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Edexcel AEA 2015 June Q10
Challenging +1.8
10
- 3 \end{array} \right) + \lambda \left( \begin{array} { r } 2
- 5
4 \end{array} \right)
& L _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 1
2
3 \end{array} \right) + \mu \left( \begin{array} { l } 1
2
2 \end{array} \right) \end{aligned}$$
  1. Show that \(L _ { 1 }\) and \(L _ { 2 }\) are perpendicular.
  2. Show that \(L _ { 1 }\) and \(L _ { 2 }\) are skew lines. The point \(A\) with position vector \(- \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) lies on \(L _ { 2 }\) and the point \(X\) lies on \(L _ { 1 }\) such that \(\overrightarrow { A X }\) is perpendicular to \(L _ { 1 }\)
  3. Find the position vector of \(X\) .
  4. Find \(| \overrightarrow { A X } |\) The point \(B\)(distinct from \(A\) )also lies on \(L _ { 2 }\) and \(| \overrightarrow { B X } | = | \overrightarrow { A X } |\)
  5. Find the position vector of \(B\) .
  6. Find the cosine of angle \(A X B\) . 7.(a)Use the substitution \(x = \sec \theta\) to show that $$\int _ { \sqrt { 2 } } ^ { 2 } \frac { 1 } { \left( x ^ { 2 } - 1 \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { \sqrt { 6 } - 2 } { \sqrt { 3 } }$$
  7. Use integration by parts to show that $$\int \operatorname { cosec } \theta \cot ^ { 2 } \theta \mathrm {~d} \theta = \frac { 1 } { 2 } [ \ln | \operatorname { cosec } \theta + \cot \theta | - \operatorname { cosec } \theta \cot \theta ] + c$$ (6) \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{3e18cb7c-1a67-4152-8628-76847e368882-6_592_1196_772_420} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \frac { 1 } { \left( x ^ { 2 } - 1 \right) ^ { \frac { 3 } { 2 } } }\) for \(x > 1\)\\ The region \(R\) ,shown shaded in Figure 2,is bounded by the curve,the \(x\)-axis and the lines \(x = \sqrt { 2 }\) and \(x = 2\)\\ The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  8. Show that the volume of the solid formed is $$\pi \left[ \frac { 3 } { 8 } \ln \left( \frac { 1 + \sqrt { 2 } } { \sqrt { 3 } } \right) + \frac { 7 } { 36 } - \frac { \sqrt { 2 } } { 8 } \right]$$
Edexcel AEA 2016 June Q1
7 marks Standard +0.8
1.The function f is given by $$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 9 \quad x \in \mathbb { R } , x \geqslant 3$$
  1. Find the range of f . The function g is given by $$\operatorname { g } ( x ) = \frac { 10 } { x + 1 } \quad x \in \mathbb { R } , x \geqslant 4$$
  2. Find an expression for \(\operatorname { gf } ( x )\) .
  3. Find the domain and range of gf.
Edexcel AEA 2016 June Q2
7 marks Challenging +1.8
2.Find the value of $$\arccos \left( \frac { 1 } { \sqrt { 2 } } \right) + \arcsin \left( \frac { 1 } { 3 } \right) + 2 \arctan \left( \frac { 1 } { \sqrt { 2 } } \right)$$ Give your answer as a multiple of \(\pi\) . $$\text { (arccos } x \text { is an alternative notion for } \cos ^ { - 1 } x \text { etc.) }$$
Edexcel AEA 2016 June Q3
9 marks Challenging +1.2
3.The points \(A , B , C , D\) and \(E\) are five of the vertices of a rectangular cuboid and \(A E\) is a diagonal of the cuboid.With respect to a fixed origin \(O\) ,the position vectors of \(A , B , C\) and \(D\) are \(\mathbf { a , b , c }\) and d respectively,where $$\mathbf { a } = \left( \begin{array} { c } 1
2
- 1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { c } 0
- 3
- 8 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { c }
Edexcel AEA 2016 June Q4
11 marks Challenging +1.8
4
- 1
- 10 \end{array} \right) \text { and } \mathbf { d } = \left( \begin{array} { c } - 4
2
- 11 \end{array} \right)$$
  1. Find the position vector of \(E\) . The volume of a tetrahedron is given by the formula $$\text { volume } = \frac { 1 } { 3 } ( \text { area of base } ) \times ( \text { height } )$$
  2. Find the volume of the tetrahedron \(A B C D\) . 4.(a)Given that \(x > 0 , y > 0 , x \neq 1\) and \(n > 0\) ,show that $$\log _ { x } y = \log _ { x ^ { n } } y ^ { n }$$
  3. Solve the following,leaving your answers in the form \(2 ^ { p }\) ,where \(p\) is a rational number.
    1. \(\log _ { 2 } u + \log _ { 4 } u ^ { 2 } + \log _ { 8 } u ^ { 3 } + \log _ { 16 } u ^ { 4 } = 5\)
    2. \(\log _ { 2 } v + \log _ { 4 } v + \log _ { 8 } v + \log _ { 16 } v = 5\)
    3. \(\log _ { 4 } w ^ { 2 } + \frac { 3 \log _ { 8 } 64 } { \log _ { 2 } w } = 5\)
Edexcel AEA 2016 June Q5
13 marks Challenging +1.8
5.(a)Show that $$\sum _ { r = 0 } ^ { n } x ^ { - r } = \frac { x } { x - 1 } - \frac { x ^ { - n } } { x - 1 } \quad \text { where } x \neq 0 \text { and } x \neq 1$$ (b)Hence find an expression in terms of \(x\) and \(n\) for \(\sum _ { r = 0 } ^ { n } r x ^ { - ( r + 1 ) }\) for \(x \neq 0\) and \(x \neq 1\) Simplify your answer.
(c)Find \(\sum _ { r = 0 } ^ { n } \left( \frac { 3 + 5 r } { 2 ^ { r } } \right)\) Give your answer in the form \(a - \frac { b + c n } { 2 ^ { n } }\) ,where \(a , b\) and \(c\) are integers.
Edexcel AEA 2016 June Q6
22 marks Challenging +1.2
6. \includegraphics[max width=\textwidth, alt={}, center]{0214eebf-93f2-4338-9222-443000115225-4_346_1040_303_548} \section*{Figure 1} Figure 1 shows a sketch of the curve \(C _ { 1 }\) with equation $$y = \cos ( \cos x ) \sin x \quad \text { for } \quad 0 \leqslant x \leqslant \pi$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Hence verify that the turning point is at \(x = \frac { \pi } { 2 }\) and find the \(y\) coordinate of this point.
  3. Find the area of the region bounded by \(C _ { 1 }\) and the positive \(x\)-axis between \(x = 0\) and \(x = \pi\) Figure 2 shows a sketch of the curve \(C _ { 1 }\) and the curve \(C _ { 2 }\) with equation $$y = \sin ( \cos x ) \sin x \quad \text { for } \quad 0 \leqslant x \leqslant \pi$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0214eebf-93f2-4338-9222-443000115225-4_519_1065_1631_484} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the origin and the point \(A ( a , b )\) ,where \(a < \pi\)
  4. Find \(a\) and \(b\) ,giving \(b\) in a form not involving trigonometric functions.
  5. Find the area of the shaded region between \(C _ { 1 }\) and \(C _ { 2 }\)
Edexcel AEA 2016 June Q7
24 marks Challenging +1.8
7. (a) Find the set of values of \(k\) for which the equation $$\frac { x ^ { 2 } + 3 x + 8 } { x ^ { 2 } + x - 2 } = k$$ has no real roots. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0214eebf-93f2-4338-9222-443000115225-5_718_869_511_603} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 x + 8 } { x ^ { 2 } + x - 2 }\) The curve has asymptotes \(x = a , x = b\) and \(y = c\), where \(a , b\) and \(c\) are integers.
(b) Find the value of \(a\), the value of \(b\) and the value of \(c\).
(c) Find the coordinates of the points of intersection of \(C _ { 1 }\) with the line \(y = 2\) (d) Find all the integer pairs \(( r , s )\) that satisfy \(s = \frac { r ^ { 2 } + 3 r + 8 } { r ^ { 2 } + r - 2 }\) The curve \(C _ { 2 }\) has equation \(y = \mathrm { g } ( x )\) where \(\mathrm { g } ( x ) = \frac { 2 x ^ { 2 } - 4 x + 6 } { x ^ { 2 } - 3 x }\) (e) Show that, for suitable integers \(m\) and \(n , \mathrm {~g} ( x )\) can be written in the form \(\mathrm { f } ( x + m ) + n\).
(f) Sketch \(C _ { 2 }\) showing any asymptotes and stating their equations.
Edexcel AEA 2017 June Q1
7 marks Standard +0.8
1.The function f is given by $$\mathrm { f } ( x ) = \sqrt { x + 2 } \quad \text { for } \quad x \in \mathbb { R } , x \geqslant 0$$
  1. Find \(\mathrm { f } ^ { - 1 } ( x )\) and state the domain of \(\mathrm { f } ^ { - 1 }\) The function g is given by $$\mathrm { g } ( x ) = x ^ { 2 } - 4 x + 5 \text { for } x \in \mathbb { R } , x \geqslant 0$$
  2. Find the range of g .
  3. Solve the equation \(\operatorname { fg } ( x ) = x\) .
Edexcel AEA 2017 June Q2
9 marks Challenging +1.8
2.(a)Show that the equation $$\tan x = \frac { \sqrt { 3 } } { 1 + 4 \cos x }$$ can be written in the form $$\sin 2 x = \sin \left( 60 ^ { \circ } - x \right)$$ (b)Solve,for \(0 < x < 180 ^ { \circ }\) $$\tan x = \frac { \sqrt { 3 } } { 1 + 4 \cos x }$$
Edexcel AEA 2017 June Q3
13 marks Challenging +1.3
  1. 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 \(90 ^ { \circ }\)
  2. Find the position vector of \(C\). The point \(D\) also lies on \(L _ { 1 }\) and triangle \(A B D\) is isosceles with \(A B = A D\).
  3. Find the area of triangle \(A B D\).
Edexcel AEA 2017 June Q4
13 marks Challenging +1.8
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 \(L M N\) .
  1. Show that \(x y = 2\)
  2. 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,
  3. 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
14 marks Challenging +1.2
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.
  1. Write down the value of \(a\) and the value of \(b\).
    (2)
  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. \(y = \mathrm { f } ( | x | ) - 3\)
Edexcel AEA 2017 June Q6
16 marks Challenging +1.8
6.(a)Show that $$\frac { \mathrm { d } } { \mathrm {~d} u } \ln \left( u + \sqrt { u ^ { 2 } - 1 } \right) = \frac { 1 } { \sqrt { u ^ { 2 } - 1 } }$$ (b)Use the result from part(a)and the substitution \(x + 3 = \frac { 1 } { t }\) to find $$\int \frac { 1 } { ( x + 3 ) \sqrt { 2 x + 7 } } \mathrm {~d} x$$ (6)
(c)Express \(\frac { 1 } { 2 x ^ { 2 } + 13 x + 21 }\) in partial fractions.
(d)Find $$\int _ { 1 } ^ { 9 } \frac { 1 } { \left( 2 x ^ { 2 } + 13 x + 21 \right) \sqrt { 2 x + 7 } } \mathrm {~d} x$$ giving your answer in the form \(\ln r - s\) where \(r\) and \(s\) are rational numbers.
Edexcel AEA 2017 June Q7
21 marks Challenging +1.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15e3f7f2-a77c-4ee4-8f0a-ac739e9fede5-7_583_1198_217_440} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows part of the curve \(C\) with equation \(y = x ^ { 4 } - 10 x ^ { 3 } + 33 x ^ { 2 } - 34 x\) and the line \(L\) with equation \(y = m x + c\) . The line \(L\) touches \(C\) at the points \(P\) and \(Q\) with \(x\) coordinates \(p\) and \(q\) respectively.
  1. Explain why $$x ^ { 4 } - 10 x ^ { 3 } + 33 x ^ { 2 } - ( 34 + m ) x - c = ( x - p ) ^ { 2 } ( x - q ) ^ { 2 }$$ The finite region \(R\) ,shown shaded in Figure 3,is bounded by \(C\) and \(L\) .
  2. Use integration by parts to show that the area of \(R\) is \(\frac { ( q - p ) ^ { 5 } } { 30 }\)
  3. Show that $$( x - p ) ^ { 2 } ( x - q ) ^ { 2 } = x ^ { 4 } - 2 ( p + q ) x ^ { 3 } + S x ^ { 2 } - T x + U$$ where \(S , T\) and \(U\) are expressions to be found in terms of \(p\) and \(q\) .
  4. Using part(a)and part(c)find the value of \(p\) ,the value of \(q\) and the equation of \(L\) .
Edexcel C1 2014 June Q1
3 marks Easy -1.8
  1. Find
$$\int \left( 8 x ^ { 3 } + 4 \right) d x$$ giving each term in its simplest form.
Edexcel C1 2014 June Q2
4 marks Easy -1.3
2.(a)Write down the value of \(32 ^ { \frac { 1 } { 5 } }\) (b)Simplify fully \(\left( 32 x ^ { 5 } \right) ^ { - \frac { 2 } { 5 } }\) I敖
䋤
Edexcel C1 2014 June Q3
7 marks Moderate -0.8
3. Find the set of values of \(x\) for which
  1. \(3 x - 7 > 3 - x\)
  2. \(x ^ { 2 } - 9 x \leqslant 36\)
  3. both \(3 x - 7 > 3 - x\) and \(x ^ { 2 } - 9 x \leqslant 36\)
Edexcel C1 2014 June Q4
5 marks Moderate -0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-05_945_1026_269_466} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \frac { 1 } { x } + 1 , \quad x \neq 0$$ The curve \(C\) crosses the \(x\)-axis at the point \(A\).
  1. State the \(x\) coordinate of the point \(A\). The curve \(D\) has equation \(y = x ^ { 2 } ( x - 2 )\), for all real values of \(x\).
  2. A copy of Figure 1 is shown on page 7. On this copy, sketch a graph of curve \(D\).
    Show on the sketch the coordinates of each point where the curve \(D\) crosses the coordinate axes.
  3. Using your sketch, state, giving a reason, the number of real solutions to the equation $$x ^ { 2 } ( x - 2 ) = \frac { 1 } { x } + 1$$ \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-06_942_1026_516_466} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}
Edexcel C1 2014 June Q5
5 marks Moderate -0.8
5. A sequence of numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } \ldots\) is defined by $$a _ { n + 1 } = 5 a _ { n } - 3 , \quad n \geqslant 1$$ Given that \(a _ { 2 } = 7\),
  1. find the value of \(a _ { 1 }\)
  2. Find the value of \(\sum _ { r = 1 } ^ { 4 } a _ { r }\)
Edexcel C1 2014 June Q6
5 marks Easy -1.2
6
  1. Write \(\sqrt { } 80\) in the form \(c \sqrt { } 5\), where \(c\) is a positive constant. A rectangle \(R\) has a length of ( \(1 + \sqrt { } 5\) ) cm and an area of \(\sqrt { 80 } \mathrm {~cm} ^ { 2 }\).
  2. Calculate the width of \(R\) in cm . Express your answer in the form \(p + q \sqrt { 5 }\), where \(p\) and \(q\) are integers to be found.
Edexcel C1 2014 June Q7
7 marks Easy -1.2
7. Differentiate with respect to \(x\), giving each answer in its simplest form.
  1. \(( 1 - 2 x ) ^ { 2 }\)
  2. \(\frac { x ^ { 5 } + 6 \sqrt { } x } { 2 x ^ { 2 } }\)
Edexcel C1 2014 June Q8
9 marks Moderate -0.3
8. In the year 2000 a shop sold 150 computers. Each year the shop sold 10 more computers than the year before, so that the shop sold 160 computers in 2001, 170 computers in 2002, and so on forming an arithmetic sequence.
  1. Show that the shop sold 220 computers in 2007.
  2. Calculate the total number of computers the shop sold from 2000 to 2013 inclusive. In the year 2000, the selling price of each computer was \(\pounds 900\). The selling price fell by \(\pounds 20\) each year, so that in 2001 the selling price was \(\pounds 880\), in 2002 the selling price was \(\pounds 860\), and so on forming an arithmetic sequence.
  3. In a particular year, the selling price of each computer in \(\pounds s\) was equal to three times the number of computers the shop sold in that year. By forming and solving an equation, find the year in which this occurred.
Edexcel C1 2014 June Q9
10 marks Moderate -0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{64f015bf-29fb-4374-af34-3745ea49aced-12_675_863_267_552} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The line \(l _ { 1 }\), shown in Figure 2 has equation \(2 x + 3 y = 26\) The line \(l _ { 2 }\) passes through the origin \(O\) and is perpendicular to \(l _ { 1 }\)
  1. Find an equation for the line \(l _ { 2 }\) The line \(l _ { 2 }\) intersects the line \(l _ { 1 }\) at the point \(C\).
    Line \(l _ { 1 }\) crosses the \(y\)-axis at the point \(B\) as shown in Figure 2.
  2. Find the area of triangle \(O B C\). Give your answer in the form \(\frac { a } { b }\), where \(a\) and \(b\) are integers to be determined.
Edexcel C1 2014 June Q10
10 marks Moderate -0.8
10. A curve with equation \(y = \mathrm { f } ( x )\) passes through the point (4,25). Given that $$f ^ { \prime } ( x ) = \frac { 3 } { 8 } x ^ { 2 } - 10 x ^ { - \frac { 1 } { 2 } } + 1 , \quad x > 0$$
  1. find \(\mathrm { f } ( x )\), simplifying each term.
  2. Find an equation of the normal to the curve at the point ( 4,25 ). Give your answer in the form \(a x + b y + c = 0\), where \(a\), \(b\) and \(c\) are integers to be found.