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Edexcel F3 2021 October Q2
6 marks Standard +0.8
2. Given that $$\cosh y = x \quad \text { and } \quad y < 0$$ use the definition of coshy in terms of exponential functions to prove that $$y = \ln \left( x - \sqrt { x ^ { 2 } - 1 } \right)$$
Edexcel F3 2021 October Q3
9 marks Challenging +1.8
3. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$$ The line \(l\) is the normal to \(E\) at the point \(P ( 8 \cos \theta , 6 \sin \theta )\).
  1. Using calculus, show that an equation for \(l\) is $$4 x \sin \theta - 3 y \cos \theta = 14 \sin \theta \cos \theta$$ The line \(l\) meets the \(x\)-axis at the point \(A\) and meets the \(y\)-axis at the point \(B\).
    The point \(M\) is the midpoint of \(A B\).
  2. Determine a Cartesian equation for the locus of \(M\) as \(\theta\) varies, giving your answer in the form \(a x ^ { 2 } + b y ^ { 2 } = c\) where \(a , b\) and \(c\) are integers.
Edexcel F3 2021 October Q4
11 marks Challenging +1.2
4. The matrix \(\mathbf { M }\) is given by $$\left( \begin{array} { r r r } 2 & 0 & - 1 \\ k & 3 & 2 \\ - 2 & 1 & k \end{array} \right)$$
  1. Show that \(\operatorname { det } \mathbf { M } = 5 k - 10\) Given that \(k \neq 2\)
  2. find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\). The points \(O ( 0,0,0 ) , A ( 4 , - 8,3 ) , B ( - 2,5 , - 4 )\) and \(C ( 4 , - 6,8 )\) are the vertices of a tetrahedron \(T\). The transformation represented by matrix \(\mathbf { M }\) transforms \(T\) to a tetrahedron with volume 50
  3. Determine the possible values of \(k\).
Edexcel F3 2021 October Q5
10 marks Standard +0.8
  1. The skew lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations
$$l _ { 1 } : \mathbf { r } = ( \mathbf { i } + 2 \mathbf { j } - 5 \mathbf { k } ) + \lambda ( 5 \mathbf { i } + \mathbf { j } )$$ and $$l _ { 2 } : \mathbf { r } = ( 2 \mathbf { i } - 4 \mathbf { j } + 4 \mathbf { k } ) + \mu ( 8 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Determine a vector that is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\)
  2. Determine an equation of the plane parallel to \(l _ { 1 }\) that contains \(l _ { 2 }\)
    1. in the form \(\mathbf { r } = \mathbf { a } + s \mathbf { b } + t \mathbf { c }\)
    2. in the form r.n \(= p\)
  3. Determine the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\) Give your answer in simplest form.
Edexcel F3 2021 October Q6
9 marks Challenging +1.8
6. $$I _ { n } = \int _ { 0 } ^ { \sqrt { \frac { \pi } { 2 } } } x ^ { n } \cos \left( x ^ { 2 } \right) \mathrm { d } x \quad n \geqslant 1$$
  1. Prove that, for \(n \geqslant 5\) $$I _ { n } = \frac { 1 } { 2 } \left( \frac { \pi } { 2 } \right) ^ { \frac { n - 1 } { 2 } } - \frac { 1 } { 4 } ( n - 1 ) ( n - 3 ) I _ { n - 4 }$$
  2. Hence, determine the exact value of \(I _ { 5 }\), giving your answer in its simplest form.
Edexcel F3 2021 October Q7
11 marks Challenging +1.8
7. A hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 25 } = 1$$ where \(a\) is a positive constant.
The eccentricity of \(H\) is \(e\).
  1. Determine an expression for \(e ^ { 2 }\) in terms of \(a\). The line \(l\) is the directrix of \(H\) for which \(x > 0\) The points \(A\) and \(A ^ { \prime }\) are the points of intersection of \(l\) with the asymptotes of \(H\).
  2. Determine, in terms of \(e\), the length of the line segment \(A A ^ { \prime }\). The point \(F\) is the focus of \(H\) for which \(x < 0\) Given that the area of triangle \(A F A ^ { \prime }\) is \(\frac { 164 } { 3 }\)
  3. show that \(a\) is a solution of the equation $$30 a ^ { 3 } - 164 a ^ { 2 } + 375 a - 4100 = 0$$
  4. Hence, using algebra and making your reasoning clear, show that the only possible value of \(a\) is \(\frac { 20 } { 3 }\)
Edexcel F3 2021 October Q8
13 marks Challenging +1.2
8. $$y = \arccos ( 2 \sqrt { x } )$$
  1. Determine \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
  2. Show that $$\int y \mathrm {~d} x = x \arccos ( 2 \sqrt { x } ) + \int \frac { \sqrt { x } } { \sqrt { 1 - 4 x } } \mathrm {~d} x$$
  3. Use the substitution \(\sqrt { x } = \frac { 1 } { 2 } \cos \theta\) to show that $$\int _ { 0 } ^ { \frac { 1 } { 8 } } \frac { \sqrt { x } } { \sqrt { 1 - 4 x } } \mathrm {~d} x = \frac { 1 } { 4 } \int _ { a } ^ { b } \cos ^ { 2 } \theta \mathrm {~d} \theta$$ where \(a\) and \(b\) are limits to be determined.
  4. Hence, determine the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 8 } } \arccos ( 2 \sqrt { x } ) d x$$
Edexcel F3 2018 Specimen Q1
6 marks Standard +0.3
  1. The curve \(C\) has equation
$$y = 9 \cosh x + 3 \sinh x + 7 x$$ Use differentiation to find the exact \(x\) coordinate of the stationary point of \(C\), giving your answer as a natural logarithm.
VIIIV SIHI NI IIIHM ION OCVIUV SIHI NI I II HM I ON OOVERV SIHI NI JIIIM ION OO
Edexcel F3 2018 Specimen Q2
11 marks Challenging +1.2
2. An ellipse has equation $$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 4 } = 1$$ The point \(P\) lies on the ellipse and has coordinates \(( 5 \cos \theta , 2 \sin \theta ) , 0 < \theta < \frac { \pi } { 2 }\) The line \(L\) is a normal to the ellipse at the point \(P\).
  1. Show that an equation for \(L\) is $$5 x \sin \theta - 2 y \cos \theta = 21 \sin \theta \cos \theta$$ Given that the line \(L\) crosses the \(y\)-axis at the point \(Q\) and that \(M\) is the midpoint of \(P Q\),
  2. find the exact area of triangle \(O P M\), where \(O\) is the origin, giving your answer as a multiple of \(\sin 2 \theta\)
    VIIIV SIHI NI JAIIM ION OCVIIIV SIHI NI JIHMM ION OOVI4V SIHI NI JIIYM IONOO
Edexcel F3 2018 Specimen Q3
12 marks Standard +0.8
3. Without using a calculator, find
  1. \(\int _ { - 2 } ^ { 1 } \frac { 1 } { x ^ { 2 } + 4 x + 13 } \mathrm {~d} x\), giving your answer as a multiple of \(\pi\),
  2. \(\int _ { - 1 } ^ { 4 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 12 x + 34 } } \mathrm {~d} x\), giving your answer in the form \(p \ln ( q + r \sqrt { 2 } )\),
    where \(p , q\) and \(r\) are rational numbers to be found.
    VIIIV SIHI NI J14M 10N OCVIIN SIHI NI III HM ION OOVERV SIHI NI JIIIM ION OO
Edexcel F3 2018 Specimen Q5
7 marks Challenging +1.8
  1. 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.
    VIIIV SIHI NI IIIYM ION OCVIIV SIHI NI JIIIM ION OCVEXV SIHIL NI JIIIM ION OO
Edexcel F3 2018 Specimen Q6
9 marks Standard +0.8
  1. The coordinates of the points \(A , B\) and \(C\) relative to a fixed origin \(O\) are \(( 1,2,3 )\),
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,
(b) find the value of \(k\). \section*{\(( - 1,3,4 )\) and \(( 2,1,6 )\) respectively. The plane \(\Pi\) contains the points \(A , B\) and \(C\).
(a) Find a cartesian equation of the plane \(\Pi\).
6. \(( - 1,3,4 )\) and \(( 2,1,6 )\) respectively. The plane (a) Find a cartesian equation of the plane \(\Pi\).}
Edexcel F3 2018 Specimen Q7
11 marks
  1. 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 FP3 Q2
7 marks Standard +0.8
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4255ef1b-2186-4a7e-adf3-a963601c95b2-04_333_360_328_794} \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 $$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = 3 \mathbf { i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k } .$$ Calculate
  1. \(\mathbf { b } \times \mathbf { c }\),
  2. a.(b \(\times \mathbf { c ) }\),
  3. the area of triangle \(O B C\),
  4. the volume of the tetrahedron \(O A B C\).
Edexcel FP3 2009 June Q1
5 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 2009 June Q2
8 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8b3dd4a1-b270-4bd7-88d6-fe10601f9d74-03_333_360_328_794} \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 $$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = 3 \mathbf { i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k } .$$ Calculate
  1. \(\mathbf { b } \times \mathbf { c }\),
  2. a.(b \(\times \mathbf { c ) }\),
  3. the area of triangle \(O B C\),
  4. the volume of the tetrahedron \(O A B C\).
Edexcel FP3 2009 June Q3
9 marks Standard +0.3
3. $$\mathbf { M } = \left( \begin{array} { r r r } 6 & 1 & - 1 \\ 0 & 7 & 0 \\ 3 & - 1 & 2 \end{array} \right)$$
  1. Show that 7 is an eigenvalue of the matrix \(\mathbf { M }\) and find the other two eigenvalues of \(\mathbf { M }\).
  2. Find an eigenvector corresponding to the eigenvalue 7.
Edexcel FP3 2009 June Q4
9 marks Challenging +1.3
  1. 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 2009 June Q5
11 marks Challenging +1.3
5. $$I _ { n } = \int _ { 0 } ^ { 5 } \frac { x ^ { n } } { \sqrt { } \left( 25 - x ^ { 2 } \right) } d x , \quad n \geqslant 0$$
  1. Find an expression for \(\int \frac { x } { \sqrt { } \left( 25 - x ^ { 2 } \right) } \mathrm { d } x , \quad 0 \leqslant x \leqslant 5\).
  2. Using your answer to part (a), or otherwise, show that $$I _ { n } = \frac { 25 ( n - 1 ) } { n } I _ { n - 2 } \quad n \geqslant 2$$
  3. Find \(I _ { 4 }\) in the form \(k \pi\), where \(k\) is a fraction.
Edexcel FP3 2009 June Q6
11 marks Challenging +1.2
  1. 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 2009 June Q7
11 marks Standard +0.3
7. The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 3 \\ 4 \end{array} \right) \text { and } \quad \mathbf { r } = \left( \begin{array} { r } \alpha \\ - 4 \\ 0 \end{array} \right) + \mu \left( \begin{array} { l } 0 \\ 3 \\ 2 \end{array} \right) .$$ 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 2009 June Q8
11 marks Challenging +1.8
  1. A curve, which is part of an ellipse, has parametric equations
$$x = 3 \cos \theta , \quad y = 5 \sin \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 } .$$ 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 $$k \pi \int _ { 0 } ^ { \alpha } \sqrt { } \left( 16 c ^ { 2 } + 9 \right) \mathrm { d } c , \quad \text { where } c = \cos \theta$$ 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 2010 June Q1
5 marks Challenging +1.2
  1. The line \(x = 8\) is a directrix of the ellipse with equation
$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , \quad a > 0 , b > 0$$ and the point \(( 2,0 )\) is the corresponding focus.
Find the value of \(a\) and the value of \(b\).
Edexcel FP3 2010 June Q2
5 marks Standard +0.3
2. Use calculus to find the exact value of \(\int _ { - 2 } ^ { 1 } \frac { 1 } { x ^ { 2 } + 4 x + 13 } \mathrm {~d} x\).
Edexcel FP3 2010 June Q3
8 marks Standard +0.3
3. (a) Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, prove that $$\cosh 2 x = 1 + 2 \sinh ^ { 2 } x$$ (b) Solve the equation $$\cosh 2 x - 3 \sinh x = 15$$ giving your answers as exact logarithms.