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Edexcel F3 2017 June Q7
8 marks Standard +0.8
  1. Find $$\int \frac { 5 + x } { \sqrt { 4 - 3 x ^ { 2 } } } \mathrm {~d} x$$
  2. Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { 5 + x } { \sqrt { 4 - 3 x ^ { 2 } } } d x$$ giving your answer in the form \(p \pi \sqrt { 3 } + q\), where \(p\) and \(q\) are rational numbers to be found.
Edexcel F3 2017 June Q8
10 marks Challenging +1.8
8. The curve \(C\) has parametric equations $$x = \theta - \sin \theta , \quad y = 1 - \cos \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. The area of the curved surface generated is given by \(S\).
  1. Show that $$S = 2 \pi \sqrt { 2 } \int _ { 0 } ^ { 2 \pi } ( 1 - \cos \theta ) ^ { \frac { 3 } { 2 } } \mathrm {~d} \theta$$
  2. Hence find the exact value of \(S\).
Edexcel F3 2017 June Q9
12 marks Challenging +1.2
9 With respect to a fixed origin \(O\), the points \(A ( - 1,5,1 ) , B ( 1,0,3 ) , C ( 2 , - 1,2 )\) and \(D ( 3,6 , - 1 )\) are the vertices of a tetrahedron.
  1. Find the volume of the tetrahedron \(A B C D\). The plane \(\Pi\) contains the points \(A , B\) and \(C\).
  2. Find a cartesian equation of \(\Pi\). The point \(T\) lies on the plane \(\Pi\). The line \(D T\) is perpendicular to \(\Pi\).
  3. Find the exact coordinates of the point \(T\).
Edexcel F3 2018 June Q1
6 marks Standard +0.3
  1. Solve the equation
$$15 \operatorname { sech } ^ { 2 } x + 7 \tanh x = 13$$ Give your answers in terms of simplified natural logarithms.
Edexcel F3 2018 June Q2
9 marks Standard +0.3
2. $$\mathbf { A } = \left( \begin{array} { l l } 3 & 2 \\ 2 & 6 \end{array} \right)$$
  1. Find the eigenvalues and corresponding normalised eigenvectors of the matrix \(\mathbf { A }\).
  2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { \mathrm { T } } \mathbf { A P } = \mathbf { D }\).
Edexcel F3 2018 June Q3
6 marks Challenging +1.2
3. Given that $$y = \arctan \left( \frac { \sin x } { \cos x - 1 } \right) \quad x \neq 2 n \pi , \quad n \in \mathbb { Z }$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k$$ where \(k\) is a constant to be found. \(\_\_\_\_\) "
Edexcel F3 2018 June Q4
12 marks Challenging +1.2
4. The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ The line \(l\) is a normal to \(H\) at the point \(P ( a \sec \theta , b \tan \theta ) , 0 < \theta < \frac { \pi } { 2 }\)
  1. Using calculus, show that an equation for \(l\) is $$a x \sin \theta + b y = \left( a ^ { 2 } + b ^ { 2 } \right) \tan \theta$$ The line \(l\) meets the \(x\)-axis at the point \(Q\), and the point \(M\) is the midpoint of \(P Q\).
  2. Find the coordinates of \(M\).
  3. Hence find the cartesian equation of the locus of \(M\) as \(\theta\) varies, giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
Edexcel F3 2018 June Q5
11 marks Challenging +1.2
5. $$\mathbf { M } = \left( \begin{array} { r r r } 4 & - 5 & 0 \\ k & 2 & 0 \\ - 3 & - 5 & k \end{array} \right) \text {, where } k \text { is a real constant, } k \neq 0 , k \neq - \frac { 8 } { 5 }$$
  1. Find, in terms of \(k\), the inverse of the matrix \(\mathbf { M }\). A transformation \(T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix $$\left( \begin{array} { r r r } 4 & - 5 & 0 \\ - 1 & 2 & 0 \\ - 3 & - 5 & - 1 \end{array} \right)$$ The transformation \(T\) maps the plane \(\Pi _ { 1 }\) onto the plane \(\Pi _ { 2 }\) Given that the plane \(\Pi _ { 2 }\) has equation \(2 x - z = 4\)
  2. find a cartesian equation of the plane \(\Pi _ { 1 }\)
Edexcel F3 2018 June Q6
7 marks Challenging +1.8
6. The curve \(C\) has parametric equations $$x = \theta - \tanh \theta , \quad y = \operatorname { sech } \theta , \quad 0 \leqslant \theta \leqslant \ln 3$$
  1. Find
    1. \(\frac { \mathrm { d } x } { \mathrm {~d} \theta }\)
    2. \(\frac { \mathrm { d } y } { \mathrm {~d} \theta }\) The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  2. Find the exact area of the curved surface formed, giving your answer as a multiple of \(\pi\).
Edexcel F3 2018 June Q7
12 marks Standard +0.8
7. The plane \(\Pi _ { 1 }\) has equation \(x + y + z = 3\) and the plane \(\Pi _ { 2 }\) has equation \(2 x + 3 y - z = 4\) The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line \(L\).
  1. Find a cartesian equation for the line \(L\). The plane \(\Pi _ { 3 }\) has equation $$\text { r. } \left( \begin{array} { r } 5 \\ - 4 \\ 4 \end{array} \right) = 12$$ The line \(L\) meets the plane \(\Pi _ { 3 }\) at the point \(A\).
  2. Find the coordinates of \(A\).
  3. Find the acute angle between \(\overrightarrow { O A }\) and the line \(L\), where \(O\) is the origin. Give your answer in degrees to one decimal place.
Edexcel F3 2018 June Q8
12 marks Challenging +1.8
8. $$I _ { n } = \int \frac { x ^ { n } } { \sqrt { \left( x ^ { 2 } + k ^ { 2 } \right) } } \mathrm { d } x \quad \text { where } k \text { is a constant and } n \in \mathbb { Z } ^ { + }$$
  1. Show that, for \(n \geqslant 2\) $$I _ { n } = \frac { x ^ { n - 1 } } { n } \left( x ^ { 2 } + k ^ { 2 } \right) ^ { \frac { 1 } { 2 } } - \frac { ( n - 1 ) } { n } k ^ { 2 } I _ { n - 2 }$$
  2. Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { \sqrt { \left( x ^ { 2 } + 1 \right) } } \mathrm { d } x$$
Edexcel F2 2021 October Q1
4 marks Standard +0.3
  1. Solve the equation
$$z ^ { 5 } - 32 i = 0$$ giving each answer in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 < \theta < 2 \pi\)
Edexcel F2 2021 October Q2
8 marks Standard +0.3
2. Use algebra to determine the set of values of \(x\) for which $$\frac { x } { 2 - x } \leqslant \frac { x + 3 } { x }$$ (Solutions relying entirely on graphical methods are not acceptable.)
(8)
Edexcel F2 2021 October Q3
6 marks Challenging +1.2
3. A transformation maps points from the \(z\)-plane, where \(z = x + \mathrm { i } y\), to the \(w\)-plane, where \(w = u + \mathrm { i } v\). The transformation is given by $$w = \frac { ( 2 + \mathrm { i } ) z + 4 } { z - \mathrm { i } } \quad z \neq \mathrm { i }$$ The transformation maps the imaginary axis in the \(z\)-plane onto the line \(l\) in the \(w\)-plane.
Determine a Cartesian equation of \(l\), giving your answer in the form \(a u + b v + c = 0\) where \(a , b\) and \(c\) are integers to be found.
(6)
Edexcel F2 2021 October Q4
9 marks Standard +0.3
4.
  1. Determine the general solution of the differential equation $$( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } - x y = \mathrm { e } ^ { 3 x } \quad x > - 1$$ giving your answer in the form \(y = \mathrm { f } ( x )\).
  2. Determine the particular solution of the differential equation for which \(y = 5\) when \(x = 0\)
Edexcel F2 2021 October Q5
8 marks Standard +0.8
5. Given that \(y = \tan ^ { 2 } x\)
  1. show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = 8 \tan x \sec ^ { 2 } x \left( p \sec ^ { 2 } x + q \right)$$ where \(p\) and \(q\) are integers to be determined.
  2. Hence determine the Taylor series expansion about \(\frac { \pi } { 3 }\) of \(\tan ^ { 2 } x\) in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 3 }\), giving each coefficient in simplest form.
    \includegraphics[max width=\textwidth, alt={}, center]{8fa1e7da-009f-4b7f-9fa8-21a1768bfd73-19_33_407_306_258} \includegraphics[max width=\textwidth, alt={}, center]{8fa1e7da-009f-4b7f-9fa8-21a1768bfd73-19_58_458_2752_150}
Edexcel F2 2021 October Q6
9 marks Challenging +1.2
6. The complex number \(z\) on an Argand diagram is represented by the point \(P\) where $$| z + 1 - 13 i | = 3 | z - 7 - 5 i |$$ Given that the locus of \(P\) is a circle,
  1. determine the centre and radius of this circle. The complex number \(w\), on the same Argand diagram, is represented by the point \(Q\), where $$\arg ( w - 8 - 6 \mathrm { i } ) = - \frac { 3 \pi } { 4 }$$ Given that the locus of \(P\) intersects the locus of \(Q\) at the point \(R\),
  2. determine the complex number representing \(R\).
Edexcel F2 2021 October Q7
11 marks Challenging +1.2
7.
  1. Show that the transformation \(x = t ^ { 2 }\) transforms the differential equation $$4 x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 ( 1 + 2 \sqrt { x } ) \frac { \mathrm { d } y } { \mathrm {~d} x } - 15 y = 15 x$$ into the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} t } - 15 y = 15 t ^ { 2 }$$
  2. Solve differential equation (II) to determine \(y\) in terms of \(t\).
  3. Hence determine the general solution of differential equation (I). \includegraphics[max width=\textwidth, alt={}, center]{8fa1e7da-009f-4b7f-9fa8-21a1768bfd73-24_2258_53_308_1980}
Edexcel F2 2021 October Q8
11 marks Challenging +1.8
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8fa1e7da-009f-4b7f-9fa8-21a1768bfd73-28_735_892_264_529} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curve \(C\) shown in Figure 1 has polar equation $$r = 1 + \sin \theta \quad - \frac { \pi } { 2 } < \theta \leqslant \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) such that the tangent to \(C\) at \(P\) is perpendicular to the initial line.
  1. Use calculus to determine the polar coordinates of \(P\). The tangent to \(C\) at the point \(Q\) where \(\theta = \frac { \pi } { 2 }\) is parallel to the initial line.
    The tangent to \(C\) at \(Q\) meets the tangent to \(C\) at \(P\) at the point \(S\), as shown in Figure 1.
    The finite region \(R\), shown shaded in Figure 1, is bounded by the line segments \(Q S , S P\) and the curve \(C\).
  2. Use algebraic integration to show that the area of \(R\) is $$\frac { 1 } { 32 } ( a \sqrt { 3 } + b \pi )$$ where \(a\) and \(b\) are integers to be determined.
    (6)
Edexcel F2 2021 October Q9
9 marks Standard +0.8
  1. Show that $$n ^ { 5 } - ( n - 1 ) ^ { 5 } \equiv 5 n ^ { 4 } - 10 n ^ { 3 } + 10 n ^ { 2 } - 5 n + 1$$
  2. Hence, using the method of differences, show that for all integer values of \(n\), $$\sum _ { r = 1 } ^ { n } r ^ { 4 } = \frac { 1 } { 30 } n ( n + 1 ) ( 2 n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are integers to be determined.
Edexcel F2 2018 Specimen Q2
5 marks Standard +0.8
  1. Express \(\frac { 1 } { ( r + 6 ) ( r + 8 ) }\) in partial fractions.
  2. Hence show that $$\sum _ { r = 1 } ^ { n } \frac { 2 } { ( r + 6 ) ( r + 8 ) } = \frac { n ( a n + b ) } { 56 ( n + 7 ) ( n + 8 ) }$$ where \(a\) and \(b\) are integers to be found.
    VIIN SIHI NI JIIIM IONOOVIIV SIHI NI III HM ION OOVI4V SIHI NI JIIIM IONOO
Edexcel F2 2018 Specimen Q3
10 marks Challenging +1.2
  1. Show that the substitution \(z = y ^ { - 2 }\) transforms the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 2 x y = x \mathrm { e } ^ { - x ^ { 2 } } y ^ { 3 }$$ into the differential equation $$\frac { \mathrm { d } z } { \mathrm {~d} x } - 4 x z = - 2 x \mathrm { e } ^ { - x ^ { 2 } }$$
  2. Solve differential equation (II) to find \(z\) as a function of \(x\).
  3. Hence find the general solution of differential equation (I), giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
    VIIIV SIHI NI J14M 10N OCVIIN SIHI NI III HM ION OOVERV SIHI NI JIIIM ION OO
Edexcel F2 2018 Specimen Q4
9 marks Challenging +1.2
  1. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by
$$w = \frac { z - 1 } { z + 1 } , \quad z \neq - 1$$ The line in the \(z\)-plane with equation \(y = 2 x\) is mapped by \(T\) onto the curve \(C\) in the \(w\)-plane.
  1. Show that \(C\) is a circle and find its centre and radius. The region \(y < 2 x\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
  2. Sketch circle \(C\) on an Argand diagram and shade and label region \(R\).
    VIIIV SIHI NI IIIYM ION OCVIIV SIHI NI JIIIM ION OCVEXV SIHIL NI JIIIM ION OO
Edexcel F2 2018 Specimen Q5
9 marks Challenging +1.2
Given that \(y = \cot x\),
  1. show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 \cot x + 2 \cot ^ { 3 } x$$
  2. Hence show that $$\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = p \cot ^ { 4 } x + q \cot ^ { 2 } x + r$$ where \(p , q\) and \(r\) are integers to be found.
  3. Find the Taylor series expansion of \(\cot x\) in ascending powers of \(\left( x - \frac { \pi } { 3 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 3 } \right) ^ { 3 }\).
    VIIIV SIHI NI J14M 10N OCVIIN SIHI NI III HM ION OOVERV SIHI NI JIIIM ION OO
Edexcel F2 2018 Specimen Q6
13 marks Challenging +1.2
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y = 2 \sin x$$ Given that \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 0\)
  2. find the particular solution of differential equation (I).