Questions — Edexcel F3 (138 questions)

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Edexcel F3 2020 June Q1
7 marks Standard +0.8
  1. (a) Use the definition of \(\sinh x\) in terms of exponentials to show that
$$\sinh 3 x \equiv 4 \sinh ^ { 3 } x + 3 \sinh x$$ (b) Hence determine the exact coordinates of the points of intersection of the curve with equation \(y = \sinh 3 x\) and the curve with equation \(y = 19 \sinh x\), giving your answers as simplified logarithms where necessary.
Edexcel F3 2020 June Q2
8 marks Standard +0.3
2. Determine
  1. \(\int \frac { 1 } { 3 x ^ { 2 } + 12 x + 24 } \mathrm {~d} x\)
  2. \(\int \frac { 1 } { \sqrt { 27 - 6 x - x ^ { 2 } } } \mathrm {~d} x\)
Edexcel F3 2020 June Q3
9 marks Standard +0.3
3. $$\mathbf { M } = \left( \begin{array} { c c c } 3 & - 4 & k \\ 1 & - 2 & k \\ 1 & - 5 & 5 \end{array} \right) \text { where } k \text { is a constant }$$ Given that 3 is an eigenvalue of \(\mathbf { M }\),
  1. find the value of \(k\).
  2. Hence find the other two eigenvalues of \(\mathbf { M }\).
  3. Find a normalised eigenvector corresponding to the eigenvalue 3
    .
    VIIIV SIHI NI JIIHM ION OCVARV SHAL NI ALIAM LON OOVERV SIHI NI JIIIM ION OO
Edexcel F3 2020 June Q4
9 marks Challenging +1.2
4.
  1. Show that, for \(n \geqslant 2\)
  2. Hence find the functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) such that $$\int x ^ { 4 } \cos x \mathrm {~d} x = \mathrm { f } ( x ) \sin x + \mathrm { g } ( x ) \cos x + c$$ where \(c\) is an arbitrary constant. $$I _ { n } = \int x ^ { n } \cos x \mathrm {~d} x$$
    1. Show that, for \(n \geqslant 2\) $$I _ { n } = x ^ { n } \sin x + n x ^ { n - 1 } \cos x - n ( n - 1 ) I _ { n - 2 }$$
    2. Hence find the functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) such that
Edexcel F3 2020 June Q5
12 marks Challenging +1.2
5. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 4 } = 1\) The line \(l\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants. Given that \(l\) is a tangent to \(H\),
  1. show that \(25 m ^ { 2 } = 4 + c ^ { 2 }\)
  2. Hence find the equations of the tangents to \(H\) that pass through the point ( 1,2 ).
  3. Find the coordinates of the point of contact each of these tangents makes with \(H\).
Edexcel F3 2020 June Q6
8 marks Challenging +1.2
6. $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & a \end{array} \right) \quad a \neq 1$$
  1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).
    . The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\mathbf { B }\). $$\mathbf { B } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & 4 \end{array} \right)$$ The equation of \(l _ { 2 }\) is $$( \mathbf { r } - ( 12 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } ) ) \times ( - 6 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) = \mathbf { 0 }$$
  2. Find a vector equation for the line \(l _ { 1 }\)
Edexcel F3 2020 June Q7
12 marks Challenging +1.3
7. The curve \(C\) has parametric equations $$x = \cosh t + t , \quad y = \cosh t - t \quad 0 \leqslant t \leqslant \ln 3$$
  1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = 2 \cosh ^ { 2 } t$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. The area of the curved surface generated is given by \(S\).
  2. Show that $$S = 2 \pi \sqrt { 2 } \int _ { 0 } ^ { \ln 3 } \left( \cosh ^ { 2 } t - t \cosh t \right) d t$$
  3. Hence find the value of \(S\), giving your answer in the form $$\frac { \pi \sqrt { 2 } } { 9 } ( a + b \ln 3 )$$ where \(a\) and \(b\) are constants to be determined.
Edexcel F3 2020 June Q8
10 marks Standard +0.8
8. The plane \(\Pi _ { 1 }\) has equation $$x - 5 y + 3 z = 11$$ The plane \(\Pi _ { 2 }\) has equation $$3 x - 2 y + 2 z = 7$$ The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line \(l\).
  1. Find a vector equation for \(l\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\) where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a scalar parameter. The point \(P ( 2,0,3 )\) lies on \(\Pi _ { 1 }\) The line \(m\), which passes through \(P\), is parallel to \(l\). The point \(Q ( 3,2,1 )\) lies on \(\Pi _ { 2 }\) The line \(n\), which passes through \(Q\), is also parallel to \(l\).
  2. Find, in exact simplified form, the shortest distance between \(m\) and \(n\).
    VIIV STHI NI JINM ION OCVIAV SIHI NI JMAM/ION OCVIAV SIHL NI JIIYM ION OO
Edexcel F3 2022 June Q1
7 marks Standard +0.3
  1. (a) Use the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials to show that
$$\cosh A \cosh B + \sinh A \sinh B \equiv \cosh ( A + B )$$ (b) Hence find the value of \(x\) for which $$\cosh ( x + \ln 2 ) = 5 \sinh x$$ giving your answer in the form \(\frac { 1 } { 2 } \ln k\), where \(k\) is a rational number to be determined.
(5)
Edexcel F3 2022 June Q2
9 marks Standard +0.8
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Determine $$\int \frac { 1 } { \sqrt { 5 + 4 x - x ^ { 2 } } } d x$$
  2. Use the substitution \(x = 3 \sec \theta\) to determine the exact value of $$\int _ { 2 \sqrt { 3 } } ^ { 6 } \frac { 18 } { \left( x ^ { 2 } - 9 \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$ Give your answer in the form \(A + B \sqrt { 3 }\) where \(A\) and \(B\) are constants to be found.
Edexcel F3 2022 June Q3
8 marks Challenging +1.2
3. $$\mathbf { M } = \left( \begin{array} { r r r } - 2 & 5 & 0 \\ 5 & 1 & - 3 \\ 0 & - 3 & 6 \end{array} \right)$$ Given that \(\mathbf { i } + \mathbf { j } + \mathbf { k }\) is an eigenvector of \(\mathbf { M }\),
  1. determine the corresponding eigenvalue. Given that 8 is an eigenvalue of \(\mathbf { M }\),
  2. determine a corresponding eigenvector.
  3. Determine a diagonal matrix \(\mathbf { D }\) and an orthogonal matrix \(\mathbf { P }\) such that $$\mathbf { D } = \mathbf { P } ^ { \mathrm { T } } \mathbf { M P }$$
Edexcel F3 2022 June Q4
4 marks Challenging +1.2
4. $$y = \operatorname { artanh } \left( \frac { \cos x + a } { \cos x - a } \right)$$ where \(a\) is a non-zero constant.
Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k \tan x$$ where \(k\) is a constant to be determined.
Edexcel F3 2022 June Q5
7 marks Challenging +1.2
  1. A curve has parametric equations
$$x = 4 \mathrm { e } ^ { \frac { 1 } { 2 } t } \quad y = \mathrm { e } ^ { t } - t \quad 0 \leqslant t \leqslant 4$$ The curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
Show that the area of the curved surface generated is $$\pi \left( \mathrm { e } ^ { 8 } + A \mathrm { e } ^ { 4 } + B \right)$$ where \(A\) and \(B\) are constants to be determined.
Edexcel F3 2022 June Q6
8 marks Standard +0.8
6. $$\mathbf { A } = \left( \begin{array} { r r r } x & 1 & 3 \\ 2 & 4 & x \\ - 4 & - 2 & - 1 \end{array} \right)$$
  1. Show that \(\mathbf { A }\) is non-singular for all real values of \(x\).
  2. Determine, in terms of \(x , \mathbf { A } ^ { - 1 }\)
Edexcel F3 2022 June Q7
10 marks Challenging +1.8
7. $$I _ { n } = \int \frac { x ^ { n } } { \sqrt { 10 - x ^ { 2 } } } \mathrm {~d} x \quad n \in \mathbb { N } \quad | x | < \sqrt { 10 }$$
  1. Show that $$n I _ { n } = 10 ( n - 1 ) I _ { n - 2 } - x ^ { n - 1 } \left( 10 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \quad n \geqslant 2$$
  2. Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { \sqrt { 10 - x ^ { 2 } } } \mathrm {~d} x$$ giving your answer in the form \(\frac { 1 } { 15 } ( p \sqrt { 10 } + q )\) where \(p\) and \(q\) are integers to be determined.
Edexcel F3 2022 June Q8
12 marks Challenging +1.2
  1. The plane \(\Pi\) has equation
$$3 x + 4 y - z = 17$$ The line \(l _ { 1 }\) is perpendicular to \(\Pi\) and passes through the point \(P ( - 4 , - 5,3 )\) The line \(l _ { 1 }\) intersects \(\Pi\) at the point \(Q\)
  1. Determine the coordinates of \(Q\) Given that the point \(R ( - 1,6,4 )\) lies on \(\Pi\)
  2. determine a Cartesian equation of the plane containing \(P Q R\) The line \(l _ { 2 }\) passes through \(P\) and \(R\) The line \(l _ { 3 }\) is the reflection of \(l _ { 2 }\) in \(\Pi\)
  3. Determine a vector equation for \(l _ { 3 }\)
Edexcel F3 2022 June Q9
10 marks Standard +0.8
  1. The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$ The line \(l\) has equation \(y = k x - 3\), where \(k\) is a constant.
Given that \(E\) and \(l\) meet at 2 distinct points \(P\) and \(Q\)
  1. show that the \(x\) coordinates of \(P\) and \(Q\) are solutions of the equation $$\left( 9 k ^ { 2 } + 4 \right) x ^ { 2 } - 54 k x + 45 = 0$$ The point \(M\) is the midpoint of \(P Q\)
  2. Determine, in simplest form in terms of \(k\), the coordinates of \(M\)
  3. Hence show that, as \(k\) varies, \(M\) lies on the curve with equation $$x ^ { 2 } + p y ^ { 2 } = q y$$ where \(p\) and \(q\) are constants to be determined.
Edexcel F3 2023 June Q1
5 marks Standard +0.3
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Solve the equation
$$7 \cosh x + 3 \sinh x = 2 \mathrm { e } ^ { x } + 7$$ Give your answers as simplified natural logarithms.
Edexcel F3 2023 June Q2
8 marks Standard +0.8
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & 0 & 0 \\ 0 & 1 & 4 \\ 3 & - 2 & - 3 \end{array} \right)$$
  1. Determine \(\mathbf { M } ^ { - 1 }\) The transformation represented by \(\mathbf { M }\) maps the plane \(\Pi _ { 1 }\) to the plane \(\Pi _ { 2 }\) The point \(( x , y , z )\) on \(\Pi _ { 1 }\) maps to the point \(( u , v , w )\) on \(\Pi _ { 2 }\)
  2. Determine \(x , y\) and \(z\) in terms of \(u , v\) and \(w\) as appropriate. The plane \(\Pi _ { 1 }\) has equation $$3 x - 7 y + 2 z = - 3$$
  3. Find a Cartesian equation for \(\Pi _ { 2 }\) Give your answer in the form \(a u + b v + c w = d\) where \(a , b , c\) and \(d\) are integers to be determined.
Edexcel F3 2023 June Q3
11 marks Challenging +1.8
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f1efd9b3-d604-4088-a4b5-8680711aa8f1-08_353_474_301_781} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \frac { 1 } { 2 } ( \tan x + \cot x ) \quad \frac { \pi } { 6 } \leqslant x \leqslant \frac { \pi } { 3 }$$
  1. Show that the length of \(C\) is given by $$\frac { 1 } { 2 } \int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 3 } } \left( \tan ^ { 2 } x + \cot ^ { 2 } x \right) d x$$
  2. Hence determine the exact length of \(C\), giving your answer in simplest form.
Edexcel F3 2023 June Q4
12 marks Challenging +1.2
  1. The plane \(\Pi _ { 1 }\) contains the point \(A ( 2,4 , - 5 )\) and is normal to the vector \(\left( \begin{array} { r } - 1 \\ 3 \\ 3 \end{array} \right)\)
The plane \(\Pi _ { 2 }\) contains the point \(B ( 3,6 , - 2 )\) and is normal to the vector \(\left( \begin{array} { r } 2 \\ 0 \\ - 5 \end{array} \right)\) The line \(l\) is the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)
  1. Determine a vector equation for \(l\). The points \(C\) and \(D\) both lie on \(l\).
    Given that \(C\) and \(D\) are 5 units apart,
  2. determine the exact volume of the tetrahedron \(A B C D\).
Edexcel F3 2023 June Q5
7 marks Challenging +1.2
5. $$\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & k \\ - 1 & - 3 & 4 \\ 2 & 6 & - 8 \end{array} \right) \quad \text { where } k \text { is a constant }$$ Given that \(\mathbf { M }\) has a repeated eigenvalue, determine
  1. the possible values of \(k\),
  2. all corresponding eigenvalues of \(\mathbf { M }\) for each value of \(k\).
Edexcel F3 2023 June Q6
13 marks Challenging +1.3
  1. The ellipse \(E\) has equation \(\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1\)
The point \(P ( 4 \cos \theta , 3 \sin \theta )\) lies on \(E\).
  1. Use calculus to show that an equation of the tangent to \(E\) at \(P\) is $$3 x \cos \theta + 4 y \sin \theta = 12$$
  2. Determine an equation for the normal to \(E\) at \(P\). The tangent to \(E\) at \(P\) meets the \(x\)-axis at the point \(A\).
    The normal to \(E\) at \(P\) meets the \(y\)-axis at the point \(B\).
  3. Show that the locus of the midpoint of \(A\) and \(B\) as \(\theta\) varies has equation $$x ^ { 2 } \left( p - q y ^ { 2 } \right) = r$$ where \(p , q\) and \(r\) are integers to be determined.
Edexcel F3 2023 June Q7
9 marks Challenging +1.8
7. $$I _ { n } = \int \cosh ^ { n } 2 x \mathrm {~d} x \quad n \geqslant 0$$
  1. Show that, for \(n \geqslant 2\) $$I _ { n } = \frac { \cosh ^ { n - 1 } 2 x \sinh 2 x } { 2 n } + \frac { n - 1 } { n } I _ { n - 2 }$$
  2. Hence determine $$\int ( 1 + \cosh 2 x ) ^ { 3 } d x$$ collecting any like terms in your answer.
Edexcel F3 2023 June Q8
10 marks Challenging +1.2
  1. (a) Differentiate \(x \operatorname { arcosh } 5 x\) with respect to \(x\) (b) Hence, or otherwise, show that
$$\int _ { \frac { 1 } { 4 } } ^ { \frac { 3 } { 5 } } \operatorname { arcosh } 5 x \mathrm {~d} x = \frac { 3 } { 20 } - \frac { 2 \sqrt { 2 } } { 5 } + \ln ( p + q \sqrt { 2 } ) ^ { k } - \frac { 1 } { 4 } \ln r$$ where \(p , q , r\) and \(k\) are rational numbers to be determined.