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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.
Edexcel F3 2024 June Q1
6 marks Standard +0.3
  1. The hyperbola \(H\) has
  • foci with coordinates \(\left( \pm \frac { 13 } { 2 } , 0 \right)\)
  • directrices with equations \(x = \pm \frac { 72 } { 13 }\)
  • eccentricity e
Determine
  1. the value of \(e\)
  2. an equation for \(H\), giving your answer in the form \(p x ^ { 2 } - q y ^ { 2 } = r\), where \(p , q\) and \(r\) are integers.
Edexcel F3 2024 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.
$$\mathbf { M } = \left( \begin{array} { r r r } 2 & 0 & 3 \\ 0 & - 4 & - 3 \\ 0 & - 4 & 0 \end{array} \right)$$ Given that \(\mathbf { M }\) has exactly two distinct eigenvalues \(\lambda _ { 1 }\) and \(\lambda _ { 2 }\) where \(\lambda _ { 1 } < \lambda _ { 2 }\)
  1. determine a normalised eigenvector corresponding to the eigenvalue \(\lambda _ { 1 }\) The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 4 \\ - 1 \\ 0 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ 0 \\ - 1 \end{array} \right)\), where \(\mu\) is a scalar parameter.
    The transformation \(T\) is represented by \(\mathbf { M }\).
    The line \(l _ { 1 }\) is transformed by \(T\) to the line \(l _ { 2 }\)
  2. Determine a vector equation for \(l _ { 2 }\), giving your answer in the form \(\mathbf { r } \times \mathbf { b } = \mathbf { c }\) where \(\mathbf { b }\) and \(\mathbf { c }\) are constant vectors.
Edexcel F3 2024 June Q3
7 marks Standard +0.8
  1. \(\quad y = \operatorname { arsinh } \left( \sqrt { x ^ { 2 } - 1 } \right) \quad x > 1\)
    1. Prove that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { x ^ { 2 } - 1 } }\)
    $$\mathrm { f } ( x ) = \frac { 1 } { 3 } \operatorname { arsinh } \left( \sqrt { x ^ { 2 } - 1 } \right) - \arctan x \quad x > 1$$
  2. Determine the exact values of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) = 0\)
Edexcel F3 2024 June Q4
9 marks Standard +0.3
  1. (a) Use the definitions of hyperbolic functions in terms of exponentials to show that
$$\sinh ( A + B ) \equiv \sinh A \cosh B + \cosh A \sinh B$$ (b) Hence express \(10 \sinh x + 8 \cosh x\) in the form \(R \sinh ( x + \alpha )\) where \(R > 0\), giving \(\alpha\) in the form \(\ln p\) where \(p\) is an integer.
(c) Hence solve the equation $$10 \sinh x + 8 \cosh x = 18 \sqrt { 7 }$$ giving your answer in the form \(\ln ( \sqrt { 7 } + q )\) where \(q\) is a rational number to be determined.
Edexcel F3 2024 June Q5
8 marks Standard +0.8
5. $$4 x ^ { 2 } + 4 x + 17 \equiv ( 2 x + p ) ^ { 2 } + q$$ where \(p\) and \(q\) are integers.
  1. Determine the value of \(p\) and the value of \(q\) Given that $$\frac { 8 x + 5 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } \equiv \frac { 1 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } + \frac { A x + B } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } }$$ where \(A\) and \(B\) are integers,
  2. write down the value of \(A\) and the value of \(B\)
  3. Hence use algebraic integration to show that $$\int _ { \frac { 1 } { 3 } } ^ { 1 } \frac { 8 x + 5 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } \mathrm {~d} x = k + \frac { 1 } { 2 } \ln k$$ where \(k\) is a rational number to be determined.
Edexcel F3 2024 June Q6
9 marks Challenging +1.3
  1. The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$$ The line \(l\) is the normal to \(E\) at the point \(P ( 5 \cos \theta , 3 \sin \theta )\) where \(0 < \theta < \frac { \pi } { 2 }\)
  1. Using calculus, show that an equation for \(l\) is $$5 x \sin \theta - 3 y \cos \theta = 16 \sin \theta \cos \theta$$ Given that
    • \(\quad l\) intersects the \(y\)-axis at the point \(Q\)
    • the midpoint of the line segment \(P Q\) is \(M\)
    • determine the exact maximum area of triangle \(O M P\) as \(\theta\) varies, where \(O\) is the origin.
    You must justify your answer.
Edexcel F3 2024 June Q7
8 marks Challenging +1.8
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{7e38e2ed-ab5f-4906-940e-4b02c6992164-22_568_1192_376_440} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve with equation $$y = \ln \left( \tanh \frac { x } { 2 } \right) \quad 1 \leqslant x \leqslant 2$$
  1. Show that the length, \(s\), of the curve is given by $$s = \int _ { 1 } ^ { 2 } \operatorname { coth } x \mathrm {~d} x$$
  2. Hence show that $$s = \ln \left( \mathrm { e } + \frac { 1 } { \mathrm { e } } \right)$$
Edexcel F3 2024 June Q8
9 marks Challenging +1.8
8. $$I _ { n } = \int _ { 0 } ^ { k } x ^ { n } ( k - x ) ^ { \frac { 1 } { 2 } } \mathrm {~d} x \quad n \geqslant 0$$ where \(k\) is a positive constant.
  1. Show that $$I _ { n } = \frac { 2 k n } { 3 + 2 n } I _ { n - 1 } \quad n \geqslant 1$$ Given that $$\int _ { 0 } ^ { k } x ^ { 2 } ( k - x ) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = \frac { 9 \sqrt { 3 } } { 280 }$$
  2. use the result in part (a) to determine the exact value of \(k\).
Edexcel F3 2024 June Q9
10 marks Standard +0.8
  1. The plane \(\Pi _ { 1 }\) has vector equation
$$\mathbf { r } = \left( \begin{array} { l } 5 \\ 3 \\ 0 \end{array} \right) + s \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 2 \\ 2 \end{array} \right)$$ where \(s\) and \(t\) are scalar parameters.
  1. Determine a Cartesian equation for \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has vector equation \(\mathbf { r } . \left( \begin{array} { r } 5 \\ - 2 \\ 3 \end{array} \right) = 1\)
  2. Determine a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) Give 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 plane \(\Pi _ { 3 }\) has Cartesian equation \(4 x - 3 y - z = 0\)
  3. Use the answer to part (b) to determine the coordinates of the point of intersection of \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\)
Edexcel F3 2021 October Q1
6 marks Challenging +1.2
  1. The curve \(C\) has equation
$$y = \frac { 1 } { 2 } \operatorname { arcosh } ( 2 x ) \quad \frac { 7 } { 2 } \leqslant x \leqslant 13$$ Using calculus, determine the exact length of the curve \(C\).
Give your answer in the form \(p \sqrt { q }\), where \(p\) and \(q\) are constants to be found.