Questions FP3 (473 questions)

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Edexcel FP3 2011 June Q1
  1. The curve \(C\) has equation \(y = 2 x ^ { 3 } , 0 \leqslant x \leqslant 2\).
The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
Using calculus, find the area of the surface generated, giving your answer to 3 significant figures.
Edexcel FP3 2011 June Q2
2. (a) Given that \(y = x \arcsin x , 0 \leqslant x \leqslant 1\), find
  1. an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
  2. the exact value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = \frac { 1 } { 2 }\).
    (b) Given that \(y = \arctan \left( 3 \mathrm { e } ^ { 2 x } \right)\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 } { 5 \cosh 2 x + 4 \sinh 2 x }$$
Edexcel FP3 2011 June Q3
  1. Show that
    1. \(\int _ { 5 } ^ { 8 } \frac { 1 } { x ^ { 2 } - 10 x + 34 } \mathrm {~d} x = k \pi\), giving the value of the fraction \(k\),
    2. \(\int _ { 5 } ^ { 8 } \frac { 1 } { \sqrt { } \left( x ^ { 2 } - 10 x + 34 \right) } \mathrm { d } x = \ln ( A + \sqrt { } n )\), giving the values of the integers \(A\) and \(n\).
    $$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } x ^ { 2 } ( \ln x ) ^ { n } \mathrm {~d} x , \quad n \geqslant 0$$
  2. Prove that, for \(n \geqslant 1\), $$I _ { n } = \frac { \mathrm { e } ^ { 3 } } { 3 } - \frac { n } { 3 } I _ { n - 1 }$$
  3. Find the exact value of \(I _ { 3 }\).
Edexcel FP3 2011 June Q5
  1. The curve \(C _ { 1 }\) has equation \(y = 3 \sinh 2 x\), and the curve \(C _ { 2 }\) has equation \(y = 13 - 3 \mathrm { e } ^ { 2 x }\).
    1. Sketch the graph of the curves \(C _ { 1 }\) and \(C _ { 2 }\) on one set of axes, giving the equation of any asymptote and the coordinates of points where the curves cross the axes.
    2. Solve the equation \(3 \sinh 2 x = 13 - 3 \mathrm { e } ^ { 2 x }\), giving your answer in the form \(\frac { 1 } { 2 } \ln k\), where \(k\) is an integer.
Edexcel FP3 2011 June Q6
  1. The plane \(P\) has equation
$$\mathbf { r } = \left( \begin{array} { l } 3
1
2 \end{array} \right) + \lambda \left( \begin{array} { r } 0
2
- 1 \end{array} \right) + \mu \left( \begin{array} { l } 3
2
2 \end{array} \right)$$
  1. Find a vector perpendicular to the plane \(P\). The line \(l\) passes through the point \(A ( 1,3,3 )\) and meets \(P\) at \(( 3,1,2 )\). The acute angle between the plane \(P\) and the line \(l\) is \(\alpha\).
  2. Find \(\alpha\) to the nearest degree.
  3. Find the perpendicular distance from \(A\) to the plane \(P\).
Edexcel FP3 2011 June Q7
  1. The matrix \(\mathbf { M }\) is given by
$$\mathbf { M } = \left( \begin{array} { r r r } k & - 1 & 1
1 & 0 & - 1
3 & - 2 & 1 \end{array} \right) , \quad k \neq 1$$
  1. Show that \(\operatorname { det } \mathbf { M } = 2 - 2 k\).
  2. Find \(\mathbf { M } ^ { - 1 }\), in terms of \(k\). The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\left( \begin{array} { r r r } 2 & - 1 & 1
    1 & 0 & - 1
    3 & - 2 & 1 \end{array} \right)\). The equation of \(l _ { 2 }\) is \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { b } = 0\), where \(\mathbf { a } = 4 \mathbf { i } + \mathbf { j } + 7 \mathbf { k }\) and \(\mathbf { b } = 4 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }\).
  3. Find a vector equation for the line \(l _ { 1 }\).
Edexcel FP3 2011 June Q8
  1. The hyperbola \(H\) has equation
$$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$
  1. Use calculus to show that the equation of the tangent to \(H\) at the point \(( a \cosh \theta , b \sinh \theta )\) may be written in the form $$x b \cosh \theta - y a \sinh \theta = a b$$ The line \(l _ { 1 }\) is the tangent to \(H\) at the point \(( a \cosh \theta , b \sinh \theta ) , \theta \neq 0\).
    Given that \(l _ { 1 }\) meets the \(x\)-axis at the point \(P\),
  2. find, in terms of \(a\) and \(\theta\), the coordinates of \(P\). The line \(l _ { 2 }\) is the tangent to \(H\) at the point ( \(a , 0\) ).
    Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(Q\),
  3. find, in terms of \(a , b\) and \(\theta\), the coordinates of \(Q\).
  4. Show that, as \(\theta\) varies, the locus of the mid-point of \(P Q\) has equation $$x \left( 4 y ^ { 2 } + b ^ { 2 } \right) = a b ^ { 2 }$$
Edexcel FP3 2012 June Q1
  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
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
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
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
  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
  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
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
  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
  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
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
  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
  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
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
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
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
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
  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
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.