Questions — Edexcel (10514 questions)

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Edexcel AEA 2009 June Q7
18 marks Challenging +1.8
7.Relative to a fixed origin \(O\) the points \(A , B\) and \(C\) have position vectors $$\mathbf { a } = - \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 7 \mathbf { k } , \quad \mathbf { b } = 4 \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { and } \mathbf { c } = 6 \mathbf { i } + \frac { 16 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { respectively. }$$
  1. Find the cosine of angle \(A B C\) . The quadrilateral \(A B C D\) is a kite \(K\) .
  2. Find the area of \(K\) . A circle is drawn inside \(K\) so that it touches each of the 4 sides of \(K\) .
  3. Find the radius of the circle,giving your answer in the form \(p \sqrt { } ( q ) - q \sqrt { } ( p )\) ,where \(p\) and \(q\) are positive integers.
  4. Find the position vector of the point \(D\) .
    (Total 18 marks)
Edexcel AEA 2010 June Q1
12 marks Standard +0.8
1.(a)Solve the equation $$\sqrt { } ( 3 x + 16 ) = 3 + \sqrt { } ( x + 1 )$$ (b)Solve the equation $$\log _ { 3 } ( x - 7 ) - \frac { 1 } { 2 } \log _ { 3 } x = 1 - \log _ { 3 } 2$$
Edexcel AEA 2010 June Q2
11 marks Challenging +1.2
2.The sum of the first \(p\) terms of an arithmetic series is \(q\) and the sum of the first \(q\) terms of the same arithmetic series is \(p\) ,where \(p\) and \(q\) are positive integers and \(p \neq q\) . Giving simplified answers in terms of \(p\) and \(q\) ,find
  1. the common difference of the terms in this series,
  2. the first term of the series,
  3. the sum of the first \(( p + q )\) terms of the series.
Edexcel AEA 2010 June Q3
11 marks Challenging +1.2
3.The curve \(C\) has equation $$x ^ { 2 } + y ^ { 2 } + f x y = g ^ { 2 }$$ where \(f\) and \(g\) are constants and \(g \neq 0\) .
  1. Find an expression in terms of \(\alpha , \beta\) and \(f\) for the gradient of \(C\) at the point \(( \alpha , \beta )\) . Given that \(f < 2\) and \(f \neq - 2\) and that the gradient of \(C\) at the point \(( \alpha , \beta )\) is 1 ,
  2. show that \(\alpha = - \beta = \frac { \pm g } { \sqrt { } ( 2 - f ) }\) . Given that \(f = - 2\) ,
  3. sketch \(C\) .
Edexcel AEA 2010 June Q4
16 marks Challenging +1.2
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-3_643_332_246_870} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a cuboid \(O A B C D E F G\), where \(O\) is the origin, \(A\) has position vector \(5 \mathbf { i } , C\) has position vector \(10 \mathbf { j }\) and \(D\) has position vector \(20 \mathbf { k }\).
  1. Find the cosine of angle \(C A F\). Given that the point \(X\) lies on \(A C\) and that \(F X\) is perpendicular to \(A C\),
  2. find the position vector of point \(X\) and the distance \(F X\). The line \(l _ { 1 }\) passes through \(O\) and through the midpoint of the face \(A B F E\). The line \(l _ { 2 }\) passes through \(A\) and through the midpoint of the edge \(F G\).
  3. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of the point of intersection.
Edexcel AEA 2010 June Q5
12 marks Challenging +1.8
5. $$I = \int \frac { 1 } { ( x - 1 ) \sqrt { } \left( x ^ { 2 } - 1 \right) } \mathrm { d } x , \quad x > 1$$
  1. Use the substitution \(x = 1 + u ^ { - 1 }\) to show that $$I = - \left( \frac { x + 1 } { x - 1 } \right) ^ { \frac { 1 } { 2 } } + c$$
  2. Hence show that $$\int _ { \sec \alpha } ^ { \sec \beta } \frac { 1 } { ( x - 1 ) \sqrt { } \left( x ^ { 2 } - 1 \right) } \mathrm { d } x = \cot \left( \frac { \alpha } { 2 } \right) - \cot \left( \frac { \beta } { 2 } \right) , \quad 0 < \alpha < \beta < \frac { \pi } { 2 }$$
Edexcel AEA 2010 June Q6
10 marks Challenging +1.8
6.(a)Given that \(x ^ { 4 } + y ^ { 4 } = 1\) ,prove that \(x ^ { 2 } + y ^ { 2 }\) is a maximum when \(x = \pm y\) ,and find the maximum and minimum values of \(x ^ { 2 } + y ^ { 2 }\) .
(b)On the same diagram,sketch the curves \(C _ { 1 }\) and \(C _ { 2 }\) with equations \(x ^ { 4 } + y ^ { 4 } = 1\) and \(x ^ { 2 } + y ^ { 2 } = 1\) respectively.
(c)Write down the equation of the circle \(C _ { 3 }\) ,centre the origin,which touches the curve \(C _ { 1 }\) at the points where \(x = \pm y\) .
Edexcel AEA 2010 June Q7
21 marks Challenging +1.2
7. $$\mathrm { f } ( x ) = \left[ 1 + \cos \left( x + \frac { \pi } { 4 } \right) \right] \left[ 1 + \sin \left( x + \frac { \pi } { 4 } \right) \right] , \quad 0 \leqslant x \leqslant 2 \pi$$
  1. Show that \(\mathrm { f } ( x )\) may be written in the form $$f ( x ) = \left( \frac { 1 } { \sqrt { 2 } } + \cos x \right) ^ { 2 } , \quad 0 \leqslant x \leqslant 2 \pi$$
  2. Find the range of the function \(\mathrm { f } ( x )\). The graph of \(y = \mathrm { f } ( x )\) is shown in Figure 2. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_426_938_849_591} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure}
  3. Find the coordinates of all the maximum and minimum points on this curve. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{0396f61a-b844-40ed-98d1-82ee2d8a6807-5_432_942_1535_589} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} The region \(R\), bounded by \(y = 2\) and \(y = \mathrm { f } ( x )\), is shown shaded in Figure 3.
  4. Find the area of \(R\).
Edexcel AEA 2012 June Q1
8 marks Challenging +1.2
1.The function f is given by $$\mathrm { f } ( x ) = x ^ { 2 } - 2 x + 6 , \quad x \geqslant 0$$
  1. Find the range of \(f\) . The function \(g\) is given by $$\mathrm { g } ( x ) = 3 + \sqrt { } ( x + 4 ) , \quad x \geqslant 2$$
  2. Find \(\operatorname { gf } ( x )\) .
  3. Find the domain and range of gf.
Edexcel AEA 2012 June Q2
10 marks Challenging +1.8
2.(a)Show that $$\sin 3 x = 3 \sin x - 4 \sin ^ { 3 } x$$ Hence find
(b) \(\int \cos x ( 6 \sin x - 2 \sin 3 x ) ^ { \frac { 2 } { 3 } } \mathrm {~d} x\) (c) \(\int ( 3 \sin 2 x - 2 \sin 3 x \cos x ) ^ { \frac { 1 } { 3 } } \mathrm {~d} x\)
Edexcel AEA 2012 June Q3
10 marks Hard +2.3
3.The angle \(\theta , 0 < \theta < \frac { \pi } { 2 }\) ,satisfies $$\tan \theta \tan 2 \theta = \sum _ { r = 0 } ^ { \infty } 2 \cos ^ { r } 2 \theta$$
  1. Show that \(\tan \theta = 3 ^ { p }\) ,where \(p\) is a rational number to be found.
  2. Hence show that \(\frac { \pi } { 6 } < \theta < \frac { \pi } { 4 }\)
Edexcel AEA 2012 June Q6
16 marks Challenging +1.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fc5d0d07-b750-4646-bdcb-419a290200c9-4_433_1011_221_529} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve with equation \(y = ( x + a ) ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are positive constants. The curve cuts the \(x\)-axis at \(P\) and has a maximum point at \(S\) and a minimum point at \(Q\).
  1. Write down the coordinates of \(P\) and \(Q\) in terms of \(a\) and \(b\).
  2. Show that \(G\), the area of the shaded region between the curve \(P S Q\) and the \(x\)-axis, is given by \(G = \frac { ( a + b ) ^ { 4 } } { 12 }\). The rectangle \(P Q R S T\) has \(R S T\) parallel to \(Q P\) and both \(P T\) and \(Q R\) are parallel to the \(y\)-axis.
  3. Show that \(\frac { G } { \text { Area of } P Q R S T } = k\), where \(k\) is a constant independent of \(a\) and \(b\) and find the value of \(k\).
Edexcel AEA 2013 June Q1
6 marks Challenging +1.2
1.In the binomial expansion of $$\left( 1 + \frac { 12 n } { 5 } x \right) ^ { n }$$ the coefficients of \(x ^ { 2 }\) and \(x ^ { 3 }\) are equal and non-zero.
  1. Find the possible values of \(n\) .
    (4)
  2. State,giving a reason,which value of \(n\) gives a valid expansion when \(x = \frac { 1 } { 2 }\) (2)
Edexcel AEA 2013 June Q2
8 marks Standard +0.8
2.(a)Use the formula for \(\sin ( A - B )\) to show that \(\sin \left( 90 ^ { \circ } - x \right) = \cos x\) (b)Solve for \(0 < \theta < 360 ^ { \circ }\) $$2 \sin \left( \theta + 17 ^ { \circ } \right) = \frac { \cos \left( \theta + 8 ^ { \circ } \right) } { \cos \left( \theta + 17 ^ { \circ } \right) }$$
Edexcel AEA 2013 June Q3
13 marks Challenging +1.2
3.The lines \(L _ { 1 }\) and \(L _ { 2 }\) have equations given by \(L _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { c } - 7 \\ 7 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { c } 2 \\ 0 \\ - 3 \end{array} \right)\) and \(L _ { 2 } : \quad \mathbf { r } = \left( \begin{array} { c } 7 \\ p \\ - 6 \end{array} \right) + \mu \left( \begin{array} { c } 10 \\ - 4 \\ - 1 \end{array} \right)\) where \(\lambda\) and \(\mu\) are parameters and \(p\) is a constant.
The two lines intersect at the point \(C\) .
  1. Find
    1. the value of \(p\) ,
    2. the position vector of \(C\) .
  2. Show that the point \(B\) with position vector \(\left( \begin{array} { c } - 13 \\ 11 \\ - 4 \end{array} \right)\) lies on \(L _ { 2 }\) . The point \(A\) with position vector \(\left( \begin{array} { c } - 7 \\ 7 \\ 1 \end{array} \right)\) lies on \(L _ { 1 }\) .
  3. Find \(\cos ( \angle A C B )\) ,giving your answer as an exact fraction. The line \(L _ { 3 }\) bisects the angle \(A C B\) .
  4. Find a vector equation of \(L _ { 3 }\) .
Edexcel AEA 2013 June Q4
13 marks Challenging +1.2
4.A sequence of positive integers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) has \(r\) th term given by $$a _ { r } = 2 ^ { r } - 1$$
  1. Write down the first 6 terms of this sequence.
  2. Verify that \(a _ { r + 1 } = 2 a _ { r } + 1\)
  3. Find \(\sum _ { r = 1 } ^ { n } a _ { r }\)
  4. Show that \(\frac { 1 } { a _ { r + 1 } } < \frac { 1 } { 2 } \times \frac { 1 } { a _ { r } }\)
  5. Hence show that \(1 + \frac { 1 } { 3 } + \frac { 1 } { 7 } + \frac { 1 } { 15 } + \frac { 1 } { 31 } + \ldots < 1 + \frac { 1 } { 3 } + \left( \frac { 1 } { 7 } + \frac { \frac { 1 } { 2 } } { 7 } + \frac { \frac { 1 } { 4 } } { 7 } + \ldots \right)\)
  6. Show that \(\frac { 31 } { 21 } < \sum _ { r = 1 } ^ { \infty } \frac { 1 } { a _ { r } } < \frac { 34 } { 21 }\)
Edexcel AEA 2013 June Q5
15 marks Hard +2.3
5.In this question u and v are functions of \(x\) .Given that \(\int \mathrm { u } \mathrm { d } x , \int \mathrm { v } \mathrm { d } x\) and \(\int \mathrm { uv } \mathrm { d } x\) satisfy $$\int \text { uv } \mathrm { d } x = \left( \int \mathrm { u } \mathrm {~d} x \right) \times \left( \int \mathrm { v } \mathrm {~d} x \right) \quad \text { uv } \neq 0$$
  1. show that \(1 = \frac { \int \mathrm { u } \mathrm { d } x } { \mathrm { u } } + \frac { \int \mathrm { v } \mathrm { d } x } { \mathrm { v } }\) Given also that \(\frac { \int \mathrm { u } \mathrm { d } x } { \mathrm { u } } = \mathrm { sin } ^ { 2 } x\),
  2. use part(a)to write down an expression,in terms of \(x\) ,for \(\frac { \int \mathrm { v } \mathrm { d } x } { \mathrm { v } }\) ,
  3. show that $$\frac { 1 } { \mathrm { u } } \frac { \mathrm { du } } { \mathrm {~d} x } = \frac { 1 - 2 \sin x \cos x } { \sin ^ { 2 } x }$$
  4. hence use integration to show that \(\mathrm { u } = A \mathrm { e } ^ { - \cot x } \operatorname { cosec } ^ { 2 } x\) ,where \(A\) is an arbitrary constant.
  5. By differentiating \(\mathrm { e } ^ { \tan x }\) find a similar expression for v .
Edexcel AEA 2013 June Q6
16 marks Hard +2.3
6.(a)Starting from \([ \mathrm { f } ( x ) - \lambda \mathrm { g } ( x ) ] ^ { 2 } \geqslant 0\) show that \(\lambda\) satisfies the quadratic inequality $$\left( \int _ { a } ^ { b } [ \operatorname { g } ( x ) ] ^ { 2 } \mathrm {~d} x \right) \lambda ^ { 2 } - 2 \left( \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { g } ( x ) \mathrm { d } x \right) \lambda + \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x \geqslant 0$$ where \(a\) and \(b\) are constants and \(\lambda\) can take any real value.
(2)
(b)Hence prove that $$\left[ \int _ { a } ^ { b } \mathrm { f } ( x ) \mathrm { g } ( x ) \mathrm { d } x \right] ^ { 2 } \leqslant \left[ \int _ { a } ^ { b } [ \mathrm { f } ( x ) ] ^ { 2 } \mathrm {~d} x \right] \times \left[ \int _ { a } ^ { b } [ \mathrm {~g} ( x ) ] ^ { 2 } \mathrm {~d} x \right]$$ (c)By letting \(\mathrm { f } ( x ) = 1\) and \(\mathrm { g } ( x ) = \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 2 } }\) show that $$\int _ { - 1 } ^ { 2 } \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x \leqslant \frac { 9 } { 2 }$$ (d)Show that \(\int _ { - 1 } ^ { 2 } x ^ { 2 } \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 4 } } \mathrm {~d} x = \frac { 12 \sqrt { } 3 } { 5 }\) (e)Hence show that $$\frac { 144 } { 55 } \leqslant \int _ { - 1 } ^ { 2 } \left( 1 + x ^ { 3 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x$$
Edexcel AEA 2013 June Q7
22 marks Challenging +1.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8bd0bc33-e69e-4e51-aae7-288810c5db07-6_643_1374_173_351} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { x } { 3 } + \frac { 12 } { x } \quad x \neq 0$$ The lines \(x = 0\) and \(y = \frac { x } { 3 }\) are asymptotes to \(C _ { 1 }\). The point \(A\) on \(C _ { 1 }\) is a minimum and the point \(B\) on \(C _ { 1 }\) is a maximum.
  1. Find the coordinates of \(A\) and \(B\). There is a normal to \(C _ { 1 }\), which does not intersect \(C _ { 1 }\) a second time, that has equation \(x = k\), where \(k > 0\).
  2. Write down the value of \(k\). The point \(P ( \alpha , \beta ) , \alpha > 0\) and \(\alpha \neq k\), lies on \(C _ { 1 }\). The normal to \(C _ { 1 }\) at \(P\) does not intersect \(C _ { 1 }\) a second time.
  3. Find the value of \(\alpha\), leaving your answer in simplified surd form.
  4. Find the equation of this normal. The curve \(C _ { 2 }\) has equation \(y = | \mathrm { f } ( x ) |\)
  5. Sketch \(C _ { 2 }\) stating the coordinates of any turning points and the equations of any asymptotes. The line with equation \(y = m x + 1\) does not touch or intersect \(C _ { 2 }\).
  6. Find the set of possible values for \(m\).
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 \(\mathbf{d}\) respectively,where $$\mathbf { a } = \left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { c } 0 \\ - 3 \\ - 8 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { c } 4 \\ - 1 \\ - 10 \end{array} \right)$$
Edexcel AEA 2016 June Q4
11 marks Challenging +1.8
\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 $$\log _ { x } y = \log _ { x ^ { n } } y ^ { n }$$
(b)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 \(C _ { 2 }\)