Questions FP1 (1385 questions)

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CAIE FP1 2019 June Q10
Standard +0.3
10 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$y = \frac { a x } { x + 5 } \quad \text { and } \quad y = \frac { x ^ { 2 } + ( a + 10 ) x + 5 a + 26 } { x + 5 }$$ respectively, where \(a\) is a constant and \(a > 2\).
  1. Find the equations of the asymptotes of \(C _ { 1 }\).
  2. Find the equation of the oblique asymptote of \(C _ { 2 }\).
  3. Show that \(C _ { 1 }\) and \(C _ { 2 }\) do not intersect.
  4. Find the coordinates of the stationary points of \(C _ { 2 }\).
  5. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on a single diagram. [You do not need to calculate the coordinates of any points where \(C _ { 2 }\) crosses the axes.]
CAIE FP1 2019 June Q11 EITHER
Challenging +1.8
The curve \(C _ { 1 }\) has polar equation \(r ^ { 2 } = 2 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  1. The point on \(C _ { 1 }\) furthest from the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(P\). Show that, at \(P\), $$2 \theta \tan \theta = 1$$ and verify that this equation has a root between 0.6 and 0.7 .
    The curve \(C _ { 2 }\) has polar equation \(r ^ { 2 } = \theta \sec ^ { 2 } \theta\), for \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole, denoted by \(O\), and at another point \(Q\).
  2. Find the exact value of \(\theta\) at \(Q\).
  3. The diagram below shows the curve \(C _ { 2 }\). Sketch \(C _ { 1 }\) on this diagram.
  4. Find, in exact form, the area of the region \(O P Q\) enclosed by \(C _ { 1 }\) and \(C _ { 2 }\).
CAIE FP1 2019 June Q11 OR
Challenging +1.8
The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix $$\mathbf { M } = \left( \begin{array} { r r r r } - 1 & 2 & 3 & 4 \\ 1 & 0 & 1 & - 1 \\ 1 & - 2 & - 3 & a \\ 1 & 2 & 5 & 2 \end{array} \right) .$$
  1. For \(a \neq - 4\), the range space of T is denoted by \(V\).
    (a) Find the dimension of \(V\) and show that $$\left( \begin{array} { r } - 1 \\ 1 \\ 1 \\ 1 \end{array} \right) , \quad \left( \begin{array} { r } 2 \\ 0 \\ - 2 \\ 2 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } 4 \\ - 1 \\ a \\ 2 \end{array} \right)$$ form a basis for \(V\).
    (b) Show that if \(\left( \begin{array} { l } x \\ y \\ z \\ t \end{array} \right)\) belongs to \(V\) then \(x + 2 y = t\).
  2. For \(a = - 4\), find the general solution of $$\mathbf { M } \mathbf { x } = \left( \begin{array} { r } - 1 \\ 1 \\ 1 \\ 1 \end{array} \right)$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE FP1 2019 June Q1
Standard +0.3
1 Prove by mathematical induction that \(3 ^ { 3 n } - 1\) is divisible by 13 for every positive integer \(n\).
CAIE FP1 2019 June Q2
Standard +0.8
2 The curve \(C\) has polar equation \(r ^ { 2 } = \ln ( 1 + \theta )\), for \(0 \leqslant \theta \leqslant 2 \pi\).
  1. Sketch \(C\).
  2. Using the substitution \(u = 1 + \theta\), or otherwise, find the area of the region bounded by \(C\) and the initial line, leaving your answer in an exact form.
CAIE FP1 2019 June Q3
Moderate -0.3
3
  1. Write down the fifth roots of unity.
  2. Find all the roots of the equation $$z ^ { 10 } + z ^ { 5 } + 1 = 0$$ giving each root in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\).
CAIE FP1 2019 June Q4
Challenging +1.8
4
  1. Use the method of differences to show that \(\sum _ { r = 1 } ^ { N } \frac { 1 } { ( 3 r + 1 ) ( 3 r - 2 ) } = \frac { 1 } { 3 } - \frac { 1 } { 3 ( 3 N + 1 ) }\).
  2. Find the limit, as \(N \rightarrow \infty\), of \(\sum _ { r = N + 1 } ^ { N ^ { 2 } } \frac { N } { ( 3 r + 1 ) ( 3 r - 2 ) }\).
CAIE FP1 2019 June Q5
Challenging +1.2
5 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & 2 & 0 & 4 \\ 5 & 2 & 1 & - 3 \\ 4 & 0 & 1 & - 7 \\ - 2 & 4 & - 1 & \alpha \end{array} \right)$$ It is given that the rank of \(\mathbf { M }\) is 2 .
  1. Find the value of \(\alpha\) and state a basis for the range space of T .
  2. Obtain a basis for the null space of T .
CAIE FP1 2019 June Q6
Standard +0.8
6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { k x - 1 }$$ where \(k\) is a positive constant.
  1. Obtain the equations of the asymptotes of \(C\).
  2. Find the coordinates of the stationary points of \(C\).
  3. Sketch \(C\).
CAIE FP1 2019 June Q7
Standard +0.8
7 The line \(l _ { 1 }\) passes through the points \(A ( - 3,1,4 )\) and \(B ( - 1,5,9 )\). The line \(l _ { 2 }\) passes through the points \(C ( - 2,6,5 )\) and \(D ( - 1,7,5 )\).
  1. Find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\).
  2. Find the acute angle between the line \(l _ { 2 }\) and the plane containing \(A , B\) and \(D\).
CAIE FP1 2019 June Q8
Standard +0.8
8 Find the particular solution of the differential equation $$9 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = 50 \sin t$$ given that when \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\).
CAIE FP1 2019 June Q9
Challenging +1.2
9 A cubic equation \(x ^ { 3 } + b x ^ { 2 } + c x + d = 0\) has real roots \(\alpha , \beta\) and \(\gamma\) such that $$\begin{aligned} \frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } & = - \frac { 5 } { 12 } \\ \alpha \beta \gamma & = - 12 \\ \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = 90 \end{aligned}$$
  1. Find the values of \(c\) and \(d\).
  2. Express \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) in terms of \(b\).
  3. Show that \(b ^ { 3 } - 15 b + 126 = 0\).
  4. Given that \(3 + \mathrm { i } \sqrt { } ( 12 )\) is a root of \(y ^ { 3 } - 15 y + 126 = 0\), deduce the value of \(b\).
CAIE FP1 2019 June Q10
Challenging +1.8
10 Let \(I _ { n } = \int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \cot ^ { n } x \mathrm {~d} x\), where \(n \geqslant 0\).
  1. By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \cot ^ { n + 1 } x \right)\), or otherwise, show that $$I _ { n + 2 } = \frac { 1 } { n + 1 } - I _ { n }$$ The curve \(C\) has equation \(y = \cot x\), for \(\frac { 1 } { 4 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
  2. Find, in an exact form, the \(y\)-coordinate of the centroid of the region enclosed by \(C\), the line \(x = \frac { 1 } { 4 } \pi\) and the \(x\)-axis.
CAIE FP1 2019 June Q11 EITHER
A \(3 \times 3\) matrix \(\mathbf { A }\) has distinct eigenvalues 2, 1, 3, with corresponding eigenvectors $$\left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ 0 \\ b \end{array} \right) , \quad \left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right)$$ respectively, where \(b\) is a positive constant.
  1. Find \(\mathbf { A }\) in terms of \(b\).
  2. Find \(\mathbf { A } ^ { - 1 } \left( \begin{array} { r } 0 \\ 2 \\ - 2 \end{array} \right)\).
  3. It is given that $$\mathbf { A } ^ { n } \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) = \left( \begin{array} { l } 4 \\ 4 \\ 0 \end{array} \right) \quad \text { and } \quad \mathbf { A } ^ { n } \left( \begin{array} { r } - 1 \\ 0 \\ b \end{array} \right) = \left( \begin{array} { c } - 1 \\ 0 \\ b ^ { - 1 } \end{array} \right) .$$ Find the values of \(n\) and \(b\).
CAIE FP1 2019 June Q11 OR
Challenging +1.2
The positive variables \(y\) and \(t\) are related by $$y = a ^ { t }$$ where \(a\) is a positive constant.
  1. (a) By differentiating \(\ln y\) with respect to \(t\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} t } = a ^ { t } \ln a\).
    (b) Write down \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
  2. Determine the set of values of \(a\) for which the infinite series $$y + \frac { \mathrm { d } y } { \mathrm {~d} t } + \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + \frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} t ^ { 3 } } + \ldots$$ is convergent.
    A curve has parametric equations $$x = t ^ { a } , \quad y = a ^ { t }$$
  3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(a\) and \(t\), and show that, when \(t = 2\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 ^ { 1 - 2 a } ( 1 - a + 2 \ln a ) \ln a$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE FP1 2002 November Q1
Standard +0.3
1 Given that $$u _ { n } = \mathrm { e } ^ { n x } - \mathrm { e } ^ { ( n + 1 ) x }$$ find \(\sum _ { n = 1 } ^ { N } \| _ { n }\) in terms of \(N\) and \(x\). Hence determine the set of values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity for cases where this exists.
CAIE FP1 2002 November Q2
Standard +0.8
2 The equation $$x ^ { 4 } + x ^ { 3 } + A x ^ { 2 } + 4 x - 2 = 0$$ where \(A\) is a constant, has roots \(\alpha , \beta , \gamma , \delta\). Find a polynomial equation whose roots are $$\frac { 1 } { \alpha } , \frac { 1 } { \beta } , \frac { 1 } { \gamma } , \frac { 1 } { \delta }$$ Given that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = \frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }$$ find the value of \(A\).
CAIE FP1 2002 November Q3
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
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
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
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
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
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
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
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 \(\mathbf { x } = \left( \begin{array} { c } x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \\ x _ { 4 } \end{array} \right) , | \mathbf { x } | = \sqrt { } \left( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 3 } ^ { 2 } + x _ { 4 } ^ { 2 } \right)\).]