Questions — Edexcel FP3 (136 questions)

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Edexcel FP3 2013 June Q3
3. The curve with parametric equations $$x = \cosh 2 \theta , \quad y = 4 \sinh \theta , \quad 0 \leqslant \theta \leqslant 1$$ is rotated through \(2 \pi\) radians about the \(x\)-axis.
Show that the area of the surface generated is \(\lambda \left( \cosh ^ { 3 } \alpha - 1 \right)\), where \(\alpha = 1\) and \(\lambda\) is a constant to be found.
Edexcel FP3 2013 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bd4cd798-61ae-49b6-a297-bb4b9ed15fb1-05_384_1040_226_438} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows part of the curve with equation $$y = 40 \operatorname { arcosh } x - 9 x , \quad x \geqslant 1$$ Use calculus to find the exact coordinates of the turning point of the curve, giving your answer in the form \(\left( \frac { p } { q } , r \ln 3 + s \right)\), where \(p , q , r\) and \(s\) are integers.
Edexcel FP3 2013 June Q5
  1. The matrix \(\mathbf { M }\) is given by
$$\mathbf { M } = \left( \begin{array} { r r r } 1 & 1 & a
2 & b & c
- 1 & 0 & 1 \end{array} \right) , \text { where } a , b \text { and } c \text { are constants. }$$
  1. Given that \(\mathbf { j } + \mathbf { k }\) and \(\mathbf { i } - \mathbf { k }\) are two of the eigenvectors of \(\mathbf { M }\), find
    1. the values of \(a , b\) and \(c\),
    2. the eigenvalues which correspond to the two given eigenvectors.
  2. The matrix \(\mathbf { P }\) is given by $$\mathbf { P } = \left( \begin{array} { r r r } 1 & 1 & 0
    2 & 1 & d
    - 1 & 0 & 1 \end{array} \right) \text {, where } d \text { is constant, } d \neq - 1$$ Find
    1. the determinant of \(\mathbf { P }\) in terms of \(d\),
    2. the matrix \(\mathbf { P } ^ { - 1 }\) in terms of \(d\).
Edexcel FP3 2013 June Q6
  1. Given that
$$I _ { n } = \int _ { 0 } ^ { 4 } x ^ { n } \sqrt { } \left( 16 - x ^ { 2 } \right) \mathrm { d } x , \quad n \geqslant 0$$
  1. prove that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = 16 ( n - 1 ) I _ { n - 2 }$$
  2. Hence, showing each step of your working, find the exact value of \(I _ { 5 }\)
Edexcel FP3 2013 June Q7
  1. The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , \quad a > b > 0$$ The line \(l\) is a normal to \(E\) at a point \(P ( a \cos \theta , b \sin \theta ) , \quad 0 < \theta < \frac { \pi } { 2 }\)
  1. Using calculus, show that an equation for \(l\) is $$a x \sin \theta - b y \cos \theta = \left( a ^ { 2 } - b ^ { 2 } \right) \sin \theta \cos \theta$$ The line \(l\) meets the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
  2. Show that the area of the triangle \(O A B\), where \(O\) is the origin, may be written as \(k \sin 2 \theta\), giving the value of the constant \(k\) in terms of \(a\) and \(b\).
  3. Find, in terms of \(a\) and \(b\), the exact coordinates of the point \(P\), for which the area of the triangle \(O A B\) is a maximum.
Edexcel FP3 2013 June Q8
  1. The plane \(\Pi _ { 1 }\) has vector equation
$$\mathbf { r } . ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) = 5$$
  1. Find the perpendicular distance from the point \(( 6,2,12 )\) to the plane \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has vector equation $$\mathbf { r } = \lambda ( 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } ) + \mu ( \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) , \text { where } \lambda \text { and } \mu \text { are scalar parameters. }$$
  2. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) giving your answer to the nearest degree.
  3. Find an equation of the line of intersection of the two planes in the form \(\mathbf { r } \times \mathbf { a } = \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors.
Edexcel FP3 2014 June Q1
  1. Solve the equation
$$5 \tanh x + 7 = 5 \operatorname { sech } x$$ Give each answer in the form \(\ln k\) where \(k\) is a rational number.
Edexcel FP3 2014 June Q2
2. $$9 x ^ { 2 } + 6 x + 5 \equiv a ( x + b ) ^ { 2 } + c$$
  1. Find the values of the constants \(a\), \(b\) and \(c\). Hence, or otherwise, find
  2. \(\int \frac { 1 } { 9 x ^ { 2 } + 6 x + 5 } d x\)
  3. \(\int \frac { 1 } { \sqrt { 9 x ^ { 2 } + 6 x + 5 } } \mathrm {~d} x\)
Edexcel FP3 2014 June Q3
  1. The curve \(C\) has equation
$$y = \frac { 1 } { 2 } \ln ( \operatorname { coth } x ) , \quad x > 0$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosech } 2 x$$ The points \(A\) and \(B\) lie on \(C\). The \(x\) coordinates of \(A\) and \(B\) are \(\ln 2\) and \(\ln 3\) respectively.
  2. Find the length of the arc \(A B\), giving your answer in the form \(p \ln q\), where \(p\) and \(q\) are rational numbers.
    (6)
Edexcel FP3 2014 June Q4
4. $$I _ { n } = \int _ { 0 } ^ { \sqrt { 3 } } \left( 3 - x ^ { 2 } \right) ^ { n } \mathrm {~d} x , \quad n \geqslant 0$$
  1. Show that, for \(n \geqslant 1\) $$I _ { n } = \frac { 6 n } { 2 n + 1 } I _ { n - 1 }$$
  2. Hence find the exact value of \(I _ { 4 }\), giving your answer in the form \(k \sqrt { 3 }\) where \(k\) is a rational number to be found.
Edexcel FP3 2014 June Q5
5. The ellipse \(E\) has equation $$x ^ { 2 } + 9 y ^ { 2 } = 9$$ The point \(P ( a \cos \theta , b \sin \theta )\) is a general point on the ellipse \(E\).
  1. Write down the value of \(a\) and the value of \(b\). The line \(L\) is a tangent to \(E\) at the point \(P\).
  2. Show that an equation of the line \(L\) is given by $$3 y \sin \theta + x \cos \theta = 3$$ The line \(L\) meets the \(x\)-axis at the point \(Q\) and meets the \(y\)-axis at the point \(R\).
  3. Show that the area of the triangle \(O Q R\), where \(O\) is the origin, is given by $$k \operatorname { cosec } 2 \theta$$ where \(k\) is a constant to be found. The point \(M\) is the midpoint of \(Q R\).
  4. Find a cartesian equation of the locus of \(M\), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
Edexcel FP3 2014 June Q6
6. The symmetric matrix \(\mathbf { M }\) has eigenvectors \(\left( \begin{array} { l } 2
2
1 \end{array} \right) , \left( \begin{array} { r } - 2
1
2 \end{array} \right)\) and \(\left( \begin{array} { r } 1
- 2
2 \end{array} \right)\) with eigenvalues 5, 2 and - 1 respectively.
  1. Find an orthogonal matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { \mathrm { T } } \mathbf { M } \mathbf { P } = \mathbf { D }$$ Given that \(\mathbf { P } ^ { - 1 } = \mathbf { P } ^ { \mathrm { T } }\)
  2. show that $$\mathbf { M } = \mathbf { P D P } ^ { - 1 }$$
  3. Hence find the matrix \(\mathbf { M }\).
Edexcel FP3 2014 June Q7
7. The curve \(C\) has equation $$y = \mathrm { e } ^ { - x } , \quad x \in \mathbb { R }$$ The part of the curve \(C\) between \(x = 0\) and \(x = \ln 3\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  1. Show that the area \(S\) of the curved surface generated is given by $$S = 2 \pi \int _ { 0 } ^ { \ln 3 } \mathrm { e } ^ { - x } \sqrt { 1 + \mathrm { e } ^ { - 2 x } } \mathrm {~d} x$$
  2. Use the substitution \(\mathrm { e } ^ { - x } = \sinh u\) to show that $$S = 2 \pi \int _ { \operatorname { arsinh } \alpha } ^ { \operatorname { arsinh } \beta } \cosh ^ { 2 } u \mathrm {~d} u$$ where \(\alpha\) and \(\beta\) are constants to be determined.
  3. Show that $$2 \int \cosh ^ { 2 } u \mathrm {~d} u = \frac { 1 } { 2 } \sinh 2 u + u + k$$ where \(k\) is an arbitrary constant.
  4. Hence find the value of \(S\), giving your answer to 3 decimal places.
Edexcel FP3 2014 June Q8
8. The plane \(\Pi _ { 1 }\) has vector equation \(\mathbf { r }\). \(\left( \begin{array} { l } 2
1
3 \end{array} \right) = 5\) The plane \(\Pi _ { 2 }\) has vector equation \(\mathbf { r } . \left( \begin{array} { r } - 1
2
4 \end{array} \right) = 7\)
  1. Find a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), 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 plane \(\Pi _ { 3 }\) has cartesian equation $$x - y + 2 z = 31$$
  2. Using your answer to part (a), or otherwise, find the coordinates of the point of intersection of the planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\)
    \includegraphics[max width=\textwidth, alt={}, center]{393fd7be-c8f5-4b83-a5c7-2de04987a039-16_104_77_2469_1804}
Edexcel FP3 2014 June Q1
  1. The line \(l\) passes through the point \(P ( 2,1,3 )\) and is perpendicular to the plane \(\Pi\) whose vector equation is
$$\mathbf { r } . ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } ) = 3$$ Find
  1. a vector equation of the line \(l\),
  2. the position vector of the point where \(l\) meets \(\Pi\).
  3. Hence find the perpendicular distance of \(P\) from \(\Pi\).
Edexcel FP3 2014 June Q2
2. $$\mathbf { M } = \left( \begin{array} { l l l } 1 & 0 & 2
0 & 4 & 1
0 & 5 & 0 \end{array} \right)$$
  1. Show that matrix \(\mathbf { M }\) is not orthogonal.
  2. Using algebra, show that 1 is an eigenvalue of \(\mathbf { M }\) and find the other two eigenvalues of \(\mathbf { M }\).
  3. Find an eigenvector of \(\mathbf { M }\) which corresponds to the eigenvalue 1 The transformation \(M : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { M }\).
  4. Find a cartesian equation of the image, under this transformation, of the line $$x = \frac { y } { 2 } = \frac { z } { - 1 }$$
Edexcel FP3 2014 June Q3
  1. Using calculus, find the exact value of
    1. \(\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt { \left( x ^ { 2 } - 2 x + 3 \right) } } \mathrm { d } x\)
    2. \(\int _ { 0 } ^ { 1 } \mathrm { e } ^ { 2 x } \sinh x \mathrm {~d} x\)
Edexcel FP3 2014 June Q4
  1. Using the definitions of hyperbolic functions in terms of exponentials,
    1. show that
    $$\operatorname { sech } ^ { 2 } x = 1 - \tanh ^ { 2 } x$$
  2. solve the equation $$4 \sinh x - 3 \cosh x = 3$$
Edexcel FP3 2014 June Q5
  1. Given that \(y = \operatorname { artanh } \frac { x } { \sqrt { } \left( 1 + x ^ { 2 } \right) }\)
    show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { } \left( 1 + x ^ { 2 } \right) }\)
  2. \hspace{0pt} [In this question you may use the appropriate trigonometric identities on page 6 of the pink Mathematical Formulae and Statistical Tables.]
The points \(P ( 3 \cos \alpha , 2 \sin \alpha )\) and \(Q ( 3 \cos \beta , 2 \sin \beta )\), where \(\alpha \neq \beta\), lie on the ellipse with equation $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$
  1. Show the equation of the chord \(P Q\) is $$\frac { x } { 3 } \cos \frac { ( \alpha + \beta ) } { 2 } + \frac { y } { 2 } \sin \frac { ( \alpha + \beta ) } { 2 } = \cos \frac { ( \alpha - \beta ) } { 2 }$$
  2. Write down the coordinates of the mid-point of \(P Q\). Given that the gradient, \(m\), of the chord \(P Q\) is a constant,
  3. show that the centre of the chord lies on a line $$y = - k x$$ expressing \(k\) in terms of \(m\).
Edexcel FP3 2014 June Q7
7. A circle \(C\) with centre \(O\) and radius \(r\) has cartesian equation \(x ^ { 2 } + y ^ { 2 } = r ^ { 2 }\) where \(r\) is a constant.
  1. Show that \(1 + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = \frac { r ^ { 2 } } { r ^ { 2 } - x ^ { 2 } }\)
  2. Show that the surface area of the sphere generated by rotating \(C\) through \(\pi\) radians about the \(x\)-axis is \(4 \pi r ^ { 2 }\).
  3. Write down the length of the arc of the curve \(y = \sqrt { } \left( 1 - x ^ { 2 } \right)\) from \(x = 0\) to \(x = 1\)
Edexcel FP3 2014 June Q8
8. The position vectors of the points \(A , B\) and \(C\) from a fixed origin \(O\) are $$\mathbf { a } = \mathbf { i } - \mathbf { j } , \quad \mathbf { b } = \mathbf { i } + \mathbf { j } + \mathbf { k } , \quad \mathbf { c } = 2 \mathbf { j } + \mathbf { k }$$ respectively.
  1. Using vector products, find the area of the triangle \(A B C\).
  2. Show that \(\frac { 1 } { 6 } \mathbf { a } . ( \mathbf { b } \times \mathbf { c } ) = 0\)
  3. Hence or otherwise, state what can be deduced about the vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\).
Edexcel FP3 2014 June Q9
9. $$I _ { n } = \int \left( x ^ { 2 } + 1 \right) ^ { - n } \mathrm {~d} x , \quad n > 0$$
  1. Show that, for \(n > 0\) $$I _ { n + 1 } = \frac { x \left( x ^ { 2 } + 1 \right) ^ { - n } } { 2 n } + \frac { 2 n - 1 } { 2 n } I _ { n }$$
  2. Find \(I _ { 2 }\)
Edexcel FP3 2015 June Q1
  1. Solve the equation
$$2 \cosh ^ { 2 } x - 3 \sinh x = 1$$ giving your answers in terms of natural logarithms.
Edexcel FP3 2015 June Q2
2. A curve has equation $$y = \cosh x , \quad 1 \leqslant x \leqslant \ln 5$$ Find the exact length of this curve. Give your answer in terms of e .
Edexcel FP3 2015 June Q3
3. $$\mathbf { A } = \left( \begin{array} { l l l } 2 & 1 & 0
1 & 2 & 1
0 & 1 & 2 \end{array} \right)$$
  1. Find the eigenvalues of \(\mathbf { A }\).
  2. Find a normalised eigenvector for each of the eigenvalues of \(\mathbf { A }\).
  3. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { \mathrm { T } } \mathbf { A P } = \mathbf { D }\).