Questions F3 (135 questions)

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Edexcel F3 2020 June Q1
  1. (a) Use the definition of \(\sinh x\) in terms of exponentials to show that
$$\sinh 3 x \equiv 4 \sinh ^ { 3 } x + 3 \sinh x$$ (b) Hence determine the exact coordinates of the points of intersection of the curve with equation \(y = \sinh 3 x\) and the curve with equation \(y = 19 \sinh x\), giving your answers as simplified logarithms where necessary.
Edexcel F3 2020 June Q2
2. Determine
  1. \(\int \frac { 1 } { 3 x ^ { 2 } + 12 x + 24 } \mathrm {~d} x\)
  2. \(\int \frac { 1 } { \sqrt { 27 - 6 x - x ^ { 2 } } } \mathrm {~d} x\)
Edexcel F3 2020 June Q3
3. $$\mathbf { M } = \left( \begin{array} { c c c } 3 & - 4 & k
1 & - 2 & k
1 & - 5 & 5 \end{array} \right) \text { where } k \text { is a constant }$$ Given that 3 is an eigenvalue of \(\mathbf { M }\),
  1. find the value of \(k\).
  2. Hence find the other two eigenvalues of \(\mathbf { M }\).
  3. Find a normalised eigenvector corresponding to the eigenvalue 3
    .
    VIIIV SIHI NI JIIHM ION OCVARV SHAL NI ALIAM LON OOVERV SIHI NI JIIIM ION OO
Edexcel F3 2020 June Q4
4.
  1. Show that, for \(n \geqslant 2\)
  2. Hence find the functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) such that $$\int x ^ { 4 } \cos x \mathrm {~d} x = \mathrm { f } ( x ) \sin x + \mathrm { g } ( x ) \cos x + c$$ where \(c\) is an arbitrary constant. $$I _ { n } = \int x ^ { n } \cos x \mathrm {~d} x$$
  3. Show that, for \(n \geqslant 2\) $$I _ { n } = x ^ { n } \sin x + n x ^ { n - 1 } \cos x - n ( n - 1 ) I _ { n - 2 }$$
  4. Hence find the functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) such that
Edexcel F3 2020 June Q5
5. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 4 } = 1\) The line \(l\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants. Given that \(l\) is a tangent to \(H\),
  1. show that \(25 m ^ { 2 } = 4 + c ^ { 2 }\)
  2. Hence find the equations of the tangents to \(H\) that pass through the point ( 1,2 ).
  3. Find the coordinates of the point of contact each of these tangents makes with \(H\).
Edexcel F3 2020 June Q6
6. $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 1 & 1
1 & 1 & 1
1 & 2 & a \end{array} \right) \quad a \neq 1$$
  1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).
    . The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\mathbf { B }\). $$\mathbf { B } = \left( \begin{array} { r r r } 1 & - 1 & 1
    1 & 1 & 1
    1 & 2 & 4 \end{array} \right)$$ The equation of \(l _ { 2 }\) is $$( \mathbf { r } - ( 12 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } ) ) \times ( - 6 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) = \mathbf { 0 }$$
  2. Find a vector equation for the line \(l _ { 1 }\)
Edexcel F3 2020 June Q7
7. The curve \(C\) has parametric equations $$x = \cosh t + t , \quad y = \cosh t - t \quad 0 \leqslant t \leqslant \ln 3$$
  1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = 2 \cosh ^ { 2 } t$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. The area of the curved surface generated is given by \(S\).
  2. Show that $$S = 2 \pi \sqrt { 2 } \int _ { 0 } ^ { \ln 3 } \left( \cosh ^ { 2 } t - t \cosh t \right) d t$$
  3. Hence find the value of \(S\), giving your answer in the form $$\frac { \pi \sqrt { 2 } } { 9 } ( a + b \ln 3 )$$ where \(a\) and \(b\) are constants to be determined.
Edexcel F3 2020 June Q8
8. The plane \(\Pi _ { 1 }\) has equation $$x - 5 y + 3 z = 11$$ The plane \(\Pi _ { 2 }\) has equation $$3 x - 2 y + 2 z = 7$$ The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line \(l\).
  1. Find a vector equation for \(l\), giving 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 point \(P ( 2,0,3 )\) lies on \(\Pi _ { 1 }\) The line \(m\), which passes through \(P\), is parallel to \(l\). The point \(Q ( 3,2,1 )\) lies on \(\Pi _ { 2 }\)
    The line \(n\), which passes through \(Q\), is also parallel to \(l\).
  2. Find, in exact simplified form, the shortest distance between \(m\) and \(n\).
    VIIV STHI NI JINM ION OCVIAV SIHI NI JMAM/ION OCVIAV SIHL NI JIIYM ION OO
Edexcel F3 2021 June Q1
  1. (a) Using the definitions of hyperbolic functions in terms of exponentials, show that
$$1 - \tanh ^ { 2 } x \equiv \operatorname { sech } ^ { 2 } x$$ (b) Solve the equation $$2 \operatorname { sech } ^ { 2 } x + 3 \tanh x = 3$$ giving your answer as an exact logarithm.
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Edexcel F3 2021 June Q2
2. A curve has equation $$y = \sqrt { 9 - x ^ { 2 } } \quad 0 \leqslant x \leqslant 3$$
  1. Using calculus, show that the length of the curve is \(\frac { 3 \pi } { 2 }\) The curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
  2. Using calculus, find the exact area of the surface generated.
    \includegraphics[max width=\textwidth, alt={}, center]{d7a92540-e36c-4e00-bbe4-b253e09962f8-07_2647_1840_118_111}
Edexcel F3 2021 June Q3
3. \(\mathbf { M } = \left( \begin{array} { r r r } 3 & 1 & p
1 & 1 & 2
- 1 & p & 2 \end{array} \right)\) where \(p\) is a real constant (a) Find the exact values of \(p\) for which \(\mathbf { M }\) has no inverse. Given that \(\mathbf { M }\) does have an inverse, (b) find \(\mathbf { M } ^ { - 1 }\) in terms of \(p\).
3. \(\mathbf { M } = \left( \begin{array} { r c c } 3 & 1 & p
1 & 1 & 2
- 1 & p & 2 \end{array} \right)\) where \(p\) is a real constant
\includegraphics[max width=\textwidth, alt={}, center]{d7a92540-e36c-4e00-bbe4-b253e09962f8-11_2647_1840_118_111}
Edexcel F3 2021 June Q4
4. (i) $$f ( x ) = x \arccos x \quad - 1 \leqslant x \leqslant 1$$ Find the exact value of \(f ^ { \prime } ( 0.5 )\).
(ii) $$\mathrm { g } ( x ) = \arctan \left( \mathrm { e } ^ { 2 x } \right)$$ Show that $$\mathrm { g } ^ { \prime \prime } ( x ) = k \operatorname { sech } ( 2 x ) \tanh ( 2 x )$$ where \(k\) is a constant to be found.
\includegraphics[max width=\textwidth, alt={}, center]{d7a92540-e36c-4e00-bbe4-b253e09962f8-15_2647_1840_119_114}
  1. Prove that for \(n \geqslant 2\) $$( n - 1 ) I _ { n } = \tan x \sec ^ { n - 2 } x + ( n - 2 ) I _ { n - 2 }$$
  2. Hence, showing each step of your working, find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 4 } } \sec ^ { 6 } x d x$$ $$I _ { n } = \int \sec ^ { n } x \mathrm {~d} x \quad n \geqslant 0$$ Prove that for \(n > 2\)
Edexcel F3 2021 June Q6
  1. The line \(l _ { 1 }\) has equation
$$\mathbf { r } = \mathbf { i } + \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } + 3 \mathbf { k } )$$ and the line \(l _ { 2 }\) has equation $$\mathrm { r } = 2 \mathbf { i } + s \mathbf { j } + \mu ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$ where \(s\) is a constant and \(\lambda\) and \(\mu\) are scalar parameters.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) both lie in a common plane \(\Pi _ { 1 }\)
  1. show that an equation for \(\Pi _ { 1 }\) is \(3 x + y - z = 3\)
  2. find the value of \(s\). The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 3\)
  3. Find an equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)
  4. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) giving your answer in degrees to 3 significant figures.
Edexcel F3 2021 June Q7
  1. Using calculus, find the exact values of
    1. \(\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 2 } - 4 x + 5 } \mathrm {~d} x\)
    2. \(\int _ { \sqrt { 3 } } ^ { 3 } \frac { \sqrt { x ^ { 2 } - 3 } } { x ^ { 2 } } \mathrm {~d} x\)
    \includegraphics[max width=\textwidth, alt={}, center]{d7a92540-e36c-4e00-bbe4-b253e09962f8-27_2647_1840_118_111}
Edexcel F3 2021 June Q8
8. The hyperbola \(H\) has equation $$4 x ^ { 2 } - y ^ { 2 } = 4$$
  1. Write down the equations of the asymptotes of \(H\).
  2. Find the coordinates of the foci of \(H\). The point \(P ( \sec \theta , 2 \tan \theta )\) lies on \(H\).
  3. Using calculus, show that the equation of the tangent to \(H\) at the point \(P\) is $$y \tan \theta = 2 x \sec \theta - 2$$ The point \(V ( - 1,0 )\) and the point \(W ( 1,0 )\) both lie on \(H\).
    The point \(Q ( \sec \theta , - 2 \tan \theta )\) also lies on \(H\).
    Given that \(P , Q , V\) and \(W\) are distinct points on \(H\) and that the lines \(V P\) and \(W Q\) intersect at the point \(S\),
  4. show that, as \(\theta\) varies, \(S\) lies on an ellipse with equation $$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(a\) and \(b\) are integers to be found.
    \includegraphics[max width=\textwidth, alt={}]{d7a92540-e36c-4e00-bbe4-b253e09962f8-32_2647_1835_118_116}
Edexcel F3 2022 June Q1
  1. (a) Use the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials to show that
$$\cosh A \cosh B + \sinh A \sinh B \equiv \cosh ( A + B )$$ (b) Hence find the value of \(x\) for which $$\cosh ( x + \ln 2 ) = 5 \sinh x$$ giving your answer in the form \(\frac { 1 } { 2 } \ln k\), where \(k\) is a rational number to be determined.
(5)
Edexcel F3 2022 June Q2
  1. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
  1. Determine $$\int \frac { 1 } { \sqrt { 5 + 4 x - x ^ { 2 } } } d x$$
  2. Use the substitution \(x = 3 \sec \theta\) to determine the exact value of $$\int _ { 2 \sqrt { 3 } } ^ { 6 } \frac { 18 } { \left( x ^ { 2 } - 9 \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$ Give your answer in the form \(A + B \sqrt { 3 }\) where \(A\) and \(B\) are constants to be found.
Edexcel F3 2022 June Q3
3. $$\mathbf { M } = \left( \begin{array} { r r r } - 2 & 5 & 0
5 & 1 & - 3
0 & - 3 & 6 \end{array} \right)$$ Given that \(\mathbf { i } + \mathbf { j } + \mathbf { k }\) is an eigenvector of \(\mathbf { M }\),
  1. determine the corresponding eigenvalue. Given that 8 is an eigenvalue of \(\mathbf { M }\),
  2. determine a corresponding eigenvector.
  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 2022 June Q4
4. $$y = \operatorname { artanh } \left( \frac { \cos x + a } { \cos x - a } \right)$$ where \(a\) is a non-zero constant.
Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k \tan x$$ where \(k\) is a constant to be determined.
Edexcel F3 2022 June Q5
  1. A curve has parametric equations
$$x = 4 \mathrm { e } ^ { \frac { 1 } { 2 } t } \quad y = \mathrm { e } ^ { t } - t \quad 0 \leqslant t \leqslant 4$$ The curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
Show that the area of the curved surface generated is $$\pi \left( \mathrm { e } ^ { 8 } + A \mathrm { e } ^ { 4 } + B \right)$$ where \(A\) and \(B\) are constants to be determined.
Edexcel F3 2022 June Q6
6. $$\mathbf { A } = \left( \begin{array} { r r r } x & 1 & 3
2 & 4 & x
- 4 & - 2 & - 1 \end{array} \right)$$
  1. Show that \(\mathbf { A }\) is non-singular for all real values of \(x\).
  2. Determine, in terms of \(x , \mathbf { A } ^ { - 1 }\)
Edexcel F3 2022 June Q7
7. $$I _ { n } = \int \frac { x ^ { n } } { \sqrt { 10 - x ^ { 2 } } } \mathrm {~d} x \quad n \in \mathbb { N } \quad | x | < \sqrt { 10 }$$
  1. Show that $$n I _ { n } = 10 ( n - 1 ) I _ { n - 2 } - x ^ { n - 1 } \left( 10 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \quad n \geqslant 2$$
  2. Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { \sqrt { 10 - x ^ { 2 } } } \mathrm {~d} x$$ giving your answer in the form \(\frac { 1 } { 15 } ( p \sqrt { 10 } + q )\) where \(p\) and \(q\) are integers to be determined.
Edexcel F3 2022 June Q8
  1. The plane \(\Pi\) has equation
$$3 x + 4 y - z = 17$$ The line \(l _ { 1 }\) is perpendicular to \(\Pi\) and passes through the point \(P ( - 4 , - 5,3 )\)
The line \(l _ { 1 }\) intersects \(\Pi\) at the point \(Q\)
  1. Determine the coordinates of \(Q\) Given that the point \(R ( - 1,6,4 )\) lies on \(\Pi\)
  2. determine a Cartesian equation of the plane containing \(P Q R\) The line \(l _ { 2 }\) passes through \(P\) and \(R\)
    The line \(l _ { 3 }\) is the reflection of \(l _ { 2 }\) in \(\Pi\)
  3. Determine a vector equation for \(l _ { 3 }\)
Edexcel F3 2022 June Q9
  1. The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$ The line \(l\) has equation \(y = k x - 3\), where \(k\) is a constant.
Given that \(E\) and \(l\) meet at 2 distinct points \(P\) and \(Q\)
  1. show that the \(x\) coordinates of \(P\) and \(Q\) are solutions of the equation $$\left( 9 k ^ { 2 } + 4 \right) x ^ { 2 } - 54 k x + 45 = 0$$ The point \(M\) is the midpoint of \(P Q\)
  2. Determine, in simplest form in terms of \(k\), the coordinates of \(M\)
  3. Hence show that, as \(k\) varies, \(M\) lies on the curve with equation $$x ^ { 2 } + p y ^ { 2 } = q y$$ where \(p\) and \(q\) are constants to be determined.
Edexcel F3 2023 June Q1
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable. Solve the equation
$$7 \cosh x + 3 \sinh x = 2 \mathrm { e } ^ { x } + 7$$ Give your answers as simplified natural logarithms.