Questions — Edexcel F3 (135 questions)

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Edexcel F3 2021 January Q1
  1. Relative to a fixed origin \(O\), the points \(A\), \(B\), \(C\) and \(D\) have coordinates \(( 0,4,1 ) , ( 4,0,0 )\), \(( 3,5,2 )\) and \(( 2,2 , k )\) respectively, where \(k\) is a constant.
    1. Determine the exact area of triangle \(A B C\).
    2. Determine in terms of \(k\), the volume of the tetrahedron \(A B C D\), simplifying your answer. \(( 3,5,2 )\) and \(( 2,2 , k )\) respectively, where \(k\) is a constant.
    3. Determine the exact area of triangle \(A B C\).
    $$\text { etrahedron } A B C D \text {, simplifying }$$
Edexcel F3 2021 January Q2
2. $$y = \ln ( \tanh 2 x ) \quad x > 0$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = p \operatorname { cosech } 4 x$$ where \(p\) is a constant to be determined.
  2. Hence determine, in simplest form, the exact value of \(x\) for which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\)
Edexcel F3 2021 January Q3
3. $$\mathbf { A } = \left( \begin{array} { l l l } 2 & k & 2
2 & 2 & k
1 & 2 & 2 \end{array} \right) \quad \text { where } k \text { is a constant }$$
  1. Determine the values of \(k\) for which \(\mathbf { A }\) is singular. Given that \(\mathbf { A }\) is non-singular,
  2. find \(\mathbf { A } ^ { - 1 }\), giving your answer in terms of \(k\).
    3.
Edexcel F3 2021 January Q4
4. Using the substitution \(x = 4 \cosh \theta\) show that $$\int \frac { 1 } { \left( x ^ { 2 } - 16 \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x = \frac { a x } { \sqrt { x ^ { 2 } - 16 } } + c \quad | x | > 4$$ where \(a\) is a constant to be determined and \(c\) is an arbitrary constant.
(6)
Edexcel F3 2021 January Q5
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
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
  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
  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
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
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
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
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
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
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
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
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
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
  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
  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
  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
  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
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
  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
  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
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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