Questions — CAIE FP1 (594 questions)

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CAIE FP1 2002 November Q3
6 marks Standard +0.8
3 It is given that, for \(n = 0,1,2,3 , \ldots\), $$a _ { n } = 17 ^ { 2 n } + 3 ( 9 ) ^ { n } + 20$$ Simplify \(a _ { n + 1 } - a _ { n }\), and hence prove by induction that \(a _ { n }\) is divisible by 24 for all \(n \geqslant 0\).
CAIE FP1 2002 November Q4
7 marks Challenging +1.2
4 It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } e ^ { - x ^ { 2 } } d x$$
  1. Find \(I _ { 1 }\) in terms of c .
  2. Show that $$I _ { n + 2 } = \frac { n + 1 } { 2 } I _ { n } - \frac { 1 } { 2 \mathrm { e } }$$
  3. Find \(I _ { 5 }\) in terms of \(e\).
CAIE FP1 2002 November Q5
8 marks Challenging +1.2
5 The curve \(C\) has polar equation \(r \theta = 1\), for \(0 < \theta \leqslant 2 \pi\).
  1. Use the fact that \(\frac { \sin \theta } { \theta }\) tends to 1 as \(\theta\) tends to 0 to show that the line with carresian equation \(y = 1\) is an asymptote to \(C\).
  2. Sketch \(C\). The points \(P\) and \(Q\) on \(C\) correspond to \(\theta = \frac { 1 } { 6 } \pi\) and \(\theta = \frac { 1 } { 3 } \pi\) respectively.
  3. Find the area of the sector \(O P Q\), where \(O\) is the origin.
  4. Show that the length of the are \(P Q\) is $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { \sqrt { } \left( 1 + \theta ^ { 2 } \right) } { \theta ^ { 2 } } \mathrm {~d} \theta$$
CAIE FP1 2002 November Q6
9 marks Standard +0.8
6 A curve has equation \(x ^ { 3 } + x y ^ { 2 } - y ^ { 3 } = 3\).
  1. Show that there is no point of the curve at which \(\frac { d y } { d x } = 0\).
  2. Find the values of \(\frac { d y } { d x }\) and \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) at the point \(( 1 , - 1 )\).
CAIE FP1 2002 November Q7
9 marks Challenging +1.2
7 Given that \(z = \cos \theta + \mathrm { i } \sin \theta\), show that
  1. \(z - \frac { 1 } { z } = 2 \mathrm { i } \sin \theta\).
  2. \(z ^ { n } + z ^ { - n } = 2 \cos n \theta\). Hence show that $$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta )$$ Find a similar expression for \(\cos ^ { 6 } \theta\), and hence express \(\cos ^ { 6 } \theta - \sin ^ { 6 } \theta\) in the fom \(a \cos 2 \theta + b \cos 6 \theta\).
CAIE FP1 2002 November Q8
12 marks Challenging +1.2
8 The value of the assets of a large commercial organisation at time \(t\), measured in years, is \(\\) \left( 10 ^ { 8 } y + 10 ^ { 9 } \right)\(. The variables \)y\( and \)t$ are related by the differential equation $$\frac { d ^ { 2 } y } { d t ^ { 2 } } + 5 \frac { d y } { d t } + 6 y = 15 \cos 3 t - 3 \sin 3 t$$ Find \(y\) in terms of \(t\), given that \(y = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = - 2\) when \(t = 0\). Show that, for large values of \(t\), the value of the assets is less than \(\\) 9.5 \times 10 ^ { 8 }$ for about a third of the time.
CAIE FP1 2002 November Q9
12 marks Challenging +1.2
9 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), which meet in the line \(/\), have vector equations $$\begin{aligned} & \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 1 } ( 2 \mathbf { i } + 3 \mathbf { k } ) + \phi _ { 1 } ( - 4 \mathbf { j } + 5 \mathbf { k } ) , \\ & \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 2 } ( 3 \mathbf { j } + \mathbf { k } ) + \phi _ { 2 } ( - \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) , \end{aligned}$$ respectively. Find a vector equation of the line \(l\) in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\). Find a vector equation of the plane \(\Pi _ { 3 }\) which contains \(l\) and which passes through the point with position vector \(4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\). Find also the equation of \(\Pi _ { 3 }\) in the form \(a x + b y + c z = d\). Deduce, or prove otherwise, that the system of equations $$\begin{aligned} & 6 x - 5 y - 4 z = - 32 \\ & 5 x - y + 3 z = 24 \\ & 9 x - 2 y + 5 z = 40 \end{aligned}$$ has an infinite number of solutions.
CAIE FP1 2002 November Q10
13 marks Challenging +1.8
10 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { H }\), where $$\mathbf { H } = \left( \begin{array} { r r r r } 1 & 2 & - 3 & - 5 \\ - 1 & 4 & 5 & 1 \\ 2 & 3 & 0 & - 3 \\ - 3 & 5 & 7 & 2 \end{array} \right)$$
  1. Find the dimension of the range space of T .
  2. Find a basis for the null space of \(T\).
  3. It is given that \(\mathbf { x }\) satisfies the equation $$\mathbf { H } \mathbf { x } = \left( \begin{array} { r } 2 \\ - 10 \\ - 1 \\ - 15 \end{array} \right)$$ Using the fact that $$\mathbf { H } \left( \begin{array} { r } 1 \\ - 3 \\ 1 \\ - 2 \end{array} \right) = \left( \begin{array} { r } 2 \\ - 10 \\ - 1 \\ - 15 \end{array} \right) ,$$ find the least possible value of \(| \mathbf { x } |\).
    [0pt] [For the vector \(\mathbf { x } = \left( \begin{array} { c } x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \\ x _ { 4 } \end{array} \right) , | \mathbf { x } | = \sqrt { } \left( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 3 } ^ { 2 } + x _ { 4 } ^ { 2 } \right)\).]
CAIE FP1 2002 November Q11 EITHER
Standard +0.8
The vector \(\mathbf { e }\) is an eigenvector of the square matrix \(\mathbf { G }\). Show that
  1. \(\mathbf { e }\) is an eigenvector of \(\mathbf { G } + k \mathbf { I }\), where \(k\) is a scalar and \(\mathbf { I }\) is an identity matrix,
  2. \(\mathbf { e }\) is an cigenvector of \(\mathbf { G } ^ { 2 }\). Find the eigenvalues, and corresponding eigenvectors, of the matrices \(\mathbf { A }\) and \(\mathbf { B } ^ { 2 }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 3 & 0 \\ 1 & 0 & 1 \\ - 1 & 3 & 2 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r r } - 5 & - 3 & 0 \\ 1 & - 8 & 1 \\ - 1 & 3 & - 6 \end{array} \right)$$
CAIE FP1 2002 November Q11 OR
Standard +0.8
The curve \(C\) has equation $$y = \frac { ( x - a ) ( x - b ) } { x - c }$$ where \(a , b , c\) are constants, and it is given that \(0 < a < b < c\).
  1. Express \(y\) in the form $$x + P + \frac { Q } { x - c }$$ giving the constants \(P\) and \(Q\) in terms of \(a , b\) and \(c\).
  2. Find the equations of the asymptotes of \(C\).
  3. Show that \(C\) has two stationary points.
  4. Given also that \(a + b > c\), sketch \(C\), showing the asymptotes and the coordinates of the points of intersection of \(C\) with the axes.
CAIE FP1 2004 November Q1
4 marks Challenging +1.2
1 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix $$\left( \begin{array} { r r r r } 1 & 5 & 2 & 6 \\ 2 & 0 & - 1 & 7 \\ 3 & - 1 & - 2 & 10 \\ 4 & 10 & 13 & 29 \end{array} \right)$$ Find the dimension of the null space of T .
CAIE FP1 2004 November Q2
5 marks Challenging +1.8
2 The curve \(C\) is defined parametrically by $$x = a \cos ^ { 3 } t , \quad y = a \sin ^ { 3 } t , \quad 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$$ where \(a\) is a positive constant. Find the area of the surface generated when \(C\) is rotated through one complete revolution about the \(x\)-axis.
CAIE FP1 2004 November Q3
6 marks Standard +0.8
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
6 marks Standard +0.8
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
7 marks Challenging +1.2
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
8 marks Standard +0.3
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
8 marks Standard +0.8
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
9 marks Challenging +1.8
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
10 marks Challenging +1.8
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
11 marks Standard +0.8
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
12 marks Challenging +1.8
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
Challenging +1.8
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
Challenging +1.2
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 2006 November Q1
5 marks Moderate -0.5
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
5 marks Challenging +1.2
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 \(I _ { 0 }\).