Questions - Edexcel AEA - A-Level Maths

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4

Edexcel AEA 2017 June Q3

The line $L _ { 1 }$ has equation $\mathbf { r } = \left( \begin{array} { c } - 13
7
- 1 \end{array} \right) + t \left( \begin{array} { c } 6
- 2
3 \end{array} \right)$. The line $L _ { 2 }$ passes through the point $A$ with position vector $\left( \begin{array} { c } 1
p
10 \end{array} \right)$ and is parallel to $\left( \begin{array} { c } - 2
11
- 5 \end{array} \right)$, where $p$
is a constant. The lines $L _ { 1 }$ and $L _ { 2 }$ intersect at the point $B$.
1. Find
  1. the value of $p$,
  2. the position vector of $B$.
The point $C$ lies on $L _ { 1 }$ and angle $A C B$ is $90 ^ { \circ }$
Find the position vector of $C$. The point $D$ also lies on $L _ { 1 }$ and triangle $A B D$ is isosceles with $A B = A D$.
Find the area of triangle $A B D$.

Edexcel AEA 2017 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{15e3f7f2-a77c-4ee4-8f0a-ac739e9fede5-4_332_454_201_810} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the equilateral triangle $L M N$ of side 2 cm ．The point $P$ lies on $L M$ such that $L P = x \mathrm {~cm}$ and the point $Q$ lies on $L N$ such that $L Q = y \mathrm {~cm}$ ．The points $P$ and $Q$ are chosen so that the area of triangle $L P Q$ is half the area of triangle $L M N$ ．
（a）Show that $x y = 2$
（b）Find the shortest possible length of $P Q$ ，justifying your answer． Mathematicians know that for any closed curve or polygon enclosing a fixed area，the ratio $\frac { \text { area enclosed } } { \text { perimeter } }$ is a maximum when the closed curve is a circle． By considering 6 copies of triangle $L M N$ suitably arranged，
（c）find the length of the shortest line or curve that can be drawn from a point on $L M$ to a point on $L N$ to divide the area of triangle $L M N$ in half．Justify your answer．
（6）

Edexcel AEA 2017 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{15e3f7f2-a77c-4ee4-8f0a-ac739e9fede5-5_946_1498_210_287} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve with equation $y = \mathrm { f } ( x )$ where $$f ( x ) = \frac { 4 ( x - 1 ) } { x ( x - 3 ) }$$ The curve cuts the $x$-axis at $( a , 0 )$. The lines $y = 0 , x = 0$ and $x = b$ are asymptotes to the curve.

Write down the value of $a$ and the value of $b$.
(2)
On separate axes, sketch the curves with the following equations. On your sketches, you should mark the coordinates of any intersections with the coordinate axes and state the equations of any asymptotes.
1. $y = \mathrm { f } ( x + 2 ) - 4$
2. $y = \mathrm { f } ( | x | ) - 3$

Write down $\overrightarrow { A C }$ in terms of $\mathbf { p } , \mathbf { q }$ and $k$ as appropriate.
Show that $\overrightarrow { C D } = - \mathbf { p } - \frac { 1 } { k } \mathbf { q }$
Hence find the value of $k$ By considering triangle $D B C$, or otherwise,
find the exact value of $\sin 54 ^ { \circ }$

Edexcel AEA 2017 Specimen Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-12_428_897_251_593} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A particle of weight $W$ lies on a rough plane．The plane is inclined to the horizontal at an angle $\alpha$ where $\tan \alpha = \frac { 3 } { 4 }$ ．The coefficient of friction between the particle and the plane is $\frac { 1 } { 2 }$ The particle is held in equilibrium by a force of magnitude 1.2 W ．The force makes an angle $\theta$ with the plane，where $0 < \theta < \pi$ ，and acts in a vertical plane containing a line of greatest slope of the plane，as shown in Figure 2.
（a）Find the value of $\theta$ for which there is no frictional force acting on the particle． The minimum value of $\theta$ for the particle to remain in equilibrium is $\beta$
（b）Show that $$\beta = \arccos \left( \frac { \sqrt { 5 } } { 3 } \right) - \arctan \left( \frac { 1 } { 2 } \right)$$ （c）Find the range of values of $\theta$ for which the particle remains in equilibrium with the frictional force acting up the plane．

Edexcel AEA 2017 Specimen Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-16_745_862_258_667} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} Show that the area of the finite region between the curves $y = \tan ^ { 2 } x$ and $y = 4 \cos 2 x - 1$ in the interval $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$, shown shaded in Figure 3, is given by $$2 \sqrt { 2 \sqrt { 3 } } - 2 \sqrt { 2 \sqrt { 3 } - 3 }$$

\includegraphics[max width=\textwidth, alt={}]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-16_2255_51_315_1987}

Edexcel AEA 2017 Specimen Q6

6．（i）Eden，who is confused about the laws of logarithms，states that $$\left( \log _ { 5 } p \right) ^ { 2 } = \log _ { 5 } \left( p ^ { 2 } \right)$$ and $\log _ { 5 } ( q - p ) = \log _ { 5 } q - \log _ { 5 } p$
However，there is a value of $p$ and a value of $q$ for which both statements are correct．
Determine these values．
（ii）（a）Let $r \in \mathbb { R } ^ { + } , r \neq 1$ ．Prove that $$\log _ { r } A = \log _ { r ^ { 2 } } B \Rightarrow A ^ { 2 } = B$$ （b）Solve $$\log _ { 4 } \left( 3 x ^ { 3 } + 26 x ^ { 2 } + 40 x \right) = 2 + \log _ { 2 } ( x + 2 )$$

\includegraphics[max width=\textwidth, alt={}]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-20_2261_53_317_1977}

Edexcel AEA 2017 Specimen Q7

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{05b21c5d-5958-4267-b1e6-3d1ed20d5609-25_670_682_301_694} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} A circular tower of radius 1 metre stands in a large horizontal field of grass．A goat is attached to one end of a rope and the other end of the rope is attached to a fixed point $O$ at the base of the tower．The goat cannot enter the tower． Taking the point $O$ as the origin（ 0,0 ），the centre of the base of the tower is at the point $T ( 0,1 )$ ，where the unit of length is the metre． The rope has length $\pi$ metres and you may ignore the size of the goat．
The curve $C$ shown in Figure 4 represents the edge of the region that the goat can reach．
（a）Write down the equation of $C$ for $y < 0$ When the goat is at the point $G ( x , y )$ ，with $x > 0$ and $y > 0$ ，as shown in Figure 4 ，the rope lies along $O A G$ where $O A$ is an arc of the circle with angle $O T A = \theta$ radians and $A G$ is a tangent to the circle at $A$ ．
（b）With the aid of a suitable diagram show that $$\begin{aligned} & x = \sin \theta + ( \pi - \theta ) \cos \theta
& y = 1 - \cos \theta + ( \pi - \theta ) \sin \theta \end{aligned}$$ (c) By considering $\int y \frac { \mathrm {~d} x } { \mathrm {~d} \theta } \mathrm {~d} \theta$, show that the area, in the first quadrant, between $C$, the positive $x$-axis and the positive $y$-axis can be expressed in the form $$\int _ { 0 } ^ { \pi } u \sin u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u$$ (d) Show that $\int _ { 0 } ^ { \pi } u ^ { 2 } \sin ^ { 2 } u \mathrm {~d} u = \frac { \pi ^ { 3 } } { 6 } + \int _ { 0 } ^ { \pi } u \sin u \cos u \mathrm {~d} u$
(e) Hence find the area of grass that can be reached by the goat.

Edexcel AEA 2012 June Q4

4. $$\mathbf { a } = \left( \begin{array} { r } - 3
1
4 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { r } 5
- 2
9 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { r } 8
- 4
3 \end{array} \right)$$ The points $A , B$ and $C$ with position vectors $\mathbf { a } , \mathbf { b }$ and $\mathbf { c }$ ，respectively，are 3 vertices of a cube．
（a）Find the volume of the cube． The points $P , Q$ and $R$ are vertices of a second cube with $\overrightarrow { P Q } = \left( \begin{array} { l } 3
4
\alpha \end{array} \right) , \overrightarrow { P R } = \left( \begin{array} { l } 7
1
0 \end{array} \right)$ and $\alpha$ a positive constant．
（b）Given that angle $Q P R = 60 ^ { \circ }$ ，find the value of $\alpha$ ．
（c）Find the length of a diagonal of the second cube．

Edexcel AEA 2012 June Q5

5．［In this question the values of $a , x$ ，and $n$ are such that $a$ and $x$ are positive real numbers，with $a > 1 , x \neq a , x \neq 1$ and $n$ is an integer with $n > 1$ ］ Sam was confused about the rules of logarithms and thought that $$\log _ { a } x ^ { n } = \left( \log _ { a } x \right) ^ { n }$$ （a）Given that $x$ satisfies statement（1）find $x$ in terms of $a$ and $n$ ． Sam also thought that $$\log _ { a } x + \log _ { a } x ^ { 2 } + \ldots + \log _ { a } x ^ { n } = \log _ { a } x + \left( \log _ { a } x \right) ^ { 2 } + \ldots + \left( \log _ { a } x \right) ^ { n }$$ （b）For $n = 3 , x _ { 1 }$ and $x _ { 2 } \left( x _ { 1 } > x _ { 2 } \right)$ are the two values of $x$ that satisfy statement（2）．
（i）Find，in terms of $a$ ，an expression for $x _ { 1 }$ and an expression for $x _ { 2 }$ ．
（ii）Find the exact value of $\log _ { a } \left( \frac { x _ { 1 } } { x _ { 2 } } \right)$ ．
（c）Show that if $\log _ { a } x$ satisfies statement（2）then $$2 \left( \log _ { a } x \right) ^ { n } - n ( n + 1 ) \log _ { a } x + \left( n ^ { 2 } + n - 2 \right) = 0$$

Edexcel AEA 2012 June Q7

7． $\left[ \arccos x \right.$ and $\arctan x$ are alternative notation for $\cos ^ { - 1 } x$ and $\tan ^ { - 1 } x$ respectively $]$ \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fc5d0d07-b750-4646-bdcb-419a290200c9-5_387_935_322_566} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the curve $C _ { 1 }$ with equation $y = \cos ( \cos x ) , 0 \leqslant x < 2 \pi$ ．
The curve has turning points at $( 0 , \cos 1 ) , P , Q$ and $R$ as shown in Figure 2.
（a）Find the coordinates of the points $P , Q$ and $R$ ． The curve $C _ { 2 }$ has equation $y = \sin ( \cos x ) , 0 \leqslant x < 2 \pi$ ．The curves $C _ { 1 }$ and $C _ { 2 }$ intersect at the points $S$ and $T$ ．
（b）Copy Figure 2 and on this diagram sketch $C _ { 2 }$ stating the coordinates of the minimum point on $C _ { 2 }$ and the points where $C _ { 2 }$ meets or crosses the coordinate axes． The coordinates of $S$ are $( \alpha , d )$ where $0 < \alpha < \pi$ ．
（c）Show that $\alpha = \arccos \left( \frac { \pi } { 4 } \right)$ ．
（d）Find the value of $d$ in surd form and write down the coordinates of $T$ ． The tangent to $C _ { 1 }$ at the point $S$ has gradient $\tan \beta$ ．
（e）Show that $\beta = \arctan \sqrt { } \left( \frac { 16 - \pi ^ { 2 } } { 32 } \right)$ ．
（f）Find，in terms of $\beta$ ，the obtuse angle between the tangent to $C _ { 1 }$ at $S$ and the tangent to $C _ { 2 }$ at $S$ ．

Edexcel AEA 2002 June Q5

the possible values of $n _ { 1 }$ and $n _ { 2 }$,
the exact value of the smallest possible area between $C _ { 1 }$ and $C _ { 2 }$, simplifying your answer,
(8)
the largest value of $x$ for which the gradients of the two curves can be the same. Leave your answer in surd form.

Edexcel AEA 2005 June Q6

Find the coordinates of the points $P , Q$ and $R$.
Sketch, on separate diagrams, the graphs of
1. $y = \mathrm { f } ( 2 x )$,
2. $y = \mathrm { f } ( | x | + 1 )$,
  indicating on each sketch the coordinates of any maximum points and the intersections with the $x$-axis.
  (6) \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{f9d3e02c-cef2-435b-9cda-76c43fcac575-5_1015_1464_232_337}
  \end{figure} Figure 2 shows a sketch of part of the curve $C$, with equation $y = \mathrm { f } ( x - v ) + w$, where $v$ and $w$ are constants. The $x$-axis is a tangent to $C$ at the minimum point $T$, and $C$ intersects the $y$-axis at $S$. The line joining $S$ to the maximum point $U$ is parallel to the $x$-axis.
Find the value of $v$ and the value of $w$ and hence find the roots of the equation $$f ( x - v ) + w = 0$$

Edexcel AEA 2006 June Q6

（a）Show that the point $P ( 1,0 )$ lies on $C$ ．
（b）Find the coordinates of the point $Q$ ．
（c）Find the area of the shaded region between $C$ and the line $P Q$ ．

Edexcel AEA 2007 June Q6

Find an expression, in terms of $x$, for the area $A$ of $R$.
Show that $\frac { \mathrm { d } A } { \mathrm {~d} x } = \frac { 1 } { 4 } ( \pi - 2 x - 2 \sin x ) \sec ^ { 2 } \frac { x } { 2 }$.
Prove that the maximum value of $A$ occurs when $\frac { \pi } { 4 } < x < \frac { \pi } { 3 }$.
Prove that $\tan \frac { \pi } { 8 } = \sqrt { } 2 - 1$.
Show that the maximum value of $A > \frac { \pi } { 4 } ( \sqrt { } 2 - 1 )$.

Edexcel AEA 2014 June Q7

Find the value of $p$, the value of $m$ and the value of $n$.
Show that the equation of $C$ can be written in the form $y = r + \mathrm { f } ( x - h )$ and specify the function f and the constants $r$ and $h$. The region bounded by $C$, the $x$-axis and the lines $x = \frac { \pi } { 6 }$ and $x = \frac { \pi } { 3 }$ is rotated through $2 \pi$ radians about the $x$-axis.
Find the volume of the solid formed.