Questions — Edexcel (9685 questions)

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Edexcel FP1 Q2
3 marks Moderate -0.5
2. A point \(P\) with coordinates \(( x , y )\) moves so that its distance from the point \(( - 3,0 )\) is equal to its distance from the line \(x = 3\). Find a cartesian equation for the locus of \(P\).
Edexcel FP1 Q3
6 marks Moderate -0.3
3. \(z = 1 + \mathrm { i } \sqrt { 3 }\) Express in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real.
  1. \(z ^ { 2 } + z\),
  2. \(\frac { z } { 3 - z }\),
    giving the exact values of \(a\) and \(b\) in each part.
Edexcel FP1 Q4
9 marks Moderate -0.3
4. \(f ( x ) = x ^ { 3 } - 4 x ^ { 2 } + 5 x - 3\) The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval ( 2,3 ).
  1. Use linear interpolation on the end points of this interval to obtain an approximation for \(\alpha\).
  2. Taking 2.5 as a first approximation to \(\alpha\), apply the Newton - Raphson procedure once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\). Give your answer to 2 decimal places.
Edexcel FP1 Q5
7 marks Moderate -0.3
5. Given that \(a\) and \(b\) are non-zero constants and that $$\mathbf { X } = \left( \begin{array} { r r } a & 2 b \\ - a & 3 b \end{array} \right) ,$$
  1. find \(\mathbf { X } ^ { - 1 }\), giving your answer in terms of \(a\) and \(b\). Given also that \(\mathbf { Z X } = \mathbf { Y }\), where \(\mathbf { Y } = \left( \begin{array} { c c } 3 a & b \\ a & 2 b \end{array} \right)\),
  2. find \(\mathbf { Z }\), simplifying your answer.
Edexcel FP1 Q6
6 marks Standard +0.3
6. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r\) and for \(\sum _ { r = 1 } ^ { n } r ^ { 3 }\) to show that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r \left( 2 r ^ { 2 } - 6 \right) = \frac { 1 } { 2 } n ( n + 1 ) ( n + 3 ) ( n - 2 ) .$$ (b) Hence calculate the value of \(\sum _ { r = 10 } ^ { 50 } r \left( 2 r ^ { 2 } - 6 \right)\).
Edexcel FP1 Q7
12 marks Moderate -0.8
7. The quadratic equation $$z ^ { 2 } + 10 z + 169 = 0$$ has complex roots \(z _ { 1 }\) and \(z _ { 2 }\).
  1. Find each of these roots in the form \(a + b \mathrm { i }\).
  2. Find the modulus and argument of \(z _ { 1 }\) and of \(z _ { 2 }\). Give the arguments in radians to 3 significant figures.
  3. Illustrate the two roots on a single Argand diagram.
  4. Find the value of \(\left| z _ { 1 } - z _ { 2 } \right|\).
Edexcel FP1 Q8
13 marks Standard +0.8
8. The rectangular hyperbola \(H\) has equation \(x y = c ^ { 2 }\). The point ( \(3 t , \frac { 3 } { t }\) ) is a general point on this hyperbola.
  1. Find the value of \(c ^ { 2 }\).
  2. Show that an equation of the normal to \(H\) at the point ( \(3 t , \frac { 3 } { t }\) ) is $$y = t ^ { 2 } x + \left( \frac { 3 } { t } - 3 t ^ { 3 } \right)$$ The point \(P\) on \(H\) has coordinates (6, 1.5). The tangent to \(H\) at \(P\) meets the curve again at the point \(Q\).
  3. Find the coordinates of the point \(Q\).
Edexcel FP1 Q9
14 marks Standard +0.3
9. (a) A sequence of numbers is defined by $$u _ { 1 } = 3 \text { and } u _ { n + 1 } = 3 u _ { n } + 4 \text { for } n \geqslant 1 .$$ Prove by induction that $$u _ { n } = 3 ^ { n } + 2 \left( 3 ^ { n - 1 } - 1 \right) \text { for } n \in \mathbb { Z } ^ { + } \text {. }$$ (b) $$\mathbf { A } = \left( \begin{array} { l l } 4 & 0 \\ 9 & 1 \end{array} \right)$$
  1. Prove by induction that $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 4 ^ { n } & 0 \\ 3 \left( 4 ^ { n } - 1 \right) & 1 \end{array} \right) \text { for } n \in \mathbb { Z } ^ { + } .$$
  2. Determine whether the result \(\mathbf { A } ^ { n } = \left( \begin{array} { c c } 4 ^ { n } & 0 \\ 3 \left( 4 ^ { n } - 1 \right) & 1 \end{array} \right)\) is also valid for \(n = - 1\).
Edexcel FP1 Specimen Q1
6 marks Moderate -0.8
1. $$f ( x ) = x ^ { 3 } - 3 x ^ { 2 } + 5 x - 4$$
  1. Use differentiation to find \(\mathrm { f } ^ { \prime } ( x )\). The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval \(1.4 < x < 1.5\)
  2. Taking 1.4 as a first approximation to \(\alpha\), use the Newton-Raphson procedure once to obtain a second approximation to \(\alpha\). Give your answer to 3 decimal places.
Edexcel FP1 Specimen Q2
7 marks Moderate -0.5
2. The rectangle \(R\) has vertices at the points \(( 0,0 ) , ( 1,0 ) , ( 1,2 )\) and \(( 0,2 )\).
  1. Find the coordinates of the vertices of the image of \(R\) under the transformation given by the matrix \(\mathbf { A } = \left( \begin{array} { c c } a & 4 \\ - 1 & 1 \end{array} \right)\), where \(a\) is a constant.
  2. Find det \(\mathbf { A }\), giving your answer in terms of \(a\). Given that the area of the image of \(R\) is 18 ,
  3. find the value of \(a\).
Edexcel FP1 Specimen Q3
5 marks Standard +0.3
3. The matrix \(\mathbf { R }\) is given by \(\mathbf { R } = \left( \begin{array} { c c } \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \\ - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } } \end{array} \right)\)
  1. Find \(\mathbf { R } ^ { 2 }\).
  2. Describe the geometrical transformation represented by \(\mathbf { R } ^ { 2 }\).
  3. Describe the geometrical transformation represented by \(\mathbf { R }\).
Edexcel FP1 Specimen Q4
3 marks Moderate -0.8
4. $$f ( x ) = 2 ^ { x } - 6 x$$ The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\) in the interval [4,5]. Using the end points of this interval find, by linear interpolation, an approximation to \(\alpha\).
Edexcel FP1 Specimen Q5
9 marks Standard +0.3
5. (a) Show that \(\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r - 1 \right) = \frac { 1 } { 3 } ( n - 2 ) n ( n + 2 )\).
(b) Hence calculate the value of \(\sum _ { r = 10 } ^ { 40 } \left( r ^ { 2 } - r - 1 \right)\).
Edexcel FP1 Specimen Q6
10 marks Moderate -0.8
6. Given that \(z = - 3 + 4 \mathrm { i }\),
  1. find the modulus of \(z\),
  2. the argument of \(z\) in radians to 2 decimal places. Given also that \(w = \frac { - 14 + 2 \mathrm { i } } { z }\),
  3. use algebra to find \(w\), giving your answers in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real. The complex numbers \(z\) and \(w\) are represented by points \(A\) and \(B\) on an Argand diagram.
  4. Show the points \(A\) and \(B\) on an Argand diagram.
Edexcel FP1 Specimen Q7
12 marks Standard +0.8
7. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a constant. The point \(\left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).
  1. Find the value of \(a\).
  2. Show that the equation for the tangent to \(C\) at the point \(\left( 4 t ^ { 2 } , 8 t \right)\) is $$y t = x + 4 t ^ { 2 } .$$ The tangent to \(C\) at the point \(A\) meets the tangent to \(C\) at the point \(B\) on the directrix of \(C\) when \(y = 15\).
  3. Find the coordinates of \(A\) and the coordinates of \(B\).
Edexcel FP1 Specimen Q8
9 marks Moderate -0.3
8. $$\mathrm { f } ( x ) \equiv 2 x ^ { 3 } - 5 x ^ { 2 } + p x - 5 , p \in \mathbb { R }$$ Given that \(1 - 2 \mathrm { i }\) is a complex solution of \(\mathrm { f } ( x ) = 0\),
  1. write down the other complex solution of \(\mathrm { f } ( x ) = 0\),
  2. solve the equation \(\mathrm { f } ( x ) = 0\),
  3. find the value of \(p\).
Edexcel FP1 Specimen Q9
14 marks Standard +0.3
9. Use the method of mathematical induction to prove that, for \(n \in \mathbb { Z } ^ { + }\),
  1. \(\left( \begin{array} { c c } 2 & 1 \\ - 1 & 0 \end{array} \right) ^ { n } = \left( \begin{array} { c c } n + 1 & n \\ - n & 1 - n \end{array} \right)\)
  2. \(\mathrm { f } ( n ) = 4 ^ { n } + 6 n - 1\) is divisible by 3 .
Edexcel F2 2021 January Q1
3 marks Standard +0.8
  1. The transformation \(T\) from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\), is given by
$$w = \frac { z + p \mathrm { i } } { \mathrm { i } z + 3 } \quad z \neq 3 \mathrm { i } \quad p \in \mathbb { Z }$$ The point representing \(\mathrm { i } ( 1 + \sqrt { 3 } )\) is invariant under \(T\).
Determine the value of \(p\).
Edexcel F2 2021 January Q2
6 marks Standard +0.3
2. (a) Show that, for \(r > 0\) $$\frac { r + 2 } { r ( r + 1 ) } - \frac { r + 3 } { ( r + 1 ) ( r + 2 ) } = \frac { r + 4 } { r ( r + 1 ) ( r + 2 ) }$$ (b) Hence show that $$\sum _ { r = 1 } ^ { n } \frac { r + 4 } { r ( r + 1 ) ( r + 2 ) } = \frac { n ( a n + b ) } { c ( n + 1 ) ( n + 2 ) }$$ where \(a\), \(b\) and \(c\) are integers to be determined.
Edexcel F2 2021 January Q3
7 marks Standard +0.8
3. Use algebra to obtain the set of values of \(x\) for which $$\left| x ^ { 2 } + x - 2 \right| < \frac { 1 } { 2 } ( x + 5 )$$
Edexcel F2 2021 January Q4
9 marks Challenging +1.2
4. (a) Show that the substitution \(y ^ { 2 } = \frac { 1 } { z }\) transforms the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 y = 3 x y ^ { 3 } \quad y \neq 0$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - 4 z = - 6 x$$ (b) Obtain the general solution of differential equation (II).
(c) Hence obtain the general solution of differential equation (I), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\)
Edexcel F2 2021 January Q5
9 marks Challenging +1.2
5. Given that $$\left( 2 - x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } = 3 y$$
  1. show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \frac { 1 } { \left( 2 - x ^ { 2 } \right) } \left( 2 x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \left( 1 - 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) - 5 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } \right)$$ Given also that \(y = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 4 }\) at \(x = 0\)
  2. obtain a series solution for \(y\) in ascending powers of \(x\) with simplified coefficients, up to and including the term in \(x ^ { 3 }\)
Edexcel F2 2021 January Q6
12 marks Challenging +1.2
6. (a) Determine the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 6 \cos x$$ (b) Find the particular solution for which \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) at \(x = 0\)
Edexcel F2 2021 January Q7
13 marks Challenging +1.2
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4d3e1c8e-c659-4cfe-82ac-5bfce0f58ba3-24_445_597_248_676} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of curve \(C\) with polar equation $$r = 3 \sin 2 \theta \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ The point \(P\) on \(C\) has polar coordinates \(( R , \phi )\). The tangent to \(C\) at \(P\) is perpendicular to the initial line.
  1. Show that \(\tan \phi = \frac { 1 } { \sqrt { 2 } }\)
  2. Determine the exact value of \(R\). The region \(S\), shown shaded in Figure 1, is bounded by \(C\) and the line \(O P\), where \(O\) is the pole.
  3. Use calculus to show that the exact area of \(S\) is $$p \arctan \frac { 1 } { \sqrt { 2 } } + q \sqrt { 2 }$$ where \(p\) and \(q\) are constants to be determined. Solutions relying entirely on calculator technology are not acceptable.
Edexcel F2 2021 January Q8
16 marks Challenging +1.2
8. Given that \(z = e ^ { \mathrm { i } \theta }\)
  1. show that \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\) where \(n\) is a positive integer.
  2. Show that $$\cos ^ { 6 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10 )$$
  3. Hence solve the equation $$\cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta = 0 \quad 0 \leqslant \theta \leqslant \pi$$ Give your answers to 3 significant figures.
  4. Use calculus to determine the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \left( 32 \cos ^ { 6 } \theta - 4 \cos ^ { 2 } \theta \right) d \theta$$ Solutions relying entirely on calculator technology are not acceptable.