Questions — CAIE FP1 (549 questions)

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CAIE FP1 2004 November Q3
3 Given that $$\alpha + \beta + \gamma = 0 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 14 , \quad \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 18$$ find a cubic equation whose roots are \(\alpha , \beta , \gamma\). Hence find possible values for \(\alpha , \beta , \gamma\).
CAIE FP1 2004 November Q4
4 The curve \(C\) has polar equation $$r = \mathrm { e } ^ { \frac { 1 } { 5 } \theta } , \quad 0 \leqslant \theta \leqslant \frac { 3 } { 2 } \pi$$
  1. Draw a sketch of \(C\).
  2. Find the length of \(C\), correct to 3 significant figures.
CAIE FP1 2004 November Q5
5 Let $$S _ { N } = \sum _ { n = 1 } ^ { N } ( - 1 ) ^ { n - 1 } n ^ { 3 }$$ Find \(S _ { 2 N }\) in terms of \(N\), simplifying your answer as far as possible. Hence write down an expression for \(S _ { 2 N + 1 }\) and find the limit, as \(N \rightarrow \infty\), of \(\frac { S _ { 2 N + 1 } } { N ^ { 3 } }\).
CAIE FP1 2004 November Q6
6 Write down all the 8th roots of unity. Verify that $$\left( z - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z - \mathrm { e } ^ { - \mathrm { i } \theta } \right) \equiv z ^ { 2 } - ( 2 \cos \theta ) z + 1$$ Hence express \(z ^ { 8 } - 1\) as the product of two linear factors and three quadratic factors, where all coefficients are real and expressed in a non-trigonometric form.
CAIE FP1 2004 November Q7
7 The curve \(C\) has equation $$x y + ( x + y ) ^ { 5 } = 1$$
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 5 } { 6 }\) at the point \(A ( 1,0 )\) on \(C\).
  2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
CAIE FP1 2004 November Q8
8 The sequence of real numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that \(a _ { 1 } = 1\) and $$a _ { n + 1 } = \left( a _ { n } + \frac { 1 } { a _ { n } } \right) ^ { \lambda }$$ where \(\lambda\) is a constant greater than 1 . Prove by mathematical induction that, for \(n \geqslant 2\), $$a _ { n } \geqslant 2 ^ { \mathrm { g } ( n ) }$$ where \(g ( n ) = \lambda ^ { n - 1 }\). Prove also that, for \(n \geqslant 2 , \frac { a _ { n + 1 } } { a _ { n } } > 2 ^ { ( \lambda - 1 ) \mathrm { g } ( n ) }\).
CAIE FP1 2004 November Q9
9 It is given that $$I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 3 } \right) ^ { - n } \mathrm {~d} x$$ where \(n > 0\).
  1. Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ x \left( 1 + x ^ { 3 } \right) ^ { - n } \right] = - ( 3 n - 1 ) \left( 1 + x ^ { 3 } \right) ^ { - n } + 3 n \left( 1 + x ^ { 3 } \right) ^ { - n - 1 }$$ and hence, or otherwise, show that $$I _ { n + 1 } = \frac { 2 ^ { - n } } { 3 n } + \left( 1 - \frac { 1 } { 3 n } \right) I _ { n }$$
  2. By considering the graph of \(y = \frac { 1 } { 1 + x ^ { 3 } }\), show that \(I _ { 1 } < 1\).
  3. Deduce that \(I _ { 3 } < \frac { 53 } { 72 }\).
CAIE FP1 2004 November Q10
10 The curve \(C\) has equation $$y = \frac { x ^ { 2 } + 2 x - 3 } { ( \lambda x + 1 ) ( x + 4 ) }$$ where \(\lambda\) is a constant.
  1. Find the equations of the asymptotes of \(C\) for the case where \(\lambda = 0\).
  2. Find the equations of the asymptotes of \(C\) for the case where \(\lambda\) is not equal to any of \(- 1,0 , \frac { 1 } { 4 } , \frac { 1 } { 3 }\).
  3. Sketch \(C\) for the case where \(\lambda = - 1\). Show, on your diagram, the equations of the asymptotes and the coordinates of the points of intersection of \(C\) with the coordinate axes.
CAIE FP1 2004 November Q11
11 The line \(l _ { 1 }\) passes through the point \(A\), whose position vector is \(3 \mathbf { i } - 5 \mathbf { j } - 4 \mathbf { k }\), and is parallel to the vector \(3 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\). The line \(l _ { 2 }\) passes through the point \(B\), whose position vector is \(2 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k }\), and is parallel to the vector \(\mathbf { i } - \mathbf { j } - 4 \mathbf { k }\). The point \(P\) on \(l _ { 1 }\) and the point \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\). The plane \(\Pi _ { 1 }\) contains \(P Q\) and \(l _ { 1 }\), and the plane \(\Pi _ { 2 }\) contains \(P Q\) and \(l _ { 2 }\).
  1. Find the length of \(P Q\).
  2. Find a vector perpendicular to \(\Pi _ { 1 }\).
  3. Find the perpendicular distance from \(B\) to \(\Pi _ { 1 }\).
  4. Find the angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
CAIE FP1 2004 November Q12 EITHER
The variable \(y\) depends on \(x\), and the variables \(x\) and \(t\) are related by \(x = \mathrm { e } ^ { t }\). Show that $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { \mathrm { d } y } { \mathrm {~d} t } \quad \text { and } \quad x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } - \frac { \mathrm { d } y } { \mathrm {~d} t } .$$
  1. Given that \(y\) satisfies the differential equation $$4 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 16 x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 25 y = 50 ( \ln x ) - 1$$ find a differential equation involving only \(t\) and \(y\).
  2. Show that the complementary function of the differential equation in \(t\) and \(y\) may be written in the form $$R \mathrm { e } ^ { - \frac { 3 } { 2 } t } \sin ( 2 t + \phi )$$ where \(R\) and \(\phi\) are arbitrary constants.
  3. Find a particular integral of the differential equation in \(t\) and \(y\).
  4. Hence find the general solution of the differential equation in \(x\) and \(y\).
CAIE FP1 2004 November Q12 OR
The matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that if \(\mathbf { A }\) is non-singular then
  1. \(\lambda \neq 0\),
  2. the matrix \(\mathbf { A } ^ { - 1 }\) has \(\lambda ^ { - 1 }\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 1 & 2
    0 & - 2 & 4
    0 & 0 & - 3 \end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + 4 \mathbf { I } ) ^ { - 1 }$$ Find a non-singular matrix \(\mathbf { P }\), and a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { B } = \mathbf { P D P } ^ { - 1 }\).
CAIE FP1 2005 November Q1
1 Write down the fifth roots of unity. Hence, or otherwise, find all the roots of the equation $$z ^ { 5 } = - 16 + ( 16 \sqrt { } 3 ) i$$ giving each root in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\).
CAIE FP1 2005 November Q2
2 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } = 1\) and $$u _ { n + 1 } = - 1 + \sqrt { } \left( u _ { n } + 7 \right)$$
  1. Prove by induction that \(u _ { n } < 2\) for all \(n \geqslant 1\).
  2. Show that if \(u _ { n } = 2 - \varepsilon\), where \(\varepsilon\) is small, then $$u _ { n + 1 } \approx 2 - \frac { 1 } { 6 } \varepsilon$$
CAIE FP1 2005 November Q3
3 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { x + \lambda }$$ where \(\lambda\) is a non-zero constant. Obtain the equations of the asymptotes of \(C\). In separate diagrams, sketch \(C\) for the cases where
  1. \(\lambda > 0\),
  2. \(\lambda < 0\).
CAIE FP1 2005 November Q4
4 Solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 24 \mathrm { e } ^ { 2 x }$$ given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 9\) when \(x = 0\).
CAIE FP1 2005 November Q5
5 In the equation $$x ^ { 3 } + a x ^ { 2 } + b x + c = 0$$ the coefficients \(a , b\) and \(c\) are real. It is given that all the roots are real and greater than 1 .
  1. Prove that \(a < - 3\).
  2. By considering the sum of the squares of the roots, prove that \(a ^ { 2 } > 2 b + 3\).
  3. By considering the sum of the cubes of the roots, prove that \(a ^ { 3 } < - 9 b - 3 c - 3\).
CAIE FP1 2005 November Q6
6 Let $$I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 2 } \right) ^ { - n } \mathrm {~d} x$$ where \(n \geqslant 1\). By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x \left( 1 + x ^ { 2 } \right) ^ { - n } \right)\), or otherwise, prove that $$2 n I _ { n + 1 } = ( 2 n - 1 ) I _ { n } + 2 ^ { - n }$$ Deduce that \(I _ { 3 } = \frac { 3 } { 32 } \pi + \frac { 1 } { 4 }\).
\(7 \quad\) Write down an expression in terms of \(z\) and \(N\) for the sum of the series $$\sum _ { n = 1 } ^ { N } 2 ^ { - n } z ^ { n }$$ Use de Moivre's theorem to deduce that $$\sum _ { n = 1 } ^ { 10 } 2 ^ { - n } \sin \left( \frac { 1 } { 10 } n \pi \right) = \frac { 1025 \sin \left( \frac { 1 } { 10 } \pi \right) } { 2560 - 2048 \cos \left( \frac { 1 } { 10 } \pi \right) }$$
CAIE FP1 2005 November Q8
8 Find the coordinates of the centroid of the finite region bounded by the \(x\)-axis and the curve whose equation is $$y = x ^ { 2 } ( 1 - x )$$ Deduce the coordinates of the centroid of the finite region bounded by the \(x\)-axis and the curve whose equation is $$y = x ( 1 - x ) ^ { 2 }$$
CAIE FP1 2005 November Q9
9 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have vector equations $$\mathbf { r } = \lambda _ { 1 } ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) + \mu _ { 1 } ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = \lambda _ { 2 } ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) + \mu _ { 2 } ( 3 \mathbf { i } + \mathbf { j } - \mathbf { k } )$$ respectively. The line \(l\) passes through the point with position vector \(4 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\) and is parallel to both \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find a vector equation for \(l\). Find also the shortest distance between \(l\) and the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
CAIE FP1 2005 November Q10
10 It is given that the eigenvalues of the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r } 4 & 1 & - 1
- 4 & - 1 & 4
0 & - 1 & 5 \end{array} \right)$$ are \(1,3,4\). Find a set of corresponding eigenvectors. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { M } ^ { n } = \mathbf { P } \mathbf { D } \mathbf { P } ^ { - 1 }$$ where \(n\) is a positive integer. Find \(\mathbf { P } ^ { - 1 }\) and deduce that $$\lim _ { n \rightarrow \infty } 4 ^ { - n } \mathbf { M } ^ { n } = \left( \begin{array} { r r r } - \frac { 1 } { 3 } & 0 & - \frac { 1 } { 3 }
\frac { 4 } { 3 } & 0 & \frac { 4 } { 3 }
\frac { 4 } { 3 } & 0 & \frac { 4 } { 3 } \end{array} \right)$$
CAIE FP1 2005 November Q11
11 Find the rank of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & 1 & 2 & 3
4 & 3 & 5 & 16
6 & 6 & 13 & 13
14 & 12 & 23 & 45 \end{array} \right)$$ Find vectors \(\mathbf { x } _ { 0 }\) and \(\mathbf { e }\) such that any solution of the equation $$\mathbf { A x } = \left( \begin{array} { r } 0
2
- 1
3 \end{array} \right)$$ can be expressed in the form \(\mathbf { x } _ { 0 } + \lambda \mathbf { e }\), where \(\lambda \in \mathbb { R }\). Hence show that there is no vector which satisfies (*) and has all its elements positive.
CAIE FP1 2005 November Q12 EITHER
Show that \(\left( n + \frac { 1 } { 2 } \right) ^ { 3 } - \left( n - \frac { 1 } { 2 } \right) ^ { 3 } \equiv 3 n ^ { 2 } + \frac { 1 } { 4 }\). Use this result to prove that \(\sum _ { n = 1 } ^ { N } n ^ { 2 } = \frac { 1 } { 6 } N ( N + 1 ) ( 2 N + 1 )\). The sums \(S , T\) and \(U\) are defined as follows: $$\begin{aligned} & S = 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + 4 ^ { 2 } + \ldots + ( 2 N ) ^ { 2 } + ( 2 N + 1 ) ^ { 2 } ,
& T = 1 ^ { 2 } + 3 ^ { 2 } + 5 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 2 N - 1 ) ^ { 2 } + ( 2 N + 1 ) ^ { 2 } ,
& U = 1 ^ { 2 } - 2 ^ { 2 } + 3 ^ { 2 } - 4 ^ { 2 } + \ldots - ( 2 N ) ^ { 2 } + ( 2 N + 1 ) ^ { 2 } . \end{aligned}$$ Find and simplify expressions in terms of \(N\) for each of \(S , T\) and \(U\). Hence
  1. describe the behaviour of \(\frac { S } { T }\) as \(N \rightarrow \infty\),
  2. prove that if \(\frac { S } { U }\) is an integer then \(\frac { T } { U }\) is an integer.
CAIE FP1 2005 November Q12 OR
The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$r = 4 \cos \theta \quad \text { and } \quad r = 1 + \cos \theta$$ respectively, where \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  1. Show that \(C _ { 1 }\) and \(C _ { 2 }\) meet at the points \(A \left( \frac { 4 } { 3 } , \alpha \right)\) and \(B \left( \frac { 4 } { 3 } , - \alpha \right)\), where \(\alpha\) is the acute angle such that \(\cos \alpha = \frac { 1 } { 3 }\).
  2. In a single diagram, draw sketch graphs of \(C _ { 1 }\) and \(C _ { 2 }\).
  3. Show that the area of the region bounded by the arcs \(O A\) and \(O B\) of \(C _ { 1 }\), and the \(\operatorname { arc } A B\) of \(C _ { 2 }\), is $$4 \pi - \frac { 1 } { 3 } \sqrt { } 2 - \frac { 13 } { 2 } \alpha .$$
CAIE FP1 2006 November Q1
1 It is given that $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 1 & - 2
0 & 2 & 1
0 & 0 & - 3 \end{array} \right)$$ Write down the eigenvalues of \(\mathbf { A }\) and find corresponding eigenvectors.
CAIE FP1 2006 November Q2
2 The integral \(I _ { n }\), where \(n\) is a non-negative integer, is defined by $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x ^ { 3 } } \mathrm {~d} x$$ By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ^ { n + 1 } \mathrm { e } ^ { - x ^ { 3 } } \right)\) or otherwise, show that $$3 I _ { n + 3 } = ( n + 1 ) I _ { n } - \mathrm { e } ^ { - 1 }$$ Hence find \(I _ { 6 }\) in terms of e and \(I _ { 0 }\).