Questions — Edexcel F3 (135 questions)

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Edexcel F3 2023 June Q2
  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
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
  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
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
  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
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
  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
  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
  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
  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
  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
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
  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
  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
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
  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
  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.
Edexcel F3 2021 October Q2
2. Given that $$\cosh y = x \quad \text { and } \quad y < 0$$ use the definition of coshy in terms of exponential functions to prove that $$y = \ln \left( x - \sqrt { x ^ { 2 } - 1 } \right)$$
Edexcel F3 2021 October Q3
3. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$$ The line \(l\) is the normal to \(E\) at the point \(P ( 8 \cos \theta , 6 \sin \theta )\).
  1. Using calculus, show that an equation for \(l\) is $$4 x \sin \theta - 3 y \cos \theta = 14 \sin \theta \cos \theta$$ The line \(l\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\).
    The point \(M\) is the midpoint of \(A B\).
  2. Determine a Cartesian equation for the locus of \(M\) as \(\theta\) varies, giving your answer in the form \(a x ^ { 2 } + b y ^ { 2 } = c\) where \(a , b\) and \(c\) are integers.
Edexcel F3 2021 October Q4
4. The matrix \(\mathbf { M }\) is given by $$\left( \begin{array} { r r r } 2 & 0 & - 1
k & 3 & 2
- 2 & 1 & k \end{array} \right)$$
  1. Show that \(\operatorname { det } \mathbf { M } = 5 k - 10\) Given that \(k \neq 2\)
  2. find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\). The points \(O ( 0,0,0 ) , A ( 4 , - 8,3 ) , B ( - 2,5 , - 4 )\) and \(C ( 4 , - 6,8 )\) are the vertices of a tetrahedron \(T\). The transformation represented by matrix \(\mathbf { M }\) transforms \(T\) to a tetrahedron with volume 50
  3. Determine the possible values of \(k\).
Edexcel F3 2021 October Q5
  1. The skew lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations
$$l _ { 1 } : \mathbf { r } = ( \mathbf { i } + 2 \mathbf { j } - 5 \mathbf { k } ) + \lambda ( 5 \mathbf { i } + \mathbf { j } )$$ and $$l _ { 2 } : \mathbf { r } = ( 2 \mathbf { i } - 4 \mathbf { j } + 4 \mathbf { k } ) + \mu ( 8 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Determine a vector that is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\)
  2. Determine an equation of the plane parallel to \(l _ { 1 }\) that contains \(l _ { 2 }\)
    1. in the form \(\mathbf { r } = \mathbf { a } + s \mathbf { b } + t \mathbf { c }\)
    2. in the form r.n \(= p\)
  3. Determine the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\) Give your answer in simplest form.
Edexcel F3 2021 October Q6
6. $$I _ { n } = \int _ { 0 } ^ { \sqrt { \frac { \pi } { 2 } } } x ^ { n } \cos \left( x ^ { 2 } \right) \mathrm { d } x \quad n \geqslant 1$$
  1. Prove that, for \(n \geqslant 5\) $$I _ { n } = \frac { 1 } { 2 } \left( \frac { \pi } { 2 } \right) ^ { \frac { n - 1 } { 2 } } - \frac { 1 } { 4 } ( n - 1 ) ( n - 3 ) I _ { n - 4 }$$
  2. Hence, determine the exact value of \(I _ { 5 }\), giving your answer in its simplest form.
Edexcel F3 2021 October Q7
7. A hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 25 } = 1$$ where \(a\) is a positive constant.
The eccentricity of \(H\) is \(e\).
  1. Determine an expression for \(e ^ { 2 }\) in terms of \(a\). The line \(l\) is the directrix of \(H\) for which \(x > 0\)
    The points \(A\) and \(A ^ { \prime }\) are the points of intersection of \(l\) with the asymptotes of \(H\).
  2. Determine, in terms of \(e\), the length of the line segment \(A A ^ { \prime }\). The point \(F\) is the focus of \(H\) for which \(x < 0\)
    Given that the area of triangle \(A F A ^ { \prime }\) is \(\frac { 164 } { 3 }\)
  3. show that \(a\) is a solution of the equation $$30 a ^ { 3 } - 164 a ^ { 2 } + 375 a - 4100 = 0$$
  4. Hence, using algebra and making your reasoning clear, show that the only possible value of \(a\) is \(\frac { 20 } { 3 }\)
Edexcel F3 2021 October Q8
8. $$y = \arccos ( 2 \sqrt { x } )$$
  1. Determine \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Show that $$\int y \mathrm {~d} x = x \arccos ( 2 \sqrt { x } ) + \int \frac { \sqrt { x } } { \sqrt { 1 - 4 x } } \mathrm {~d} x$$
  3. Use the substitution \(\sqrt { x } = \frac { 1 } { 2 } \cos \theta\) to show that $$\int _ { 0 } ^ { \frac { 1 } { 8 } } \frac { \sqrt { x } } { \sqrt { 1 - 4 x } } \mathrm {~d} x = \frac { 1 } { 4 } \int _ { a } ^ { b } \cos ^ { 2 } \theta \mathrm {~d} \theta$$ where \(a\) and \(b\) are limits to be determined.
  4. Hence, determine the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 8 } } \arccos ( 2 \sqrt { x } ) d x$$
Edexcel F3 2018 Specimen Q1
  1. The curve \(C\) has equation
$$y = 9 \cosh x + 3 \sinh x + 7 x$$ Use differentiation to find the exact \(x\) coordinate of the stationary point of \(C\), giving your answer as a natural logarithm.
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