Questions FP1 (1491 questions)

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CAIE FP1 2012 November Q8
10 marks Challenging +1.2
8 The curve \(C\) has parametric equations $$x = \frac { 1 } { 3 } t ^ { 3 } - \ln t , \quad y = \frac { 4 } { 3 } t ^ { \frac { 3 } { 2 } }$$ for \(1 \leqslant t \leqslant 3\). Find the arc length of \(C\). Find also the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
CAIE FP1 2012 November Q9
12 marks Standard +0.3
9 The plane \(\Pi\) has equation $$\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) + \mu ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )$$ The line \(l\), which does not lie in \(\Pi\), has equation $$\mathbf { r } = 3 \mathbf { i } + 6 \mathbf { j } + 12 \mathbf { k } + t ( 8 \mathbf { i } + 5 \mathbf { j } - 8 \mathbf { k } )$$ Show that \(l\) is parallel to \(\Pi\). Find the position vector of the point at which the line with equation \(\mathbf { r } = 5 \mathbf { i } - 4 \mathbf { j } + 7 \mathbf { k } + s ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\) meets \(\Pi\). Find the perpendicular distance from the point with position vector \(9 \mathbf { i } + 11 \mathbf { j } + 2 \mathbf { k }\) to \(l\).
CAIE FP1 2012 November Q10
13 marks Standard +0.8
10 Write down the eigenvalues of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 4 & - 16 \\ 0 & 2 & 3 \\ 0 & 0 & 3 \end{array} \right)$$ Find corresponding eigenvectors. Let \(n\) be a positive integer. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { A } ^ { n } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$$ Find \(\mathbf { P } ^ { - 1 }\) and \(\mathbf { A } ^ { n }\). Hence find \(\lim _ { n \rightarrow \infty } \left( 3 ^ { - n } \mathbf { A } ^ { n } \right)\).
CAIE FP1 2012 November Q11 EITHER
Challenging +1.3
The roots of the equation \(x ^ { 4 } - 3 x ^ { 2 } + 5 x - 2 = 0\) are \(\alpha , \beta , \gamma , \delta\), and \(\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } + \delta ^ { n }\) is denoted by \(S _ { n }\). Show that $$S _ { n + 4 } - 3 S _ { n + 2 } + 5 S _ { n + 1 } - 2 S _ { n } = 0$$ Find the values of
  1. \(S _ { 2 }\) and \(S _ { 4 }\),
  2. \(S _ { 3 }\) and \(S _ { 5 }\). Hence find the value of $$\alpha ^ { 2 } \left( \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } \right) + \beta ^ { 2 } \left( \gamma ^ { 3 } + \delta ^ { 3 } + \alpha ^ { 3 } \right) + \gamma ^ { 2 } \left( \delta ^ { 3 } + \alpha ^ { 3 } + \beta ^ { 3 } \right) + \delta ^ { 2 } \left( \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \right) .$$
CAIE FP1 2012 November Q11 OR
Challenging +1.2
The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 2 & 1 & - 1 & 4 \\ 3 & 4 & 6 & 1 \\ - 1 & 2 & 8 & - 7 \end{array} \right)$$ The range space of T is \(R\). In any order,
  1. show that the dimension of \(R\) is 2 ,
  2. find a basis for \(R\) and obtain a cartesian equation for \(R\),
  3. find a basis for the null space of T . The vector \(\left( \begin{array} { l } 8 \\ 7 \\ k \end{array} \right)\) belongs to \(R\). Find the value of \(k\) and, with this value of \(k\), find the general solution of $$\mathbf { M } \mathbf { x } = \left( \begin{array} { l } 8 \\ 7 \\ k \end{array} \right) .$$
CAIE FP1 2013 November Q6
9 marks Challenging +1.2
6 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r }
CAIE FP1 2013 November Q10
12 marks Standard +0.3
10
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 }$$ where \(\lambda\) and \(\mu\) are real numbers and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\). 7 The square 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. Find the eigenvalues of the matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0
2 & 0 & 2
1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of \(\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. 8 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)\). Find a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has equation \(2 x - y + z = 10\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\). 9 The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the coordinates of the centroid of the region bounded by \(C\), the \(x\)-axis and the line \(x = 1\). 10 The curve \(C\) has equation $$y = \frac { p x ^ { 2 } + 4 x + 1 } { x + 1 }$$ where \(p\) is a positive constant and \(p \neq 3\).
  1. Obtain the equations of the asymptotes of \(C\).
  2. Find the value of \(p\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
  3. For the case \(p = 1\), show that \(C\) has no turning points, and sketch \(C\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.
CAIE FP1 2013 November Q16 EITHER
Standard +0.3
State the fifth roots of unity in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(- \pi < \theta \leqslant \pi\). Simplify $$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right)$$ Hence find the real factors of $$x ^ { 5 } - 1$$ Express the six roots of the equation $$x ^ { 6 } - x ^ { 3 } + 1 = 0$$ as three conjugate pairs, in the form \(\cos \theta \pm \mathrm { i } \sin \theta\). Hence find the real factors of $$x ^ { 6 } - x ^ { 3 } + 1$$
CAIE FP1 2013 November Q16 OR
Challenging +1.8
Given that $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$ and that \(v = y ^ { 3 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$ Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\).
CAIE FP1 2013 November Q1
6 marks Standard +0.8
1 Express \(\frac { 1 } { r ( r + 1 ) ( r - 1 ) }\) in partial fractions. Find $$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$ State the value of $$\sum _ { r = 2 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$
CAIE FP1 2013 November Q2
6 marks Standard +0.3
2 Show that the matrix \(\left( \begin{array} { r r r } 1 & 4 & 2 \\ 3 & 0 & - 2 \\ 3 & - 3 & - 4 \end{array} \right)\) has no inverse. Solve the system of equations $$\begin{array} { r } x + 4 y + 2 z = 0 \\ 3 x - 2 z = 4 \\ 3 x - 3 y - 4 z = 5 \end{array}$$
CAIE FP1 2013 November Q3
7 marks Standard +0.8
3 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 4 x ^ { 2 } + 8$$
CAIE FP1 2013 November Q4
7 marks Standard +0.3
4 A curve has parametric equations $$x = 2 \theta - \sin 2 \theta , \quad y = 1 - \cos 2 \theta , \quad \text { for } - 3 \pi \leqslant \theta \leqslant 3 \pi$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta$$ except for certain values of \(\theta\), which should be stated. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(\theta = \frac { 1 } { 4 } \pi\).
CAIE FP1 2013 November Q5
8 marks Standard +0.8
5 The equation $$8 x ^ { 3 } + 36 x ^ { 2 } + k x - 21 = 0$$ where \(k\) is a constant, has roots \(a - d , a , a + d\). Find the numerical values of the roots and determine the value of \(k\).
CAIE FP1 2013 November Q6
8 marks Challenging +1.2
6 [In this question you may use, without proof, the formula \(\int \sec x \mathrm {~d} x = \ln ( \sec x + \tan x ) + \operatorname { const }\).]
  1. Let \(y = \sec x\). Find the mean value of \(y\) with respect to \(x\) over the interval \(\frac { 1 } { 6 } \pi \leqslant x \leqslant \frac { 1 } { 3 } \pi\).
  2. The curve \(C\) has equation \(y = - \ln ( \cos x )\), for \(0 \leqslant x \leqslant \frac { 1 } { 3 } \pi\). Find the arc length of \(C\).
CAIE FP1 2013 November Q7
9 marks Standard +0.8
7 The curve \(C\) has equation $$y = \frac { 2 x ^ { 2 } + 5 x - 1 } { x + 2 }$$ Find the equations of the asymptotes of \(C\). Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 2\) at all points on \(C\). Sketch C.
CAIE FP1 2013 November Q8
11 marks Standard +0.3
8 The points \(A , B , C\) have position vectors $$4 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k } , \quad 5 \mathbf { i } + 7 \mathbf { j } + 8 \mathbf { k } , \quad 2 \mathbf { i } + 6 \mathbf { j } + 4 \mathbf { k }$$ respectively, relative to the origin \(O\). Find a cartesian equation of the plane \(A B C\). The point \(D\) has position vector \(6 \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }\). Find the coordinates of \(E\), the point of intersection of the line \(O D\) with the plane \(A B C\). Find the acute angle between the line \(E D\) and the plane \(A B C\).
CAIE FP1 2013 November Q9
11 marks Challenging +1.2
9 Prove by mathematical induction that, for every positive integer \(n\), $$( \cos \theta + i \sin \theta ) ^ { n } = \cos n \theta + i \sin n \theta$$ Express \(\sin ^ { 5 } \theta\) in the form \(p \sin 5 \theta + q \sin 3 \theta + r \sin \theta\), where \(p , q\) and \(r\) are rational numbers to be determined.
CAIE FP1 2013 November Q10
13 marks Standard +0.8
10 The curve \(C\) has polar equation \(r = 2 \sin \theta ( 1 - \cos \theta )\), for \(0 \leqslant \theta \leqslant \pi\). Find \(\frac { \mathrm { d } r } { \mathrm {~d} \theta }\) and hence find the polar coordinates of the point of \(C\) that is furthest from the pole. Sketch \(C\). Find the exact area of the sector from \(\theta = 0\) to \(\theta = \frac { 1 } { 4 } \pi\).
CAIE FP1 2013 November Q11 EITHER
Challenging +1.2
Let \(I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 2 } \right) ^ { n } \mathrm {~d} x\). Show that, for all integers \(n\), $$( 2 n + 1 ) I _ { n } = 2 n I _ { n - 1 } + 2 ^ { n }$$ Evaluate \(I _ { 0 }\) and hence find \(I _ { 3 }\). Given that \(I _ { - 1 } = \frac { 1 } { 4 } \pi\), find \(I _ { - 3 }\).
CAIE FP1 2013 November Q11 OR
Standard +0.3
The vector \(\mathbf { e }\) is an eigenvector of each of the \(3 \times 3\) matrices \(\mathbf { A }\) and \(\mathbf { B }\), with corresponding eigenvalues \(\lambda\) and \(\mu\) respectively. Justifying your answer, state an eigenvalue of \(\mathbf { A } + \mathbf { B }\). The matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 6 & - 1 & - 6 \\ 1 & 0 & - 2 \\ 3 & - 1 & - 3 \end{array} \right)$$ has eigenvectors \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right) , \left( \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right)\). Find the corresponding eigenvalues. The matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { r r r } 8 & - 2 & - 8 \\ 2 & 0 & - 4 \\ 4 & - 2 & - 4 \end{array} \right) ,$$ also has eigenvectors \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right) , \left( \begin{array} { l } 2 \\ 0 \\ 1 \end{array} \right)\), for which \(- 2,2,4\), respectively, are corresponding eigenvalues. The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \mathbf { A } + \mathbf { B } - 5 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. State the eigenvalues of \(\mathbf { M }\). Find matrices \(\mathbf { R }\) and \(\mathbf { S }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } ^ { 5 } = \mathbf { R D S }\).
[0pt] [You should show clearly all the elements of the matrices \(\mathbf { R } , \mathbf { S }\) and \(\mathbf { D }\).]
CAIE FP1 2014 November Q1
5 marks Standard +0.8
1 Given that $$u _ { k } = \frac { 1 } { \sqrt { } ( 2 k - 1 ) } - \frac { 1 } { \sqrt { } ( 2 k + 1 ) }$$ express \(\sum _ { k = 13 } ^ { n } u _ { k }\) in terms of \(n\). Deduce the value of \(\sum _ { k = 13 } ^ { \infty } u _ { k }\).
CAIE FP1 2014 November Q2
6 marks Challenging +1.2
2 A curve \(C\) has parametric equations $$x = \mathrm { e } ^ { t } \cos t , \quad y = \mathrm { e } ^ { t } \sin t , \quad \text { for } 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi$$ Find the arc length of \(C\).
CAIE FP1 2014 November Q3
7 marks Standard +0.8
3 It is given that \(u _ { r } = r \times r !\) for \(r = 1,2,3 , \ldots\). Let \(S _ { n } = u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots + u _ { n }\). Write down the values of $$2 ! - S _ { 1 } , \quad 3 ! - S _ { 2 } , \quad 4 ! - S _ { 3 } , \quad 5 ! - S _ { 4 }$$ Conjecture a formula for \(S _ { n }\). Prove, by mathematical induction, a formula for \(S _ { n }\), for all positive integers \(n\).
CAIE FP1 2014 November Q4
7 marks Standard +0.8
4 A curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } + x - 1 } { x - 1 }\). Find the equations of the asymptotes of \(C\). Show that there is no point on \(C\) for which \(1 < y < 9\).