Questions F3 (135 questions)

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Edexcel F3 2023 January Q9
  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
  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
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
  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.
Edexcel F3 2024 January Q4
4. $$\mathbf { M } = \left( \begin{array} { r r r } 0 & - 1 & 3
- 1 & 4 & - 1
3 & - 1 & 0 \end{array} \right)$$ Given that \(\left( \begin{array} { r } 1
- 2
1 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\)
  1. determine its corresponding eigenvalue. Given that - 3 is an eigenvalue of \(\mathbf { M }\)
  2. determine a corresponding eigenvector. Hence, given that \(\left( \begin{array} { l } 1
    1
    1 \end{array} \right)\) is also an eigenvector of \(\mathbf { M }\)
  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 2024 January Q5
  1. (a) Use the definitions of hyperbolic functions in terms of exponentials to prove that
$$\begin{gathered} 1 - \operatorname { sech } ^ { 2 } x \equiv \tanh ^ { 2 } x
I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 3 } \ln 2 } \tanh ^ { n } 3 x \mathrm {~d} x \quad n \in \mathbb { Z } \quad n \geqslant 0 \end{gathered}$$ (b) Show that $$I _ { n } = I _ { n - 2 } - \frac { p ^ { n - 1 } } { 3 ( n - 1 ) } \quad n \geqslant 2$$ where \(p\) is a rational number to be determined.
(c) Hence determine the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \ln 2 } \tanh ^ { 5 } 3 x \mathrm {~d} x$$ giving your answer in the form \(a \ln b + c\) where \(a , b\) and \(c\) are rational numbers to be found.
Edexcel F3 2024 January Q6
  1. The points \(A , B\) and \(C\) have coordinates ( \(3,2,2\) ), ( \(- 1,1,3\) ) and ( \(- 2,4,2\) ) respectively. The plane \(\Pi _ { 1 }\) contains the points \(A , B\) and \(C\)
    1. Determine a Cartesian equation of \(\Pi _ { 1 }\)
    Given that
    • point \(D\) has coordinates \(( - 1,1 , - 2 )\)
    • line \(l\) passes through \(D\) and is perpendicular to \(\Pi _ { 1 }\)
    • plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( 14 \mathbf { i } - \mathbf { j } - 17 \mathbf { k } ) = - 66\)
    • \(I\) meets \(\Pi _ { 2 }\) at the point \(E\)
    • show that \(D E = p \sqrt { 22 }\) where \(p\) is a rational number to be determined.
    The point \(F\) has coordinates ( \(4,3 , q\) ) where \(q\) is a constant.
    Given that \(A , B , C\) and \(F\) are the vertices of a tetrahedron of volume 12
  2. determine the possible values of \(q\)
Edexcel F3 2024 January Q7
7.
  1. Show that \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{469976eb-f1a9-4bdc-8f52-64ab23856109-26_1088_691_251_676} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \arccos ( \operatorname { sech } x ) + \operatorname { coth } x \quad x > 0$$ The point \(P\) is a minimum turning point of \(C\)
  2. Show that the \(x\) coordinate of \(P\) is \(\ln ( q + \sqrt { q } )\) where \(q = \frac { 1 } { 2 } ( 1 + \sqrt { k } )\) and \(k\) is an integer to be determined.
Edexcel F3 2024 January Q8
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{469976eb-f1a9-4bdc-8f52-64ab23856109-30_695_904_386_568} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation \(y ^ { 2 } = 8 x\) and part of the line \(l\) with equation \(x = 18\) The region \(R\), shown shaded in Figure 2, is bounded by \(C\) and \(l\)
  1. Show that the perimeter of \(R\) is given by $$\alpha + 2 \int _ { 0 } ^ { \beta } \sqrt { 1 + \frac { y ^ { 2 } } { 16 } } d y$$ where \(\alpha\) and \(\beta\) are positive constants to be determined.
  2. Use the substitution \(y = 4 \sinh u\) and algebraic integration to determine the exact perimeter of \(R\), giving your answer in simplest form.
Edexcel F3 2014 June Q1
  1. Given that \(y = \arctan \left( \frac { 2 x } { 3 } \right)\),
    1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in its simplest form.
    2. Use integration by parts to find
    $$\int \arctan \left( \frac { 2 x } { 3 } \right) \mathrm { d } x$$
Edexcel F3 2014 June Q2
2. The line with equation \(x = 9\) is a directrix of an ellipse with equation $$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 8 } = 1$$ where \(a\) is a positive constant. Find the two possible exact values of the constant \(a\).
Edexcel F3 2014 June Q3
3. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials,
  1. prove that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1$$
  2. find algebraically the exact solutions of the equation $$2 \sinh x + 7 \cosh x = 9$$ giving your answers as natural logarithms.
Edexcel F3 2014 June Q4
4. A non-singular matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left( \begin{array} { l l l } 3 & k & 0
k & 2 & 0
k & 0 & 1 \end{array} \right) \text {, where } k \text { is a constant. }$$
  1. Find, in terms of \(k\), the inverse of the matrix \(\mathbf { M }\). The point \(A\) is mapped onto the point ( \(- 5,10,7\) ) by the transformation represented by the matrix $$\left( \begin{array} { l l l } 3 & 1 & 0
    1 & 2 & 0
    1 & 0 & 1 \end{array} \right)$$
  2. Find the coordinates of the point \(A\).
Edexcel F3 2014 June Q5
  1. Given that
$$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 4 } } \cos ^ { n } \theta \mathrm {~d} \theta , \quad n \geqslant 0$$
  1. prove that, for \(n \geqslant 2\), $$n I _ { n } = \left( \frac { 1 } { \sqrt { 2 } } \right) ^ { n } + ( n - 1 ) I _ { n - 2 }$$
  2. Hence find the exact value of \(I _ { 5 }\), showing each step of your working.
Edexcel F3 2014 June Q6
6. The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$$ The line \(l\) is a tangent to \(H\) at the point \(P ( 4 \cosh \alpha , 2 \sinh \alpha )\), where \(\alpha\) is a constant, \(\alpha \neq 0\)
  1. Using calculus, show that an equation for \(l\) is $$2 y \sinh \alpha - x \cosh \alpha + 4 = 0$$ The line \(l\) cuts the \(y\)-axis at the point \(A\).
  2. Find the coordinates of \(A\) in terms of \(\alpha\). The point \(B\) has coordinates ( \(0,10 \sinh \alpha\) ) and the point \(S\) is the focus of \(H\) for which \(x > 0\)
  3. Show that the line segment \(A S\) is perpendicular to the line segment \(B S\).
Edexcel F3 2014 June Q7
7. The curve \(C\) has parametric equations $$x = 3 t ^ { 2 } , \quad y = 12 t , \quad 0 \leqslant t \leqslant 4$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  1. Show that the area of the surface generated is $$\pi ( a \sqrt { 5 } + b )$$ where \(a\) and \(b\) are constants to be found.
  2. Show that the length of the curve \(C\) is given by $$k \int _ { 0 } ^ { 4 } \sqrt { \left( t ^ { 2 } + 4 \right) } \mathrm { d } t$$ where \(k\) is a constant to be found.
  3. Use the substitution \(t = 2 \sinh \theta\) to show that the exact value of the length of the curve \(C\) is $$24 \sqrt { 5 } + 12 \ln ( 2 + \sqrt { 5 } )$$
Edexcel F3 2014 June Q8
8. The line \(l\) has equation $$\mathbf { r } = ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) + \lambda ( 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) , \text { where } \lambda \text { is a scalar parameter, }$$ and the plane \(\Pi\) has equation $$\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 19$$
  1. Find the coordinates of the point of intersection of \(l\) and \(\Pi\). The perpendicular to \(\Pi\) from the point \(A ( 2,1 , - 2 )\) meets \(\Pi\) at the point \(B\).
  2. Verify that the coordinates of \(B\) are \(( 4,3 , - 6 )\). The point \(A ( 2,1 , - 2 )\) is reflected in the plane \(\Pi\) to give the image point \(A ^ { \prime }\).
  3. Find the coordinates of the point \(A ^ { \prime }\).
  4. Find an equation for the line obtained by reflecting the line \(l\) in the plane \(\Pi\), giving your answer in the form $$\mathbf { r } \times \mathbf { a } = \mathbf { b } ,$$ where \(\mathbf { a }\) and \(\mathbf { b }\) are vectors to be found.
Edexcel F3 2015 June Q1
  1. Find the exact values of \(x\) for which
$$\cosh 2 x - 7 \sinh x = 5$$ giving your answers as natural logarithms.
Edexcel F3 2015 June Q2
2. The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(a\) and \(b\) are positive constants.
The hyperbola \(H\) has eccentricity \(\frac { \sqrt { 21 } } { 4 }\) and passes through the point (12, 5).
Find
  1. the value of \(a\) and the value of \(b\),
  2. the coordinates of the foci of \(H\).
Edexcel F3 2015 June Q3
  1. \(\mathbf { M } = \left( \begin{array} { r r r } 0 & 1 & 9
    1 & 4 & k
    1 & 0 & - 3 \end{array} \right)\), where \(k\) is a constant.
Given that \(\left( \begin{array} { r } 7
19
1 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { M }\),
  1. find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { r } 7
    19
    1 \end{array} \right)\),
  2. show that \(k = - 7\)
  3. find the other two eigenvalues of the matrix \(\mathbf { M }\). The image of the vector \(\left( \begin{array} { c } p
    q
    r \end{array} \right)\) under the transformation represented by \(\mathbf { M }\) is \(\left( \begin{array} { r } - 6
    21
    5 \end{array} \right)\).
  4. Find the values of the constants \(p , q\) and \(r\).
Edexcel F3 2015 June Q4
4. $$I _ { n } = \int \cosh ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$
  1. Show that, for \(n \geqslant 2\) $$n I _ { n } = \sinh x \cosh ^ { n - 1 } x + ( n - 1 ) I _ { n - 2 }$$
  2. Hence find the exact value of $$\int _ { 0 } ^ { \ln 2 } \cosh ^ { 5 } x \mathrm {~d} x$$
Edexcel F3 2015 June Q5
  1. The ellipse \(E\) has equation \(\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1\)
The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.
Given that \(L\) is a tangent to \(E\),
  1. show that $$c ^ { 2 } - 25 m ^ { 2 } = 9$$
  2. find the equations of the tangents to \(E\) which pass through the point \(( 3,4 )\).
Edexcel F3 2015 June Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ddee434-f7e1-4f56-91fc-f487112dbf6b-11_709_1269_292_349} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows the curve \(C\) with parametric equations $$x = 2 \cos \theta - \cos 2 \theta , y = 2 \sin \theta - \sin 2 \theta , \quad 0 \leqslant \theta \leqslant \pi$$
  1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = 8 ( 1 - \cos \theta )$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  2. Find the area of the surface generated, giving your answer in the form \(k \pi\), where \(k\) is a rational number.
Edexcel F3 2015 June Q7
  1. The plane \(\Pi _ { 1 }\) contains the point \(( 3,3 , - 2 )\) and the line \(\frac { x - 1 } { 2 } = \frac { y - 2 } { - 1 } = \frac { z + 1 } { 4 }\)
    1. Show that a cartesian equation of the plane \(\Pi _ { 1 }\) is
    $$3 x - 10 y - 4 z = - 13$$ The plane \(\Pi _ { 2 }\) is parallel to the plane \(\Pi _ { 1 }\)
    The point ( \(\alpha , 1,1\) ), where \(\alpha\) is a constant, lies in \(\Pi _ { 2 }\)
    Given that the shortest distance between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) is \(\frac { 1 } { \sqrt { 5 } }\)
  2. find the possible values of \(\alpha\).
Edexcel F3 2015 June Q8
  1. (a) Show that, under the substitution \(x = \frac { 3 } { 4 } \sinh u\),
$$\int \frac { x ^ { 2 } } { \sqrt { 16 x ^ { 2 } + 9 } } \mathrm {~d} x = k \int ( \cosh 2 u - 1 ) \mathrm { d } u$$ where \(k\) is a constant to be determined.
(b) Hence show that $$\int _ { 0 } ^ { 1 } \frac { 64 x ^ { 2 } } { \sqrt { 16 x ^ { 2 } + 9 } } \mathrm {~d} x = p + q \ln 3$$ where \(p\) and \(q\) are rational numbers to be found.