Questions — Edexcel F3 (138 questions)

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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.
Edexcel F3 2021 October Q2
6 marks Standard +0.8
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
9 marks Challenging +1.8
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
11 marks Challenging +1.2
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
10 marks Standard +0.8
  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
9 marks Challenging +1.8
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
11 marks Challenging +1.8
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
13 marks Challenging +1.2
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 Specimen Q4
8 marks Challenging +1.2
4. \(I _ { n } = \int _ { 0 } ^ { a } ( a - x ) ^ { n } \cos x \mathrm {~d} x , \quad a > 0 , \quad n \geqslant 0\)
  1. Show that, for \(n \geqslant 2\), $$I _ { n } = n a ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$
  2. Hence evaluate \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \left( \frac { \pi } { 2 } - x \right) ^ { 2 } \cos x d x\)
Edexcel F3 Specimen Q5
9 marks Challenging +1.2
5. Given that \(y = ( \operatorname { arcosh } 3 x ) ^ { 2 }\), where \(3 x > 1\), show that
  1. \(\left( 9 x ^ { 2 } - 1 \right) \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = 36 y\),
  2. \(\left( 9 x ^ { 2 } - 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 9 x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 18\).
Edexcel F3 Specimen Q6
13 marks Standard +0.3
6. \(\mathbf { M } = \left( \begin{array} { c c c } 1 & 0 & 3 \\ 0 & - 2 & 1 \\ k & 0 & 1 \end{array} \right)\), where \(k\) is a constant.
Given that \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\) ,
  1. find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\) ,
  2. show that \(k = 3\) ,
  3. show that \(\mathbf { M }\) has exactly two eigenvalues. A transformation \(T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by \(\mathbf { M }\) .
    The transformation \(T\) maps the line \(l _ { 1 }\) ,with cartesian equations \(\frac { x - 2 } { 1 } = \frac { y } { - 3 } = \frac { z + 1 } { 4 }\) ,onto the line \(l _ { 2 }\) .
    6. \(\mathbf { M } = \left( \begin{array} { c c c } 0 & - 2 & 1 \\ k & 0 & 1 \end{array} \right)\), where \(k\) is a constant. Given that \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\) ,
    1. find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\)
    2. Taking \(k = 3\) ,find cartesian equations of \(l _ { 2 }\) .
Edexcel F3 Specimen Q7
14 marks Standard +0.8
  1. The plane \(\Pi\) has vector equation
$$\mathbf { r } = 3 \mathbf { i } + \mathbf { k } + \lambda ( - 4 \mathbf { i } + \mathbf { j } ) + \mu ( 6 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$
  1. Find an equation of \(\Pi\) in the form \(\mathbf { r } . \mathbf { n } = p\), where \(\mathbf { n }\) is a vector perpendicular to \(\Pi\) and \(p\) is a constant. The point \(P\) has coordinates \(( 6,13,5 )\). The line \(l\) passes through \(P\) and is perpendicular to \(\Pi\). The line \(l\) intersects \(\Pi\) at the point \(N\).
  2. Show that the coordinates of \(N\) are \(( 3,1 , - 1 )\). The point \(R\) lies on \(\Pi\) and has coordinates \(( 1,0,2 )\).
  3. Find the perpendicular distance from \(N\) to the line \(P R\). Give your answer to 3 significant figures.
Edexcel F3 Specimen Q8
13 marks Challenging +1.3
  1. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1\).
The line \(l _ { 1 }\) is the tangent to \(H\) at the point \(P ( 4 \sec t , 2 \tan t )\).
  1. Use calculus to show that an equation of \(l _ { 1 }\) is $$2 y \sin t = x - 4 \cos t$$ The line \(l _ { 2 }\) passes through the origin and is perpendicular to \(l _ { 1 }\).
    The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(Q\).
  2. Show that, as \(t\) varies, an equation of the locus of \(Q\) is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 16 x ^ { 2 } - 4 y ^ { 2 }$$
Edexcel F3 2021 June Q1
6 marks Standard +0.3
  1. Using the definitions of hyperbolic functions in terms of exponentials, show that $$1 - \tanh^2 x = \operatorname{sech}^2 x$$ [3]
  2. Solve the equation $$2\operatorname{sech}^2 x + 3\tanh x = 3$$ giving your answer as an exact logarithm. [3]
Edexcel F3 2021 June Q2
7 marks Challenging +1.2
A curve has equation $$y = \sqrt{9 - x^2} \quad 0 \leq x \leq 3$$
  1. Using calculus, show that the length of the curve is \(\frac{3\pi}{2}\) [4]
The curve is rotated through \(2\pi\) radians about the \(x\)-axis.
  1. Using calculus, find the exact area of the surface generated. [3]
Edexcel F3 2021 June Q3
9 marks Standard +0.8
$$\mathbf{M} = \begin{pmatrix} 3 & 1 & p \\ 1 & 1 & 2 \\ -1 & p & 2 \end{pmatrix}$$ where \(p\) is a real constant
  1. Find the exact values of \(p\) for which \(\mathbf{M}\) has no inverse. [4]
Given that \(\mathbf{M}\) does have an inverse,
  1. find \(\mathbf{M}^{-1}\) in terms of \(p\). [5]