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Edexcel FP3 2013 June Q6
11 marks Standard +0.3
6. It is given that \(\left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { l l l } 4 & 2 & 3 \\ 2 & b & 0 \\ a & 1 & 8 \end{array} \right)$$ and \(a\) and \(b\) are constants.
  1. Find the eigenvalue of \(\mathbf { A }\) corresponding to the eigenvector \(\left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)\).
  2. Find the values of \(a\) and \(b\).
  3. Find the other eigenvalues of \(\mathbf { A }\).
Edexcel FP3 2013 June Q7
12 marks Standard +0.8
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{094b3c91-1460-44a2-b9d6-4de90d3adfa0-13_593_1292_118_328} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curves shown in Figure 1 have equations $$y = 6 \cosh x \text { and } y = 9 - 2 \sinh x$$
  1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\), find exact values for the \(x\)-coordinates of the two points where the curves intersect. The finite region between the two curves is shown shaded in Figure 1.
  2. Using calculus, find the area of the shaded region, giving your answer in the form \(a \ln b + c\), where \(a , b\) and \(c\) are integers.
Edexcel FP3 2013 June Q8
11 marks Challenging +1.2
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{094b3c91-1460-44a2-b9d6-4de90d3adfa0-15_590_855_210_548} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} The curve \(C\), shown in Figure 2, has equation $$y = 2 x ^ { \frac { 1 } { 2 } } , \quad 1 \leqslant x \leqslant 8$$
  1. Show that the length \(s\) of curve \(C\) is given by the equation $$s = \int _ { 1 } ^ { 8 } \sqrt { } \left( 1 + \frac { 1 } { x } \right) \mathrm { d } x$$
  2. Using the substitution \(x = \sinh ^ { 2 } u\), or otherwise, find an exact value for \(s\). Give your answer in the form \(a \sqrt { } 2 + \ln ( b + c \sqrt { } 2 )\) where \(a , b\) and \(c\) are integers.
Edexcel FP3 2013 June Q1
6 marks Standard +0.3
  1. A hyperbola \(H\) has equation
$$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 25 } = 1 , \quad \text { where } a \text { is a positive constant. }$$ The foci of \(H\) are at the points with coordinates \(( 13,0 )\) and \(( - 13,0 )\).
Find
  1. the value of the constant \(a\),
  2. the equations of the directrices of \(H\).
Edexcel FP3 2013 June Q2
5 marks Standard +0.3
2. (a) Find $$\int \frac { 1 } { \sqrt { } \left( 4 x ^ { 2 } + 9 \right) } d x$$ (b) Use your answer to part (a) to find the exact value of $$\int _ { - 3 } ^ { 3 } \frac { 1 } { \sqrt { \left( 4 x ^ { 2 } + 9 \right) } } d x$$ giving your answer in the form \(k \ln ( a + b \sqrt { } 5 )\), where \(a\) and \(b\) are integers and \(k\) is a constant.
Edexcel FP3 2013 June Q3
7 marks Challenging +1.2
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
7 marks
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
13 marks Standard +0.8
  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
11 marks Challenging +1.8
  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
12 marks Challenging +1.2
  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
14 marks Standard +0.3
  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
5 marks Standard +0.8
  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
7 marks Standard +0.3
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
9 marks Challenging +1.2
  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
11 marks Challenging +1.2
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
11 marks Standard +0.8
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
11 marks Standard +0.8
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
12 marks Challenging +1.2
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
9 marks Standard +0.8
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
8 marks Standard +0.3
  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
13 marks
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
8 marks Standard +0.8
  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
7 marks Standard +0.3
  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
4 marks Challenging +1.8
  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
9 marks Standard +0.3
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 \(x = 1\)