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(a)The diagram below shows the triangle t and the line l. Construct the image of the triangle t under axial symmetry in l. Show all construction lines clearly. If you use measurements, write them in an appropriate place on your solution.
Construct the reflection of triangle t in the line l — draw directly on the diagram
(b)A right circular cone has a height of h cm and a radius of h4 cm, where h ∈ ℕ is a constant. The volume of the cone is 36 000π cm3. Work out the value of h.
(b) working
The diagram below shows a belt drive system. In the diagram, the belt is the path from A to D to C to B and back to A, made up of two line segments and two major arcs of circles. The circles k and s have centres P and Q, respectively. A and B are points on k, and C and D are points on s. AD and BC are tangents to both circles, as shown, and intersect at the point O, which lies on PQ.
(a)Fill in the two missing statements and four missing reasons in the table below to prove that the triangles POA and POB are congruent.
| Statement | Reason | |
|---|---|---|
| Statement 1 | |PA| = |PB| | |
| Statement 2 | ||
| Statement 3 | ||
| Conclusion | Therefore, ΔPOA and ΔPOB are congruent |
(a)
(b)The radius of the circle k is |PA| = 12 cm. The size of the angle ∠APO is 70°.
(i)Find the distance |AO|, correct to 1 decimal place.
(b)(i)
The triangles POA and QOC are similar. The radius of circle s is |QC| = 8 cm.
(ii)Find the total length of the belt, that is, the path from A to D to C to B and back to A. Give your answer correct to the nearest cm.
(b)(ii)
(c)The gear belt is packed in a closed box. The net of the box is the shaded part in the diagram below. It can be cut from a sheet of cardboard of size 55 cm by 120 cm. The height of the box, when assembled, is x cm, as shown. Two other lengths are marked j and k in the diagram.
Find the volume of the box in cm3, when assembled. Give your answer in the form ax3 + bx2 + cx, where a, b, c ∈ ℝ are constants. As part of your solution, express the lengths marked j and k in terms of x.
(c)
(d)The gear belt is part of a robot lawnmower. It is used to cut the grass in the region shown in the diagram below. The perimeter of this region is made up of a number of line segments, plus the arc of one quarter of a circle from A to B with its centre on the side BC. BC is divided into six segments of equal length. The length, in metres, of five line segments are shown in the diagram. Each of these line segments is perpendicular to BC.
Liam uses the trapezoidal rule to estimate the total area of this region. By finding the actual total area, or otherwise, work out the error in Liam’s estimate for the total area of this region. Give your answer correct to the nearest m2.
(d)
(a)The diagram shows two triangles, ACD and BCE. C is the midpoint of [AB]. AD is parallel to EB, and the point C lies on DE.
Prove that the triangles ACD and BCE are congruent. Give a reason for each statement that you make in your proof.
(a) proof
(b)The parallelogram PQRS is shown below. The point X lies on the line PQ. P′Q′R′S′ is an enlargement of PQRS, using point X as the centre of enlargement.
(i)|XQ| = 8 cm and |QQ′| = 4 cm. Show that the scale factor of the enlargement, k, is 1·5.
(b)(i)
(ii)Y is the point where the lines SR and P′S′ intersect. The region P′QRY is shaded. |XP| = 3 cm. The area of PQRS is 20 cm2. Use this to find the area of the shaded region P′QRY, in cm2.
(b)(ii)
Below is a scaled diagram of a submarine. The body of the submarine is roughly in the shape of a cylinder, with a cone at one end and a hemisphere at the other end, as shown.
(a)(i)Measure the lengths labelled r, d, and h on the diagram above. Write the length of each, correct to the nearest cm, in the table below.
| Label | r | d | h |
|---|---|---|---|
| Length on diagram (cm) |
The actual total length of the submarine shown in the scaled diagram above is 90 m.
(a)(ii)Use the measurements from part (a)(i) to work out the actual lengths represented by r, d, and h. Give each value in metres, correct to 1 decimal place.
| Label | r | d | h |
|---|---|---|---|
| Actual length (metres) |
(a) working
(b)The diagram below shows the body of a different submarine with a similar design. The dimensions of this submarine are all given in terms of x, where x ∈ ℝ. The hemisphere, cylinder, and cone all have a radius of x.
This submarine has a volume of 6738 m3, correct to the nearest m3. By solving an equation in x, find the total length of this submarine. Give your answer in metres, correct to 1 decimal place.
(b) working
A roof is in the shape of a square-based pyramid, as shown. The square base, ABCD, has sides of length 6 m. The diagonals of ABCD meet at the point O. The top of the pyramid, P, is directly above O. The four triangular faces are congruent to each other, with |AP| = |BP| = 11 m.
(i)Use the theorem of Pythagoras to show that |OB| = 3√2 m, and hence find the value of |OP|, the vertical height of the pyramid. Give |OP| in surd form.
(a)(i)
(ii)On the triangular face PAB, the size of ∠PAB is 74·2°, correct to 1 decimal place. Using this, or otherwise, work out the total area of the four triangular faces of the roof. Give your answer correct to the nearest m2.
(a)(ii)
(iii)The diagram below shows part of a scaled diagram of the net of this pyramid. The diagram shows the square base and two of the triangular sides. Construct the rest of the scaled diagram of the net of the pyramid. Show all construction lines clearly.
Construct the remaining faces of the net directly on the diagram
(a)Construct the centroid of the triangle PQR below. Show all construction lines clearly. Where measurement is used, show all relevant measurements and calculations clearly.
Construct the centroid directly on the triangle
(b)The lines AD, BE, and CF in the diagram below are parallel. E ∈ DF and B ∈ AC. |AB| = |BC|. Prove that |DE| = |EF|. (That is, prove that if three parallel lines cut off equal segments on some transversal line, then they will cut off equal segments on any other transversal.) Give a reason for each statement that you make in your proof, as appropriate.
Add any construction lines on the diagram
(b) proof
Tommy makes ornaments from metal and glass.
(a)He makes an open metal cylinder with a height of 15 cm and a radius of 5 cm. The net of this cylinder is a rectangle. Find the dimensions of this rectangle. Give your answers in cm, correct to 1 decimal place where appropriate.
(a)
(b)Tommy makes another cylinder with a height of 22 cm and a diameter of 12 cm. This cylinder fits exactly inside a glass sphere. The top and bottom edges of the cylinder touch the sphere. Find the volume of the sphere, in cm3, correct to 1 decimal place. Use the Theorem of Pythagoras in your solution.
(b)
(c)Another ornament is made of two cones inscribed in a sphere. The top cone is upright; the bottom cone is inverted. The cones have the same base. A vertical cross-section of the ornament, taken through the centre of the sphere, shows the cones as two triangles, ABC and ADB, with a common side [AB]. ABC is the top cone. The points A, B, C, and D all lie on the circle s, which represents the cross-section of the sphere. The lines AB and CD intersect at the point E.
(i)The diagram is symmetrical about the line DC. State why |∠CBD| = 90°.
(c)(i)
(ii)Hence, or otherwise, prove that the triangles BCE and DBE are similar. Give a reason for each statement that you make, where appropriate.
(c)(ii)
(iii)The top cone has a radius of r and a height of h; that is, |EB| = r and |EC| = h. The sphere, represented by s, has a radius of 10 cm. Use the similar triangles BCE and DBE to show that r2 = 20h − h2.
(c)(iii)
(iv)Hence, write the volume of the top cone in terms of h and π, and find the value of h that gives the maximum volume for the top cone.
(c)(iv)
(b)Rohan makes a solid cuboid of dimensions 5 × 3 × 4 using his small cubes. Each small cube has sides of length 1 unit. Some of the small cubes have 1 face, 2 faces, or 3 faces on the outside of the cuboid. Other small cubes have no faces on the outside. In the diagram, the shaded small cube has 3 faces on the outside.
Fill in the table below, showing the number of small cubes with 3 faces, 2 faces, 1 face, or no faces on the outside of this cuboid. Show your working out. (One value is filled in: 1 face = 22.)
| Cubes with 3 faces outside | |
|---|---|
| Cubes with 2 faces outside | |
| Cubes with 1 face outside | 22 |
| Cubes with no faces outside |
(b)
(a)State whether the following statement is true or false: "Two angles are vertically opposite if, and only if, they are equal in size." Justify your answer.
(a)
(b)The two diagrams below show the same rectangle, ACEG (not to scale). The points B, D, and F lie on [AC], [CE], and [EG], respectively, as shown. |AB| = 20 cm, |BC| = 30 cm, and |AG| = 90 cm. |∠GFA| = |∠EFD| = |∠DBC| = θ, where θ ∈ ℝ. In Diagram B, [GE] and [BD] are extended, and they meet at the point H.
(i)Prove that |FE| = |EH|, in Diagram B. Use congruent triangles. Give a reason for each statement that you make in your proof.
(b)(i)
(ii)Hence, or otherwise, find the size of the angle θ. Give your answer correct to the nearest degree.
(b)(ii)
(a)Séamus has a basin in the shape of an inverted right circular cone with the lower part removed (a frustum), as shown (not to scale). The radius of the original cone is 15 cm, and |CE|, the slant height of the basin, is 10 cm. The base of the basin is a horizontal circle with a radius of 12 cm. The basin is open — it does not have a top.
(i)Show that |BE| = 40 cm.
(a)(i)
(ii)Find the total internal surface area of the basin, correct to 1 decimal place.
(a)(ii)
(iii)Draw a diagram of the net of the curved surface of the basin. Show your working out. Include enough measurements of lengths and/or angles so that the net could be constructed using a ruler, straight edge, compass, and protractor, without any further calculation.
(a)(iii)
(a)Construct the circumcentre of the triangle XYZ shown below, using only a compass and straight edge. Label the circumcentre C. Show your construction lines clearly.
(a)
(b)The points A, B, C, and D lie on a circle (not to scale). [AB] is a diameter of the circle. |∠DAC| = 40°, as shown. The triangle ABD is isosceles. Find |∠ADC|.
(b)
(c)The diagram shows the triangle PQR (not to scale). The angle at R is greater than 90°. k is the circumcircle of PQR, and O is the circumcentre (not shown). Prove that O cannot be inside the triangle PQR. (If proving by contradiction, your first line should be: "Assume that O is inside the triangle PQR.")
(c)
A company makes and sells candles of different shapes and sizes.
(a)A candle in the shape of a cylinder has a diameter of 10 cm and a volume of 450π cm3. Work out the height of this candle.
(a)
(b)A small cone candle has a volume of 12π cm3. A large cone candle has a volume of 150π cm3. The small candle has height h; the large candle has height 2h. The large candle has a radius that is k times that of the small candle, where k ∈ ℝ. Work out the value of k.
(b)
(c)A third conical candle has its curved surface covered in cloth. The net of the cloth (not to scale) covers the cone perfectly with no overlap. |∠BOA| = 216° and |OA| = 8 cm. Find the length of the arc from B to A, in terms of π, and hence find the radius of the cone. Give both answers in cm.
(c)
(d)(i) A spherical ball of wax is used as a candle. The radius of the sphere is 2·7 cm. Find the volume of the sphere, correct to 3 decimal places.
(d)(i)
(ii)A horizontal slice is cut off this sphere so that the candle will balance on level surfaces. The area of the circular base of this candle is 5·4 cm2. Find the value of l, the vertical distance from the top of the candle to where the cut is made (l > 2·7 cm). Give your answer in cm, correct to the nearest mm.
(d)(ii)
(e)Part of the company logo is shown. ABCD is a square with sides of length 30 cm. The points E and F are the midpoints of [AB] and [AD]. The lines EC and FB are perpendicular, and meet at the point O. Using similar triangles or trigonometry, find the length |EO|. Give your answer in the form a√b cm, where a, b ∈ ℕ.
(e)
(a)Two identical right-circular solid cones meet along their bases and fit exactly inside a sphere, as shown.
(i)Prove that the volume of the remaining space inside the sphere is exactly half the total volume of the sphere.
(a)(i)
(ii)The combined volume of the two cones is 6863π cm3. Find the radius of one of the cones.
(a)(ii)
(b)At 9.00 a.m. a delivery van leaves a factory, travelling towards its destination at an average speed of 60 km/h. One hour and 45 minutes later a second van leaves the factory on the same route, travelling at an average speed of 95 km/h. Both vans arrive at their destination at the same time. Find at what time they arrive.
(b)
(a)Prove that if two triangles ΔABC and ΔA′B′C′ are similar, then the lengths of their sides are proportional in order: |AB||A′B′| = |BC||B′C′| = |CA||C′A′|
(a)
(b)In the diagram, the lines PA, HK, and BR are parallel. Prove that |AH| × |QB| = |AP| × |HB|. Give a reason for each geometrical statement you use.
(b)
(a)A company makes biodegradable paper cups in the shape of a right circular cone. Each cup has a radius of 3·3 cm and a slant height of 9 cm, as shown.
(i)Show that the vertical height of the cup is 8·37 cm, correct to 2 decimal places.
(a)(i)
(ii)Find the curved surface area of the cup correct to 2 decimal places.
(a)(ii)
(iii)The diagram shows the net of the cup. Find, in degrees, the size of the angle θ.
(a)(iii)
(b)In order to avoid spillages each cup is marked with a dotted line at F which is 1 cm vertically below the top of the cup, as shown. Find the volume of water in the cup when it is filled as far as the dotted line. Give your answer correct to 1 decimal place.
(b)
(c)Water flows into one of these cups through a cylindrical pipe of radius 0·8 cm at a flow rate of 2·5 cm/sec. Find, to the nearest second, how long it will take to fill the cup to the line at F.
(c)
(d)The company decides to change the position of the line F in order to limit the capacity of the cup to 60 cm³. How far, vertically below the rim of the cup, should the line F be drawn? Give your answer, in cm, correct to 1 decimal place.
(d)
Two ships set sail at the same time, Ship A from Port A and Ship B from Port B. Port A is 90 km due west of Port B, as shown. Ship A is travelling due east at a speed of 15 km/h. Ship B is travelling due south at a speed of 30 km/h.
(a)Find the distance between the two ships 30 minutes after they set sail. Give your answer in km, correct to 2 decimal places.
(a)
(b)t is the time in hours after the ships set sail. Show that the distance between the ships at time t can be given by the function s(t) = (1125t2 − 2700t + 8100)1/2, where 0 ≤ t ≤ 6.
(b)
(c)Use calculus to find the value of t when the ships are closest to each other, and find the distance between the ships at your value of t. Give the distance in km, correct to 1 decimal place.
(c)
(a)Construct and label the orthocentre of the triangle ABC in the diagram. Show any construction lines or arcs clearly.
(a)
(b)In the diagram O is the centre of circle s. [AB] is a diameter of s. BE is a tangent to s at point B. [CD] is a chord of circle s. |CD| = 12|AB| and CD is parallel to AB. Find, with justification, |∠BEA|.
(b)
(a)A cattle feeding trough of uniform cross section and 2·5 m in length is shown (Figure 1). The front of the trough (segment ABC) is shown in Figure 2. The front of the trough is a segment of a circle of radius 90 cm. The height of the trough, |DB|, is 30 cm. |OA| = |OC| = |OB| = 90 cm. [OB] ⊥ [AC].
(i)Find |AD|. Give your answer in the form a√b cm, where a, b ∈ ℤ.
(a)(i)
(ii)Find |∠DOA|. Give your answer in radians, correct to 2 decimal places.
(a)(ii)
(iii)Find the area of the segment ABC. Give your answer in m² correct to 2 decimal places.
(a)(iii)
(iv)Find the volume of the trough. Give your answer in m³, correct to 2 decimal places.
(a)(iv)
(b)A sand timer for games is shown. Each half of the timer consists of a hemisphere, a cylinder of height 3·5 cm and a cone of height 1·5 cm. All of the parts have a radius of 1·25 cm.
(i)The upper half of the timer is full of sand. Find the volume of sand in the upper half of the timer. Give your answer in cm³ correct to 2 decimal places.
(b)(i)
(ii)Sand flows from the top half into the bottom part; the top surfaces in both parts remain level. At a certain time, 98% of the sand has flowed into the bottom half. Find h, the height of the remaining sand (in the conical part of the top of the timer). Give your answer in cm, correct to 2 decimal places.
(b)(ii)
(a)Let ΔABC be a triangle. Prove that if a line l is parallel to BC and cuts [AB] in the ratio s : t, where s, t ∈ ℕ, then it also cuts [AC] in the same ratio.
(a)
(b)In triangle ABC: |∠CAB| = 90°, |AX| = 4 cm, |AY| = 3 cm, XY ∥ BC, XZ ∥ AC, and |AX| : |XB| = 1 : 2. Find |BZ| (using similar triangles).
(b)
A section of a garden railing consists of nine cylindrical bars, labelled A to I, with a solid sphere attached to the centre of the top of each bar. The volume of each sphere from B to E is 1·75 times the volume of the previous sphere.
(a)The radius of sphere A is 3 cm. Find the sum of the volumes of the five spheres A, B, C, D, and E. Give your answer correct to the nearest cm³.
(a)
(b)(i)The surface area of sphere E is 503 cm². The height of the railing at E (sum of the heights of bar E and sphere E) is 1·2 m. Find the height of bar E, in cm, correct to 1 decimal place.
(b)(i)
(ii)The radius of each bar is 1 cm. The volume of bar A is 71·3π cm³. The heights of the bars A, B, C, D, and E form an arithmetic sequence. Find, in cm, the height of each bar.
(b)(ii)
(c)There is a wall on each side of the railing; the distance from wall to wall is 1·5 m. The distance from each wall to the nearest bar (A and I) is 20 cm. The radius of each bar is 1 cm and the gap between each bar is identical. Find the size of this gap.
(c)
(d)The sphere on bar A and the sphere on bar B are joined by a straight rod. Find the length of the shortest rod that will join sphere A to sphere B. Give your answer in cm, correct to 1 decimal place (Pythagoras).
(d)
ABCD is a rectangle. F ∈ [AB], G ∈ [BC], [FD] ∩ [AG] = {E}, and FD ⊥ AG. |AE| = 12 cm, |EG| = 27 cm, and |FE| = 5 cm.
(a)Prove that ΔAFE and ΔDAE are similar (equiangular).
(a)
(b)Find |AD|.
(b)
(c)ΔAFE and ΔAGB are similar. Show that |AB| = 36 cm.
(c)
(d)Find the area of the quadrilateral GCDE.
(d)
(a)Take the earth as a sphere with radius 6371 km. Jack stands on the Cliffs of Moher at point J, 214 m above sea level, looking out to a point H on the horizon. Taking A as the centre of the earth, find |JH|, the distance from Jack to the horizon (a tangent from an external point). Give your answer correct to the nearest km.
(a)
(b)The Cliffs of Moher, at point C, are at latitude 53° north. s1 is the circle at latitude 53°; s2 is the equator (latitude 0°); A is the centre of the earth; s1 and s2 are on parallel planes. Find the length of the circle s1, correct to the nearest km.
(b)
Two solid cones, each of radius R cm and height R cm, are welded together at their vertices and placed in the smallest possible hollow cylinder (Figure 1).
(a)Show that the capacity (volume) of the empty space in the cylinder is equal to the capacity of an empty sphere of radius R cm (Figure 2).
(a)
(b)For the remainder, R = 12 cm. Water is poured into both the cylinder and the sphere to a depth of 6 cm (Figures 3 and 4).
(i)Find |AB|, the radius of the circular surface of the water in the sphere (Figure 4). Give your answer in the form a√b cm, where a, b ∈ ℕ.
(b)(i)
(ii)Find |CD|, the radius of the cone at water level (Figure 3).
(b)(ii)
(iii)Verify that the area of the surface of the water in the sphere is equal to the area of the surface of the water in the cylinder.
(b)(iii)
(c)Cavalieri discovered that, at the same depth, the volume of water in the available space in the cylinder equals the volume of water in the sphere. Use this to find the volume of water in the sphere when the depth is 6 cm. Give your answer in terms of π.
(c)
The diagram shows a semi-circle standing on a diameter [AC], with [BD] ⊥ [AC].
(a)(i)Prove that the triangles ABD and DBC are similar.
(a)(i)
(ii)If |AB| = x, |BC| = 1, and |BD| = y, write y in terms of x.
(a)(ii)
(b)Use your result from part (a)(ii) to construct a line segment equal in length (in centimetres) to the square root of the length of the line segment [TU] drawn on the paper.
(b)
A glass Roof Lantern is a pyramid with rectangular base CDEF and apex B. The vertical height is |AB|, where A is the intersection of the diagonals of the base. |CD| = 2·5 m and |CF| = 3 m.
(a)(i)Show that |AC| = 1·95 m, correct to two decimal places (Pythagoras).
(a)(i)
(ii)The angle of elevation of B from C is 50° (|∠BCA| = 50°). Show that |AB| = 2·3 m, correct to one decimal place.
(a)(ii)
(iii)Find |BC|, correct to the nearest metre.
(a)(iii)
(iv)Find |∠BCD|, correct to the nearest degree.
(a)(iv)
(v)Find the area of glass required to glaze all four triangular sides of the pyramid. Give your answer correct to the nearest m².
(a)(v)
(b)Another Roof Lantern pyramid has a square base CDEF. The vertical height |AB| = 3 m. The angle of elevation of B from C is 60° (|∠BCA| = 60°). Find the length of the side of the square base. Give your answer in the form √a m, where a ∈ ℕ.
(b)
(a)Construct the centroid of the triangle ABC below. Show all construction lines. (Where measurement is used, show all relevant measurements and calculations clearly.)
(a)
(b)Prove that, if three parallel lines cut off equal segments on some transversal line, then they will cut off equal segments on any other transversal line. State your Given, To Prove, Construction and Proof.
(b)
A flat machine part consists of two circular ends attached to a plate (diagram not to scale). The sides of the plate, HK and PQ, are tangential to each circle. The larger circle has centre A and radius 4r cm; the smaller has centre B and radius r cm. The length of [HK] is 8r cm and |AB| = 20√73 cm.
(a)Find r, the radius of the smaller circle. (Hint: Draw BT ∥ KH, T ∈ AH.)
(a)
(b)Find the area of the quadrilateral ABKH.
(b)
(c)(i)Find |∠HAP|, in degrees, correct to one decimal place.
(c)(i)
(ii)Find the area of the machine part, correct to the nearest cm2.
(c)(ii)
Answer either 6A or 6B. This is 6A.
(a)Prove that, if two triangles ABC and A′B′C′ are similar, then their sides are proportional, in order: |AB||A′B′| = |BC||B′C′| = |CA||C′A′|. State your Given, To Prove, Construction and Proof.
(a)
(b)Given the line segment [BC], construct, without using a protractor or set square, a point A such that |∠ABC| = 60°. Show your construction lines.
(b)
Answer either 6A or 6B. This is 6B. [AB] and [CD] are chords of a circle that intersect externally at E, as shown.
(a)Name two similar triangles in the diagram above and give reasons for your answer.
(a)
(b)Prove that |EA| · |EB| = |EC| · |ED|.
(b)
(c)Given that |EB| = 6·25, |ED| = 5·94 and |CB| = 10, find |AD|.
(c)
The triangle ABC is right-angled at C. The circle s has diameter [AC] and the circle t has diameter [CB].
(i)Draw the circle u which has diameter [AB].
(i)
(ii)Prove that in any right-angled triangle ABC, the area of the circle u equals the sum of the areas of the circles s and t.
(ii)
(iii)The diagram shows the right-angled triangle ABC and arcs of the circles s, t and u. Each shaded area is a lune, a crescent-shaped area bounded by arcs of the circles. Prove that the sum of the areas of the two shaded lunes is equal to the area of the triangle ABC.
(iii)
Questions reproduced from State Examinations Commission Leaving Certificate examination papers,
© State Examinations Commission (examinations.ie). Gathered here for study use.
© State Examinations Commission (examinations.ie). Gathered here for study use.