| Date | May 2011 | Marks available | 3 | Reference code | 11M.2.sl.TZ1.5 |
| Level | SL only | Paper | 2 | Time zone | TZ1 |
| Command term | Show that | Question number | 5 | Adapted from | N/A |
Question
Pauline owns a piece of land ABCD in the shape of a quadrilateral. The length of BC is \(190{\text{ m}}\) , the length of CD is \(120{\text{ m}}\) , the length of AD is \(70{\text{ m}}\) , the size of angle BCD is \({75^ \circ }\) and the size of angle BAD is \({115^ \circ }\) .
Pauline decides to sell the triangular portion of land ABD . She first builds a straight fence from B to D .
Calculate the length of the fence.
The fence costs \(17\) USD per metre to build.
Calculate the cost of building the fence. Give your answer correct to the nearest USD.
Show that the size of angle ABD is \({18.8^ \circ }\) , correct to three significant figures.
Calculate the area of triangle ABD .
She sells the land for \(120\) USD per square metre.
Calculate the value of the land that Pauline sells. Give your answer correct to the nearest USD.
Pauline invests \(300 000\) USD from the sale of the land in a bank that pays compound interest compounded annually.
Find the interest rate that the bank pays so that the investment will double in value in 15 years.
Markscheme
\({\text{B}}{{\text{D}}^2} = {190^2} + {120^2} - 2(190)(120)\cos {75^ \circ }\) (M1)(A1)
Note: Award (M1) for substituted cosine formula, (A1) for correct substitution.
\(= 197\) m (A1)(G2)
Note: If radians are used award a maximum of (M1)(A1)(A0).
[3 marks]
\({\text{cost}} = 196.717 \ldots \times 17\) (M1)
\( = 3344{\text{ USD}}\) (A1)(ft)(G2)
Note: Accept \(3349\) from \(197\).
[2 marks]
\(\frac{{\sin ({\text{ABD}})}}{{70}} = \frac{{\sin ({{115}^ \circ })}}{{196.7}}\) (M1)(A1)
Note: Award (M1) for substituted sine formula, (A1) for correct substitution.
\( = {18.81^ \circ } \ldots \) (A1)(ft)
\( = {18.8^ \circ } \) (AG)
Notes: Both the unrounded and rounded answers must be seen for the final (A1) to be awarded. Follow through from their (a). If 197 is used the unrounded answer is \( = {18.78^ \circ } \ldots \)
[3 marks]
\({\text{angle BDA}} = {46.2^ \circ }\) (A1)
\({\text{Area}} = \frac{{70 \times (196.717 \ldots ) \times \sin ({{46.2}^ \circ })}}{2}\) (M1)(A1)
Note: Award (M1) for substituted area formula, (A1) for correct substitution.
\({\text{Area ABD}} = 4970{\text{ }}{{\text{m}}^2}\) (A1)(ft)(G2)
Notes: If \(197\) used answer is \(4980\).
Notes: Follow through from (a) only. Award (G2) if there is no working shown and \({46.2^ \circ }\) not seen. If \({46.2^ \circ }\) seen without subsequent working, award (A1)(G2).
[4 marks]
\(4969.38 \ldots \times 120\) (M1)
\( = 596 327{\text{ USD}}\) (A1)(ft)(G2)
Notes: Follow through from their (d).
[2 marks]
\(300000{\left( {1 + \frac{r}{{100}}} \right)^{15}} = 600000\) or equivalent (A1)(M1)(A1)
Notes: Award (A1) for \(600 000\) seen or implied by alternative formula, (M1) for substituted CI formula, (A1) for correct substitutions.
\(r = 4.73\) (A1)(ft)(G3)
Notes: Award (G3) for \(4.73\) with no working. Award (G2) for \(4.7\) with no working.
[4 marks]
Examiners report
Most candidates were able to recognise cosine rule, and substitute correctly. Where the final answer was not attained, this was mainly due to further unnecessary manipulation; the GDC should be used efficiently in such a case. Some students used the answer given and sine rule – this gained no credit.
Most candidates were able to recognise cosine rule, and substitute correctly. Where the final answer was not attained, this was mainly due to further unnecessary manipulation; the GDC should be used efficiently in such a case. Some students used the answer given and sine rule – this gained no credit.
Most candidates were able to recognise cosine rule, and substitute correctly. Where the final answer was not attained, this was mainly due to further unnecessary manipulation; the GDC should be used efficiently in such a case. Some students used the answer given and sine rule – this gained no credit.
Again, most candidates used the appropriate area formula – however, some did not appreciate the purpose of the given answer and were unable to complete the question accurately.
Again, most candidates used the appropriate area formula – however, some did not appreciate the purpose of the given answer and were unable to complete the question accurately.
The final part, in which compound interest was again asked for, tested most candidates but there were many successful attempts using either the GDC's finance package or correct use of the formula. Care must be taken with the former to show some indication of the values to be used in the context of the question. With the latter approach marks were again lost due to a lack of appreciation of the difference between interest and value.
