Questions — Edexcel FP3 (160 questions)

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Edexcel FP3 2012 June Q1
5 marks Standard +0.3
  1. The hyperbola \(H\) has equation
$$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ Find
  1. the coordinates of the foci of \(H\),
  2. the equations of the directrices of \(H\).
Edexcel FP3 2012 June Q2
6 marks Challenging +1.2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb604886-6671-441a-b03d-427b5176df6e-03_606_1271_212_335} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\), shown in Figure 1, has equation $$y = \frac { 1 } { 3 } \cosh 3 x , \quad 0 \leqslant x \leqslant \ln a$$ where \(a\) is a constant and \(a > 1\) Using calculus, show that the length of curve \(C\) is $$k \left( a ^ { 3 } - \frac { 1 } { a ^ { 3 } } \right)$$ and state the value of the constant \(k\).
Edexcel FP3 2012 June Q3
8 marks Standard +0.3
3. The position vectors of the points \(A , B\) and \(C\) relative to an origin \(O\) are \(\mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } , 7 \mathbf { i } - 3 \mathbf { k }\) and \(4 \mathbf { i } + 4 \mathbf { j }\) respectively. Find
  1. \(\overrightarrow { A C } \times \overrightarrow { B C }\),
  2. the area of triangle \(A B C\),
  3. an equation of the plane \(A B C\) in the form \(\mathbf { r } . \mathbf { n } = p\)
Edexcel FP3 2012 June Q4
11 marks Challenging +1.2
4. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 4 } } x ^ { n } \sin 2 x \mathrm {~d} x , \quad n \geqslant 0$$
  1. Prove that, for \(n \geqslant 2\), $$I _ { n } = \frac { 1 } { 4 } n \left( \frac { \pi } { 4 } \right) ^ { n - 1 } - \frac { 1 } { 4 } n ( n - 1 ) I _ { n - 2 }$$
  2. Find the exact value of \(I _ { 2 }\)
  3. Show that \(I _ { 4 } = \frac { 1 } { 64 } \left( \pi ^ { 3 } - 24 \pi + 48 \right)\)
Edexcel FP3 2012 June Q5
10 marks Standard +0.8
  1. (a) Differentiate \(x \operatorname { arsinh } 2 x\) with respect to \(x\).
    (b) Hence, or otherwise, find the exact value of
$$\int _ { 0 } ^ { \sqrt { 2 } } \operatorname { arsinh } 2 x \mathrm {~d} x$$ giving your answer in the form \(A \ln B + C\), where \(A , B\) and \(C\) are real.
Edexcel FP3 2012 June Q6
11 marks Challenging +1.2
  1. The ellipse \(E\) has equation
$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ The line \(l _ { 1 }\) is a tangent to \(E\) at the point \(P ( a \cos \theta , b \sin \theta )\).
  1. Using calculus, show that an equation for \(l _ { 1 }\) is $$\frac { x \cos \theta } { a } + \frac { y \sin \theta } { b } = 1$$ The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$$ The line \(l _ { 2 }\) is a tangent to \(C\) at the point \(Q ( a \cos \theta , a \sin \theta )\).
  2. Find an equation for the line \(l _ { 2 }\). Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(R\),
  3. find, in terms of \(a , b\) and \(\theta\), the coordinates of \(R\).
  4. Find the locus of \(R\), as \(\theta\) varies.
Edexcel FP3 2012 June Q7
11 marks Standard +0.3
7. $$\mathrm { f } ( x ) = 5 \cosh x - 4 \sinh x , \quad x \in \mathbb { R }$$
  1. Show that \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + 9 \mathrm { e } ^ { - x } \right)\) Hence
  2. solve \(\mathrm { f } ( x ) = 5\)
  3. show that \(\int _ { \frac { 1 } { 2 } \ln 3 } ^ { \ln 3 } \frac { 1 } { 5 \cosh x - 4 \sinh x } \mathrm {~d} x = \frac { \pi } { 18 }\)
Edexcel FP3 2012 June Q8
13 marks Challenging +1.2
  1. The matrix \(\mathbf { M }\) is given by
$$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 0 \\ 1 & 2 & 0 \\ - 1 & 0 & 4 \end{array} \right)$$
  1. Show that 4 is an eigenvalue of \(\mathbf { M }\), and find the other two eigenvalues.
  2. For the eigenvalue 4, find a corresponding eigenvector. The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\mathbf { M }\). The equation of \(l _ { 1 }\) is \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { b } = 0\), where \(\mathbf { a } = 3 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\).
  3. Find a vector equation for the line \(l _ { 2 }\).
Edexcel FP3 2013 June Q1
7 marks Standard +0.8
  1. The hyperbola \(H\) has foci at \(( 5,0 )\) and \(( - 5,0 )\) and directrices with equations \(x = \frac { 9 } { 5 }\) and \(x = - \frac { 9 } { 5 }\).
Find a cartesian equation for \(H\).
Edexcel FP3 2013 June Q2
7 marks Standard +0.8
2. Two skew lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \lambda ( 4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 3 \mathbf { i } + 7 \mathbf { j } + 2 \mathbf { k } ) + \mu ( - 4 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } ) \end{aligned}$$ respectively, where \(\lambda\) and \(\mu\) are real parameters.
  1. Find a vector in the direction of the common perpendicular to \(l _ { 1 }\) and \(l _ { 2 }\)
  2. Find the shortest distance between these two lines.
Edexcel FP3 2013 June Q3
8 marks Challenging +1.2
  1. The point \(P\) lies on the ellipse \(E\) with equation
$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 9 } = 1$$ \(N\) is the foot of the perpendicular from point \(P\) to the line \(x = 8\) \(M\) is the midpoint of \(P N\).
  1. Sketch the graph of the ellipse \(E\), showing also the line \(x = 8\) and a possible position for the line \(P N\).
  2. Find an equation of the locus of \(M\) as \(P\) moves around the ellipse.
  3. Show that this locus is a circle and state its centre and radius.
Edexcel FP3 2013 June Q4
9 marks Challenging +1.2
  1. The plane \(\Pi _ { 1 }\) has vector equation
$$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) + t \left( \begin{array} { r } 1 \\ 2 \\ - 2 \end{array} \right) ,$$ where \(s\) and \(t\) are real parameters. The plane \(\Pi _ { 1 }\) is transformed to the plane \(\Pi _ { 2 }\) by the transformation represented by the matrix \(\mathbf { T }\), where $$\mathbf { T } = \left( \begin{array} { r r r } 2 & 0 & 3 \\ 0 & 2 & - 1 \\ 0 & 1 & 2 \end{array} \right)$$ Find an equation of the plane \(\Pi _ { 2 }\) in the form r.n=p
Edexcel FP3 2013 June Q5
10 marks Challenging +1.8
5. $$I _ { n } = \int _ { 1 } ^ { 5 } x ^ { n } ( 2 x - 1 ) ^ { - \frac { 1 } { 2 } } \mathrm {~d} x , \quad n \geqslant 0$$
  1. Prove that, for \(n \geqslant 1\), $$( 2 n + 1 ) I _ { n } = n I _ { n - 1 } + 3 \times 5 ^ { n } - 1$$
  2. Using the reduction formula given in part (a), find the exact value of \(I _ { 2 }\)
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 Standard +0.8
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.