Questions — Edexcel FP3 (160 questions)

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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 2015 June Q1
6 marks Standard +0.3
  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
5 marks Standard +0.8
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
12 marks Standard +0.3
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 \(\mathbf { P } ^ { \mathrm { T } } \mathbf { A P } = \mathbf { D }\).
Edexcel FP3 2015 June Q4
7 marks Challenging +1.2
  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
10 marks Standard +0.3
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
10 marks Challenging +1.2
  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
11 marks Challenging +1.2
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
14 marks Challenging +1.2
  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
4 marks Moderate -0.3
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
7 marks Challenging +1.2
  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
8 marks Standard +0.3
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
12 marks Standard +0.8
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
11 marks Challenging +1.2
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
11 marks Standard +0.8
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
10 marks Challenging +1.8
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
12 marks Standard +0.8
  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
5 marks Standard +0.8
  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
9 marks Challenging +1.2
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 \(E\).