Questions FP1 (1385 questions)

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CAIE FP1 2008 November Q1
1 The curve \(C\) is defined parametrically by $$x = t ^ { 4 } - 4 \ln t , \quad y = 4 t ^ { 2 }$$ Show that the length of the arc of \(C\) from the point where \(t = 2\) to the point where \(t = 4\) is $$240 + 4 \ln 2 .$$
CAIE FP1 2008 November Q2
2 Let \(y = \mathrm { e } ^ { x }\). Find the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant 2\). Show that the mean value of \(x\) with respect to \(y\) over the interval \(1 \leqslant y \leqslant \mathrm { e } ^ { 2 }\) is \(\frac { \mathrm { e } ^ { 2 } + 1 } { \mathrm { e } ^ { 2 } - 1 }\).
CAIE FP1 2008 November Q3
3 The curve \(C\) has polar equation $$r = \left( \frac { 1 } { 2 } \pi - \theta \right) ^ { 2 } ,$$ where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Draw a sketch of \(C\). Find the area of the region bounded by \(C\) and the initial line, leaving your answer in terms of \(\pi\).
CAIE FP1 2008 November Q4
4 The matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 2 }\) and state the corresponding eigenvalue. Given that one eigenvalue of \(\mathbf { A }\) is 3 , find an eigenvalue of the matrix \(\mathbf { A } ^ { 4 } + 3 \mathbf { A } ^ { 2 } + 2 \mathbf { I }\), justifying your answer.
CAIE FP1 2008 November Q5
5 The curve \(C\) has equation $$x ^ { 2 } - x y - 2 y ^ { 2 } = 4 .$$ Show that, at the point \(A ( 2,0 )\) on \(C , \frac { \mathrm {~d} y } { \mathrm {~d} x } = 2\). Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
CAIE FP1 2008 November Q6
6 The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & - 1 & - 2 & - 3
- 2 & 1 & 7 & 2
- 3 & 3 & 6 & \alpha
7 & - 6 & - 17 & - 17 \end{array} \right) .$$
  1. Show that if \(\alpha = 9\) then the rank of \(\mathbf { A }\) is 2, and find a basis for the null space of \(\mathbf { A }\) in this case.
  2. Find the rank of \(\mathbf { A }\) when \(\alpha \neq 9\).
CAIE FP1 2008 November Q7
7 Let \(I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 4 } \right) ^ { n } } \mathrm {~d} x\). By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \frac { x } { \left( 1 + x ^ { 4 } \right) ^ { n } } \right)\), show that $$4 n I _ { n + 1 } = \frac { 1 } { 2 ^ { n } } + ( 4 n - 1 ) I _ { n }$$ Given that \(I _ { 1 } = 0.86697\), correct to 5 decimal places, find \(I _ { 3 }\).
CAIE FP1 2008 November Q8
8 Find \(y\) in terms of \(t\), given that $$5 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 5 y = 15 + 12 t + 5 t ^ { 2 }$$ and that \(y = \frac { \mathrm { d } y } { \mathrm {~d} t } = 0\) when \(t = 0\).
CAIE FP1 2008 November Q9
9 Use induction to prove that $$\sum _ { n = 1 } ^ { N } \frac { 4 n + 1 } { n ( n + 1 ) ( 2 n - 1 ) ( 2 n + 1 ) } = 1 - \frac { 1 } { ( N + 1 ) ( 2 N + 1 ) }$$ Show that $$\sum _ { n = N + 1 } ^ { 2 N } \frac { 4 n + 1 } { n ( n + 1 ) ( 2 n - 1 ) ( 2 n + 1 ) } < \frac { 3 } { 8 N ^ { 2 } }$$
CAIE FP1 2008 November Q10
10 Use de Moivre's theorem to express \(\cos 8 \theta\) as a polynomial in \(\cos \theta\). Hence
  1. express \(\cos 8 \theta\) as a polynomial in \(\sin \theta\),
  2. find the exact value of $$4 x ^ { 4 } - 8 x ^ { 3 } + 5 x ^ { 2 } - x$$ where \(x = \cos ^ { 2 } \left( \frac { 1 } { 8 } \pi \right)\).
CAIE FP1 2008 November Q11
11 The plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \theta ( 2 \mathbf { j } - \mathbf { k } ) + \phi ( 3 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )$$ Find a vector normal to \(\Pi _ { 1 }\) and hence show that the equation of \(\Pi _ { 1 }\) can be written as \(2 x + 3 y + 6 z = 14\). The line \(l\) has equation $$\mathbf { r } = 3 \mathbf { i } + 8 \mathbf { j } + 2 \mathbf { k } + t ( 4 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k } )$$ The point on \(l\) where \(t = \lambda\) is denoted by \(P\). Find the set of values of \(\lambda\) for which the perpendicular distance of \(P\) from \(\Pi _ { 1 }\) is not greater than 4 . The plane \(\Pi _ { 2 }\) contains \(l\) and the point with position vector \(\mathbf { i } + 2 \mathbf { j } + \mathbf { k }\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
CAIE FP1 2008 November Q12 EITHER
The curve \(C\) has equation $$y = \frac { ( x - 2 ) ( x - a ) } { ( x - 1 ) ( x - 3 ) } ,$$ where \(a\) is a constant not equal to 1,2 or 3 .
  1. Write down the equations of the asymptotes of \(C\).
  2. Show that \(C\) meets the asymptote parallel to the \(x\)-axis at the point where \(x = \frac { 2 a - 3 } { a - 2 }\).
  3. Show that the \(x\)-coordinates of any stationary points on \(C\) satisfy $$( a - 2 ) x ^ { 2 } + ( 6 - 4 a ) x + ( 5 a - 6 ) = 0$$ and hence find the set of values of \(a\) for which \(C\) has stationary points.
  4. Sketch the graph of \(C\) for
    (a) \(a > 3\),
    (b) \(2 < a < 3\).
CAIE FP1 2008 November Q12 OR
The roots of the equation $$x ^ { 4 } - 5 x ^ { 2 } + 2 x - 1 = 0$$ are \(\alpha , \beta , \gamma , \delta\). Let \(S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } + \delta ^ { n }\).
  1. Show that $$S _ { n + 4 } - 5 S _ { n + 2 } + 2 S _ { n + 1 } - S _ { n } = 0 .$$
  2. Find the values of \(S _ { 2 }\) and \(S _ { 4 }\).
  3. Find the value of \(S _ { 3 }\) and hence find the value of \(S _ { 6 }\).
  4. Hence find the value of $$\alpha ^ { 2 } \left( \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 } \right) + \beta ^ { 2 } \left( \gamma ^ { 4 } + \delta ^ { 4 } + \alpha ^ { 4 } \right) + \gamma ^ { 2 } \left( \delta ^ { 4 } + \alpha ^ { 4 } + \beta ^ { 4 } \right) + \delta ^ { 2 } \left( \alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } \right) .$$ \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE FP1 2009 November Q1
1 Given that $$y = x ^ { 2 } \sin x$$
  1. show that the mean value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\) is \(\frac { 1 } { 2 } \pi\),
  2. find the mean value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
CAIE FP1 2009 November Q2
2 Relative to an origin \(O\), the points \(A , B , C\) have position vectors $$\mathbf { i } , \quad \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + \mathbf { j } + \theta \mathbf { k }$$ respectively. The shortest distance between the lines \(A B\) and \(O C\) is \(\frac { 1 } { \sqrt { 2 } }\). Find the value of \(\theta\).
CAIE FP1 2009 November Q3
3 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 5 x + 4 } { x + 1 }$$
  1. Obtain the coordinates of the points of intersection of \(C\) with the axes.
  2. Obtain the equation of each of the asymptotes of \(C\).
  3. Draw a sketch of \(C\).
CAIE FP1 2009 November Q4
4 It is given that $$x = t + \sin t , \quad y = t ^ { 2 } + 2 \cos t$$ where \(- \pi < t < \pi\). Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). Show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 2 t \sin t } { ( 1 + \cos t ) ^ { 3 } }$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) increases with \(x\) over the given interval of \(t\).
CAIE FP1 2009 November Q5
5 The equation $$x ^ { 3 } + 5 x + 3 = 0$$ has roots \(\alpha , \beta , \gamma\). Use the substitution \(x = - \frac { 3 } { y }\) to find a cubic equation in \(y\) and show that the roots of this equation are \(\beta \gamma , \gamma \alpha , \alpha \beta\). Find the exact values of \(\beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } + \alpha ^ { 2 } \beta ^ { 2 }\) and \(\beta ^ { 3 } \gamma ^ { 3 } + \gamma ^ { 3 } \alpha ^ { 3 } + \alpha ^ { 3 } \beta ^ { 3 }\).
CAIE FP1 2009 November Q6
6 Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ x ^ { n - 1 } \sqrt { } \left( 4 - x ^ { 2 } \right) \right] = \frac { 4 ( n - 1 ) x ^ { n - 2 } } { \sqrt { } \left( 4 - x ^ { 2 } \right) } - \frac { n x ^ { n } } { \sqrt { } \left( 4 - x ^ { 2 } \right) }$$ Let $$I _ { n } = \int _ { 0 } ^ { 1 } \frac { x ^ { n } } { \sqrt { } \left( 4 - x ^ { 2 } \right) } \mathrm { d } x$$ where \(n \geqslant 0\). Prove that $$n I _ { n } = 4 ( n - 1 ) I _ { n - 2 } - \sqrt { } 3$$ for \(n \geq 2\). Given that \(I _ { 0 } = \frac { 1 } { 6 } \pi\), find \(I _ { 4 }\), leaving your answer in an exact form.
CAIE FP1 2009 November Q7
7 Use de Moivre's theorem to express \(\sin ^ { 6 } \theta\) in the form $$a + b \cos 2 \theta + c \cos 4 \theta + d \cos 6 \theta$$ where \(a , b , c , d\) are constants to be found. Hence evaluate $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 6 } 2 x d x$$ leaving your answer in terms of \(\pi\).
CAIE FP1 2009 November Q8
8
  1. The curve \(C _ { 1 }\) has equation \(y = - \ln ( \cos x )\). Show that the length of the arc of \(C _ { 1 }\) from the point where \(x = 0\) to the point where \(x = \frac { 1 } { 3 } \pi\) is \(\ln ( 2 + \sqrt { 3 } )\).
  2. The curve \(C _ { 2 }\) has equation \(y = 2 \sqrt { } ( x + 3 )\). The arc of \(C _ { 2 }\) joining the point where \(x = 0\) to the point where \(x = 1\) is rotated through one complete revolution about the \(x\)-axis. Show that the area of the surface generated is $$\frac { 8 } { 3 } \pi ( 5 \sqrt { } 5 - 8 )$$
CAIE FP1 2009 November Q9
9 Show that if \(y\) depends on \(x\) and \(x = \mathrm { e } ^ { u }\) then $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} u ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} u } .$$ Given that \(y\) satisfies the differential equation $$x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = 30 x ^ { 2 }$$ use the substitution \(x = \mathrm { e } ^ { u }\) to show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} u ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} u } + 3 y = 30 \mathrm { e } ^ { 2 u }$$ Hence find the general solution for \(y\) in terms of \(x\).
CAIE FP1 2009 November Q10
10 The curve \(C\) has polar equation $$r = a \sin 3 \theta$$ where \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\).
  1. Show that the area of the region enclosed by \(C\) is \(\frac { 1 } { 12 } \pi a ^ { 2 }\).
  2. Show that, at the point of \(C\) at maximum distance from the initial line, $$\tan 3 \theta + 3 \tan \theta = 0 .$$
  3. Use the formula $$\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }$$ to find this maximum distance.
  4. Draw a sketch of \(C\).
CAIE FP1 2009 November Q11 EITHER
Prove by induction that $$\sum _ { n = 1 } ^ { N } n ^ { 3 } = \frac { 1 } { 4 } N ^ { 2 } ( N + 1 ) ^ { 2 }$$ Use this result, together with the formula for \(\sum _ { n = 1 } ^ { N } n ^ { 2 }\), to show that $$\sum _ { n = 1 } ^ { N } \left( 20 n ^ { 3 } + 36 n ^ { 2 } \right) = N ( N + 1 ) ( N + 3 ) ( 5 N + 2 ) .$$ Let $$S _ { N } = \sum _ { n = 1 } ^ { N } \left( 20 n ^ { 3 } + 36 n ^ { 2 } + \mu n \right)$$ Find the value of the constant \(\mu\) such that \(S _ { N }\) is of the form \(N ^ { 2 } ( N + 1 ) ( a N + b )\), where the constants \(a\) and \(b\) are to be determined. Show that, for this value of \(\mu\), $$5 + \frac { 22 } { N } < N ^ { - 4 } S _ { N } < 5 + \frac { 23 } { N }$$ for all \(N \geqslant 18\).
CAIE FP1 2009 November Q11 OR
One of the eigenvalues of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 4 & 6
2 & - 4 & 2
- 3 & 4 & a \end{array} \right)$$ is - 2 . Find the value of \(a\). Another eigenvalue of \(\mathbf { A }\) is - 5 . Find eigenvectors \(\mathbf { e } _ { 1 }\) and \(\mathbf { e } _ { 2 }\) corresponding to the eigenvalues - 2 and - 5 respectively. The linear space spanned by \(\mathbf { e } _ { 1 }\) and \(\mathbf { e } _ { 2 }\) is denoted by \(V\).
  1. Prove that, for any vector \(\mathbf { x }\) belonging to \(V\), the vector \(\mathbf { A x }\) also belongs to \(V\).
  2. Find a non-zero vector which is perpendicular to every vector in \(V\), and determine whether it is an eigenvector of \(\mathbf { A }\).