Questions FP3 (473 questions)

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Edexcel FP3 2015 June Q4
  1. The curve \(C\) has equation
$$y = \frac { 1 } { \sqrt { x ^ { 2 } + 2 x - 3 } } , \quad x > 1$$
  1. Find \(\int y \mathrm {~d} x\) The region \(R\) is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = 2\) and \(x = 3\). The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  2. Find the volume of the solid generated. Give your answer in the form \(p \pi \ln q\), where \(p\) and \(q\) are rational numbers to be found.
Edexcel FP3 2015 June Q5
5. The points \(A , B\) and \(C\) have position vectors \(\left( \begin{array} { l } 1
3
2 \end{array} \right) , \left( \begin{array} { r } - 1
0
1 \end{array} \right)\) and \(\left( \begin{array} { l } 2
1
0 \end{array} \right)\) respectively.
  1. Find a vector equation of the straight line \(A B\).
  2. Find a cartesian form of the equation of the straight line \(A B\). The plane \(\Pi\) contains the points \(A , B\) and \(C\).
  3. Find a vector equation of \(\Pi\) in the form r.n \(= p\).
  4. Find the perpendicular distance from the origin to \(\Pi\).
Edexcel FP3 2015 June Q6
  1. The hyperbola \(H\) is given by the equation \(x ^ { 2 } - y ^ { 2 } = 1\)
    1. Write down the equations of the two asymptotes of \(H\).
    2. Show that an equation of the tangent to \(H\) at the point \(P ( \cosh t , \sinh t )\) is
    $$y \sinh t = x \cosh t - 1$$ The tangent at \(P\) meets the asymptotes of \(H\) at the points \(Q\) and \(R\).
  2. Show that \(P\) is the midpoint of \(Q R\).
  3. Show that the area of the triangle \(O Q R\), where \(O\) is the origin, is independent of \(t\).
Edexcel FP3 2015 June Q7
7. $$I _ { n } = \int \sin ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$
  1. Prove that for \(n \geqslant 2\) $$I _ { n } = \frac { 1 } { n } \left( - \sin ^ { n - 1 } x \cos x + ( n - 1 ) I _ { n - 2 } \right)$$ Given that \(n\) is an odd number, \(n \geqslant 3\)
  2. show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } x \mathrm {~d} x = \frac { ( n - 1 ) ( n - 3 ) \ldots 6.4 .2 } { n ( n - 2 ) ( n - 4 ) \ldots 7.5 .3 }$$
  3. Hence find \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { 5 } x \cos ^ { 2 } x d x\)
Edexcel FP3 2015 June Q8
  1. The ellipse \(E\) has equation \(x ^ { 2 } + 4 y ^ { 2 } = 4\)
    1. (i) Find the coordinates of the foci, \(F _ { 1 }\) and \(F _ { 2 }\), of \(E\).
      (ii) Write down the equations of the directrices of \(E\).
    2. Given that the point \(P\) lies on the ellipse, show that
    $$\left| P F _ { 1 } \right| + \left| P F _ { 2 } \right| = 4$$ A chord of an ellipse is a line segment joining two points on the ellipse.
    The set of midpoints of the parallel chords of \(E\) with gradient \(m\), where \(m\) is a constant, lie on a straight line \(l\).
  2. Find an equation of \(l\).
Edexcel FP3 2016 June Q1
1. $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 1 & - 3
k & 1 & 3
2 & - 1 & k \end{array} \right) \text {, where } k \text { is a constant }$$ Given that the matrix \(\mathbf { A }\) is singular, find the possible values of \(k\).
Edexcel FP3 2016 June Q2
  1. The curve \(C\) has equation
$$y = \frac { x ^ { 2 } } { 8 } - \ln x , \quad 2 \leqslant x \leqslant 3$$ Find the length of the curve \(C\) giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are rational numbers to be found.
Edexcel FP3 2016 June Q3
3. (a) Prove that $$\frac { \mathrm { d } ( \operatorname { arcoth } x ) } { \mathrm { d } x } = \frac { 1 } { 1 - x ^ { 2 } }$$ Given that \(y = ( \operatorname { arcoth } x ) ^ { 2 }\),
(b) show that $$\left( 1 - x ^ { 2 } \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } - 2 x \frac { d y } { d x } = \frac { k } { 1 - x ^ { 2 } }$$ where \(k\) is a constant to be determined.
Edexcel FP3 2016 June Q4
4. (i) Find, without using a calculator, $$\int _ { 3 } ^ { 5 } \frac { 1 } { \sqrt { 15 + 2 x - x ^ { 2 } } } d x$$ giving your answer as a multiple of \(\pi\).
(ii)
  1. Show that $$5 \cosh x - 4 \sinh x = \frac { \mathrm { e } ^ { 2 x } + 9 } { 2 \mathrm { e } ^ { x } }$$
  2. Hence, using the substitution \(u = e ^ { x }\) or otherwise, find $$\int \frac { 1 } { 5 \cosh x - 4 \sinh x } d x$$
Edexcel FP3 2016 June Q5
5. The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ The point \(P ( 4 \sec \theta , 3 \tan \theta ) , 0 < \theta < \frac { \pi } { 2 }\), lies on \(H\).
  1. Show that an equation of the normal to \(H\) at the point \(P\) is $$3 y + 4 x \sin \theta = 25 \tan \theta$$ The line \(l\) is the directrix of \(H\) for which \(x > 0\)
    The normal to \(H\) at \(P\) crosses the line \(l\) at the point \(Q\). Given that \(\theta = \frac { \pi } { 4 }\)
  2. find the \(y\) coordinate of \(Q\), giving your answer in the form \(a + b \sqrt { 2 }\), where \(a\) and \(b\) are rational numbers to be found.
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Edexcel FP3 2016 June Q6
6. $$\mathbf { M } = \left( \begin{array} { r r r } p & - 2 & 0
- 2 & 6 & - 2
0 & - 2 & q \end{array} \right)$$ where \(p\) and \(q\) are constants.
Given that \(\left( \begin{array} { r } 2
- 2
1 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { M }\),
  1. find the eigenvalue corresponding to this eigenvector,
  2. find the value of \(p\) and the value of \(q\). Given that 6 is another eigenvalue of \(\mathbf { M }\),
  3. find a corresponding eigenvector. Given that \(\left( \begin{array} { l } 1
    2
    2 \end{array} \right)\) is a third eigenvector of \(\mathbf { M }\) with eigenvalue 3
  4. find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { \mathrm { T } } \mathbf { M } \mathbf { P } = \mathbf { D }$$
Edexcel FP3 2016 June Q7
7. Given that $$I _ { n } = \int \frac { \sin n x } { \sin x } \mathrm {~d} x , \quad n \geqslant 1$$
  1. prove that, for \(n \geqslant 3\) $$I _ { n } - I _ { n - 2 } = \int 2 \cos ( n - 1 ) x \mathrm {~d} x$$
  2. Hence, showing each step of your working, find the exact value of $$\int _ { \frac { \pi } { 12 } } ^ { \frac { \pi } { 6 } } \frac { \sin 5 x } { \sin x } d x$$ giving your answer in the form \(\frac { 1 } { 12 } ( a \pi + b \sqrt { 3 } + c )\), where \(a\), \(b\) and \(c\) are integers to be found.
Edexcel FP3 2016 June Q8
  1. The plane \(\Pi _ { 1 }\) has equation
$$x - 5 y - 2 z = 3$$ The plane \(\Pi _ { 2 }\) has equation $$\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k } ) + \mu ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that \(\Pi _ { 1 }\) is perpendicular to \(\Pi _ { 2 }\)
  2. Find a cartesian equation for \(\Pi _ { 2 }\)
  3. Find an equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) giving your answer in the form \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { b } = \mathbf { 0 }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors to be found.
    (6)
Edexcel FP3 2017 June Q1
  1. Given that \(y = \operatorname { arsinh } ( \tanh x )\), show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \operatorname { sech } ^ { 2 } x } { \sqrt { 1 + \tanh ^ { 2 } x } }$$ \section*{-} \includegraphics[max width=\textwidth, alt={}, center]{64dc962a-1788-49ac-a4db-af1241b552a0-03_51_51_276_2012}
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Edexcel FP3 2017 June Q2
2. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 25 } = 1$$ The line \(l\) is the normal to \(E\) at the point \(P ( 6 \cos \theta , 5 \sin \theta )\), where \(0 < \theta < \frac { \pi } { 2 }\)
  1. Use calculus to show that an equation of \(l\) is $$6 x \sin \theta - 5 y \cos \theta = 11 \sin \theta \cos \theta$$ The line \(l\) meets the \(x\)-axis at the point \(Q\). The point \(R\) is the foot of the perpendicular from \(P\) to the \(x\)-axis.
  2. Show that \(\frac { O Q } { O R } = e ^ { 2 }\), where \(e\) is the eccentricity of the ellipse \(E\).
Edexcel FP3 2017 June Q3
3. (a) Using the definition for \(\cosh x\) in terms of exponentials, show that $$\cosh 2 x \equiv 2 \cosh ^ { 2 } x - 1$$ (b) Find the exact values of \(x\) for which $$29 \cosh x - 3 \cosh 2 x = 38$$ giving your answers in terms of natural logarithms.
Edexcel FP3 2017 June Q4
4. Use the substitution \(x + 2 = u ^ { 2 }\), where \(u > 0\), to show that $$\int _ { - 1 } ^ { 7 } \frac { ( x + 2 ) ^ { \frac { 1 } { 2 } } } { x + 5 } \mathrm {~d} x = a + b \pi \sqrt { 3 }$$ where \(a\) and \(b\) are rational numbers to be found. \includegraphics[max width=\textwidth, alt={}, center]{image-not-found}
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Edexcel FP3 2017 June Q5
5. The plane \(\Pi _ { 1 }\) has equation \(x - 2 y - 3 z = 5\) and the plane \(\Pi _ { 2 }\) has equation \(6 x + y - 4 z = 7\)
  1. Find, to the nearest degree, the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) The point \(P\) has coordinates \(( 2,3 , - 1 )\). The line \(l\) is perpendicular to \(\Pi _ { 1 }\) and passes through the point \(P\). The line \(l\) intersects \(\Pi _ { 2 }\) at the point \(Q\).
  2. Find the coordinates of \(Q\). The plane \(\Pi _ { 3 }\) passes through the point \(Q\) and is perpendicular to \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)
  3. Find an equation of the plane \(\Pi _ { 3 }\) in the form \(\mathbf { r } . \mathbf { n } = p\)
Edexcel FP3 2017 June Q6
6. The matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left( \begin{array} { r r r } 1 & k & 0
2 & - 2 & 1
- 4 & 1 & - 1 \end{array} \right) , k \in \mathbb { R } , k \neq \frac { 1 } { 2 }$$
  1. Show that \(\operatorname { det } \mathbf { M } = 1 - 2 k\).
  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 $$\left( \begin{array} { r r r } 1 & 0 & 0
    2 & - 2 & 1
    - 4 & 1 & - 1 \end{array} \right)$$ Given that \(l _ { 2 }\) has cartesian equation $$\frac { x - 1 } { 5 } = \frac { y + 2 } { 2 } = \frac { z - 3 } { 1 }$$
  3. find a cartesian equation of the line \(l _ { 1 }\)
Edexcel FP3 2017 June Q7
7. $$I _ { n } = \int _ { 0 } ^ { \ln 2 } \cosh ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$
  1. Show that, for \(n \geqslant 2\), $$I _ { n } = \frac { 3 a ^ { n - 1 } } { n b ^ { n } } + \frac { n - 1 } { n } I _ { n - 2 }$$ where \(a\) and \(b\) are integers to be found.
  2. Hence, or otherwise, find the exact value of $$\int _ { 0 } ^ { \ln 2 } \cosh ^ { 4 } x \mathrm {~d} x$$
Edexcel FP3 2017 June Q8
8. The curve \(C\) has equation $$y = \ln \left( \frac { \mathrm { e } ^ { x } + 1 } { \mathrm { e } ^ { x } - 1 } \right) , \quad \ln 2 \leqslant x \leqslant \ln 3$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } - 1 }$$
  2. Find the length of the curve \(C\), giving your answer in the form \(\ln a\), where \(a\) is a rational number.
    (6)
Edexcel FP3 2018 June Q1
  1. (a) Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, show that, for \(x \in \mathbb { R }\)
$$\tanh x = \frac { \mathrm { e } ^ { 2 x } - 1 } { \mathrm { e } ^ { 2 x } + 1 }$$ (b) Hence, given that \(- 1 < \theta < 1\), prove that $$\operatorname { artanh } \theta = \frac { 1 } { 2 } \ln \left( \frac { 1 + \theta } { 1 - \theta } \right)$$ uestion 1 continued
\(\_\_\_\_\) 7
Edexcel FP3 2018 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{38487750-8c0f-4c3d-a019-5213ed2866eb-04_616_764_246_584} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = 5 \cosh x - 6 \sinh x$$ The curve crosses the \(x\)-axis at the point \(A\).
  1. Find the exact value of the \(x\) coordinate of the point \(A\), giving your answer as a natural logarithm.
  2. Show that $$( 5 \cosh x - 6 \sinh x ) ^ { 2 } \equiv a \cosh 2 x + b \sinh 2 x + c$$ where \(a , b\) and \(c\) are constants to be found. The finite region \(R\), bounded by the curve and the coordinate axes, is shown shaded in Figure 1. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  3. Use calculus to find the volume of the solid generated, giving your answer as an exact multiple of \(\pi\).
Edexcel FP3 2018 June Q3
3. \(\mathbf { M } = \left( \begin{array} { r r r } 3 & k & 2
- 1 & 0 & 1
1 & k & 1 \end{array} \right)\), where \(k\) 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 an eigenvector corresponding to the eigenvalue 3
    3. \(\quad \mathbf { M } = \left( \begin{array} { r c c } 3 & k & 2
    - 1 & 0 & 1
    1 & k & 1 \end{array} \right)\), where \(k\) is a constant Given that 3 is an eigenvalue of \(\mathbf { M }\), (a) find the value of \(k\).
Edexcel FP3 2018 June Q4
4. The curve \(C\) has equation $$y = \operatorname { arsinh } x + x \sqrt { x ^ { 2 } + 1 } , \quad 0 \leqslant x \leqslant 1$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sqrt { x ^ { 2 } + 1 }\)
  2. Hence show that the length of the curve \(C\) is given by $$\int _ { 0 } ^ { 1 } \sqrt { 4 x ^ { 2 } + 5 } d x$$
  3. Using the substitution \(x = \frac { \sqrt { 5 } } { 2 } \sinh u\), find the exact length of the curve \(C\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are constants to be found.