Questions — CAIE FP1 (549 questions)

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CAIE FP1 2006 November Q3
3 Verify that if $$v _ { n } = n ( n + 1 ) ( n + 2 ) \ldots ( n + m )$$ then $$v _ { n + 1 } - v _ { n } = ( m + 1 ) ( n + 1 ) ( n + 2 ) \ldots ( n + m ) .$$ Given now that $$u _ { n } = ( n + 1 ) ( n + 2 ) \ldots ( n + m ) ,$$ find \(\sum _ { n = 1 } ^ { N } u _ { n }\) in terms of \(m\) and \(N\).
CAIE FP1 2006 November Q4
4 Prove by mathematical induction that, for all positive integers \(n , 10 ^ { 3 n } + 13 ^ { n + 1 }\) is divisible by 7 .
CAIE FP1 2006 November Q5
5 Show that if \(a \neq 3\) then the system of equations $$\begin{aligned} & 2 x + 3 y + 4 z = - 5
& 4 x + 5 y - z = 5 a + 15
& 6 x + 8 y + a z = b - 2 a + 21 \end{aligned}$$ has a unique solution. Given that \(a = 3\), find the value of \(b\) for which the equations are consistent.
CAIE FP1 2006 November Q6
6 The roots of the equation $$x ^ { 3 } + x + 1 = 0$$ are \(\alpha , \beta , \gamma\). Show that the equation whose roots are $$\frac { 4 \alpha + 1 } { \alpha + 1 } , \quad \frac { 4 \beta + 1 } { \beta + 1 } , \quad \frac { 4 \gamma + 1 } { \gamma + 1 }$$ is of the form $$y ^ { 3 } + p y + q = 0$$ where the numbers \(p\) and \(q\) are to be determined. Hence find the value of $$\left( \frac { 4 \alpha + 1 } { \alpha + 1 } \right) ^ { n } + \left( \frac { 4 \beta + 1 } { \beta + 1 } \right) ^ { n } + \left( \frac { 4 \gamma + 1 } { \gamma + 1 } \right) ^ { n }$$ for \(n = 2\) and for \(n = 3\).
CAIE FP1 2006 November Q7
7 The curve \(C\) has equation $$r = 10 \ln ( 1 + \theta )$$ where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Draw a sketch of \(C\). Use the substitution \(w = \ln ( 1 + \theta )\) to show that the area of the sector bounded by the line \(\theta = \frac { 1 } { 2 } \pi\) and the arc of \(C\) joining the origin to the point where \(\theta = \frac { 1 } { 2 } \pi\) is $$50 \left( b ^ { 2 } - 2 b + 2 \right) \mathrm { e } ^ { b } - 100$$ where \(b = \ln \left( 1 + \frac { 1 } { 2 } \pi \right)\).
CAIE FP1 2006 November Q8
8 Given that $$2 y ^ { 3 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 12 y ^ { 3 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 6 y ^ { 2 } \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 17 y ^ { 4 } = 13 \mathrm { e } ^ { - 4 x }$$ and that \(v = y ^ { 4 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 34 v = 26 \mathrm { e } ^ { - 4 x }$$ Hence find the general solution for \(y\) in terms of \(x\).
CAIE FP1 2006 November Q9
9 With \(O\) as origin, the points \(A , B , C\) have position vectors $$\mathbf { i } , \quad \mathbf { i } + \mathbf { j } , \quad \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$$ respectively. Find a vector equation of the common perpendicular of the lines \(A B\) and \(O C\). Show that the shortest distance between the lines \(A B\) and \(O C\) is \(\frac { 2 } { 5 } \sqrt { } 5\). Find, in the form \(a x + b y + c z = d\), an equation for the plane containing \(A B\) and the common perpendicular of the lines \(A B\) and \(O C\).
CAIE FP1 2006 November Q10
10 The curve \(C\) has equation $$y = x ^ { 2 } + \lambda \sin ( x + y ) ,$$ where \(\lambda\) is a constant, and passes through the point \(A \left( \frac { 1 } { 4 } \pi , \frac { 1 } { 4 } \pi \right)\). Show that \(C\) has no tangent which is parallel to the \(y\)-axis. Show that, at \(A\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 - \frac { 1 } { 64 } \pi ( 4 - \pi ) ( \pi + 2 ) ^ { 2 }$$
CAIE FP1 2006 November Q11
11 Prove de Moivre's theorem for a positive integral exponent: $$\text { for all positive integers } n , \quad ( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta \text {. }$$ Use de Moivre's theorem to show that $$\cos 7 \theta = 64 \cos ^ { 7 } \theta - 112 \cos ^ { 5 } \theta + 56 \cos ^ { 3 } \theta - 7 \cos \theta$$ Hence obtain the roots of the equation $$128 x ^ { 7 } - 224 x ^ { 5 } + 112 x ^ { 3 } - 14 x + 1 = 0$$ in the form \(\cos q \pi\), where \(q\) is a rational number.
CAIE FP1 2006 November Q12 EITHER
The curve \(C\) has equation $$y = \frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } ,$$ where \(q\) is a positive constant.
  1. Obtain the equations of the asymptotes of \(C\).
  2. Find the value of \(q\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
  3. Sketch \(C\) for the case \(q = 3\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.
  4. It is given that, for all values of the constant \(\lambda\), the line $$y = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$ passes through the point of intersection of the asymptotes of \(C\). Use this result, with the diagrams you have drawn, to show that if \(\lambda < \frac { 1 } { 2 }\) then the equation $$\frac { x ^ { 2 } + q x + 1 } { 2 x + 3 } = \lambda x + \frac { 3 } { 2 } \lambda + \frac { 1 } { 2 } ( q - 3 )$$ has no real solution if \(q\) has the value found in part (ii), but has 2 real distinct solutions if \(q = 3\).
CAIE FP1 2006 November Q12 OR
The curve \(C\) has equation $$y = x ^ { \frac { 1 } { 2 } } - \frac { 1 } { 3 } x ^ { \frac { 3 } { 2 } } + \lambda ,$$ where \(\lambda > 0\) and \(0 \leqslant x \leqslant 3\). The length of \(C\) is denoted by \(s\). Prove that \(s = 2 \sqrt { } 3\). The area of the surface generated when \(C\) is rotated through one revolution about the \(x\)-axis is denoted by \(S\). Find \(S\) in terms of \(\lambda\). The \(y\)-coordinate of the centroid of the region bounded by \(C\), the axes and the line \(x = 3\) is denoted by h. Given that \(\int _ { 0 } ^ { 3 } y ^ { 2 } \mathrm {~d} x = \frac { 3 } { 4 } + \frac { 8 \sqrt { } 3 } { 5 } \lambda + 3 \lambda ^ { 2 }\), show that $$\lim _ { \lambda \rightarrow \infty } \frac { S } { h s } = 4 \pi$$
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 \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).