Date | May 2008 | Marks available | 2 | Reference code | 08M.2.sl.TZ2.3 |
Level | SL only | Paper | 2 | Time zone | TZ2 |
Command term | Show that | Question number | 3 | Adapted from | N/A |
Question
Amir needs to construct an isosceles triangle ABCABC whose area is 100 cm2100 cm2. The equal sides, ABAB and BCBC, are 20 cm20 cm long.
Sylvia is making a square-based pyramid. Each triangle has a base of length 12 cm12 cm and a height of 10 cm10 cm.
Angle ABCABC is acute. Show that the angle ABCABC is 30∘30∘.
Find the length of ACAC.
Show that the height of the pyramid is 8 cm8 cm.
MM is the midpoint of the base of one of the triangles and OO is the apex of the pyramid.
Find the angle that the line MOMO makes with the base of the pyramid.
Calculate the volume of the pyramid.
Daniel wants to make a rectangular prism with the same volume as that of Sylvia’s pyramid. The base of his prism is to be a square of side 10 cm10 cm. Calculate the height of the prism.
Markscheme
12202sinB=10012202sinB=100 (M1)(A1)
B=30∘B=30∘ (AG)
Note: (M1) for correct substituted formula and (A1) for correct substitution. B=30∘B=30∘ must be seen or previous (A1) mark is lost.
[2 marks]
Unit penalty (UP) is applicable where indicated in the left hand column.
¯AC2=2×202−2×202×cos30∘¯¯¯¯¯¯¯¯AC2=2×202−2×202×cos30∘ (M1)(A1)
(UP) ¯AC=10.4 cm¯¯¯¯¯¯¯¯AC=10.4 cm (A1)(G2)
Note: (M1) for using cosine rule, (A1) for correct substitution. Last (A1) is for the correct answer. Accept use of sine rule or any correct method e.g. AC=2×20sin15∘AC=2×20sin15∘ .
[3 marks]
x2+62=102x2+62=102 (A1)(M1)
x=8 cmx=8 cm (AG)
Note: (A1) for 66 (or 3636) seen and (M1) for using Pythagoras with correct substitution. x=8x=8 must be seen or previous (M1) mark is lost.
[2 marks]
cosβ=610cosβ=610 (M1)(A1)
β=53.1∘β=53.1∘ (A1)(G2)
OR equivalent
Note: (M1) for use of trigonometric ratio with numbers from question. (A1) for use of correct numbers, and (A1) for correct answer.
[3 marks]
Unit penalty (UP) is applicable where indicated in the left hand column.
vol=122×83vol=122×83 (M1)
(UP) =384 cm3=384 cm3 (A1)(G2)
Note: (M1) for correct formula and correct substitution, (A1) for correct answer.
[2 marks]
Unit penalty (UP) is applicable where indicated in the left hand column.
Let h be the height
102h=384102h=384 (M1)
(UP) =3.84 cm=3.84 cm (A1)(ft)(G2)
Note: (M1) for correct formula and correct substitution, (A1) for correct answer. (ft) from answer to part (c).
[2 marks]
Examiners report
Many students did not write the units in their answers and were penalized with the UP in this question.
Part (a) was not very well answered. It looked as if the candidates did not understand the question. Many candidates did not draw a sketch of the triangle; this would have helped them to solve the question. Many candidates simply calculated the remaining angles of the triangle and showed that the sum was 180∘180∘ . This was a clear example of the misunderstanding of the term "show that". Part (b) was well done though some candidates lost a mark for not giving the answer to the correct accuracy.
Many students did not write the units in their answers and were penalized with the UP in this question.
Part (a) was not very well answered. It looked as if the candidates did not understand the question. Many candidates did not draw a sketch of the triangle; this would have helped them to solve the question. Many candidates simply calculated the remaining angles of the triangle and showed that the sum was 180∘180∘ . This was a clear example of the misunderstanding of the term "show that". Part (b) was well done though some candidates lost a mark for not giving the answer to the correct accuracy.
Many students did not write the units in their answers and were penalized with the UP in this question.
The weaker candidates spent a lot of time in (a) using the wrong triangle to find half of the diagonal of the base. Finally they used Pythagoras theorem with the wrong numbers. Part (b) was well answered by most of the students. For the volume of the pyramid in (c) they used the correct formula though not always with the correct substitutions. To find the height of the prism in (d) the most common error was multiplying the volume of the prism by 1313. It seemed that many did not know the term 'prism'.
Many students did not write the units in their answers and were penalized with the UP in this question.
The weaker candidates spent a lot of time in (a) using the wrong triangle to find half of the diagonal of the base. Finally they used Pythagoras theorem with the wrong numbers. Part (b) was well answered by most of the students. For the volume of the pyramid in (c) they used the correct formula though not always with the correct substitutions. To find the height of the prism in (d) the most common error was multiplying the volume of the prism by 1313. It seemed that many did not know the term 'prism'.
Many students did not write the units in their answers and were penalized with the UP in this question.
The weaker candidates spent a lot of time in (a) using the wrong triangle to find half of the diagonal of the base. Finally they used Pythagoras theorem with the wrong numbers. Part (b) was well answered by most of the students. For the volume of the pyramid in (c) they used the correct formula though not always with the correct substitutions. To find the height of the prism in (d) the most common error was multiplying the volume of the prism by 1313. It seemed that many did not know the term 'prism'.
Many students did not write the units in their answers and were penalized with the UP in this question.
The weaker candidates spent a lot of time in (a) using the wrong triangle to find half of the diagonal of the base. Finally they used Pythagoras theorem with the wrong numbers. Part (b) was well answered by most of the students. For the volume of the pyramid in (c) they used the correct formula though not always with the correct substitutions. To find the height of the prism in (d) the most common error was multiplying the volume of the prism by 1313. It seemed that many did not know the term 'prism'.