\times 4 \\mathrm{~cm}$",
"$12 \\mathrm{~cm} \\times 8 \\mathrm{~cm}$",
"$21 \\mathrm{~cm} \\times 14 \\mathrm{~cm}$",
"$24 \\mathrm{~cm} \\times 16 \\mathrm{~cm}$",
"$27 \\mathrm{~cm} \\times 18 \\mathrm{~cm}$",
"I don't know"
] | images/1659.jpg | D | Let each of the small squares in the grid have side-length $x \mathrm{~cm}$. Remove the shading and divide the dove into regions as shown. It can be seen that the regions marked $\mathrm{A}$ and $\mathrm{B}$ combine to make a square of side $2 x \mathrm{~cm}$ and hence of area $4 x^{2} \mathrm{~cm}^{2}$. Similarly, re... | 4 | combinatorial geometry | D | |
1662 | In the isosceles triangle $A B C$, points $K$ and $L$ are marked on the equal sides $A B$ and $B C$ respectively so that $A K=K L=L B$ and $K B=A C$.
<image1>
What is the size of angle $A B C$ ? | [
"$36^{\\circ}$",
"$38^{\\circ}$",
"$40^{\\circ}$",
"$42^{\\circ}$",
"$44^{\\circ}$",
"I don't know"
] | images/1662.jpg | A | Since triangle $A B C$ is isosceles with $A B=B C$ and we are given that $L B=A K$, the other parts of the equal sides must themselves be equal. Hence $L C=B K=A C$. Draw in line $K C$ as shown to form triangles $A C K$ and $L C K$. Since $A K=K L, A C=L C$ and $K C$ is common to both, triangles $A C K$ and $L C K$ are... | 5 | metric geometry - angle | A | |
1663 | Which of the diagrams below cannot be drawn without lifting your pencil off the page and without drawing along the same line twice?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1663.jpg | D | It is known that any diagram with at most two points where an odd number of lines meet can be drawn without lifting your pencil off the page and without drawing along the same line twice. Any diagram with more than two such points cannot be drawn in this way. Of the options given, only diagram $\mathrm{D}$ has more tha... | 3 | graph theory | D | |
1664 | A large square is divided into smaller squares, as shown. What fraction of the large square is shaded grey? <image1> | [
"$\\frac{2}{3}$",
"$\\frac{2}{5}$",
"$\\frac{4}{7}$",
"$\\frac{4}{9}$",
"$\\frac{5}{12}$",
"I don't know"
] | images/1664.jpg | D | The largest grey square is a quarter of the large square. The smaller grey squares are each one ninth of the size of the largest grey square. Hence the fraction of the large square which is shaded is $\frac{1}{4}+\frac{7}{9} \times \frac{1}{4}=\frac{1}{4} \times\left(1+\frac{7}{9}\right)=\frac{1}{4} \times \frac{16}{9}... | 4 | combinatorial geometry | D | |
1665 | A four-digit integer is written on each of three pieces of paper and the pieces of paper are arranged so that three of the digits are covered, as shown. The sum of the three four-digit integers is 10126 . What are the covered digits? <image1> | [
"5,6 and 7",
"4,5 and 7",
"4,6 and 7",
"4, 5 and 6",
"3,5 and 6",
"I don't know"
] | images/1665.jpg | A | The required sum can be written as shown below, with $a, b$ and $c$ as the missing digits: $$ \begin{array}{r} 1243 \\ 21 a 7 \\ +b c 26 \\ \hline 10126 \end{array} $$ The sum of the digits in the units column is 16 and hence there is a carry of 1 to the tens column. Therefore, when we consider the tens column, we ha... | 4 | algebra | A | |
1666 | In the diagram, $P Q=P R=Q S$ and $\angle Q P R=20^{\circ}$. What is $\angle R Q S$ ? <image1> | [
"$50^{\\circ}$",
"$60^{\\circ}$",
"$65^{\\circ}$",
"$70^{\\circ}$",
"$75^{\\circ}$",
"I don't know"
] | images/1666.jpg | B | Since $P Q=Q S$, triangle $P S Q$ is isosceles and hence $\angle P S Q=20^{\circ}$. Since the angles in a triangle add to $180^{\circ}$, we have $20^{\circ}+20^{\circ}+\angle S Q P=180^{\circ}$ and hence $\angle S Q P=140^{\circ}$. Since $P Q=P R$, triangle $P R Q$ is isosceles and hence $\angle P R Q=\angle R Q P$. Al... | 5 | metric geometry - angle | B | |
1667 | Which of the following $4 \times 4$ tiles cannot be formed by combining the two given pieces?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1667.jpg | E | When the two given pieces are joined together, any resulting square must have on its outside one row and one column, each of which have alternating black and white squares. Therefore tile E cannot be made. The diagrams below show how the tiles in options A, B, C and D can be made by combining the given pieces, confirmi... | 4 | combinatorial geometry | E | |
1669 | Some identical rectangles are drawn on the floor. A triangle of base $10 \mathrm{~cm}$ and height $6 \mathrm{~cm}$ is drawn over them, as shown, and the region inside the rectangles and outside the triangle is shaded. What is the area of the shaded region? <image1> | [
"$10 \\mathrm{~cm}^{2}$",
"$12 \\mathrm{~cm}^{2}$",
"$14 \\mathrm{~cm}^{2}$",
"$15 \\mathrm{~cm}^{2}$",
"$21 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1669.jpg | B | Since the length of five identical rectangles is $10 \mathrm{~cm}$, the length of one rectangle is $2 \mathrm{~cm}$. Similarly, since the height of four rectangles is $6 \mathrm{~cm}$, the height of one rectangle is $1.5 \mathrm{~cm}$. Therefore the total area of the 14 rectangles is $14 \times(2 \times 1.5) \mathrm{cm... | 4 | combinatorial geometry | B | |
1672 | Prab painted each of the eight circles in the diagram red, yellow or blue such that no two circles that are joined directly were painted the same colour. Which two circles must have been painted the same colour? <image1> | [
"5 and 8",
"1 and 6",
"2 and 7",
"4 and 5",
"3 and 6",
"I don't know"
] | images/1672.jpg | A | Since no two circles that are joined directly are painted the same colour and circles 2,5 and 6 are joined to each other, they are all painted different colours. Similarly circles 2, 6 and 8 join to each other and hence are painted different colours. Therefore circles 5 and 8 must have been painted the same colour. It ... | 3 | graph theory | A | |
1673 | The diagram shows the square $W X Y Z$. The points $P, Q$ and $R$ are the midpoints of the sides $Z W, X Y$ and $Y Z$ respectively. What fraction of the square $W X Y Z$ is shaded? <image1> | [
"$\\frac{3}{4}$",
"$\\frac{5}{8}$",
"$\\frac{1}{2}$",
"$\\frac{7}{16}$",
"$\\frac{3}{8}$",
"I don't know"
] | images/1673.jpg | E | Label the intersection of $W Q$ and $X P$ as $V$ and the midpoint of $W X$ as $U$. Let the side-length of the square be 1 unit. The area of triangle $W X R$ is $\frac{1}{2} \times 1 \times 1$ units $^{2}=\frac{1}{2}$ units $^{2}$. Consider triangle $W X P$ and triangle $U X V$. These two triangles have the same angles ... | 3 | metric geometry - area | E | |
1674 | A large square is divided into smaller squares. In one of the smaller squares a diagonal is also drawn, as shown. What fraction of the large square is shaded? <image1> | [
"$\\frac{4}{5}$",
"$\\frac{3}{8}$",
"$\\frac{4}{9}$",
"$\\frac{1}{3}$",
"$\\frac{1}{2}$",
"I don't know"
] | images/1674.jpg | E | The shaded square in the lower right corner of the large square is $\frac{1}{4}$ of the large square. The shaded triangle is half of $\frac{1}{4}$ of the large square. Hence it is $\frac{1}{8}$ of the large square. The two small shaded squares in the upper left corner together are half of $\frac{1}{4}$, or $\frac{1}{8}... | 3 | metric geometry - area | E | |
1677 | The shortest path from Atown to Cetown runs through Betown. The two signposts shown are set up at different places along this path. What distance is written on the broken sign? <image1> | [
"$1 \\mathrm{~km}$",
"$3 \\mathrm{~km}$",
"$4 \\mathrm{~km}$",
"$5 \\mathrm{~km}$",
"$9 \\mathrm{~km}$",
"I don't know"
] | images/1677.jpg | A | The information on the signs pointing to Atown and the signs pointing to Cetown both tell us that the distance between the signs is $(7-2) \mathrm{km}=(9-4) \mathrm{km}=5 \mathrm{~km}$. Therefore the distance which is written on the broken sign is $(5-4) \mathrm{km}=1 \mathrm{~km}$. | 2 | metric geometry - length | A | |
1679 | The pattern on a large square tile consists of eight congruent right-angled triangles and a small square. The area of the tile is $49 \mathrm{~cm}^{2}$ and the length of the hypotenuse $P Q$ of one of the triangles is $5 \mathrm{~cm}$. What is the area of the small square? <image1> | [
"$1 \\mathrm{~cm}^{2}$",
"$4 \\mathrm{~cm}^{2}$",
"$9 \\mathrm{~cm}^{2}$",
"$16 \\mathrm{~cm}^{2}$",
"$25 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1679.jpg | A | Since the four rectangles are congruent, the diagonal $P Q$ is also the side of a square. This square has area $(5 \times 5) \mathrm{cm}^{2}=25 \mathrm{~cm}^{2}$. Therefore the total area of the rectangles outside the square with side $P Q$ but inside the large square is $(49-25) \mathrm{cm}^{2}=24 \mathrm{~cm}^{2}$. H... | 3 | metric geometry - area | A | |
1680 | Aisha has a strip of paper with the numbers $1,2,3,4$ and 5 written in five cells as shown. She folds the strip so that the cells overlap, forming 5 layers. Which of the following configurations, from top layer to bottom layer, is it not possible to obtain? <image1> | [
"$3,5,4,2,1$",
"$3,4,5,1,2$",
"$3,2,1,4,5$",
"$3,1,2,4,5$",
"$3,4,2,1,5$",
"I don't know"
] | images/1680.jpg | E | The four figures (A) to (D) give a side-view of how the strip could be folded to give the arrangements of numbers in options A to D. Figure (E) shows that it is not possible to get option E since number 5 would end up between number 4 and number 2 (as indicated by the dashed line labelled 5) rather than below number 1 ... | 3 | topology | E | |
1681 | Bella took a square piece of paper and folded two of its sides to lie along the diagonal, as shown, to obtain a quadrilateral. What is the largest size of an angle in that quadrilateral? <image1> | [
"$112.5^{\\circ}$",
"$120^{\\circ}$",
"$125^{\\circ}$",
"$135^{\\circ}$",
"$150^{\\circ}$",
"I don't know"
] | images/1681.jpg | A | Since the quadrilateral is formed by folding the $45^{\circ}$ angles above and below the diagonal of the square in half, the size of the small angle of the quadrilateral is $2 \times\left(\frac{1}{2} \times 45^{\circ}\right)=45^{\circ}$. One angle of the quadrilateral is $90^{\circ}$ and the other two are equal from th... | 3 | transformation geometry | A | |
1687 | In the diagram, the area of the large square is $16 \mathrm{~cm}^{2}$ and the area of each small corner square is $1 \mathrm{~cm}^{2}$. What is the shaded area? <image1> | [
"$3 \\mathrm{~cm}^{2}$",
"$\\frac{7}{2} \\mathrm{~cm}^{2}$",
"$4 \\mathrm{~cm}^{2}$",
"$\\frac{11}{2} \\mathrm{~cm}^{2}$",
"$6 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1687.jpg | C | Since the area of the large square is $16 \mathrm{~cm}^{2}$ and the area of each small square is $1 \mathrm{~cm}^{2}$, their side-lengths are $4 \mathrm{~cm}$ and $1 \mathrm{~cm}$ respectively. Therefore the base of each of the four triangles is $2 \mathrm{~cm}$ and, since these triangles meet at the centre of the larg... | 3 | metric geometry - area | C | |
1690 | Three villages are connected by paths as shown. From Downend to Uphill, the detour via Middleton is $1 \mathrm{~km}$ longer than the direct path. From Downend to Middleton, the detour via Uphill is $5 \mathrm{~km}$ longer than the direct path. From Uphill to Middleton, the detour via Downend is $7 \mathrm{~km}$ longer ... | [
"$1 \\mathrm{~km}$",
"$2 \\mathrm{~km}$",
"$3 \\mathrm{~km}$",
"$4 \\mathrm{~km}$",
"$5 \\mathrm{~km}$",
"I don't know"
] | images/1690.jpg | C | Let the lengths of the direct paths from Uphill to Middleton, Middleton to Downend and Downend to Uphill be $x \mathrm{~km}, y \mathrm{~km}$ and $z \mathrm{~km}$ respectively. The information in the question tells us that $x+y=z+1, x+z=y+5$ and $y+z=x+7$. When we add these three equations, we obtain $2 x+2 y+2 z=z+y+x+... | 2 | metric geometry - length | C | |
1691 | A triangular pyramid is built with 20 cannonballs, as shown. Each cannonball is labelled with one of A, B, C, D or E. There are four cannonballs with each type of label.
<image1>
The diagrams show the labels on the cannonballs on three of the faces of the pyramid. What is the label on the hidden cannonball in the middl... | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1691.jpg | D | Note that each cannonball on the two non-horizontal edges of each pictured face appears on two of those faces, except the cannonball at the vertex which appears on all three. Hence, when the labels of the cannonballs are counted, these must only be counted once. Careful counting of the cannonballs shown gives four cann... | 4 | solid geometry | D | |
1695 | Kanga likes jumping on the number line. She always makes two large jumps of length 3 , followed by three small jumps of length 1 , as shown, and then repeats this over and over again. She starts jumping at 0 .
<image1>
Which of these numbers will Kanga land on? | [
"82",
"83",
"84",
"85",
"86",
"I don't know"
] | images/1695.jpg | C | Each time she completes a set of five jumps, Kanga moves forward 9 places on the number line. Since she started at 0 , this means she will eventually land on $9 \times 9=81$. Her next set of jumps will take her to $84,87,88,89$ and 90 . Therefore, of the numbers given, the only one Kanga will land on is 84 . | 4 | algebra | C | |
1696 | There are five big trees and three paths in a park. It has been decided to plant a sixth tree so that there are the same number of trees on either side of each path. In which region of the park should the sixth tree be planted? <image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1696.jpg | B | The path running from the top of the park to the bottom has two trees to the left of it and three trees to the right of it on the diagram. Hence the sixth tree should be planted to the left of this path. The path running from the top left of the park to the bottom right has two trees above it and three trees below it o... | 4 | logic | B | |
1700 | The area of the intersection of a triangle and a circle is $45 \%$ of the total area of the diagram. The area of the triangle outside the circle is $40 \%$ of the total area of the diagram. What percentage of the circle lies outside the triangle? <image1> | [
"$20 \\%$",
"$25 \\%$",
"$30 \\%$",
"$33 \\frac{1}{3} \\%$",
"$35 \\%$",
"I don't know"
] | images/1700.jpg | B | The area of the circle inside the triangle is $45 \%$ of the total area of the diagram. The area of the circle outside the triangle is $(100-40-45) \%=15 \%$ of the total area of the diagram. Therefore, the percentage of the circle that lies outside the triangle is $\frac{15}{15+45} \times 100=25 \%$. | 3 | metric geometry - area | B | |
1702 | The large rectangle $W X Y Z$ is divided into seven identical rectangles, as shown. What is the ratio $W X: X Y$ ? <image1> | [
"$3: 2$",
"$4: 3$",
"$8: 5$",
"$12: 7$",
"$7: 3$",
"I don't know"
] | images/1702.jpg | D | Let the longer side of each of the small rectangles be $p$ and let the shorter side be $q$. From the diagram, it can be seen that $3 p=4 q$ and hence $q=\frac{3}{4} p$. It can also be seen that the ratio $W X: X Y=3 p: p+q$. This is equal to $3 p: p+\frac{3}{4} p=3 p: \frac{7}{4} p=12 p: 7 p=12: 7$. | 2 | metric geometry - length | D | |
1706 | Which of the shapes below cannot be divided into two trapeziums by a single straight line?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1706.jpg | A | The diagrams on the right show how a rectangle, a trapezium, a regular hexagon and a square can be  divided into two trapeziums by a single straight line. However, when a single straight line ... | 4 | combinatorial geometry | A | |
1708 | Kristina has a piece of transparent paper with some lines marked on it. She folds it along the central dashed line, as indicated. What can she now see? <image1> | [
"$2: 6: 9$",
"$2: 6: 6$",
"$5: 6: 9$",
"$2: 8: 6$",
"$5: 8: 9$",
"I don't know"
] | images/1708.jpg | C | Each digit in the completed number contains three horizontal line segments. Since the markings above the fold will appear the other way up when folded, the first digit in the completed number contains a vertical line top left and a vertical line bottom right in addition to the three horizontal lines and so is a 5. The ... | 3 | transformation geometry | C | |
1711 | In the diagram shown, sides $P Q$ and $P R$ are equal. Also $\angle Q P R=40^{\circ}$ and $\angle T Q P=\angle S R Q$. What is the size of $\angle T U R$ ? <image1> | [
"$55^{\\circ}$",
"$60^{\\circ}$",
"$65^{\\circ}$",
"$70^{\\circ}$",
"$75^{\\circ}$",
"I don't know"
] | images/1711.jpg | D | Let $\angle S R Q$ be $x^{\circ}$. Since sides $P R$ and $P Q$ are equal, triangle $P Q R$ is isosceles and hence $\angle P R Q=\angle P Q R=\left(180^{\circ}-40^{\circ}\right) / 2=70^{\circ}$. Therefore, since we are given that $\angle T Q P=\angle S R Q$, we have $\angle U Q R$ is $70^{\circ}-x^{\circ}$. Hence, since... | 5 | metric geometry - angle | D | |
1717 | The diagram on the right shows a cube of side $18 \mathrm{~cm}$. A giant ant walks across the cube's surface from $\mathrm{X}$ to $\mathrm{Y}$ along the route shown. How far does it walk? <image1> | [
"$54 \\mathrm{~cm}$",
"$72 \\mathrm{~cm}$",
"$80 \\mathrm{~cm}$",
"$88 \\mathrm{~cm}$",
"$90 \\mathrm{~cm}$",
"I don't know"
] | images/1717.jpg | E | From the diagram, the ant walks the equivalent of five edges. Therefore the ant walks $5 \times 18 \mathrm{~cm}=90 \mathrm{~cm}$. | 5 | solid geometry | E | |
1718 | In the diagram, five rectangles of the same size are shown with each side labelled with a number.
<image1>
These rectangles are placed in the positions I to $\mathrm{V}$ as shown so that the numbers on the sides that touch each other are equal.
<image2>
Which of the rectangles should be placed in position I? | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1718.jpg | C | Look first at the numbers labelling the left- and right-hand sides of the rectangles. It can be seen that only rectangles $A, C$ and $E$ can be arranged in a row of three with their touching sides equal and so they must form the top row of the diagram. The only common value on the right- and left-hand sides of rectangl... | 3 | combinatorics | C | |
1721 | In the diagram, the small equilateral triangles have area $4 \mathrm{~cm}^{2}$. What is the area of the shaded region? <image1> | [
"$80 \\mathrm{~cm}^{2}$",
"$90 \\mathrm{~cm}^{2}$",
"$100 \\mathrm{~cm}^{2}$",
"$110 \\mathrm{~cm}^{2}$",
"$120 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1721.jpg | B | Let $b \mathrm{~cm}$ be the length of the base and let $h \mathrm{~cm}$ be the height of the small equilateral triangles. The area of each triangle is $4 \mathrm{~cm}^{2}$ so $\frac{1}{2} \times b \times h=4$. The shaded area is a trapezium with parallel sides of length $4 h$ and $5 h$ and with distance $\frac{5}{2} b... | 2 | combinatorial geometry | B | |
1723 | The diagram shows five circles of the same radius touching each other. A square is drawn so that its vertices are at the centres of the four outer circles.
<image1>
What is the ratio of the area of the shaded parts of the circles to the area of the unshaded parts of the circles? | [
"$1: 3$",
"$1: 4$",
"$2: 5$",
"$2: 3$",
"$5: 4$",
"I don't know"
] | images/1723.jpg | D | The shaded area is equal to that of 1 circle $+4 \times \frac{1}{4}$ circles $=2$ circles. The area of the unshaded parts of the circles is equal to that of $4 \times \frac{3}{4}$ circles $=3$ circles. Hence the required ratio is $2: 3$. | 2 | metric geometry - area | D | |
1724 | A rectangular garden is surrounded by a path of constant width. The perimeter of the garden is $24 \mathrm{~m}$ shorter than the distance along the outside edge of the path. What is the width of the path? <image1> | [
"$1 \\mathrm{~m}$",
"$2 \\mathrm{~m}$",
"$3 \\mathrm{~m}$",
"$4 \\mathrm{~m}$",
"$5 \\mathrm{~m}$",
"I don't know"
] | images/1724.jpg | C | Let the length and width of the garden be $a$ metres and $b$ metres respectively and let the width of the path be $x$ metres. The perimeter of the garden is $2(a+b)$ metres and the perimeter of the larger rectangle formed by the garden and the path is $2(a+2 x+b+2 x)$ metres. Hence the difference between the distance a... | 3 | metric geometry - length | C | |
1727 | It takes 9 litres of paint to cover the surface of the cube on the left.
<image1>
How much paint would it take to cover the surface of the shape on the right? | [
"9 litres",
"8 litres",
"6 litres",
"4 litres",
"2 litres",
"I don't know"
] | images/1727.jpg | A | The surface areas of the two solids are the same. Hence the same amount of paint is required to cover them. Therefore it would take 9 litres of paint to cover the surface of the second solid. | 5 | solid geometry | A | |
1728 | Which of the following nets can be used to build the partial cube shown in the diagram?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1728.jpg | C | Nets A and D would produce cubes with holes on two edges of the same face. Net E would produce a cube with a hole in the centre of two opposite faces while net B would produce a cube with one hole on an edge and two small holes. The given partial cube has holes on two opposite edges and therefore its net will have a ho... | 5 | solid geometry | C | |
1729 | A piece of paper in the shape of a regular hexagon, as shown, is folded so that the three marked vertices meet at the centre $O$ of the hexagon. What is the shape of the figure that is formed? <image1> | [
"Six-pointed star",
"Dodecagon",
"Hexagon",
"Square",
"Equilateral Triangle",
"I don't know"
] | images/1729.jpg | E | Label vertices $A, B, C, D, E, E$ and $F$ as shown. Since the hexagon is regular, it can be divided into six equilateral triangles as shown. Therefore quadrilateral $O A B C$ is a rhombus and hence its diagonal $A C$ is a line of symmetry. Therefore, if vertex $B$ is folded onto $O$, the fold will be along $A C$. Simil... | 1 | transformation geometry | E | |
1730 | Four circles of radius $5 \mathrm{~cm}$ touch the sides of a square and each other, as shown in the diagram. On each side of the square, an equilateral triangle is drawn to form a four-pointed star.
<image1>
What is the perimeter of the star? | [
"$40 \\mathrm{~cm}$",
"$80 \\mathrm{~cm}$",
"$120 \\mathrm{~cm}$",
"$160 \\mathrm{~cm}$",
"$200 \\mathrm{~cm}$",
"I don't know"
] | images/1730.jpg | D | The radius of each of the circles is $5 \mathrm{~cm}$ and hence the diameter of each is $10 \mathrm{~cm}$. The length of the side of the square is equal to the sum of the diameters of two circles and hence is equal to $20 \mathrm{~cm}$. The length of each side of the equilateral triangle is equal to the length of the s... | 3 | metric geometry - length | D | |
1731 | The diagram shows a rectangle $A B C D$ in which $A B=1$ metre and $A D=4$ metres. The points $E$ and $G$ are the midpoints of $A D$ and $A B$ and the points $F$ and $H$ are the midpoints of $A E$ and $A G$.
<image1>
What is the area of the shaded rectangle? | [
"$\\frac{1}{16} \\mathrm{~m}^{2}$",
"$\\frac{1}{8} \\mathrm{~m}^{2}$",
"$\\frac{1}{4} \\mathrm{~m}^{2}$",
"$\\frac{1}{2} \\mathrm{~m}^{2}$",
"$1 \\mathrm{~m}^{2}$",
"I don't know"
] | images/1731.jpg | C |  Since $E$ is the midpoint of $A D$ and $F$ is the midpoint of $A E$, the length of $F E$ is $\frac{1}{2} \times \frac{1}{2} \times 4 \mathrm{~cm}=1 \mathrm{~cm}$. Similarly, since $G$ is the m... | 2 | metric geometry - area | C | |
1732 | The diameter of the circle shown is $10 \mathrm{~cm}$. The circle passes through the vertices of a large rectangle which is divided into 16 identical smaller rectangles.
<image1>
What is the perimeter of the shape drawn with a dark line? | [
"$10 \\mathrm{~cm}$",
"$16 \\mathrm{~cm}$",
"$20 \\mathrm{~cm}$",
"$24 \\mathrm{~cm}$",
"$30 \\mathrm{~cm}$",
"I don't know"
] | images/1732.jpg | C | The diagonals of a rectangle bisect each other at the midpoint of the rectangle. Hence, the midpoint of a rectangle is equidistant from all four vertices and is the centre of a circle through its vertices. In this case, the diameter of the circle is $10 \mathrm{~cm}$. This is equal to the sum of the lengths of the dia... | 3 | metric geometry - length | C | |
1734 | The square $A B C D$ consists of four congruent rectangles arranged around a central square. The perimeter of each of the rectangles is $40 \mathrm{~cm}$. What is the area of the square $A B C D$ ? <image1> | [
"$400 \\mathrm{~cm}^{2}$",
"$200 \\mathrm{~cm}^{2}$",
"$160 \\mathrm{~cm}^{2}$",
"$120 \\mathrm{~cm}^{2}$",
"$80 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1734.jpg | A | Let the length of each of the rectangles be $x \mathrm{~cm}$ and the width be $y \mathrm{~cm}$. The perimeter of each of the rectangles is $40 \mathrm{~cm}$ and hence $2 x+2 y=40$. Therefore $x+y=20$. From the diagram we can see that the length of each side of the square $A B C D$ is $(x+y) \mathrm{cm}$. Therefore the ... | 2 | metric geometry - area | A | |
1735 | The diagram shows five congruent right-angled isosceles triangles. What is the total area of the triangles? <image1> | [
"$25 \\mathrm{~cm}^{2}$",
"$30 \\mathrm{~cm}^{2}$",
"$35 \\mathrm{~cm}^{2}$",
"$45 \\mathrm{~cm}^{2}$",
"$60 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1735.jpg | D | Consider one of the right-angled isosceles triangles as shown. The longest side is $(30 / 5) \mathrm{cm}=6 \mathrm{~cm}$. The triangle can be divided into two identical right-angled isosceles triangles with base $3 \mathrm{~cm}$ and hence with height $3 \mathrm{~cm}$. Therefore the area of each of the original triangl... | 2 | metric geometry - area | D | |
1739 | A cube has diagonals drawn on three adjacent faces as shown in the diagram. Which of the following nets could Usman use to make the cube shown?
<image1>
<image2> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1739.jpg | D | On each net, label the four vertices of the right-hand square 1,2,3 and 4 as shown. Also label any vertex on any of the other squares that will meet vertices 1,2,3 or 4 when the net of the cube is assembled into a cube with the corresponding value. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f598bb42a47f3275d94g-2.... | 5 | solid geometry | D | |
1740 | Maddie has a paper ribbon of length $36 \mathrm{~cm}$. She divides it into four rectangles of different lengths. She draws two lines joining the centres of two adjacent rectangles as shown.
<image1>
What is the sum of the lengths of the lines that she draws? | [
"$18 \\mathrm{~cm}$",
"$17 \\mathrm{~cm}$",
"$20 \\mathrm{~cm}$",
"$19 \\mathrm{~cm}$",
"It depends upon the sizes of the rectangles",
"I don't know"
] | images/1740.jpg | A | Let the lengths of the four rectangles be $p \mathrm{~cm}, q \mathrm{~cm}, r \mathrm{~cm}$ and $s \mathrm{~cm}$ with $p+q+r+s=36$. The lines Maddie draws join the centres of two pairs of rectangles and hence have total length $\left(\frac{1}{2} p+\frac{1}{2} q\right) \mathrm{cm}+\left(\frac{1}{2} r+\frac{1}{2} s\right)... | 1 | transformation geometry | A | |
1742 | In the diagram, $P R S V$ is a rectangle with $P R=20 \mathrm{~cm}$ and $P V=12 \mathrm{~cm}$. Jeffrey marks points $U$ and $T$ on $V S$ and $Q$ on $P R$ as shown. What is the shaded area? <image1> | [
"More information needed",
"$60 \\mathrm{~cm}^{2}$",
"$100 \\mathrm{~cm}^{2}$",
"$110 \\mathrm{~cm}^{2}$",
"$120 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1742.jpg | E | Consider the two unshaded triangles. Each has height equal to $12 \mathrm{~cm}$ and hence their total area is $\left(\frac{1}{2} \times P Q \times 12+\frac{1}{2} \times Q R \times 12\right) \mathrm{cm}^{2}=6 \times(P Q+Q R) \mathrm{cm}^{2}=6 \times 20 \mathrm{~cm}^{2}=120 \mathrm{~cm}^{2}$. Therefore the shaded area is... | 2 | metric geometry - area | E | |
1743 | The line $P Q$ is divided into six parts by the points $V, W, X, Y$ and $Z$. Squares are drawn on $P V, V W, W X, X Y, Y Z$ and $Z Q$ as shown in the diagram. The length of line $P Q$ is $24 \mathrm{~cm}$. What is the length of the path from $P$ to $Q$ indicated by the arrows?
<image1> | [
"$48 \\mathrm{~cm}$",
"$60 \\mathrm{~cm}$",
"$66 \\mathrm{~cm}$",
"$72 \\mathrm{~cm}$",
"$96 \\mathrm{~cm}$",
"I don't know"
] | images/1743.jpg | D | The path indicated follows three sides of each of the squares shown. The sum of the lengths of one side of each square is equal to the length of $P Q$, which is $24 \mathrm{~cm}$. Therefore the length of the path is $3 \times 24 \mathrm{~cm}=72 \mathrm{~cm}$. | 3 | metric geometry - length | D | |
1744 | Henna has four hair ribbons of width $10 \mathrm{~cm}$. When she measures them, she finds that each ribbon is $25 \mathrm{~cm}$ longer than the next smallest ribbon. She then arranges the ribbons to form two different shapes as shown in the diagram. How much longer is the perimeter of the second shape than the perimete... | [
"$75 \\mathrm{~cm}$",
"$50 \\mathrm{~cm}$",
"$25 \\mathrm{~cm}$",
"$20 \\mathrm{~cm}$",
"$0 \\mathrm{~cm}$",
"I don't know"
] | images/1744.jpg | B | Let the length of the shortest ribbon be $x \mathrm{~cm}$. Therefore the lengths of the other ribbons are $(x+25) \mathrm{cm},(x+50) \mathrm{cm}$ and $(x+75) \mathrm{cm}$. The perimeter of the first shape (starting from the lower left corner and working clockwise) is $(x+10+25+10+25+10+25+10+x+75+40) \mathrm{cm}=(2 x+2... | 2 | combinatorial geometry | B | |
1745 | In the diagram, $P Q R S$ is a square of side $10 \mathrm{~cm} . T$ is a point inside the square so that $\angle S P T=75^{\circ}$ and $\angle T S P=30^{\circ}$. What is the length of $T R$ ? <image1> | [
"$8 \\mathrm{~cm}$",
"$8.5 \\mathrm{~cm}$",
"$9 \\mathrm{~cm}$",
"$9.5 \\mathrm{~cm}$",
"$10 \\mathrm{~cm}$",
"I don't know"
] | images/1745.jpg | E | Draw in lines $P T$ and $T S$ as shown. Since angles in a triangle add to $180^{\circ}$ and we are given $\angle S P T=75^{\circ}$ and $\angle T S P=30^{\circ}$, we obtain $\angle P T S=75^{\circ}$. Therefore $\triangle P T S$ is isosceles and hence $T S=P S=10 \mathrm{~cm}$. Therefore, since $R S=10 \mathrm{~cm}$ as i... | 3 | metric geometry - length | E | |
1746 | In the diagram, $P Q R S$ and $W X Y Z$ are congruent squares. The sides $P S$ and $W Z$ are parallel. The shaded area is equal to $1 \mathrm{~cm}^{2}$. What is the area of square $P Q R S$ ? <image1> | [
"$1 \\mathrm{~cm}^{2}$",
"$2 \\mathrm{~cm}^{2}$",
"$\\frac{1}{2} \\mathrm{~cm}^{2}$",
"$1 \\frac{1}{2} \\mathrm{~cm}^{2}$",
"$\\frac{3}{4} \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1746.jpg | A | Let the length of a side of $P Q R S$ and of $W X Y Z$ be $x \mathrm{~cm}$. Consider quadrilateral $Q X R W$.  The diagonals $Q R$ and $W X$ are perpendicular and of length $x \mathrm{~cm}$. T... | 2 | metric geometry - area | A | |
1748 | Three congruent isosceles trapeziums are assembled to form an equilateral triangle with a hole in the middle, as shown in the diagram.
<image1>
What is the perimeter of the hole? | [
"$3 a+6 b$",
"$3 b-6 a$",
"$6 b-3 a$",
"$6 a+3 b$",
"$6 a-3 b$",
"I don't know"
] | images/1748.jpg | E | Since the triangle formed when the trapeziums are put together is equilateral, the smaller angles in the isosceles trapeziums are both $60^{\circ}$. Consider one trapezium split into a parallelogram and a triangle as shown. ![](https://cdn.mathpix.com/cropped/2023_12_27_0f598bb42a47f3275d94g-4.jpg?height=177&width=508... | 2 | combinatorial geometry | E | |
1750 | Emily has two identical cards in the shape of equilateral triangles. She places them both onto a sheet of paper so that they touch or overlap and draws around the shape she creates. Which one of the following is it impossible for her to draw?
<image1> | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1750.jpg | E |  The dotted lines on the diagrams which complete equilateral triangles show that she can create shapes A, B, C and D. Therefore it is shape E that is impossible for her to draw. | 2 | combinatorial geometry | E | |
1751 | In the triangle $P Q R$, the lengths of sides $P Q$ and $P R$ are the same. The point $S$ lies on $Q R$ so that $Q S=P S$ and $\angle R P S=75^{\circ}$. What is the size of $\angle Q R P$ ? <image1> | [
"$35^{\\circ}$",
"$30^{\\circ}$",
"$25^{\\circ}$",
"$20^{\\circ}$",
"$15^{\\circ}$",
"I don't know"
] | images/1751.jpg | A | Let $\angle P Q S$ be $x^{\circ}$. Since $P Q=P R$, the triangle $P Q R$ is isosceles and hence $\angle Q R P=x^{\circ}$. Also, since $Q S=P S$, the triangle $P Q S$ is isosceles and hence $\angle S P Q=x^{\circ}$. Therefore, since angles in a triangle add to $180^{\circ}$, we have $x+x+x+75=180$, which has solution $x... | 3 | metric geometry - angle | A | |
1754 | The diagram below shows five rectangles, each containing some of the letters $\mathrm{P}, \mathrm{R}, \mathrm{I}, \mathrm{S}$ and $\mathrm{M}$.
<image1>
Harry wants to cross out letters so that each rectangle contains only one letter and each rectangle contains a different letter. Which letter does he not cross out in ... | [
"P",
"R",
"I",
"S",
"M",
"I don't know"
] | images/1754.jpg | B | Rectangle 4 only contains one letter and hence letter $\mathrm{S}$ must be crossed out in any other rectangle. Therefore letter $\mathrm{P}$ is the only letter left in rectangle 1 and must be crossed out in all the other rectangles. This means letter I is the only one left in rectangle 3 and must be crossed out in all ... | 2 | logic | B | |
1756 | The diagram shows a triangle joined to a square to form an irregular pentagon. The triangle has the same perimeter as the square.
<image1>
What is the ratio of the perimeter of the pentagon to the perimeter of the square? | [
"2: 1",
"3: 2",
"4: 3",
"5: 4",
"6: 5",
"I don't know"
] | images/1756.jpg | B | Let the length of the edge of the square be 1 unit. Therefore the perimeter of the square and hence the perimeter of the triangle is 4 units. Since the pentagon is made by joining the square and the triangle along one common edge, the perimeter of the pentagon is equal to the sum of their perimeters minus twice the len... | 3 | metric geometry - length | B | |
1757 | My TV screen has sides in the ratio $16: 9$. My mother's TV screen has sides in the ratio $4: 3$. A picture which exactly fills the screen of my TV only fills the width of the screen of my mother's TV.
What fraction of the screen on my mother's TV is not covered?
<image1> | [
"$\\frac{1}{6}$",
"$\\frac{1}{5}$",
"$\\frac{1}{4}$",
"$\\frac{1}{3}$",
"It depends on the size of the screen.",
"I don't know"
] | images/1757.jpg | C | The ratio $4: 3=16: 12$. Therefore the fraction of the screen not covered is $\frac{12-9}{12}=\frac{1}{4}$. | 2 | metric geometry - area | C | |
1759 | The diagram shows a plan of a town with various bus stops. There are four bus routes in the town.
Route 1 goes $\mathrm{C}-\mathrm{D}-\mathrm{E}-\mathrm{F}-\mathrm{G}-\mathrm{H}-\mathrm{C}$ and is $17 \mathrm{~km}$ long.
Route 2 goes $\mathrm{A}-\mathrm{B}-\mathrm{C}-\mathrm{F}-\mathrm{G}-\mathrm{H}-\mathrm{A}$ and is ... | [
"$10 \\mathrm{~km}$",
"$9 \\mathrm{~km}$",
"$8 \\mathrm{~km}$",
"$7 \\mathrm{~km}$",
"$6 \\mathrm{~km}$",
"I don't know"
] | images/1759.jpg | B | The roads covered in routes 1 and 2 combined are the same as the roads covered in routes 3 and 4 combined. Therefore the length of route 4 is $(17+12-20) \mathrm{km}=9 \mathrm{~km}$. | 3 | metric geometry - length | B | |
1761 | The diagram shows squares of three different sizes arranged into a rectangle. The length of each side of the smallest squares is $20 \mathrm{~cm}$. Adam Ant walks along the path marked from $P$ to $Q$. How far does Adam walk? <image1> | [
"$380 \\mathrm{~cm}$",
"$400 \\mathrm{~cm}$",
"$420 \\mathrm{~cm}$",
"$440 \\mathrm{~cm}$",
"$460 \\mathrm{~cm}$",
"I don't know"
] | images/1761.jpg | C | From the diagram, it can be seen that the sides of the larger squares are $2 \times 20 \mathrm{~cm}=40 \mathrm{~cm}$ and $3 \times 20 \mathrm{~cm}=60 \mathrm{~cm}$. Therefore the distance Adam walks is $(5 \times 20+5 \times 40+2 \times 60) \mathrm{cm}=420 \mathrm{~cm}$. | 2 | combinatorial geometry | C | |
1762 | Two rectangles of dimensions $8 \mathrm{~cm}$ by $10 \mathrm{~cm}$ and $9 \mathrm{~cm}$ by $12 \mathrm{~cm}$ overlap as shown in the diagram. The area of the black region is $37 \mathrm{~cm}^{2}$. What is the area of the grey region? <image1> | [
"$60 \\mathrm{~cm}^{2}$",
"$62 \\mathrm{~cm}^{2}$",
"$62.5 \\mathrm{~cm}^{2}$",
"$64 \\mathrm{~cm}^{2}$",
"$65 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1762.jpg | E | The areas of the two rectangles are $(8 \times 10) \mathrm{cm}^{2}=80 \mathrm{~cm}^{2}$ and $(9 \times 12) \mathrm{cm}^{2}=108 \mathrm{~cm}^{2}$. Since the area of the black region is $37 \mathrm{~cm}^{2}$, the area of the unshaded region is $(80-37) \mathrm{cm}^{2}=43 \mathrm{~cm}^{2}$. Hence the area of the grey regi... | 2 | metric geometry - area | E | |
1763 | In the quadrilateral $P Q R S$, the length of $P Q$ is $11 \mathrm{~cm}$, the length of $Q R$ is $7 \mathrm{~cm}$, the length of $R S$ is $9 \mathrm{~cm}$ and the length of $S P$ is $3 \mathrm{~cm}$. Both $\angle Q R S$ and $\angle S P Q$ are $90^{\circ}$. What is the area of the quadrilateral $P Q R S$ ? <image1> | [
"$30 \\mathrm{~cm}^{2}$",
"$48 \\mathrm{~cm}^{2}$",
"$50 \\mathrm{~cm}^{2}$",
"$52 \\mathrm{~cm}^{2}$",
"$60 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1763.jpg | B | The information in the question tells us that both triangle $S P Q$ and triangle $Q R S$ are right-angled. The area of the quadrilateral $P Q R S$ is equal to the sum of the areas of triangle $S P Q$ and triangle $Q R S$. Therefore the area of $P Q R S$ is $\frac{1}{2}(11 \times 3) \mathrm{cm}^{2}+\frac{1}{2}(7 \times... | 2 | metric geometry - area | B | |
1765 | The shape in the diagram is made up of a rectangle, a square and an equilateral triangle, all of which have the same perimeter. The length of the side of the square is $9 \mathrm{~cm}$. What is the length of the shorter sides of the rectangle? <image1> | [
"$4 \\mathrm{~cm}$",
"$5 \\mathrm{~cm}$",
"$6 \\mathrm{~cm}$",
"$7 \\mathrm{~cm}$",
"$8 \\mathrm{~cm}$",
"I don't know"
] | images/1765.jpg | C | The perimeter of the square is $4 \times 9 \mathrm{~cm}=36 \mathrm{~cm}$. Therefore, since the perimeter of the square and the equilateral triangle are the same, the side-length of the equilateral triangle is $36 \mathrm{~cm} \div 3=12 \mathrm{~cm}$. Hence the length of each of the longer sides of the rectangle is $12 ... | 3 | metric geometry - length | C | |
1766 | The diagram shows a square, an equilateral triangle and a regular pentagon. What is the size of $\angle W U V$ ? <image1> | [
"$21^{\\circ}$",
"$23^{\\circ}$",
"$25^{\\circ}$",
"$27^{\\circ}$",
"$29^{\\circ}$",
"I don't know"
] | images/1766.jpg | A | The interior angles of an equilateral triangle, a square and a regular pentagon are $180^{\circ} \div 3=60^{\circ}$, $2 \times 180^{\circ} \div 4=90^{\circ}$ and $3 \times 180^{\circ} \div 5=108^{\circ}$ respectively. Therefore the size of the obtuse $\angle U V W$ is $108^{\circ}+90^{\circ}-60^{\circ}=138^{\circ}$. Si... | 3 | metric geometry - angle | A | |
1768 | In the diagram, $P Q R S$ is a square of side $10 \mathrm{~cm}$. The distance $M N$ is $6 \mathrm{~cm}$. The square is divided into four congruent isosceles triangles, four congruent squares and the shaded region.
<image1>
What is the area of the shaded region? | [
"$42 \\mathrm{~cm}^{2}$",
"$46 \\mathrm{~cm}^{2}$",
"$48 \\mathrm{~cm}^{2}$",
"$52 \\mathrm{~cm}^{2}$",
"$58 \\mathrm{~cm}^{2}$",
"I don't know"
] | images/1768.jpg | C | Since the non-shaded squares are congruent and since $M N=6 \mathrm{~cm}$, both $S N$ and $M R$ have length $(10-6) \mathrm{cm} \div 2=2 \mathrm{~cm}$. Therefore the areas of the four non-shaded squares are each $(2 \times 2) \mathrm{cm}^{2}=4 \mathrm{~cm}^{2}$. Label the point $X$ on $S P$ as shown. Since all of the ... | 2 | metric geometry - area | C | |
1771 | Sid is colouring the cells in the grid using the four colours red, blue, yellow and green in such a way that any two cells that share a vertex are coloured differently. He has already coloured some of the cells as shown.
What colour will he use for the cell marked $X$ ?
<image1> | [
"Red",
"Blue",
"Yellow",
"Green",
"You can't be certain",
"I don't know"
] | images/1771.jpg | A | Since any two cells which share a vertex are coloured differently, the centre cell in the top row could only be coloured red or green. The cell below that cannot be coloured blue or yellow or the same colour as the centre cell in the top row and so is coloured green or red opposite to the choice of the colour to the fi... | 2 | graph theory | A | |
1773 | Andrew wants to write the letters of the word KANGAROO in the cells of a $2 \times 4$ grid such that each cell contains exactly one letter. He can write the first letter in any cell he chooses but each subsequent letter can only be written in a cell with at least one common vertex with the cell in which the previous le... | [
"A",
"B",
"C",
"D",
"E",
"I don't know"
] | images/1773.jpg | D | To produce the arrangement of diagram D, Andrew would first need to write the letters K, A, N and $\mathrm{G}$ in the top four cells as shown. He would then need to write $\mathrm{A}$ in a vacant cell next to the G. Therefore he could not write $\mathrm{O}$ and $\mathrm{O}$ in the third row. Hence arrangement D could n... | 3 | combinatorics | D |
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