Questions FP3 (473 questions)

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Edexcel FP3 2018 June Q5
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
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
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 \(O A B\).
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    Q7

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Edexcel FP3 Specimen Q1
  1. Find the eigenvalues of the matrix \(\left( \begin{array} { l l } 7 & 6
    6 & 2 \end{array} \right)\)
  2. Find the values of \(x\) for which
$$9 \cosh x - 6 \sinh x = 7$$ giving your answers as natural logarithms.
(Total 6 marks)
Edexcel FP3 Specimen Q3
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{6c256e1b-455d-42fb-81f2-a9a8ed1148bc-2_503_801_998_566}
\end{figure} The parametric equations of the curve \(C\) shown in Figure 1 are $$x = a ( t - \sin t ) , \quad y = a ( 1 - \cos t ) , \quad 0 \leq t \leq 2 \pi$$ Find, by using integration, the length of \(C\).
Edexcel FP3 Specimen Q4
4. Find \(\int \sqrt { } \left( x ^ { 2 } + 4 \right) \mathrm { d } x\).
Edexcel FP3 Specimen Q5
5. Given that \(y = \arcsin x\) prove that
  1. \(\quad \frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { \left( 1 - x ^ { 2 } \right) } }\)
  2. \(\quad \left( 1 - x ^ { 2 } \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - x \frac { \mathrm {~d} y } { \mathrm {~d} x } = 0\)
Edexcel FP3 Specimen Q6
6. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 2 } } x ^ { n } \sin x \mathrm {~d} x$$
  1. Show that for \(n \geq 2\) $$I _ { n } = n \left( \frac { \pi } { 2 } \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$
  2. Hence obtain \(I _ { 3 }\), giving your answers in terms of \(\pi\).
Edexcel FP3 Specimen Q7
7. $$\mathbf { A } ( x ) = \left( \begin{array} { c c c } 1 & x & - 1
3 & 0 & 2
1 & 1 & 0 \end{array} \right) , x \neq \frac { 5 } { 2 }$$
  1. Calculate the inverse of \(\mathbf { A } ( x )\). $$\mathbf { B } = \left( \begin{array} { c c c } 1 & 3 & - 1
    3 & 0 & 2
    1 & 1 & 0 \end{array} \right)$$ The image of the vector \(\left( \begin{array} { c } p
    q
    r \end{array} \right)\) when transformed by \(\mathbf { B }\) is \(\left( \begin{array} { l } 2
    3
    4 \end{array} \right)\)
  2. Find the values of \(p , q\) and \(r\).
Edexcel FP3 Specimen Q8
8. The points \(A , B , C\), and \(D\) have position vectors $$\mathbf { a } = 2 \mathbf { i } + \mathbf { k } , \mathrm { b } = \mathbf { i } + 3 \mathbf { j } , \mathbf { c } = \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } , \mathbf { d } = 4 \mathbf { j } + \mathbf { k }$$ respectively.
  1. Find \(\overrightarrow { A B } \times \overrightarrow { A C }\) and hence find the area of triangle \(A B C\).
  2. Find the volume of the tetrahedron \(A B C D\).
  3. Find the perpendicular distance of \(D\) from the plane containing \(A , B\) and \(C\).
Edexcel FP3 Specimen Q9
9. The hyperbola \(C\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\)
  1. Show that an equation of the normal to \(C\) at \(P ( a \sec \theta , b \tan \theta )\) is $$b y + a x \sin \theta = \left( a ^ { 2 } + b ^ { 2 } \right) \tan \theta$$ The normal at \(P\) cuts the coordinate axes at \(A\) and \(B\). The mid-point of \(A B\) is \(M\).
  2. Find, in cartesian form, an equation of the locus of \(M\) as \(\theta\) varies.
    (Total 13 marks)
OCR FP3 2007 January Q1
1
  1. Show that the set of numbers \(\{ 3,5,7 \}\), under multiplication modulo 8, does not form a group.
  2. The set of numbers \(\{ 3,5,7 , a \}\), under multiplication modulo 8 , forms a group. Write down the value of \(a\).
  3. State, justifying your answer, whether or not the group in part (ii) is isomorphic to the multiplicative group \(\left\{ e , r , r ^ { 2 } , r ^ { 3 } \right\}\), where \(e\) is the identity and \(r ^ { 4 } = e\).
OCR FP3 2007 January Q2
2 Find the equation of the line of intersection of the planes with equations $$\mathbf { r } . ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 4 \quad \text { and } \quad \mathbf { r } . ( \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } ) = 6 ,$$ giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).
OCR FP3 2007 January Q3
3
  1. Solve the equation \(z ^ { 2 } - 6 z + 36 = 0\), and give your answers in the form \(r ( \cos \theta \pm \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta \leqslant \pi\).
  2. Given that \(Z\) is either of the roots found in part (i), deduce the exact value of \(Z ^ { - 3 }\).
OCR FP3 2007 January Q4
4 The variables \(x\) and \(y\) are related by the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 2 } - y ^ { 2 } } { x y }$$
  1. Use the substitution \(y = x z\), where \(z\) is a function of \(x\), to obtain the differential equation $$x \frac { \mathrm {~d} z } { \mathrm {~d} x } = \frac { 1 - 2 z ^ { 2 } } { z }$$
  2. Hence show by integration that the general solution of the differential equation (A) may be expressed in the form \(x ^ { 2 } \left( x ^ { 2 } - 2 y ^ { 2 } \right) = k\), where \(k\) is a constant.
OCR FP3 2007 January Q5
5 A multiplicative group \(G\) of order 9 has distinct elements \(p\) and \(q\), both of which have order 3 . The group is commutative, the identity element is \(e\), and it is given that \(q \neq p ^ { 2 }\).
  1. Write down the elements of a proper subgroup of \(G\)
    (a) which does not contain \(q\),
    (b) which does not contain \(p\).
  2. Find the order of each of the elements \(p q\) and \(p q ^ { 2 }\), justifying your answers.
  3. State the possible order(s) of proper subgroups of \(G\).
  4. Find two proper subgroups of \(G\) which are distinct from those in part (i), simplifying the elements.
OCR FP3 2007 January Q6
6 The variables \(x\) and \(y\) satisfy the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + 3 y = 2 x + 1$$ Find
  1. the complementary function,
  2. the general solution. In a particular case, it is given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).
  3. Find the solution of the differential equation in this case.
  4. Write down the function to which \(y\) approximates when \(x\) is large and positive.
OCR FP3 2007 January Q7
7 The position vectors of the points \(A , B , C , D , G\) are given by $$\mathbf { a } = 6 \mathbf { i } + 4 \mathbf { j } + 8 \mathbf { k } , \quad \mathbf { b } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad \mathbf { c } = \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } , \quad \mathbf { d } = 3 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k } , \quad \mathbf { g } = 3 \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k }$$ respectively.
  1. The line through \(A\) and \(G\) meets the plane \(B C D\) at \(M\). Write down the vector equation of the line through \(A\) and \(G\) and hence show that the position vector of \(M\) is \(2 \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k }\).
  2. Find the value of the ratio \(A G : A M\).
  3. Find the position vector of the point \(P\) on the line through \(C\) and \(G\), such that \(\overrightarrow { C P } = \frac { 4 } { 3 } \overrightarrow { C G }\).
  4. Verify that \(P\) lies in the plane \(A B D\).
OCR FP3 2007 January Q8
8
  1. Use de Moivre's theorem to find an expression for \(\tan 4 \theta\) in terms of \(\tan \theta\).
  2. Deduce that \(\cot 4 \theta = \frac { \cot ^ { 4 } \theta - 6 \cot ^ { 2 } \theta + 1 } { 4 \cot ^ { 3 } \theta - 4 \cot \theta }\).
  3. Hence show that one of the roots of the equation \(x ^ { 2 } - 6 x + 1 = 0\) is \(\cot ^ { 2 } \left( \frac { 1 } { 8 } \pi \right)\).
  4. Hence find the value of \(\operatorname { cosec } ^ { 2 } \left( \frac { 1 } { 8 } \pi \right) + \operatorname { cosec } ^ { 2 } \left( \frac { 3 } { 8 } \pi \right)\), justifying your answer.
OCR FP3 2008 January Q1
1
  1. A group \(G\) of order 6 has the combination table shown below.
    \(e\)\(a\)\(b\)\(p\)\(q\)\(r\)
    \(e\)\(e\)\(a\)\(b\)\(p\)\(q\)\(r\)
    \(a\)\(a\)\(b\)\(e\)\(r\)\(p\)\(q\)
    \(b\)\(b\)\(e\)\(a\)\(q\)\(r\)\(p\)
    \(p\)\(p\)\(q\)\(r\)\(e\)\(a\)\(b\)
    \(q\)\(q\)\(r\)\(p\)\(b\)\(e\)\(a\)
    \(r\)\(r\)\(p\)\(q\)\(a\)\(b\)\(e\)
    1. State, with a reason, whether or not \(G\) is commutative.
    2. State the number of subgroups of \(G\) which are of order 2 .
    3. List the elements of the subgroup of \(G\) which is of order 3 .
  2. A multiplicative group \(H\) of order 6 has elements \(e , c , c ^ { 2 } , c ^ { 3 } , c ^ { 4 } , c ^ { 5 }\), where \(e\) is the identity. Write down the order of each of the elements \(c ^ { 3 } , c ^ { 4 }\) and \(c ^ { 5 }\).
OCR FP3 2008 January Q2
2 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 8 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 16 y = 4 x .$$
OCR FP3 2008 January Q3
3 Two fixed points, \(A\) and \(B\), have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to the origin \(O\), and a variable point \(P\) has position vector \(\mathbf { r }\).
  1. Give a geometrical description of the locus of \(P\) when \(\mathbf { r }\) satisfies the equation \(\mathbf { r } = \lambda \mathbf { a }\), where \(0 \leqslant \lambda \leqslant 1\).
  2. Given that \(P\) is a point on the line \(A B\), use a property of the vector product to explain why \(( \mathbf { r } - \mathbf { a } ) \times ( \mathbf { r } - \mathbf { b } ) = \mathbf { 0 }\).
  3. Give a geometrical description of the locus of \(P\) when \(\mathbf { r }\) satisfies the equation \(\mathbf { r } \times ( \mathbf { a } - \mathbf { b } ) = \mathbf { 0 }\).
OCR FP3 2008 January Q4
4 The integrals \(C\) and \(S\) are defined by $$C = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \cos 3 x \mathrm {~d} x \quad \text { and } \quad S = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \sin 3 x \mathrm {~d} x$$ By considering \(C + \mathrm { i } S\) as a single integral, show that $$C = - \frac { 1 } { 13 } \left( 2 + 3 \mathrm { e } ^ { \pi } \right) ,$$ and obtain a similar expression for \(S\).
(You may assume that the standard result for \(\int \mathrm { e } ^ { k x } \mathrm {~d} x\) remains true when \(k\) is a complex constant, so that \(\left. \int \mathrm { e } ^ { ( a + \mathrm { i } b ) x } \mathrm {~d} x = \frac { 1 } { a + \mathrm { i } b } \mathrm { e } ^ { ( a + \mathrm { i } b ) x } .\right)\)
OCR FP3 2008 January Q5
5
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { y } { x } = \sin 2 x$$ expressing \(y\) in terms of \(x\) in your answer. In a particular case, it is given that \(y = \frac { 2 } { \pi }\) when \(x = \frac { 1 } { 4 } \pi\).
  2. Find the solution of the differential equation in this case.
  3. Write down a function to which \(y\) approximates when \(x\) is large and positive.
OCR FP3 2008 January Q6
6 A tetrahedron \(A B C D\) is such that \(A B\) is perpendicular to the base \(B C D\). The coordinates of the points \(A , C\) and \(D\) are \(( - 1 , - 7,2 ) , ( 5,0,3 )\) and \(( - 1,3,3 )\) respectively, and the equation of the plane \(B C D\) is \(x + 2 y - 2 z = - 1\).
  1. Find, in either order, the coordinates of \(B\) and the length of \(A B\).
  2. Find the acute angle between the planes \(A C D\) and \(B C D\).
  3. (a) Verify, without using a calculator, that \(\theta = \frac { 1 } { 8 } \pi\) is a solution of the equation \(\sin 6 \theta = \sin 2 \theta\).
    (b) By sketching the graphs of \(y = \sin 6 \theta\) and \(y = \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), or otherwise, find the other solution of the equation \(\sin 6 \theta = \sin 2 \theta\) in the interval \(0 < \theta < \frac { 1 } { 2 } \pi\).
  4. Use de Moivre's theorem to prove that $$\sin 6 \theta \equiv \sin 2 \theta \left( 16 \cos ^ { 4 } \theta - 16 \cos ^ { 2 } \theta + 3 \right) .$$
  5. Hence show that one of the solutions obtained in part (i) satisfies \(\cos ^ { 2 } \theta = \frac { 1 } { 4 } ( 2 - \sqrt { 2 } )\), and justify which solution it is.