Question 20 - A-Level Maths

Edexcel D1 2021 October — Question 20

Exam Board	Edexcel
Module	D1 (Decision Mathematics 1)
Year	2021
Session	October
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Sorting Algorithms
Type	Quick Sort Execution
Difficulty	Easy -1.2 This is a routine D1 question testing standard algorithm execution (quick sort) with clear procedural steps. Part (a) involves basic bin packing arithmetic, and part (b) requires mechanical application of quick sort algorithm—both are textbook exercises requiring recall and careful execution rather than problem-solving or insight.
Spec	7.03k Sorting: quick sort 7.03l Bin packing: next-fit, first-fit, first-fit decreasing, full bin

20
23
17
15
22
19
25
13
28
32 A lower bound for the number of bins required is 4

Determine the range of possible values of $n$. You must make your method clear.
(3)
Carry out a quick sort to produce a list of the numbers in descending order. You should show the result of each pass and identify your pivots clearly.
(4) When the first-fit bin packing algorithm is applied to the original list of numbers, the following allocation is achieved. \end{table}
\includegraphics[max width=\textwidth, alt={}]{d409aaae-811d-4eca-b118-efc927885f97-14_1193_1586_1270_185}
Shortest path from A to J: $\_\_\_\_$ Length of shortest path from A to J: $\_\_\_\_$ \section*{2. $\_\_\_\_$} \section*{3.}
A B C D E F G H
A - 36 38 40 23 39 38 35
B 36 - 35 36 35 34 41 38
C 38 35 - 39 25 32 40 40
D 40 36 39 - 37 37 26 33
E 23 35 25 37 - 42 24 43
F 39 34 32 37 42 - 45 38
G 38 41 40 26 24 45 - 40
H 35 38 40 33 43 38 40 -

A B C D E F G H
A - 36 38 40 23 39 38 35
B 36 - 35 36 35 34 41 38
C 38 35 - 39 25 32 40 40
D 40 36 39 - 37 37 26 33
E 23 35 25 37 - 42 24 43
F 39 34 32 37 42 - 45 38
G 38 41 40 26 24 45 - 40
H 35 38 40 33 43 38 40 -

A B C D E F G H
A - 36 38 40 23 39 38 35
B 36 - 35 36 35 34 41 38
C 38 35 - 39 25 32 40 40
D 40 36 39 - 37 37 26 33
E 23 35 25 37 - 42 24 43
F 39 34 32 37 42 - 45 38
G 38 41 40 26 24 45 - 40
H 35 38 40 33 43 38 40 -

A B C D E F G H
A - 36 38 40 23 39 38 35
B 36 - 35 36 35 34 41 38
C 38 35 - 39 25 32 40 40
D 40 36 39 - 37 37 26 33
E 23 35 25 37 - 42 24 43
F 39 34 32 37 42 - 45 38
G 38 41 40 26 24 45 - 40
H 35 38 40 33 43 38 40 -
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d409aaae-811d-4eca-b118-efc927885f97-22_755_1157_246_404} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure}
1. Activity
  Immediately
  preceding
  activities
  A
  B
  C
  D
  E
  
  Activity
  Immediately
  preceding
  activities
  F
  G
  H
  I
  J
  
  Activity
  Immediately
  preceding
  activities
  K
  L
  M
  $$v = \ldots \quad x = \ldots \quad y = \ldots$$ \includegraphics[max width=\textwidth, alt={}, center]{d409aaae-811d-4eca-b118-efc927885f97-23_2255_56_315_37}
  \includegraphics[max width=\textwidth, alt={}]{d409aaae-811d-4eca-b118-efc927885f97-23_1153_1338_303_310}
  \section*{Grid 2} 5. \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{d409aaae-811d-4eca-b118-efc927885f97-24_591_1433_255_260} \captionsetup{labelformat=empty} \caption{Figure 3
  [0pt] [The total weight of the network is 166]}
  \end{figure} 6.
  \includegraphics[max width=\textwidth, alt={}]{d409aaae-811d-4eca-b118-efc927885f97-26_1287_1645_301_162}
  \section*{Diagram 1} 7. $\begin{array} { l l l l l l l l l l l } 14 & 20 & 23 & 17 & 15 & 22 & 19 & 25 & 13 & 28 & 32 \end{array}$

Show mark scheme Show mark scheme source

Question 20:

M1 Find $\sin(\theta) = \frac{1}{2}$ or equivalent

A1 $\theta = 30°$ or $\theta = 150°$ or $\theta = \frac{\pi}{6}$ or $\theta = \frac{5\pi}{6}$

M1 Attempt to find all solutions in the given range

A1 All four solutions correct: $30°, 150°, 210°, 330°$ or equivalent in radians

DM1 Demonstrate correct method for solving trigonometric equation

B1 State the general solution where applicable

Question 20:

M1 Find $\sin(\theta) = \frac{1}{2}$ or equivalent

A1 $\theta = 30°$ or $\theta = 150°$ or $\theta = \frac{\pi}{6}$ or $\theta = \frac{5\pi}{6}$

M1 Attempt to find all solutions in the given range

A1 All four solutions correct: $30°, 150°, 210°, 330°$ or equivalent in radians

DM1 Demonstrate correct method for solving trigonometric equation

B1 State the general solution where applicable

Show LaTeX source

20\\
23\\
17\\
15\\
22\\
19\\
25\\
13\\
28\\
32

A lower bound for the number of bins required is 4
\begin{enumerate}[label=(\alph*)]
\item Determine the range of possible values of $n$. You must make your method clear.\\
(3)
\item Carry out a quick sort to produce a list of the numbers in descending order. You should show the result of each pass and identify your pivots clearly.\\
(4)

When the first-fit bin packing algorithm is applied to the original list of numbers, the following allocation is achieved.

\end{table}

\begin{center}
\includegraphics[max width=\textwidth, alt={}]{d409aaae-811d-4eca-b118-efc927885f97-14_1193_1586_1270_185}
\end{center}

Shortest path from A to J: $\_\_\_\_$\\
Length of shortest path from A to J: $\_\_\_\_$

\section*{2. \\

 $\_\_\_\_$}

\section*{3.}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
 & A & B & C & D & E & F & G & H \\
\hline
A & - & 36 & 38 & 40 & 23 & 39 & 38 & 35 \\
\hline
B & 36 & - & 35 & 36 & 35 & 34 & 41 & 38 \\
\hline
C & 38 & 35 & - & 39 & 25 & 32 & 40 & 40 \\
\hline
D & 40 & 36 & 39 & - & 37 & 37 & 26 & 33 \\
\hline
E & 23 & 35 & 25 & 37 & - & 42 & 24 & 43 \\
\hline
F & 39 & 34 & 32 & 37 & 42 & - & 45 & 38 \\
\hline
G & 38 & 41 & 40 & 26 & 24 & 45 & - & 40 \\
\hline
H & 35 & 38 & 40 & 33 & 43 & 38 & 40 & - \\
\hline
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
 & A & B & C & D & E & F & G & H \\
\hline
A & - & 36 & 38 & 40 & 23 & 39 & 38 & 35 \\
\hline
B & 36 & - & 35 & 36 & 35 & 34 & 41 & 38 \\
\hline
C & 38 & 35 & - & 39 & 25 & 32 & 40 & 40 \\
\hline
D & 40 & 36 & 39 & - & 37 & 37 & 26 & 33 \\
\hline
E & 23 & 35 & 25 & 37 & - & 42 & 24 & 43 \\
\hline
F & 39 & 34 & 32 & 37 & 42 & - & 45 & 38 \\
\hline
G & 38 & 41 & 40 & 26 & 24 & 45 & - & 40 \\
\hline
H & 35 & 38 & 40 & 33 & 43 & 38 & 40 & - \\
\hline
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
 & A & B & C & D & E & F & G & H \\
\hline
A & - & 36 & 38 & 40 & 23 & 39 & 38 & 35 \\
\hline
B & 36 & - & 35 & 36 & 35 & 34 & 41 & 38 \\
\hline
C & 38 & 35 & - & 39 & 25 & 32 & 40 & 40 \\
\hline
D & 40 & 36 & 39 & - & 37 & 37 & 26 & 33 \\
\hline
E & 23 & 35 & 25 & 37 & - & 42 & 24 & 43 \\
\hline
F & 39 & 34 & 32 & 37 & 42 & - & 45 & 38 \\
\hline
G & 38 & 41 & 40 & 26 & 24 & 45 & - & 40 \\
\hline
H & 35 & 38 & 40 & 33 & 43 & 38 & 40 & - \\
\hline
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|}
\hline
 & A & B & C & D & E & F & G & H \\
\hline
A & - & 36 & 38 & 40 & 23 & 39 & 38 & 35 \\
\hline
B & 36 & - & 35 & 36 & 35 & 34 & 41 & 38 \\
\hline
C & 38 & 35 & - & 39 & 25 & 32 & 40 & 40 \\
\hline
D & 40 & 36 & 39 & - & 37 & 37 & 26 & 33 \\
\hline
E & 23 & 35 & 25 & 37 & - & 42 & 24 & 43 \\
\hline
F & 39 & 34 & 32 & 37 & 42 & - & 45 & 38 \\
\hline
G & 38 & 41 & 40 & 26 & 24 & 45 & - & 40 \\
\hline
H & 35 & 38 & 40 & 33 & 43 & 38 & 40 & - \\
\hline
\end{tabular}
\end{center}

4.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{d409aaae-811d-4eca-b118-efc927885f97-22_755_1157_246_404}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}

(a)

\begin{center}
\begin{tabular}{ | c | c | }
\hline
Activity & \begin{tabular}{ c }
Immediately \\
preceding \\
activities \\
\end{tabular} \\
\hline
A &  \\
\hline
B &  \\
\hline
C &  \\
\hline
D &  \\
\hline
E &  \\
\hline
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{ | c | c | }
\hline
Activity & \begin{tabular}{ c }
Immediately \\
preceding \\
activities \\
\end{tabular} \\
\hline
F &  \\
\hline
G &  \\
\hline
H &  \\
\hline
I &  \\
\hline
J &  \\
\hline
\end{tabular}
\end{center}

\begin{center}
\begin{tabular}{ | c | c | }
\hline
Activity & \begin{tabular}{ c }
Immediately \\
preceding \\
activities \\
\end{tabular} \\
\hline
K &  \\
\hline
L &  \\
\hline
M &  \\
\hline
\end{tabular}
\end{center}

$$v = \ldots \quad x = \ldots \quad y = \ldots$$

\includegraphics[max width=\textwidth, alt={}, center]{d409aaae-811d-4eca-b118-efc927885f97-23_2255_56_315_37}

\begin{center}
\includegraphics[max width=\textwidth, alt={}]{d409aaae-811d-4eca-b118-efc927885f97-23_1153_1338_303_310}
\end{center}

\section*{Grid 2}
5.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{d409aaae-811d-4eca-b118-efc927885f97-24_591_1433_255_260}
\captionsetup{labelformat=empty}
\caption{Figure 3\\[0pt]
[The total weight of the network is 166]}
\end{center}
\end{figure}

6.

\begin{center}
\includegraphics[max width=\textwidth, alt={}]{d409aaae-811d-4eca-b118-efc927885f97-26_1287_1645_301_162}
\end{center}

\section*{Diagram 1}

7.

$\begin{array} { l l l l l l l l l l l } 14 & 20 & 23 & 17 & 15 & 22 & 19 & 25 & 13 & 28 & 32 \end{array}$
\end{enumerate}

\hfill \mbox{\textit{Edexcel D1 2021 Q20}}

This paper (8 questions)

View full paper

Q1 10 Q2 6 Q3 15 Q4 11 Q5 10 Q6 13 Q7 10 Q20

Edexcel D1 2021 October — Question 20

Question 20:

M1 Find \(\sin(\theta) = \frac{1}{2}\) or equivalent

A1 \(\theta = 30°\) or \(\theta = 150°\) or \(\theta = \frac{\pi}{6}\) or \(\theta = \frac{5\pi}{6}\)

M1 Attempt to find all solutions in the given range

A1 All four solutions correct: \(30°, 150°, 210°, 330°\) or equivalent in radians

DM1 Demonstrate correct method for solving trigonometric equation

B1 State the general solution where applicable

This paper (8 questions)

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-

	A	B	C	D	E	F	G	H
A	-	36	38	40	23	39	38	35
B	36	-	35	36	35	34	41	38
C	38	35	-	39	25	32	40	40
D	40	36	39	-	37	37	26	33
E	23	35	25	37	-	42	24	43
F	39	34	32	37	42	-	45	38
G	38	41	40	26	24	45	-	40
H	35	38	40	33	43	38	40	-