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Edexcel F3 2016 June Q5
7 marks Challenging +1.8
5. Given that \(y = \operatorname { artanh } ( \cos x )\)
  1. show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } x$$
  2. Hence find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \cos x \operatorname { artanh } ( \cos x ) d x$$ giving your answer in the form \(a \ln ( b + c \sqrt { 3 } ) + d \pi\), where \(a , b , c\) and \(d\) are rational numbers to be found.
    (5)
Edexcel F3 2016 June Q6
9 marks Standard +0.8
6. The coordinates of the points \(A , B\) and \(C\) relative to a fixed origin \(O\) are ( \(1,2,3\) ), \(( - 1,3,4 )\) and \(( 2,1,6 )\) respectively. The plane \(\Pi\) contains the points \(A , B\) and \(C\).
  1. Find a cartesian equation of the plane \(\Pi\). The point \(D\) has coordinates \(( k , 4,14 )\) where \(k\) is a positive constant.
    Given that the volume of the tetrahedron \(A B C D\) is 6 cubic units,
  2. find the value of \(k\).
    VIIIV SIHI NI IIIIM I I O N OAVI4V SIHI NI JIIIM ION OCVJYV SIHI NI JIIIM ION OO
Edexcel F3 2016 June Q7
11 marks Challenging +1.8
7. The curve \(C\) has parametric equations $$x = 3 t ^ { 4 } , \quad y = 4 t ^ { 3 } , \quad 0 \leqslant t \leqslant 1$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. The area of the curved surface generated is \(S\).
  1. Show that $$S = k \pi \int _ { 0 } ^ { 1 } t ^ { 5 } \left( t ^ { 2 } + 1 \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} t$$ where \(k\) is a constant to be found.
  2. Use the substitution \(u ^ { 2 } = t ^ { 2 } + 1\) to find the value of \(S\), giving your answer in the form \(p \pi ( 11 \sqrt { 2 } - 4 )\) where \(p\) is a rational number to be found.
Edexcel F3 2016 June Q8
10 marks Challenging +1.8
8. $$I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { 2 n } x \mathrm {~d} x , \quad n \geqslant 0$$
  1. Show that, for \(n \geqslant 1\)
  2. Hence show that $$\int _ { 0 } ^ { \ln 2 } \tanh ^ { 4 } x \mathrm {~d} x = p + \ln 2$$ where \(p\) is a rational number to be found.
    8. \(\quad I _ { n } = \int _ { 0 } ^ { \ln 2 } \tanh ^ { 2 n } x \mathrm {~d} x , \quad n \geqslant 0\)
  3. Show that, for \(n \geqslant 1\) $$I _ { n } = I _ { n - 1 } - \frac { 1 } { 2 n - 1 } \left( \frac { 3 } { 5 } \right) ^ { 2 n - 1 }$$
Edexcel F3 2017 June Q1
5 marks Standard +0.3
  1. Solve the equation
$$18 \cosh x + 14 \sinh x = 11 + \mathrm { e } ^ { x }$$ Give your answers in the form \(\ln a\), where \(a\) is rational.
Edexcel F3 2017 June Q2
6 marks Moderate -0.8
2. $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & 3 & a \\ 2 & 0 & 1 \\ 1 & - 2 & 1 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 0 & 4 \\ 3 & - 2 & 3 \\ 1 & 2 & b \end{array} \right)$$ where \(a\) and \(b\) are constants.
  1. Write down \(\mathbf { A } ^ { \mathrm { T } }\) in terms of \(a\).
  2. Calculate \(\mathbf { A B }\), giving your answer in terms of \(a\) and \(b\).
  3. Hence show that $$( \mathbf { A B } ) ^ { \mathrm { T } } = \mathbf { B } ^ { \mathrm { T } } \mathbf { A } ^ { \mathrm { T } }$$
Edexcel F3 2017 June Q3
8 marks Challenging +1.2
3. Given that $$y = x - \operatorname { artanh } \left( \frac { 2 x } { 1 + x ^ { 2 } } \right)$$
  1. show that $$1 - \frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { 1 - x ^ { 2 } }$$ where \(k\) is a constant to be found.
  2. Hence, or otherwise, show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + x \left( 1 - \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = 0$$
Edexcel F3 2017 June Q4
8 marks Standard +0.3
4. $$\mathbf { M } = \left( \begin{array} { l l l } 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{array} \right)$$
  1. Show that 6 is an eigenvalue of the matrix \(\mathbf { M }\) and find the other two eigenvalues of \(\mathbf { M }\).
  2. Find a normalised eigenvector corresponding to the eigenvalue 6
Edexcel F3 2017 June Q5
9 marks Challenging +1.8
5. $$I _ { n } = \int \operatorname { cosec } ^ { n } x \mathrm {~d} x , \quad 0 < x < \frac { \pi } { 2 } , \quad n \geqslant 0$$
  1. Show that, for \(n \geqslant 2\) $$I _ { n } = \frac { n - 2 } { n - 1 } I _ { n - 2 } - \frac { 1 } { n - 1 } \cot x \operatorname { cosec } ^ { n - 2 } x$$
  2. Hence, or otherwise, find $$\int \operatorname { cosec } ^ { 4 } x \mathrm {~d} x$$ giving your answer in terms of \(\cot x\).
Edexcel F3 2017 June Q6
9 marks Challenging +1.8
  1. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\) and the ellipse \(E\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1\) where \(a > b > 0\) The line \(l\) is a tangent to hyperbola \(H\) at the point \(P ( a \sec \theta , b \tan \theta )\), where \(0 < \theta < \frac { \pi } { 2 }\)
    1. Using calculus, show that an equation for \(l\) is
    $$b x \sec \theta - a y \tan \theta = a b$$ Given that the point \(F\) is the focus of ellipse \(E\) for which \(x > 0\) and that the line \(l\) passes through \(F\),
  2. show that \(l\) is parallel to the line \(y = x\)
Edexcel F3 2017 June Q7
8 marks Standard +0.8
  1. (a) Find
$$\int \frac { 5 + x } { \sqrt { 4 - 3 x ^ { 2 } } } \mathrm {~d} x$$ (b) 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
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. (a) 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 \(y = \mathrm { f } ( x )\).
(b) Determine the particular solution of the differential equation for which \(y = 5\) when \(x = 0\)