Questions FP1 (1491 questions)

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CAIE FP1 2014 November Q5
8 marks Challenging +1.2
5 Find the value of \(a\) for which the system of equations $$\begin{aligned} & x - y + 2 z = 4 \\ & x + a y - 3 z = b \\ & x - y + 7 z = 13 \end{aligned}$$ where \(a\) and \(b\) are constants, has no unique solution. Taking \(a\) as the value just found,
  1. find the general solution in the case \(b = - 5\),
  2. interpret the situation geometrically in the case \(b \neq - 5\).
CAIE FP1 2014 November Q6
9 marks Challenging +1.3
6 Use de Moivre's theorem to show that $$\cos 5 \theta \equiv \cos \theta \left( 16 \sin ^ { 4 } \theta - 12 \sin ^ { 2 } \theta + 1 \right)$$ By considering the equation \(\cos 5 \theta = 0\), show that the exact value of \(\sin ^ { 2 } \left( \frac { 1 } { 10 } \pi \right)\) is \(\frac { 3 - \sqrt { 5 } } { 8 }\).
CAIE FP1 2014 November Q7
10 marks Challenging +1.8
7 Let \(I _ { n } = \int _ { 0 } ^ { 1 } ( 1 - x ) ^ { n } \mathrm { e } ^ { x } \mathrm {~d} x\). Show that, for all positive integers \(n\), $$I _ { n } = n I _ { n - 1 } - 1$$ Find the exact value of \(I _ { 4 }\). By considering the area of the region enclosed by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y = ( 1 - x ) ^ { 4 } \mathrm { e } ^ { x }\) in the interval \(0 \leqslant x \leqslant 1\), show that $$\frac { 65 } { 24 } < \mathrm { e } < \frac { 11 } { 4 }$$
CAIE FP1 2014 November Q8
11 marks Challenging +1.2
8 A circle has polar equation \(r = a\), for \(0 \leqslant \theta < 2 \pi\), and a cardioid has polar equation \(r = a ( 1 - \cos \theta )\), for \(0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant. Draw sketches of the circle and the cardioid on the same diagram. Write down the polar coordinates of the points of intersection of the circle and the cardioid. Show that the area of the region that is both inside the circle and inside the cardioid is $$\left( \frac { 5 } { 4 } \pi - 2 \right) a ^ { 2 }$$
CAIE FP1 2014 November Q9
11 marks Challenging +1.3
9 Given that $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 2 x + 2 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + ( 2 - 3 x ) y = 10 \mathrm { e } ^ { 2 x }$$ and that \(v = x y\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} v } { \mathrm {~d} x } - 3 v = 10 \mathrm { e } ^ { 2 x }$$ Find the general solution for \(y\) in terms of \(x\).
CAIE FP1 2014 November Q10
12 marks Challenging +1.3
10 The line \(l _ { 1 }\) is parallel to the vector \(\mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k }\) and passes through the point \(A\), whose position vector is \(3 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\). The line \(l _ { 2 }\) is parallel to the vector \(- 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }\) and passes through the point \(B\), whose position vector is \(- 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\). 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 \(l _ { 2 }\). Find
  1. the length \(P Q\),
  2. the cartesian equation of the plane \(\Pi\) containing \(P Q\) and \(l _ { 2 }\),
  3. the perpendicular distance of \(A\) from \(\Pi\).
CAIE FP1 2014 November Q11 EITHER
Standard +0.8
The roots of the quartic equation \(x ^ { 4 } + 4 x ^ { 3 } + 2 x ^ { 2 } - 4 x + 1 = 0\) are \(\alpha , \beta , \gamma\) and \(\delta\). Find the values of
  1. \(\alpha + \beta + \gamma + \delta\),
  2. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }\),
  3. \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } + \frac { 1 } { \delta }\),
  4. \(\frac { \alpha } { \beta \gamma \delta } + \frac { \beta } { \alpha \gamma \delta } + \frac { \gamma } { \alpha \beta \delta } + \frac { \delta } { \alpha \beta \gamma }\). Using the substitution \(y = x + 1\), find a quartic equation in \(y\). Solve this quartic equation and hence find the roots of the equation \(x ^ { 4 } + 4 x ^ { 3 } + 2 x ^ { 2 } - 4 x + 1 = 0\).
CAIE FP1 2014 November Q11 OR
Standard +0.8
The square 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 \(3 \times 3\) matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 2 & - 4 \\ 0 & - 1 & 5 \\ 0 & 0 & 3 \end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + 3 \mathbf { I } ) ^ { - 1 }$$ where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. Find a non-singular matrix \(\mathbf { P }\), and a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { B } = \mathbf { P D P } ^ { - 1 }\).
CAIE FP1 2014 November Q11 OR
Standard +0.8
The square 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 \(3 \times 3\) matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 2 & - 4 \\ 0 & - 1 & 5 \\ 0 & 0 & 3 \end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + 3 \mathbf { I } ) ^ { - 1 }$$ where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. Find a non-singular matrix \(\mathbf { P }\), and a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { B } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
CAIE FP1 2016 November Q1
5 marks Standard +0.8
1 Use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r ) ^ { 2 } - 1 }\). Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r ) ^ { 2 } - 1 }\).
CAIE FP1 2016 November Q2
6 marks Standard +0.8
2 Find the cubic equation with roots \(\alpha , \beta\) and \(\gamma\) such that $$\begin{aligned} \alpha + \beta + \gamma & = 3 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 1 \\ \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = - 30 \end{aligned}$$ giving your answer in the form \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers to be found.
CAIE FP1 2016 November Q3
7 marks Standard +0.3
3 Find a matrix \(\mathbf { A }\) whose eigenvalues are \(- 1,1,2\) and for which corresponding eigenvectors are $$\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \quad \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) , \quad \left( \begin{array} { l } 0 \\ 1 \\ 1 \end{array} \right) ,$$ respectively.
CAIE FP1 2016 November Q4
6 marks Challenging +1.2
4 Using factorials, show that \(\binom { n } { r - 1 } + \binom { n } { r } = \binom { n + 1 } { r }\). Hence prove by mathematical induction that $$( a + x ) ^ { n } = \binom { n } { 0 } a ^ { n } + \binom { n } { 1 } a ^ { n - 1 } x + \ldots + \binom { n } { r } a ^ { n - r } x ^ { r } + \ldots + \binom { n } { n } x ^ { n }$$ for every positive integer \(n\).
CAIE FP1 2016 November Q5
8 marks Challenging +1.2
5 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & 3 & 5 & 7 \\ 2 & 8 & 7 & 9 \\ 3 & 13 & 9 & 11 \\ 6 & 24 & 21 & 27 \end{array} \right)$$ Find
  1. the rank of \(\mathbf { A }\),
  2. a basis for the range space of T ,
  3. a basis for the null space of T .
CAIE FP1 2016 November Q6
9 marks Standard +0.8
6 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 7 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 10 x = 116 \sin 2 t$$ State an approximate solution for large positive values of \(t\).
CAIE FP1 2016 November Q7
11 marks Standard +0.3
7 The curve \(C\) has equation \(y = \mathrm { e } ^ { - 2 x }\). Find, giving your answers correct to 3 significant figures,
  1. the mean value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) over the interval \(0 \leqslant x \leqslant 2\),
  2. the coordinates of the centroid of the region bounded by \(C\), \(x = 0\), \(x = 2\) and \(y = 0\).
CAIE FP1 2016 November Q8
11 marks Challenging +1.2
8 A curve \(C\) has equation \(x ^ { 2 } + 4 x y - y ^ { 2 } + 20 = 0\). Show that, at stationary points on \(C , x = - 2 y\). Find the coordinates of the stationary points on \(C\), and determine their nature by considering the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the stationary points.
CAIE FP1 2016 November Q9
11 marks Challenging +1.2
9 Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \sin x \mathrm {~d} x\). Given that \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x\), prove that, for \(n > 1\), $$I _ { n } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$ By first using the substitution \(x = \cos ^ { - 1 } u\), find the value of $$\int _ { 0 } ^ { 1 } \left( \cos ^ { - 1 } u \right) ^ { 3 } \mathrm {~d} u$$ giving your answer in an exact form.
CAIE FP1 2016 November Q10
12 marks Challenging +1.2
10 Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta \quad \text { and } \quad z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ By considering \(\left( z - \frac { 1 } { z } \right) ^ { 4 } \left( z + \frac { 1 } { z } \right) ^ { 2 }\), show that $$\sin ^ { 4 } \theta \cos ^ { 2 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta - 2 \cos 4 \theta - \cos 2 \theta + 2 ) .$$ Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 4 } \theta \cos ^ { 2 } \theta d \theta\).
[0pt] [Question 11 is printed on the next page.]
CAIE FP1 2016 November Q11 EITHER
Challenging +1.8
The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = 6 \mathbf { i } - 3 \mathbf { j } + s ( 3 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } - \mathbf { j } - 4 \mathbf { k } + t ( \mathbf { i } - 3 \mathbf { j } - \mathbf { k } )$$ respectively. 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 \(l _ { 2 }\). Show that the position vector of \(P\) is \(3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\) and find the position vector of \(Q\). Find, in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\), an equation of the plane \(\Pi\) which passes through \(P\) and is perpendicular to \(l _ { 1 }\). The plane \(\Pi\) meets the plane \(\mathbf { r } = p \mathbf { i } + q \mathbf { j }\) in the line \(l _ { 3 }\). Find a vector equation of \(l _ { 3 }\).
CAIE FP1 2016 November Q11 OR
Challenging +1.2
A curve \(C\) has parametric equations $$x = 1 - 3 t ^ { 2 } , \quad y = t \left( 1 - 3 t ^ { 2 } \right) , \quad \text { for } 0 \leqslant t \leqslant \frac { 1 } { \sqrt { 3 } }$$ Show that \(\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + 9 t ^ { 2 } \right) ^ { 2 }\). Hence find
  1. the arc length of \(C\),
  2. the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Use the fact that \(t = \frac { y } { x }\) to find a cartesian equation of \(C\). Hence show that the polar equation of \(C\) is \(r = \sec \theta \left( 1 - 3 \tan ^ { 2 } \theta \right)\), and state the domain of \(\theta\). Find the area of the region enclosed between \(C\) and the initial line. {www.cie.org.uk} after the live examination series. }
CAIE FP1 2017 November Q1
4 marks Standard +0.3
1 Find \(\sum _ { r = 1 } ^ { n } ( 4 r - 3 ) ( 4 r + 1 )\), giving your answer in its simplest form.
CAIE FP1 2017 November Q2
6 marks Standard +0.8
2 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 4 - 5 t ^ { 2 }$$
CAIE FP1 2017 November Q3
7 marks Challenging +1.8
3
  1. Show that \(\frac { \mathrm { d } ^ { n + 1 } } { \mathrm {~d} x ^ { n + 1 } } \left( x ^ { n + 1 } \ln x \right) = \frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( x ^ { n } + ( n + 1 ) x ^ { n } \ln x \right)\).
  2. Prove by mathematical induction that, for all positive integers \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( x ^ { n } \ln x \right) = n ! \left( \ln x + 1 + \frac { 1 } { 2 } + \ldots + \frac { 1 } { n } \right)$$
CAIE FP1 2017 November Q4
8 marks Standard +0.8
4 The cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + 4 x - 10 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).
  1. Find the value of \(( \alpha + 1 ) ( \beta + 1 ) ( \gamma + 1 )\).
  2. Find the value of \(( \beta + \gamma ) ( \gamma + \alpha ) ( \alpha + \beta )\).