Questions — Edexcel FP3 (160 questions)

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Edexcel FP3 2017 June Q3
9 marks Standard +0.3
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
9 marks Challenging +1.2
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
11 marks Standard +0.3
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
12 marks Challenging +1.2
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
10 marks Challenging +1.8
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
10 marks Challenging +1.8
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
5 marks Standard +0.3
  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
10 marks Challenging +1.2
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
9 marks Standard +0.3
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
12 marks Challenging +1.3
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.
Edexcel FP3 2018 June Q5
11 marks Challenging +1.2
5. Given that $$I _ { n } = \int x ^ { n } \sqrt { ( x + 8 ) } \mathrm { d } x , \quad n \geqslant 0 , x \geqslant 0$$
  1. show that, for \(n \geqslant 1\) $$I _ { n } = \frac { p x ^ { n } ( x + 8 ) ^ { \frac { 3 } { 2 } } } { 2 n + 3 } - \frac { q n } { 2 n + 3 } I _ { n - 1 }$$ where \(p\) and \(q\) are constants to be found.
  2. Use part (a) to find the exact value of $$\int _ { 0 } ^ { 10 } x ^ { 2 } \sqrt { ( x + 8 ) } d x$$ giving your answer in the form \(k \sqrt { 2 }\), where \(k\) is rational.
Edexcel FP3 2018 June Q6
13 marks Standard +0.8
6. The line \(l _ { 1 }\) has equation $$\mathbf { r } = \mathbf { i } + 2 \mathbf { k } + \lambda ( 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } )$$ where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) has equation $$\frac { x + 1 } { 1 } = \frac { y - 4 } { 1 } = \frac { z - 1 } { 3 }$$
  1. Prove that the lines \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
  2. Find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\) The plane \(\Pi\) contains \(l _ { 1 }\) and intersects \(l _ { 2 }\) at the point \(( 3,8,13 )\).
  3. Find a cartesian equation for the plane \(\Pi\).
Edexcel FP3 2018 June Q7
15 marks Challenging +1.2
7. The ellipse \(E\) has foci at the points \(( \pm 3,0 )\) and has directrices with equations \(x = \pm \frac { 25 } { 3 }\)
  1. Find a cartesian equation for the ellipse \(E\). The straight line \(l\) has equation \(y = m x + c\), where \(m\) and \(c\) are positive constants.
  2. Show that the \(x\) coordinates of any points of intersection of \(l\) and \(E\) satisfy the equation $$\left( 16 + 25 m ^ { 2 } \right) x ^ { 2 } + 50 m c x + 25 \left( c ^ { 2 } - 16 \right) = 0$$ Given that the line \(l\) is a tangent to \(E\),
  3. show that \(c ^ { 2 } = p m ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found. The line \(l\) intersects the \(x\)-axis at the point \(A\) and intersects the \(y\)-axis at the point \(B\).
  4. Show that the area of triangle \(O A B\), where \(O\) is the origin, is $$\frac { 25 m ^ { 2 } + 16 } { 2 m }$$
  5. Find the minimum area of triangle \(O A B\).
    Leave
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    Q7
Edexcel FP3 Q1
6 marks Standard +0.8
  1. Solve the equation
$$7 \operatorname { sech } x - \tanh x = 5$$ Give your answers in the form \(\ln a\), where \(a\) is a rational number.
Edexcel FP3 Q2
7 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{045545c7-06d9-40b6-9d01-fc792ab0aa07-01_222_241_525_2042} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to a fixed origin \(O\), as shown in Figure 1. It is given that $$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = \mathbf { 3 i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = \mathbf { 2 i } + \mathbf { j } - \mathbf { k } .$$ Calculate
  1. \(\mathbf { b } \times \mathbf { c }\),
  2. \(\mathbf { a . } ( \mathbf { b } \times \mathbf { c } )\),
  3. the area of triangle \(O B C\),
  4. the volume of the tetrahedron \(O A B C\).
Edexcel FP3 Q4
8 marks Challenging +1.2
4. Given that \(y = \operatorname { arsinh } ( \sqrt { } x ) , x > 0\),
  1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer as a simplified fraction.
  2. Hence, or otherwise, find $$\int _ { \frac { 1 } { 4 } } ^ { 4 } \frac { 1 } { \sqrt { [ x ( x + 1 ) ] } } \mathrm { d } x$$ giving your answer in the form \(\ln \left( \frac { a + b \sqrt { } 5 } { 2 } \right)\), where \(a\) and \(b\) are integers.
Edexcel FP3 Q5
4 marks Challenging +1.3
5. $$I _ { n } = \int _ { 0 } ^ { 5 } \frac { x ^ { n } } { \sqrt { } \left( 25 - x ^ { 2 } \right) } \mathrm { d } x , \quad n \geq 0$$
  1. Find an expression for \(\int \frac { x } { \sqrt { \left( 25 - x ^ { 2 } \right) } } \mathrm { d } x , \quad 0 \leq x \leq 5\).
  2. Using your answer to part (a), or otherwise, show that $$I _ { n } = \frac { 25 ( n - 1 ) } { n } I _ { n - 2 } , \quad n \geq 2$$
  3. Find \(I _ { 4 }\) in the form \(k \pi\), where \(k\) is a fraction.
Edexcel FP3 Q6
10 marks Challenging +1.2
6. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\), where \(a\) and \(b\) are constants. The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.
  1. Given that \(L\) and \(H\) meet, show that the \(x\)-coordinates of the points of intersection are the roots of the equation $$\left( a ^ { 2 } m ^ { 2 } - b ^ { 2 } \right) x ^ { 2 } + 2 a ^ { 2 } m c x + a ^ { 2 } \left( c ^ { 2 } + b ^ { 2 } \right) = 0$$ Hence, given that \(L\) is a tangent to \(H\),
  2. show that \(a ^ { 2 } m ^ { 2 } = b ^ { 2 } + c ^ { 2 }\). The hyperbola \(H ^ { \prime }\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 16 } = 1\).
  3. Find the equations of the tangents to \(H ^ { \prime }\) which pass through the point \(( 1,4 )\).
Edexcel FP3 Q7
9 marks Standard +0.3
7. The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 3 \\ 4 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } \alpha \\ - 4 \\ 0 \end{array} \right) + \mu \left( \begin{array} { l } 0 \\ 3 \\ 2 \end{array} \right) .$$ If the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect, find
  1. the value of \(\alpha\),
  2. an equation for the plane containing the lines \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in the form \(a x + b y + c z + d = 0\), where \(a , b , c\) and \(d\) are constants. For other values of \(\alpha\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect and are skew lines.
    Given that \(\alpha = 2\),
  3. find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\).
Edexcel FP3 Q8
8 marks Challenging +1.8
8. A curve, which is part of an ellipse, has parametric equations $$x = 3 \cos \theta , \quad y = 5 \sin \theta , \quad 0 \leq \theta \leq \frac { \pi } { 2 }$$ The curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
  1. Show that the area of the surface generated is given by the integral $$k \pi \int _ { 0 } ^ { a } \sqrt { } \left( 16 c ^ { 2 } + 9 \right) \mathrm { d } c , \text { where } c = \cos \theta$$ and where \(k\) and \(\alpha\) are constants to be found.
  2. Using the substitution \(c = \frac { 3 } { 4 } \sinh u\), or otherwise, evaluate the integral, showing all of your working and giving the final answer to 3 significant figures.
Edexcel FP3 Q9
8 marks Challenging +1.8
9. $$I _ { n } = \int \left( x ^ { 2 } + 1 \right) ^ { - n } \mathrm {~d} x , \quad n > 0$$
  1. Show that, for \(n > 0\), $$I _ { n + 1 } = \frac { x \left( x ^ { 2 } + 1 \right) ^ { - n } } { 2 n } + \frac { 2 n - 1 } { 2 n } I _ { n }$$
  2. Find \(I _ { 2 }\).
Edexcel FP3 2011 June Q1
5 marks Challenging +1.2
The curve \(C\) has equation \(y = 2x^3\), \(0 \leq x \leq 2\). 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. [5]
Edexcel FP3 2011 June Q2
8 marks Standard +0.8
  1. Given that \(y = x \arcsin x\), \(0 \leq x \leq 1\), find
    1. an expression for \(\frac{dy}{dx}\),
    2. the exact value of \(\frac{dy}{dx}\) when \(x = \frac{1}{2}\).
    [3]
  2. Given that \(y = \arctan(3e^{2x})\), show that $$\frac{dy}{dx} = \frac{3}{5\cosh 2x + 4\sinh 2x}.$$ [5]
Edexcel FP3 2011 June Q3
9 marks Challenging +1.2
Show that
  1. \(\int_5^8 \frac{1}{x^2 - 10x + 34} dx = k\pi\), giving the value of the fraction \(k\), [5]
  2. \(\int_5^8 \frac{1}{\sqrt{x^2 - 10x + 34}} dx = \ln(A + \sqrt{n})\), giving the values of the integers \(A\) and \(n\). [4]
Edexcel FP3 2011 June Q4
8 marks Challenging +1.2
$$I_n = \int_1^e x^2 (\ln x)^n dx, \quad n \geq 0$$
  1. Prove that, for \(n \geq 1\), $$I_n = \frac{e^3}{3} - \frac{n}{3} I_{n-1}$$ [4]
  2. Find the exact value of \(I_3\). [4]