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CAIE FP1 2008 June Q4
7 marks Standard +0.3
4 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$r = \theta + 2 \quad \text { and } \quad r = \theta ^ { 2 }$$ respectively, where \(0 \leqslant \theta \leqslant \pi\).
  1. Find the polar coordinates of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
  2. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
  3. Find the area bounded by \(C _ { 1 } , C _ { 2 }\) and the line \(\theta = 0\).
CAIE FP1 2008 June Q5
7 marks Challenging +1.2
5 The equation $$x ^ { 3 } + x - 1 = 0$$ has roots \(\alpha , \beta , \gamma\). Show that the equation with roots \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 }\) is $$y ^ { 3 } - 3 y ^ { 2 } + 4 y - 1 = 0$$ Hence find the value of \(\alpha ^ { 6 } + \beta ^ { 6 } + \gamma ^ { 6 }\).
CAIE FP1 2008 June Q6
8 marks Challenging +1.2
6 The curve \(C\) is defined parametrically by $$x = 4 t - t ^ { 2 } \quad \text { and } \quad y = 1 - \mathrm { e } ^ { - t }$$ where \(0 \leqslant t < 2\). Show that at all points of \(C\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { ( t - 1 ) \mathrm { e } ^ { - t } } { 4 ( 2 - t ) ^ { 3 } }$$ Show that the mean value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 7 } { 4 }\) is $$\frac { 4 e ^ { - \frac { 1 } { 2 } } - 3 } { 21 }$$
CAIE FP1 2008 June Q7
8 marks Standard +0.8
7 Prove by induction that $$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 5 } + r ^ { 3 } \right) = \frac { 1 } { 2 } n ^ { 3 } ( n + 1 ) ^ { 3 }$$ for all \(n \geqslant 1\). Use this result together with the List of Formulae (MF10) to prove that $$\sum _ { r = 1 } ^ { n } r ^ { 5 } = \frac { 1 } { 12 } n ^ { 2 } ( n + 1 ) ^ { 2 } \mathrm { Q } ( n )$$ where \(\mathrm { Q } ( n )\) is a quadratic function of \(n\) which is to be determined.
CAIE FP1 2008 June Q8
10 marks Challenging +1.8
8
  1. Given that $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } t ^ { n } \sin t \mathrm {~d} t$$ show that, for \(n \geqslant 2\), $$I _ { n } = n \left( \frac { \pi } { 2 } \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 } .$$
  2. A curve \(C\) in the \(x - y\) plane is defined parametrically in terms of \(t\). It is given that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = t ^ { 4 } ( 1 - \cos 2 t ) \quad \text { and } \quad \frac { \mathrm { d } y } { \mathrm {~d} t } = t ^ { 4 } \sin 2 t .$$ Find the length of the arc of \(C\) from the point where \(t = 0\) to the point where \(t = \frac { 1 } { 2 } \pi\).
CAIE FP1 2008 June Q9
10 marks Standard +0.8
9 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 2 x + \lambda } { x + 1 }$$ where \(\lambda\) is a constant. Show that the equations of the asymptotes of \(C\) are independent of \(\lambda\). Find the value of \(\lambda\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case. Sketch \(C\) in the case \(\lambda = - 4\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.
CAIE FP1 2008 June Q10
10 marks Challenging +1.8
10 By considering \(\sum _ { n = 1 } ^ { N } z ^ { 2 n - 1 }\), where \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), show that $$\sum _ { n = 1 } ^ { N } \cos ( 2 n - 1 ) \theta = \frac { \sin ( 2 N \theta ) } { 2 \sin \theta }$$ where \(\sin \theta \neq 0\). Deduce that $$\sum _ { n = 1 } ^ { N } ( 2 n - 1 ) \sin \left[ \frac { ( 2 n - 1 ) \pi } { N } \right] = - N \operatorname { cosec } \frac { \pi } { N }$$
CAIE FP1 2008 June Q11
11 marks Challenging +1.3
11 Show that, with a suitable value of the constant \(\alpha\), the substitution \(y = x ^ { \alpha } w\) reduces the differential equation $$2 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \left( 3 x ^ { 2 } + 8 x \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + \left( x ^ { 2 } + 6 x + 4 \right) y = \mathrm { f } ( x )$$ to $$2 \frac { \mathrm {~d} ^ { 2 } w } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} w } { \mathrm {~d} x } + w = \mathrm { f } ( x )$$ Find the general solution for \(y\) in the case where \(\mathrm { f } ( x ) = 6 \sin 2 x + 7 \cos 2 x\).
CAIE FP1 2008 June Q12 EITHER
Challenging +1.8
The position vectors of the points \(A , B , C , D\) are
\(7 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\),
\(3 \mathbf { i } + 5 \mathbf { j } - 2 \mathbf { k }\),
\(2 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }\),
\(2 \mathbf { i } + 7 \mathbf { j } + \lambda \mathbf { k }\)
respectively. It is given that the shortest distance between the line \(A B\) and the line \(C D\) is 3 .
  1. Show that \(\lambda ^ { 2 } - 5 \lambda + 4 = 0\).
  2. Find the acute angle between the planes through \(A , B , D\) corresponding to the values of \(\lambda\) satisfying the equation in part (i).
CAIE FP1 2008 June Q12 OR
Challenging +1.8
The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix $$\left( \begin{array} { r r r r } 1 & 2 & - 1 & - 1 \\ 1 & 3 & - 1 & 0 \\ 1 & 0 & 3 & 1 \\ 0 & 3 & - 4 & - 1 \end{array} \right) .$$ The range space of T is denoted by \(V\).
  1. Determine the dimension of \(V\).
  2. Show that the vectors \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \\ 0 \end{array} \right) , \left( \begin{array} { l } 2 \\ 3 \\ 0 \\ 3 \end{array} \right) , \left( \begin{array} { r } - 1 \\ - 1 \\ 3 \\ - 4 \end{array} \right)\) are linearly independent.
  3. Write down a basis of \(V\). The set of elements of \(\mathbb { R } ^ { 4 }\) which do not belong to \(V\) is denoted by \(W\).
  4. State, with a reason, whether \(W\) is a vector space.
  5. Show that if the vector \(\left( \begin{array} { l } x \\ y \\ z \\ t \end{array} \right)\) belongs to \(W\) then \(y - z - t \neq 0\).
CAIE FP1 2009 June Q1
5 marks Standard +0.8
1 The equation $$x ^ { 4 } - x ^ { 3 } - 1 = 0$$ has roots \(\alpha , \beta , \gamma , \delta\). By using the substitution \(y = x ^ { 3 }\), or by any other method, find the exact value of \(\alpha ^ { 6 } + \beta ^ { 6 } + \gamma ^ { 6 } + \delta ^ { 6 }\).
CAIE FP1 2009 June Q2
6 marks Standard +0.8
2 Verify that, for all positive values of \(n\), $$\frac { 1 } { ( n + 2 ) ( 2 n + 3 ) } - \frac { 1 } { ( n + 3 ) ( 2 n + 5 ) } = \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) } .$$ For the series $$\sum _ { n = 1 } ^ { N } \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) }$$ find
  1. the sum to \(N\) terms,
  2. the sum to infinity.
CAIE FP1 2009 June Q3
6 marks Challenging +1.2
3 The equation of a curve is \(y = \lambda x ^ { 2 }\), where \(\lambda > 0\). The region bounded by the curve, the \(x\)-axis and the line \(x = a\), where \(a > 0\), is denoted by \(R\). The \(y\)-coordinate of the centroid of \(R\) is \(a\). Show that \(\lambda = \frac { 10 } { 3 a }\).
CAIE FP1 2009 June Q4
6 marks Standard +0.8
4 A curve has equation $$y = \frac { 1 } { 3 } x ^ { 3 } + 1$$ The length of the arc of the curve joining the point where \(x = 0\) to the point where \(x = 1\) is denoted by \(s\). Show that $$s = \int _ { 0 } ^ { 1 } \sqrt { } \left( 1 + x ^ { 4 } \right) \mathrm { d } x$$ The surface area generated when this arc is rotated through one complete revolution about the \(x\)-axis is denoted by \(S\). Show that $$S = \frac { 1 } { 9 } \pi ( 18 s + 2 \sqrt { } 2 - 1 )$$ [Do not attempt to evaluate \(s\) or \(S\).]
CAIE FP1 2009 June Q5
7 marks Standard +0.8
5 Draw a sketch of the curve \(C\) whose polar equation is \(r = \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). On the same diagram draw the line \(\theta = \alpha\), where \(0 < \alpha < \frac { 1 } { 2 } \pi\). The region bounded by \(C\) and the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(R\). Find the exact value of \(\alpha\) for which the line \(\theta = \alpha\) divides \(R\) into two regions of equal area.
CAIE FP1 2009 June Q6
7 marks Standard +0.8
6 A curve has equation $$( x + y ) \left( x ^ { 2 } + y ^ { 2 } \right) = 1$$ Find the values of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(( 0,1 )\).
CAIE FP1 2009 June Q7
8 marks Challenging +1.2
7 Let $$I _ { n } = \int _ { 0 } ^ { 1 } t ^ { n } \mathrm { e } ^ { - t } \mathrm {~d} t$$ where \(n \geqslant 0\). Show that, for all \(n \geqslant 1\), $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 }$$ Hence prove by induction that, for all positive integers \(n\), $$I _ { n } < n ! .$$
CAIE FP1 2009 June Q8
8 marks Challenging +1.2
8 Find the general solution of the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 65 y = 65 x ^ { 2 } + 8 x + 73$$ Show that, whatever the initial conditions, \(\frac { y } { x ^ { 2 } } \rightarrow 1\) as \(x \rightarrow \infty\).
CAIE FP1 2009 June Q9
11 marks Standard +0.8
9 The matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & 4 \\ 1 & 5 & - 1 \\ 2 & 1 & 5 \end{array} \right)$$ has eigenvalues \(1,5,7\). Find a set of corresponding eigenvectors. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { n } = \mathbf { P D P } ^ { - 1 }\).
[0pt] [The evaluation of \(\mathbf { P } ^ { - 1 }\) is not required.]
Determine the set of values of the real constant \(k\) such that \(k ^ { n } \mathbf { A } ^ { n }\) tends to the zero matrix as \(n \rightarrow \infty\).
CAIE FP1 2009 June Q10
11 marks Standard +0.8
10 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { x + \lambda }$$ where \(\lambda\) is a non-zero constant. Obtain the equation of each of the asymptotes of \(C\). In separate diagrams, sketch \(C\) for the cases \(\lambda > 0\) and \(\lambda < 0\). In both cases the coordinates of the turning points must be indicated.
CAIE FP1 2009 June Q11
12 marks Challenging +1.8
11 The line \(l _ { 1 }\) is parallel to the vector \(4 \mathbf { j } - \mathbf { k }\) and passes through the point \(A\) whose position vector is \(2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\). The variable line \(l _ { 2 }\) is parallel to the vector \(\mathbf { i } - ( 2 \sin t ) \mathbf { j }\), where \(0 \leqslant t < 2 \pi\), and passes through the point \(B\) whose position vector is \(\mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\). The points \(P\) and \(Q\) are on \(l _ { 1 }\) and \(l _ { 2 }\), respectively, and \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).
  1. Find the length of \(P Q\) in terms of \(t\).
  2. Hence find the values of \(t\) for which \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
  3. For the case \(t = \frac { 1 } { 4 } \pi\), find the perpendicular distance from \(A\) to the plane \(B P Q\), giving your answer correct to 3 decimal places.
CAIE FP1 2009 June Q12 EITHER
Challenging +1.8
By considering \(\sum _ { k = 0 } ^ { n - 1 } ( 1 + \mathrm { i } \tan \theta ) ^ { k }\), show that $$\sum _ { k = 0 } ^ { n - 1 } \cos k \theta \sec ^ { k } \theta = \cot \theta \sin n \theta \sec ^ { n } \theta$$ provided \(\theta\) is not an integer multiple of \(\frac { 1 } { 2 } \pi\). Hence or otherwise show that $$\sum _ { k = 0 } ^ { n - 1 } 2 ^ { k } \cos \left( \frac { 1 } { 3 } k \pi \right) = \frac { 2 ^ { n } } { \sqrt { 3 } } \sin \left( \frac { 1 } { 3 } n \pi \right)$$ Given that \(0 < x < 1\), show that $$\sum _ { k = 0 } ^ { n - 1 } \frac { \cos \left( k \cos ^ { - 1 } x \right) } { x ^ { k } } = \frac { \sin \left( n \cos ^ { - 1 } x \right) } { x ^ { n - 1 } \sqrt { } \left( 1 - x ^ { 2 } \right) }$$
CAIE FP1 2009 June Q12 OR
Challenging +1.8
The linear transformations \(\mathrm { T } _ { 1 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) and \(\mathrm { T } _ { 2 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) are represented by the matrices \(\mathbf { M } _ { 1 }\) and \(\mathbf { M } _ { 2 }\), respectively, where $$\mathbf { M } _ { 1 } = \left( \begin{array} { r r r r } 1 & 1 & 1 & 2 \\ 1 & 4 & 7 & 8 \\ 1 & 7 & 11 & 13 \\ 1 & 2 & 5 & 5 \end{array} \right) , \quad \mathbf { M } _ { 2 } = \left( \begin{array} { r r r r } 2 & 0 & - 1 & - 1 \\ 5 & 1 & - 3 & - 3 \\ 3 & - 1 & - 1 & - 1 \\ 13 & - 1 & - 6 & - 6 \end{array} \right) .$$
  1. Find a basis for \(R _ { 1 }\), the range space of \(\mathrm { T } _ { 1 }\).
  2. Find a basis for \(K _ { 2 }\), the null space of \(\mathrm { T } _ { 2 }\), and hence show that \(K _ { 2 }\) is a subspace of \(R _ { 1 }\). The set of vectors which belong to \(R _ { 1 }\) but do not belong to \(K _ { 2 }\) is denoted by \(W\).
  3. State whether \(W\) is a vector space, justifying your answer. The linear transformation \(\mathrm { T } _ { 3 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is the result of applying \(\mathrm { T } _ { 1 }\) and then \(\mathrm { T } _ { 2 }\), in that order.
  4. Find the dimension of the null space of \(\mathrm { T } _ { 3 }\).
CAIE FP1 2010 June Q1
5 marks Standard +0.8
1 The variables \(x\) and \(y\) are such that \(y = - 1\) when \(x = 1\) and $$x ^ { 2 } + y ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 3 } = 29$$ Find the values of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 1\).
CAIE FP1 2010 June Q2
7 marks Standard +0.8
2 The curve \(C\) has polar equation $$r = a \left( 1 - \mathrm { e } ^ { - \theta } \right)$$ where \(a\) is a positive constant and \(0 \leqslant \theta < 2 \pi\).
  1. Draw a sketch of \(C\).
  2. Show that the area of the region bounded by \(C\) and the lines \(\theta = \ln 2\) and \(\theta = \ln 4\) is $$\frac { 1 } { 2 } a ^ { 2 } \left( \ln 2 - \frac { 13 } { 32 } \right)$$