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Edexcel F3 2021 January Q5
9 marks Standard +0.8
5. $$\mathbf { M } = \left( \begin{array} { r r r } 6 & - 2 & - 1 \\ - 2 & 6 & - 1 \\ - 1 & - 1 & 5 \end{array} \right)$$ Given that 8 is an eigenvalue of \(\mathbf { M }\)
  1. determine an eigenvector corresponding to the eigenvalue 8
  2. Determine the other two eigenvalues of \(\mathbf { M }\).
  3. Hence find an orthogonal matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { T } \mathbf { M P } = \mathbf { D }\) 5.
Edexcel F3 2021 January Q6
10 marks Challenging +1.2
6. $$I _ { n } = \int \frac { x ^ { n } } { \sqrt { x ^ { 2 } + 3 } } \mathrm {~d} x \quad n \in \mathbb { N }$$
  1. Show that $$I _ { n } = \frac { x ^ { n - 1 } } { n } \left( x ^ { 2 } + 3 \right) ^ { \frac { 1 } { 2 } } - \frac { 3 ( n - 1 ) } { n } I _ { n - 2 } \quad n \geqslant 3$$
  2. Hence show that $$\int \frac { x ^ { 5 } } { \sqrt { x ^ { 2 } + 3 } } \mathrm {~d} x = \frac { 1 } { 5 } \left( x ^ { 2 } + 3 \right) ^ { \frac { 1 } { 2 } } \left( x ^ { 4 } + p x ^ { 2 } + q \right) + k$$ where \(p\) and \(q\) are integers to be determined and \(k\) is an arbitrary constant.
Edexcel F3 2021 January Q7
11 marks Standard +0.8
  1. The point \(P\) has coordinates \(( 1,2,1 )\)
The line \(l\) has Cartesian equation $$\frac { x - 3 } { 5 } = \frac { y + 1 } { 3 } = \frac { z + 5 } { - 8 }$$ The plane \(\Pi _ { 1 }\) contains the point \(P\) and the line \(l\).
  1. Show that a Cartesian equation for \(\Pi _ { 1 }\) is $$6 x - 2 y + 3 z = 5$$ The point \(Q\) has coordinates \(( 2 , k , - 7 )\), where \(k\) is a constant.
  2. Show that the shortest distance between \(\Pi _ { 1 }\) and \(Q\) is $$\frac { 2 } { 7 } | k + 7 |$$ The plane \(\Pi _ { 2 }\) has Cartesian equation \(8 x - 4 y + z = - 3\) Given that the shortest distance between \(\Pi _ { 1 }\) and \(Q\) is the same as the shortest distance between \(\Pi _ { 2 }\) and \(Q\),
  3. determine the possible values of \(k\).
Edexcel F3 2021 January Q8
9 marks Challenging +1.2
  1. The curve \(C\) has equation
$$y = 2 + \ln \left( 1 - x ^ { 2 } \right) \quad \frac { 1 } { 2 } \leqslant x \leqslant \frac { 3 } { 4 }$$
  1. Show that the length of the curve \(C\) is given by $$\int _ { \frac { 1 } { 2 } } ^ { \frac { 3 } { 4 } } \left( \frac { 1 + x ^ { 2 } } { 1 - x ^ { 2 } } \right) \mathrm { d } x$$
  2. Hence, using algebraic integration, show that the length of the curve \(C\) is \(p + \ln q\) where \(p\) and \(q\) are rational numbers to be determined.
Edexcel F3 2021 January Q9
12 marks Challenging +1.2
9. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$$ The point \(P\) lies on the ellipse and has coordinates \(( 5 \cos \theta , 4 \sin \theta )\) where \(0 < \theta < \frac { \pi } { 2 }\) The line \(l\) is the normal to the ellipse at the point \(P\).
  1. Show that an equation for \(l\) is $$5 x \sin \theta - 4 y \cos \theta = 9 \sin \theta \cos \theta$$ The point \(F\) is the focus of \(E\) that lies on the positive \(x\)-axis.
  2. Determine the coordinates of \(F\). The line \(l\) crosses the \(x\)-axis at the point \(Q\).
  3. Show that $$\frac { | Q F | } { | P F | } = e$$ where \(e\) is the eccentricity of \(E\).
    END
Edexcel F3 2022 January Q1
8 marks Challenging +1.2
1
  1. Use the definitions of hyperbolic functions in terms of exponentials to prove that $$8 \cosh ^ { 4 } x = \cosh 4 x + p \cosh 2 x + q$$ where \(p\) and \(q\) are constants to be determined.
  2. Hence, or otherwise, solve the equation $$\cosh 4 x - 17 \cosh 2 x + 9 = 0$$ giving your answers in exact simplified form in terms of natural logarithms.
Edexcel F3 2022 January Q2
8 marks Challenging +1.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cfc4afbd-3353-4f9f-b954-cb5178ebcf6c-06_624_872_210_543} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the curve \(C\) with parametric equations $$x = \ln ( \sec \theta + \tan \theta ) - \sin \theta \quad y = \cos \theta \quad 0 \leqslant \theta \leqslant \frac { \pi } { 4 }$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis and is used to form a solid of revolution \(S\). Using calculus, show that the total surface area of \(S\) is given by $$\frac { \pi } { 2 } ( p + q \sqrt { 2 } )$$ where \(p\) and \(q\) are integers to be determined.
Edexcel F3 2022 January Q3
9 marks Challenging +1.2
3. (a) Given that \(y = \operatorname { arsech } \left( \frac { x } { 2 } \right)\), where \(0 < x \leqslant 2\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { p } { x \sqrt { q - x ^ { 2 } } }$$ where \(p\) and \(q\) are constants to be determined. In part (b) solutions based entirely on calculator technology are not acceptable. $$\mathrm { f } ( x ) = \operatorname { artanh } ( x ) + \operatorname { arsech } \left( \frac { x } { 2 } \right) \quad 0 < x \leqslant 1$$ (b) Determine, in simplest form, the exact value of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) = 0\)
Edexcel F3 2022 January Q4
9 marks Standard +0.8
4. $$\mathbf { M } = \left( \begin{array} { l l l } 6 & k & 2 \\ k & 5 & 0 \\ 2 & 0 & 7 \end{array} \right)$$ where \(k\) is a constant. Given that 3 is an eigenvalue of \(\mathbf { M }\),
  1. determine the possible values of \(k\). Given that \(k < 0\)
  2. determine the other eigenvalues of \(\mathbf { M }\).
  3. Determine a normalised eigenvector corresponding to the eigenvalue 3
Edexcel F3 2022 January Q5
7 marks Standard +0.8
5. Determine
  1. \(\int \frac { 1 } { \sqrt { x ^ { 2 } - 3 x + 5 } } \mathrm {~d} x\)
  2. \(\int \frac { 1 } { \sqrt { 63 + 4 x - 4 x ^ { 2 } } } \mathrm {~d} x\)
Edexcel F3 2022 January Q6
10 marks Challenging +1.8
6. $$I _ { n } = \int \mathrm { e } ^ { x } \sin ^ { n } x \mathrm {~d} x \quad n \in \mathbb { Z } \quad n \geqslant 0$$
  1. Show that $$I _ { n } = \frac { \mathrm { e } ^ { x } \sin ^ { n - 1 } x } { n ^ { 2 } + 1 } ( \sin x - n \cos x ) + \frac { n ( n - 1 ) } { n ^ { 2 } + 1 } I _ { n - 2 } \quad n \geqslant 2$$
  2. Hence find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } e ^ { x } \sin ^ { 4 } x d x$$ giving your answer in the form \(A \mathrm { e } ^ { \frac { \pi } { 2 } } + B\) where \(A\) and \(B\) are rational numbers to be determined.
Edexcel F3 2022 January Q7
11 marks Standard +0.8
7. The line \(l _ { 1 }\) has equation $$\frac { x - 3 } { 4 } = \frac { y - 5 } { - 2 } = \frac { z - 4 } { 7 }$$ The plane \(\Pi\) has equation $$2 x + 4 y - z = 1$$ The line \(l _ { 1 }\) intersects the plane \(\Pi\) at the point \(P\)
  1. Determine the coordinates of \(P\) The acute angle between \(l _ { 1 }\) and \(\Pi\) is \(\theta\) degrees.
  2. Determine, to one decimal place, the value of \(\theta\) The line \(l _ { 2 }\) lies in \(\Pi\) and passes through \(P\) Given that the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is also \(\theta\) degrees,
  3. determine a vector equation for \(l _ { 2 }\)
Edexcel F3 2022 January Q8
13 marks Challenging +1.2
8. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$
  1. Determine the eccentricity of \(E\)
  2. Hence, for this ellipse, determine
    1. the coordinates of the foci,
    2. the equations of the directrices. The point \(P\) lies on \(E\) and has coordinates \(( 3 \cos \theta , 2 \sin \theta )\). The line \(l _ { 1 }\) is the tangent to \(E\) at the point \(P\)
  3. Using calculus, show that an equation for \(l _ { 1 }\) is $$2 x \cos \theta + 3 y \sin \theta = 6$$ The line \(l _ { 2 }\) passes through the origin and is perpendicular to \(l _ { 1 }\) The line \(l _ { 1 }\) intersects the line \(l _ { 2 }\) at the point \(Q\)
  4. Determine the coordinates of \(Q\)
  5. Show that, as \(\theta\) varies, the point \(Q\) lies on the curve with equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = \alpha x ^ { 2 } + \beta y ^ { 2 }$$ where \(\alpha\) and \(\beta\) are constants to be determined.
    \includegraphics[max width=\textwidth, alt={}]{cfc4afbd-3353-4f9f-b954-cb5178ebcf6c-36_2817_1962_105_105}
Edexcel F3 2023 January Q1
3 marks Standard +0.8
  1. Given that
$$y = 3 x \arcsin 2 x \quad 0 \leqslant x \leqslant \frac { 1 } { 2 }$$
  1. determine an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Hence determine the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = \frac { 1 } { 4 }\), giving your answer in the form \(a \pi + b\) where \(a\) and \(b\) are fully simplified constants to be found.
Edexcel F3 2023 January Q2
6 marks Standard +0.8
  1. A hyperbola \(H\) has equation
$$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 5 } = 1 \quad \text { where } a \text { is a positive constant }$$ The line with equation \(x = \frac { 4 } { 3 }\) is a directrix of \(H\)
  1. Write down an equation of the other directrix.
  2. Determine
    1. the value of \(a\)
    2. the coordinates of each of the foci of \(H\)
Edexcel F3 2023 January Q3
6 marks Standard +0.8
  1. Solve the equation
$$4 \tanh x - \operatorname { sech } x = 1$$ giving your answer in the form \(x = \ln k\) where \(k\) is a fully simplified rational number.
(6)
Edexcel F3 2023 January Q4
5 marks Standard +0.8
  1. (a) Determine
$$\int \frac { 1 } { \sqrt { 9 x ^ { 2 } + 16 } } \mathrm {~d} x$$ (b) Hence determine the exact value of $$\int _ { - 2 } ^ { 2 } \frac { 1 } { \sqrt { 9 x ^ { 2 } + 16 } } d x$$ Give your answer in the form \(a \ln ( b + c \sqrt { 13 } )\), where \(a , b\) and \(c\) are rational numbers.
Edexcel F3 2023 January Q5
12 marks Standard +0.8
5. $$\mathbf { A } = \left( \begin{array} { r r r } a & a & 1 \\ - a & 4 & 0 \\ 4 & a & 5 \end{array} \right) \quad \text { where } a \text { is a positive constant }$$
  1. Determine the exact value of \(a\) for which the matrix \(\mathbf { A }\) is singular. Given that 2 is an eigenvalue of \(\mathbf { A }\)
  2. determine
    1. the value of \(a\)
    2. the other two eigenvalues of \(\mathbf { A }\) A normalised eigenvector for the eigenvalue 2 is \(\left( \begin{array} { c } \frac { 1 } { \sqrt { 6 } } \\ \frac { 1 } { \sqrt { 6 } } \\ - \frac { 2 } { \sqrt { 6 } } \end{array} \right)\)
  3. Determine a normalised eigenvector for each of the other eigenvalues of \(\mathbf { A }\)
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Edexcel F3 2023 January Q6
9 marks Challenging +1.2
  1. A curve has parametric equations
    where \(a\) is a positive constant.
$$\begin{aligned} & x = a ( \theta - \sin \theta ) \\ & y = a ( 1 - \cos \theta ) \end{aligned}$$
  1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = k a ^ { 2 } \sin ^ { 2 } \frac { \theta } { 2 }$$ where \(k\) is a constant to be determined. The part of the curve from \(\theta = 0\) to \(\theta = 2 \pi\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  2. Determine the area of the surface generated, giving your answer in terms of \(\pi\) and \(a\).
    [0pt] [Solutions relying on calculator technology are not acceptable.]
Edexcel F3 2023 January Q7
10 marks Standard +0.3
  1. The plane \(\Pi\) has equation
$$\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 0 \\ 3 \\ - 2 \end{array} \right) + \mu \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Determine a vector perpendicular to \(\Pi\) The line \(l\) meets \(\Pi\) at the point ( \(1,2,3\) ) and passes through the point ( \(1,0,1\) )
  2. Determine the size of the acute angle between \(\Pi\) and \(l\) Give your answer to the nearest degree.
  3. Determine the shortest distance between \(\Pi\) and the point \(( 6 , - 3 , - 6 )\)
Edexcel F3 2023 January Q8
11 marks Challenging +1.2
8. $$I _ { n } = \int \cos ^ { n } x \mathrm {~d} x \quad n \geqslant 0$$
  1. Prove that for \(n \geqslant 2\) $$I _ { n } = \frac { 1 } { n } \cos ^ { n - 1 } x \sin x + \frac { n - 1 } { n } I _ { n - 2 }$$
  2. Show that for positive even integers \(n\) $$\int _ { 0 } ^ { \overline { 2 } } \cos ^ { n } x d x = \frac { ( n - 1 ) ( n - 3 ) \ldots 5 \times 3 \times 1 } { n ( n - 2 ) ( n - 4 ) \ldots 6 \times 4 \times 2 } \times \overline { 2 }$$
  3. Hence determine the exact value of $$\int _ { 0 } ^ { \overline { 2 } } \cos ^ { 6 } x \sin ^ { 2 } x d x$$
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Edexcel F3 2023 January Q9
13 marks Challenging +1.2
  1. The ellipse \(E\) has equation
$$x ^ { 2 } + 9 y ^ { 2 } = 9$$ The foci of \(E\) are \(F _ { 1 }\) and \(F _ { 2 }\)
    1. Determine the coordinates of \(F _ { 1 }\) and the coordinates of \(F _ { 2 }\)
    2. Write down the equation of each of the directrices of \(E\) The point \(P\) lies on the ellipse.
  2. Show that \(\left| P F _ { 1 } \right| + \left| P F _ { 2 } \right| = 6\) The straight line through \(P\) with equation \(y = 2 x + c\) meets \(E\) again at the point \(Q\) The point \(M\) is the midpoint of \(P Q\)
  3. Show that as \(P\) varies the locus of \(M\) is a straight line passing through the origin.
Edexcel F3 2024 January Q1
7 marks Moderate -0.3
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Show that $$\int _ { 4 } ^ { 4 \sqrt { 3 } } \frac { 8 } { 16 + x ^ { 2 } } d x = p \pi$$ where \(p\) is a rational number to be determined.
  2. Determine the exact value of \(k\) for which $$\int _ { \frac { 3 } { 4 } } ^ { k } \frac { 2 } { \sqrt { 9 - 4 x ^ { 2 } } } d x = \frac { \pi } { 12 }$$
Edexcel F3 2024 January Q2
8 marks Standard +0.8
2. $$\mathbf { T } = \left( \begin{array} { l l l } 2 & 3 & 7 \\ 3 & 2 & 6 \\ a & 4 & b \end{array} \right) \quad \mathbf { U } = \left( \begin{array} { r r r } 6 & - 1 & - 4 \\ 15 & c & - 9 \\ - 8 & a & 5 \end{array} \right)$$ where \(a\), \(b\) and \(c\) are constants.
Given that \(\mathbf { T U } = \mathbf { I }\)
  1. determine the value of \(a\), the value of \(b\) and the value of \(c\) The transformation represented by the matrix \(\mathbf { T }\) transforms the line \(l _ { 1 }\) to the line \(l _ { 2 }\) Given that \(l _ { 2 }\) has equation $$\frac { x - 1 } { 3 } = \frac { y } { - 4 } = z + 2$$
  2. determine a Cartesian equation for \(l _ { 1 }\)
Edexcel F3 2024 January Q3
11 marks Challenging +1.2
  1. The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(b\) is a constant and \(0 < b < 7\) The eccentricity of the ellipse is \(e\)
  1. Write down, in terms of \(e\) only,
    1. the coordinates of the foci of \(E\)
    2. the equations of the directrices of \(E\) Given that
      • the point \(P ( x , y )\) lies on \(E\) where \(x > 0\)
  2. the point \(S\) is the focus of \(E\) on the positive \(x\)-axis
  3. the line \(l\) is the directrix of \(E\) which crosses the positive \(x\)-axis
  4. the point \(M\) lies on \(l\) such that the line through \(P\) and \(M\) is parallel to the \(x\)-axis
  5. determine an expression for
    1. \(P S ^ { 2 }\) in terms of \(e , x\) and \(y\)
    2. \(P M ^ { 2 }\) in terms of \(e\) and \(x\)
  6. Hence show that
    7. $$b ^ { 2 } = 49 \left( 1 - e ^ { 2 } \right)$$ Given that \(E\) crosses the \(y\)-axis at the points with coordinates \(( 0 , \pm 4 \sqrt { 3 } )\)
  7. determine the value of \(e\) Given that the \(x\) coordinate of \(P\) is \(\frac { 7 } { 2 }\)
  8. determine the area of triangle \(O P M\), where \(O\) is the origin.