Questions — Edexcel (10514 questions)

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Edexcel C4 2018 June Q1
8 marks Standard +0.3
  1. Find the binomial series expansion of $$\sqrt { 4 - 9 x } , | x | < \frac { 4 } { 9 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) Give each coefficient in its simplest form.
  2. Use the expansion from part (a), with a suitable value of \(x\), to find an approximate value for \(\sqrt { 310 }\) Show all your working and give your answer to 3 decimal places.
Edexcel C4 2018 June Q2
10 marks Standard +0.3
  1. The curve \(C\) has equation
$$x ^ { 2 } + x y + y ^ { 2 } - 4 x - 5 y + 1 = 0$$
  1. Use implicit differentiation to find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
  2. Find the \(x\) coordinates of the two points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) Give exact answers in their simplest form.
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C4 2018 June Q3
14 marks Standard +0.3
3. (i) Given that $$\frac { 13 - 4 x } { ( 2 x + 1 ) ^ { 2 } ( x + 3 ) } \equiv \frac { A } { ( 2 x + 1 ) } + \frac { B } { ( 2 x + 1 ) ^ { 2 } } + \frac { C } { ( x + 3 ) }$$
  1. find the values of the constants \(A , B\) and \(C\).
  2. Hence find $$\int \frac { 13 - 4 x } { ( 2 x + 1 ) ^ { 2 } ( x + 3 ) } \mathrm { d } x , \quad x > - \frac { 1 } { 2 }$$ (ii) Find $$\int \left( \mathrm { e } ^ { x } + 1 \right) ^ { 3 } \mathrm {~d} x$$ (iii) Using the substitution \(u ^ { 3 } = x\), or otherwise, find $$\int \frac { 1 } { 4 x + 5 x ^ { \frac { 1 } { 3 } } } \mathrm {~d} x , \quad x > 0$$
Edexcel C4 2018 June Q4
6 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-12_978_1264_121_411} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A water container is made in the shape of a hollow inverted right circular cone with semi-vertical angle of \(30 ^ { \circ }\), as shown in Figure 1. The height of the container is 50 cm . When the depth of the water in the container is \(h \mathrm {~cm}\), the surface of the water has radius \(r \mathrm {~cm}\) and the volume of water is \(V \mathrm {~cm} ^ { 3 }\).
  1. Show that \(V = \frac { 1 } { 9 } \pi h ^ { 3 }\) [0pt] [You may assume the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.] Given that the volume of water in the container increases at a constant rate of \(200 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\),
  2. find the rate of change of the depth of the water, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), when \(h = 15\) Give your answer in its simplest form in terms of \(\pi\).
Edexcel C4 2018 June Q5
7 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-16_938_1257_125_486} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of the curve \(C\) with parametric equations $$x = 1 + t - 5 \sin t , \quad y = 2 - 4 \cos t , \quad - \pi \leqslant t \leqslant \pi$$ The point \(A\) lies on the curve \(C\). Given that the coordinates of \(A\) are ( \(k , 2\) ), where \(k > 0\)
  1. find the exact value of \(k\), giving your answer in a fully simplified form.
  2. Find the equation of the tangent to \(C\) at the point \(A\). Give your answer in the form \(y = p x + q\), where \(p\) and \(q\) are exact real values.
Edexcel C4 2018 June Q6
6 marks Standard +0.3
  1. Given that \(y = 2\) when \(x = - \frac { \pi } { 8 }\), solve the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } } { 3 \cos ^ { 2 } 2 x } \quad - \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
Edexcel C4 2018 June Q7
15 marks Standard +0.8
7. The point \(A\) with coordinates ( \(- 3,7,2\) ) lies on a line \(l _ { 1 }\) The point \(B\) also lies on the line \(l _ { 1 }\) Given that \(\quad \overrightarrow { A B } = \left( \begin{array} { r } 4 \\ - 6 \\ 2 \end{array} \right)\),
  1. find the coordinates of point \(B\). The point \(P\) has coordinates ( \(9,1,8\) )
  2. Find the cosine of the angle \(P A B\), giving your answer as a simplified surd.
  3. Find the exact area of triangle \(P A B\), giving your answer in its simplest form. The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
  4. Find a vector equation for the line \(l _ { 2 }\) The point \(Q\) lies on the line \(l _ { 2 }\) Given that the line segment \(A P\) is perpendicular to the line segment \(B Q\),
  5. find the coordinates of the point \(Q\).
Edexcel C4 2018 June Q8
9 marks Challenging +1.2
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-28_680_1266_118_482} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}
  1. Find \(\int x \cos 4 x d x\) Figure 3 shows part of the curve with equation \(y = \sqrt { x } \sin 2 x , \quad x \geqslant 0\) The finite region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \frac { \pi } { 4 }\) The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  2. Find the exact value of the volume of this solid of revolution, giving your answer in its simplest form.
    (Solutions based entirely on graphical or numerical methods are not acceptable.) \includegraphics[max width=\textwidth, alt={}, center]{0c4a3759-ecaa-47c3-a071-ce25fd11159f-32_2630_1828_121_121}
Edexcel C4 Specimen Q1
5 marks Moderate -0.3
Use the binomial theorem to expand \(( 4 - 3 x ) ^ { - \frac { 1 } { 2 } }\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.
Edexcel C4 Specimen Q3
8 marks Standard +0.3
3. Use the substitution \(x = \tan \theta\) to show that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 2 } \right) ^ { 2 } } \mathrm {~d} x = \frac { \pi } { 8 } + \frac { 1 } { 4 }$$ (8)
Edexcel C4 Specimen Q4
9 marks Moderate -0.3
4. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{0191bf56-a59e-44fe-af8c-bad796156f63-3_458_1552_415_223}
\end{figure} Figure 1 shows part of the curve with parametric equations $$x = \tan t , \quad y = \sin 2 t , \quad - \frac { \pi } { 2 } < t < \frac { \pi } { 2 } .$$
  1. Find the gradient of the curve at the point \(P\) where \(t = \frac { \pi } { 3 }\).
  2. Find an equation of the normal to the curve at \(P\).
  3. Find an equation of the normal to the curve at the point \(Q\) where \(t = \frac { \pi } { 4 }\).
Edexcel C4 Specimen Q5
11 marks Standard +0.3
5. The vector equations of two straight lines are $$\begin{aligned} & \mathbf { r } = 5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) \quad \text { and } \\ & \mathbf { r } = 2 \mathbf { i } - 11 \mathbf { j } + a \mathbf { k } + \mu ( - 3 \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k } ) . \end{aligned}$$ Given that the two lines intersect, find
  1. the coordinates of the point of intersection,
  2. the value of the constant \(a\),
  3. the acute angle between the two lines.
Edexcel C4 Specimen Q6
11 marks Standard +0.3
6. Given that $$\frac { 11 x - 1 } { ( 1 - x ) ^ { 2 } ( 2 + 3 x ) } \equiv \frac { A } { ( 1 - x ) ^ { 2 } } + \frac { B } { ( 1 - x ) } + \frac { C } { ( 2 + 3 x ) }$$
  1. find the values of \(A , B\) and \(C\).
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 11 x - 1 } { ( 1 - x ) ^ { 2 } ( 2 + 3 x ) } \mathrm { d } x\), giving your answer in the form \(k + \ln a\), where \(k\) is an integer and \(a\) is a simplified fraction.
Edexcel C4 Specimen Q7
12 marks Challenging +1.2
7.
  1. Given that \(u = \frac { x } { 2 } - \frac { 1 } { 8 } \sin 4 x\), show that \(\frac { \mathrm { d } u } { \mathrm {~d} x } = \sin ^ { 2 } 2 x\). \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{0191bf56-a59e-44fe-af8c-bad796156f63-5_697_1239_587_367}
    \end{figure} Figure 2 shows the finite region bounded by the curve \(y = x ^ { \frac { 1 } { 2 } } \sin 2 x\), the line \(x = \frac { \pi } { 4 }\) and the \(x\)-axis. This region is rotated through \(2 \pi\) radians about the \(x\)-axis.
  2. Using the result in part (a), or otherwise, find the exact value of the volume generated.
    (8)
Edexcel C4 Specimen Q8
13 marks Standard +0.3
8. A circular stain grows in such a way that the rate of increase of its radius is inversely proportional to the square of the radius. Given that the area of the stain at time \(t\) seconds is \(A \mathrm {~cm} ^ { 2 }\),
  1. show that \(\frac { \mathrm { d } A } { \mathrm {~d} t } \propto \frac { 1 } { \sqrt { A } }\).
    (6) Another stain, which is growing more quickly, has area \(S \mathrm {~cm} ^ { 2 }\) at time \(t\) seconds. It is given that $$\frac { \mathrm { d } S } { \mathrm {~d} t } = \frac { 2 \mathrm { e } ^ { 2 t } } { \sqrt { S } }$$ Given that, for this second stain, \(S = 9\) at time \(t = 0\),
  2. solve the differential equation to find the time at which \(S = 16\). Give your answer to 2 significant figures. \section*{END}
Edexcel F1 2014 January Q1
10 marks Standard +0.3
1. $$\mathrm { f } ( x ) = 6 \sqrt { x } - x ^ { 2 } - \frac { 1 } { 2 x } , \quad x > 0$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \([ 3,4 ]\).
  2. Taking 3 as a first approximation to \(\alpha\), apply the Newton-Raphson process once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
    [0pt]
  3. Use linear interpolation once on the interval [3,4] to find another approximation to \(\alpha\). Give your answer to 3 decimal places.
Edexcel F1 2014 January Q2
8 marks Moderate -0.3
2. The quadratic equation $$5 x ^ { 2 } - 4 x + 2 = 0$$ has roots \(\alpha\) and \(\beta\).
  1. Write down the value of \(\alpha + \beta\) and the value of \(\alpha \beta\).
  2. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 }\).
  3. Find a quadratic equation which has roots $$\frac { 1 } { \alpha ^ { 2 } } \text { and } \frac { 1 } { \beta ^ { 2 } }$$ giving your answer in the form \(p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers. \includegraphics[max width=\textwidth, alt={}, center]{4da2bb2c-a51b-493c-a9f2-f4ff008a3aac-07_70_51_2663_1896}
Edexcel F1 2014 January Q3
6 marks Standard +0.3
3. $$\mathbf { A } = \left( \begin{array} { l l } 6 & 4 \\ 1 & 1 \end{array} \right)$$
  1. Show that \(\mathbf { A }\) is non-singular. The triangle \(R\) is transformed to the triangle \(S\) by the matrix \(\mathbf { A }\).
    Given that the area of triangle \(R\) is 10 square units,
  2. find the area of triangle \(S\). Given that $$\mathbf { B } = \mathbf { A } ^ { 4 }$$ and that the triangle \(R\) is transformed to the triangle \(T\) by the matrix \(\mathbf { B }\),
  3. find, without evaluating \(\mathbf { B }\), the area of triangle \(T\).
Edexcel F1 2014 January Q4
9 marks Standard +0.3
4. $$f ( x ) = x ^ { 4 } + 3 x ^ { 3 } - 5 x ^ { 2 } - 19 x - 60$$
  1. Given that \(x = - 4\) and \(x = 3\) are roots of the equation \(\mathrm { f } ( x ) = 0\), use algebra to solve \(\mathrm { f } ( x ) = 0\) completely.
  2. Show the four roots of \(\mathrm { f } ( x ) = 0\) on a single Argand diagram.
Edexcel F1 2014 January Q5
8 marks Standard +0.3
5.
  1. Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } \left( 9 r ^ { 2 } - 4 r \right) = \frac { 1 } { 2 } n ( n + 1 ) ( 6 n - 1 )$$ for all positive integers \(n\). Given that $$\sum _ { r = 1 } ^ { 12 } \left( 9 r ^ { 2 } - 4 r + k \left( 2 ^ { r } \right) \right) = 6630$$
  2. find the exact value of the constant \(k\).
Edexcel F1 2014 January Q6
8 marks Standard +0.3
6.
  1. $$\mathbf { B } = \left( \begin{array} { r r } - 1 & 2 \\ 3 & - 4 \end{array} \right) , \quad \mathbf { Y } = \left( \begin{array} { r r } 4 & - 2 \\ 1 & 0 \end{array} \right)$$
    1. Find \(\mathbf { B } ^ { - 1 }\). The transformation represented by \(\mathbf { Y }\) is equivalent to the transformation represented by \(\mathbf { B }\) followed by the transformation represented by the matrix \(\mathbf { A }\).
    2. Find \(\mathbf { A }\).
    3. $$\mathbf { M } = \left( \begin{array} { r r } - \sqrt { 3 } & - 1 \\ 1 & - \sqrt { 3 } \end{array} \right)$$ The matrix \(\mathbf { M }\) represents an enlargement scale factor \(k\), centre ( 0,0 ), where \(k > 0\), followed by a rotation anti-clockwise through an angle \(\theta\) about \(( 0,0 )\).
      (a) Find the value of \(k\).
      (b) Find the value of \(\theta\).
Edexcel F1 2014 January Q7
8 marks Standard +0.3
7.
  1. Given that $$\frac { 2 w - 3 } { 10 } = \frac { 4 + 7 i } { 4 - 3 i }$$ find \(w\), giving your answer in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real constants. You must show your working.
  2. Given that $$z = ( 2 + \lambda i ) ( 5 + i )$$ where \(\lambda\) is a real constant, and that $$\arg z = \frac { \pi } { 4 }$$ find the value of \(\lambda\).
    □
Edexcel F1 2014 January Q8
12 marks Standard +0.8
8. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant. The point \(P \left( a p ^ { 2 } , 2 a p \right)\) lies on the parabola \(C\).
  1. Show that an equation of the tangent to \(C\) at \(P\) is $$p y = x + a p ^ { 2 }$$ The tangent to \(C\) at the point \(P\) intersects the directrix of \(C\) at the point \(B\) and intersects the \(x\)-axis at the point \(D\). Given that the \(y\)-coordinate of \(B\) is \(\frac { 5 } { 6 } a\) and \(p > 0\),
  2. find, in terms of \(a\), the \(x\)-coordinate of \(D\). Given that \(O\) is the origin,
  3. find, in terms of \(a\), the area of the triangle \(O P D\), giving your answer in its simplest form.
Edexcel F1 2014 January Q9
6 marks Moderate -0.3
  1. Prove by induction that, for \(n \in \mathbb { Z } ^ { + }\),
$$f ( n ) = 7 ^ { n } - 2 ^ { n } \text { is divisible by } 5$$
Edexcel F1 2015 January Q1
7 marks Standard +0.3
1. $$f ( x ) = x ^ { 4 } - x ^ { 3 } - 9 x ^ { 2 } + 29 x - 60$$ Given that \(x = 1 + 2 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( x ) = 0\), use algebra to find the three other roots of the equation \(\mathrm { f } ( x ) = 0\)