Edexcel P1 2018 Specimen — Question 10 12 marks

Exam BoardEdexcel
ModuleP1 (Pure Mathematics 1)
Year2018
SessionSpecimen
Marks12
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Mark schemeDownload PDF ↗
TopicSine and Cosine Rules
TypeShaded region with arc
DifficultyModerate -0.3 This is a straightforward multi-part question testing standard applications of cosine rule, sector area, and triangle area formulas. Part (a) is routine cosine rule calculation, parts (b-d) involve standard sector and triangle area computations with clear given values. While it requires multiple steps (12 marks total), each individual step uses direct formula application without requiring problem-solving insight or novel approaches, making it slightly easier than average.
Spec1.05b Sine and cosine rules: including ambiguous case1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta
\includegraphics{figure_4} The triangle \(XYZ\) in Figure 4 has \(XY = 6\) cm, \(YZ = 9\) cm, \(ZX = 4\) cm and angle \(ZXY = a\). The point \(W\) lies on the line \(XY\). The circular arc \(ZW\), in Figure 4, is a major arc of the circle with centre \(X\) and radius 4 cm.
  1. Show that, to 3 significant figures, \(a = 2.22\) radians. [2]
  2. Find the area, in cm\(^2\), of the major sector \(XZWX\). [3]
The region, shown shaded in Figure 4, is to be used as a design for a logo. Calculate
  1. the area of the logo [3]
  2. the perimeter of the logo. [4]
Question 10:
AnswerMarks
10(a)Correct use of cosine rule
92 (cid:32)42 (cid:14)62 (cid:16)2(cid:117)4(cid:117)6cos(cid:68)(cid:159)cos(cid:68)(cid:32).....
leading to a value for cos (cid:68)
42 (cid:14)62 (cid:16)92 (cid:167) 29 (cid:183)
cos(cid:68)(cid:32) (cid:168)(cid:32)(cid:16) (cid:32)(cid:16)0.604..(cid:184)
AnswerMarks
2(cid:117)4(cid:117)6 (cid:169) 48 (cid:185)M1
(cid:68) = 2.22 (cid:13) csoA1
(2)
Alternative
AnswerMarks
XY2 (cid:32)42 (cid:14)62 (cid:16)2(cid:117)4(cid:117)6cos2.22(cid:159) XY2 (cid:32)..Correct use of cosine rule
leading to a value for XY2M1
XY = 9.00....A1
(2)
AnswerMarks Guidance
(b)2(cid:83)(cid:16)2.22((cid:32)4.06366......) 2(cid:83)(cid:16)2.22 or 2(cid:83)(cid:16)2.2or awrt
4.06 (May be implied)B1
1(cid:117)42(cid:117)"4.06"
AnswerMarks
2Correct method for major
sector area. Allow (cid:83)(cid:16)2.22
AnswerMarks Guidance
for the major sector angle.M1
32.5Awrt 32.5 A1
(3)
Alternative – Circle Minor – sector
AnswerMarks
(cid:83)(cid:117)42Correct expression for circle
areaB1
1
(cid:83)(cid:117)42 (cid:16) (cid:117)42 (cid:117)2.22(cid:32)32.5
AnswerMarks Guidance
2Correct method for
circle - minor sector areaM1
(cid:32)32.5Awrt 32.5 A1
(3)
AnswerMarks
(c)Area of triangle =
1
(cid:117)4(cid:117)6(cid:117)sin2.22(cid:11)(cid:32)9.56(cid:12)
AnswerMarks
2Correct expression for the area of
triangle XYZ (allow 2.2 or awrt
AnswerMarks
2.22)B1
So area required = “9.56” + “32.5”Their Triangle XYZ + part (b) or
correct attempt at major sector
AnswerMarks Guidance
(Not triangle ZXW)M1
Area of logo = 42.1 cm2 or 42.0 cm2Awrt 42.1 or 42.0 (or just 42) A1
(3)
AnswerMarks
(d)Arc length(cid:32)4(cid:117)4.06(cid:11)(cid:32)16.24(cid:12)
or
AnswerMarks
8(cid:83)(cid:16)4(cid:117)2.22M1: 4(cid:117)their(cid:11)2(cid:83)(cid:16)2.22(cid:12)
or circumference – minor arc
AnswerMarks
A1: Correct ft expressionM1
A1ft
AnswerMarks Guidance
Perimeter = ZY + WY + Arc Length9 + 2 + Any Arc M1
Perimeter of logo = 27.2 or 27.3Awrt 27.2 or awrt 27.3 A1
(4)
(12 marks)
PPPMMMTTT
44 Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics, Further Mathematics and
Pure Mathematics – Sample Assessment Materials (SAMs) – Issue 3 – June 2018 © Pearson Education Limited 2018
Question 10:
--- 10(a) ---
10(a) | Correct use of cosine rule
92 (cid:32)42 (cid:14)62 (cid:16)2(cid:117)4(cid:117)6cos(cid:68)(cid:159)cos(cid:68)(cid:32).....
leading to a value for cos (cid:68)
42 (cid:14)62 (cid:16)92 (cid:167) 29 (cid:183)
cos(cid:68)(cid:32) (cid:168)(cid:32)(cid:16) (cid:32)(cid:16)0.604..(cid:184)
2(cid:117)4(cid:117)6 (cid:169) 48 (cid:185) | M1
(cid:68) = 2.22 (cid:13) cso | A1
(2)
Alternative
XY2 (cid:32)42 (cid:14)62 (cid:16)2(cid:117)4(cid:117)6cos2.22(cid:159) XY2 (cid:32).. | Correct use of cosine rule
leading to a value for XY2 | M1
XY = 9.00.... | A1
(2)
(b) | 2(cid:83)(cid:16)2.22((cid:32)4.06366......) | 2(cid:83)(cid:16)2.22 or 2(cid:83)(cid:16)2.2or awrt
4.06 (May be implied) | B1
1(cid:117)42(cid:117)"4.06"
2 | Correct method for major
sector area. Allow (cid:83)(cid:16)2.22
for the major sector angle. | M1
32.5 | Awrt 32.5 | A1
(3)
Alternative – Circle Minor – sector
(cid:83)(cid:117)42 | Correct expression for circle
area | B1
1
(cid:83)(cid:117)42 (cid:16) (cid:117)42 (cid:117)2.22(cid:32)32.5
2 | Correct method for
circle - minor sector area | M1
(cid:32)32.5 | Awrt 32.5 | A1
(3)
(c) | Area of triangle =
1
(cid:117)4(cid:117)6(cid:117)sin2.22(cid:11)(cid:32)9.56(cid:12)
2 | Correct expression for the area of
triangle XYZ (allow 2.2 or awrt
2.22) | B1
So area required = “9.56” + “32.5” | Their Triangle XYZ + part (b) or
correct attempt at major sector
(Not triangle ZXW) | M1
Area of logo = 42.1 cm2 or 42.0 cm2 | Awrt 42.1 or 42.0 (or just 42) | A1
(3)
(d) | Arc length(cid:32)4(cid:117)4.06(cid:11)(cid:32)16.24(cid:12)
or
8(cid:83)(cid:16)4(cid:117)2.22 | M1: 4(cid:117)their(cid:11)2(cid:83)(cid:16)2.22(cid:12)
or circumference – minor arc
A1: Correct ft expression | M1
A1ft
Perimeter = ZY + WY + Arc Length | 9 + 2 + Any Arc | M1
Perimeter of logo = 27.2 or 27.3 | Awrt 27.2 or awrt 27.3 | A1
(4)
(12 marks)
PPPMMMTTT
44 Pearson Edexcel International Advanced Subsidiary/Advanced Level in Mathematics, Further Mathematics and
Pure Mathematics – Sample Assessment Materials (SAMs) – Issue 3 – June 2018 © Pearson Education Limited 2018
\includegraphics{figure_4}

The triangle $XYZ$ in Figure 4 has $XY = 6$ cm, $YZ = 9$ cm, $ZX = 4$ cm and angle $ZXY = a$.

The point $W$ lies on the line $XY$.

The circular arc $ZW$, in Figure 4, is a major arc of the circle with centre $X$ and radius 4 cm.

\begin{enumerate}[label=(\alph*)]
\item Show that, to 3 significant figures, $a = 2.22$ radians. [2]
\item Find the area, in cm$^2$, of the major sector $XZWX$. [3]
\end{enumerate}

The region, shown shaded in Figure 4, is to be used as a design for a logo.

Calculate
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item the area of the logo [3]
\item the perimeter of the logo. [4]
\end{enumerate}

\hfill \mbox{\textit{Edexcel P1 2018 Q10 [12]}}
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