Questions — CAIE FP1 (594 questions)

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CAIE FP1 2017 November Q6
9 marks Standard +0.3
6 The points \(A , B\) and \(C\) have position vectors \(2 \mathbf { i } - \mathbf { j } + \mathbf { k } , 3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\) and \(- \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\) respectively.
  1. Find the area of the triangle \(A B C\).
    .................................................................................................................................... \includegraphics[max width=\textwidth, alt={}]{a0987277-06e9-451b-ae18-bb7de9e7661c-08_72_1566_484_328} .................................................................................................................................... .................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{a0987277-06e9-451b-ae18-bb7de9e7661c-08_71_1563_772_331} \includegraphics[max width=\textwidth, alt={}, center]{a0987277-06e9-451b-ae18-bb7de9e7661c-08_71_1563_868_331}
  2. Find the perpendicular distance of the point \(A\) from the line \(B C\).
  3. Find the cartesian equation of the plane through \(A , B\) and \(C\).
CAIE FP1 2017 November Q9
12 marks Standard +0.3
9 The curve \(C\) has equation $$y = \frac { 3 x - 9 } { ( x - 2 ) ( x + 1 ) }$$
  1. Find the equations of the asymptotes of \(C\). \includegraphics[max width=\textwidth, alt={}, center]{a0987277-06e9-451b-ae18-bb7de9e7661c-14_61_1566_513_328}
  2. Show that there is no point on \(C\) for which \(\frac { 1 } { 3 } < y < 3\).
  3. Find the coordinates of the turning points of \(C\).
  4. Sketch \(C\).
CAIE FP1 2017 November Q6
9 marks Standard +0.3
6 The points \(A , B\) and \(C\) have position vectors \(2 \mathbf { i } - \mathbf { j } + \mathbf { k } , 3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\) and \(- \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\) respectively.
  1. Find the area of the triangle \(A B C\).
    .................................................................................................................................... \includegraphics[max width=\textwidth, alt={}]{68e31138-756a-433a-bf42-0fdfadad091e-08_72_1566_484_328} .................................................................................................................................... .................................................................................................................................... \includegraphics[max width=\textwidth, alt={}, center]{68e31138-756a-433a-bf42-0fdfadad091e-08_71_1563_772_331} \includegraphics[max width=\textwidth, alt={}, center]{68e31138-756a-433a-bf42-0fdfadad091e-08_71_1563_868_331}
  2. Find the perpendicular distance of the point \(A\) from the line \(B C\).
  3. Find the cartesian equation of the plane through \(A , B\) and \(C\).
CAIE FP1 2017 November Q9
12 marks Standard +0.8
9 The curve \(C\) has equation $$y = \frac { 3 x - 9 } { ( x - 2 ) ( x + 1 ) }$$
  1. Find the equations of the asymptotes of \(C\). \includegraphics[max width=\textwidth, alt={}, center]{68e31138-756a-433a-bf42-0fdfadad091e-14_61_1566_513_328}
  2. Show that there is no point on \(C\) for which \(\frac { 1 } { 3 } < y < 3\).
  3. Find the coordinates of the turning points of \(C\).
  4. Sketch \(C\).
CAIE FP1 2019 November Q1
6 marks Standard +0.8
1 The curve \(C\) has equation \(y = x ^ { a }\) for \(0 \leqslant x \leqslant 1\), where \(a\) is a positive constant. Find, in terms of \(a\), the coordinates of the centroid of the region enclosed by \(C\), the line \(x = 1\) and the \(x\)-axis.
CAIE FP1 2019 November Q2
6 marks Challenging +1.2
2 It is given that \(y = \ln ( a x + 1 )\), where \(a\) is a positive constant. Prove by mathematical induction that, for every positive integer \(n\), $$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = ( - 1 ) ^ { n - 1 } \frac { ( n - 1 ) ! a ^ { n } } { ( a x + 1 ) ^ { n } }$$
CAIE FP1 2019 November Q3
7 marks Challenging +1.8
3 The integral \(I _ { n }\), where \(n\) is a positive integer, is defined by $$I _ { n } = \int _ { \frac { 1 } { 2 } } ^ { 1 } x ^ { - n } \sin \pi x \mathrm {~d} x$$
  1. Show that $$n ( n + 1 ) I _ { n + 2 } = 2 ^ { n + 1 } n + \pi - \pi ^ { 2 } I _ { n }$$
  2. Find \(I _ { 5 }\) in terms of \(\pi\) and \(I _ { 1 }\).
CAIE FP1 2019 November Q4
7 marks Standard +0.3
4 The line \(y = 2 x + 1\) is an asymptote of the curve \(C\) with equation $$y = \frac { x ^ { 2 } + 1 } { a x + b }$$
  1. Find the values of the constants \(a\) and \(b\).
  2. State the equation of the other asymptote of \(C\).
  3. Sketch C. [Your sketch should indicate the coordinates of any points of intersection with the \(y\)-axis. You do not need to find the coordinates of any stationary points.] \(5 \quad\) Let \(S _ { N } = \sum _ { r = 1 } ^ { N } ( 5 r + 1 ) ( 5 r + 6 )\) and \(T _ { N } = \sum _ { r = 1 } ^ { N } \frac { 1 } { ( 5 r + 1 ) ( 5 r + 6 ) }\).
CAIE FP1 2019 November Q6
9 marks Challenging +1.2
6 With \(O\) as the origin, the points \(A , B , C\) have position vectors $$\mathbf { i } - \mathbf { j } , \quad 2 \mathbf { i } + \mathbf { j } + 7 \mathbf { k } , \quad \mathbf { i } - \mathbf { j } + \mathbf { k }$$ respectively.
  1. Find the shortest distance between the lines \(O C\) and \(A B\).
  2. Find the cartesian equation of the plane containing the line \(O C\) and the common perpendicular of the lines \(O C\) and \(A B\).
CAIE FP1 2019 November Q7
9 marks Challenging +1.2
7 The equation \(x ^ { 3 } + 2 x ^ { 2 } + x + 7 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. Use the relation \(x ^ { 2 } = - 7 y\) to show that the equation $$49 y ^ { 3 } + 14 y ^ { 2 } - 27 y + 7 = 0$$ has roots \(\frac { \alpha } { \beta \gamma } , \frac { \beta } { \gamma \alpha } , \frac { \gamma } { \alpha \beta }\).
  2. Show that \(\frac { \alpha ^ { 2 } } { \beta ^ { 2 } \gamma ^ { 2 } } + \frac { \beta ^ { 2 } } { \gamma ^ { 2 } \alpha ^ { 2 } } + \frac { \gamma ^ { 2 } } { \alpha ^ { 2 } \beta ^ { 2 } } = \frac { 58 } { 49 }\).
  3. Find the exact value of \(\frac { \alpha ^ { 3 } } { \beta ^ { 3 } \gamma ^ { 3 } } + \frac { \beta ^ { 3 } } { \gamma ^ { 3 } \alpha ^ { 3 } } + \frac { \gamma ^ { 3 } } { \alpha ^ { 3 } \beta ^ { 3 } }\).
CAIE FP1 2019 November Q8
10 marks Standard +0.8
8 The matrix \(\mathbf { M }\) is defined by $$\mathbf { M } = \left( \begin{array} { c c c } 2 & m & 1 \\ 0 & m & 7 \\ 0 & 0 & 1 \end{array} \right) ,$$ where \(m \neq 0,1,2\).
  1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } = \mathbf { P D P } ^ { - 1 }\).
  2. Find \(\mathbf { M } ^ { 7 } \mathbf { P }\).
CAIE FP1 2019 November Q9
11 marks Challenging +1.8
9
  1. Use de Moivre's theorem to show that $$\sec 6 \theta = \frac { \sec ^ { 6 } \theta } { 32 - 48 \sec ^ { 2 } \theta + 18 \sec ^ { 4 } \theta - \sec ^ { 6 } \theta }$$
  2. Hence obtain the roots of the equation $$3 x ^ { 6 } - 36 x ^ { 4 } + 96 x ^ { 2 } - 64 = 0$$ in the form sec \(q \pi\), where \(q\) is rational.
CAIE FP1 2019 November Q10
12 marks Standard +0.8
10 The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 5 & 1 \\ 1 & - 2 & - 2 \\ 2 & 3 & \theta \end{array} \right)$$
  1. (a) Find the rank of \(\mathbf { A }\) when \(\theta \neq - 1\).
    (b) Find the rank of \(\mathbf { A }\) when \(\theta = - 1\).
    Consider the system of equations $$\begin{aligned} x + 5 y + z & = - 1 \\ x - 2 y - 2 z & = 0 \\ 2 x + 3 y + \theta z & = \theta \end{aligned}$$
  2. Solve the system of equations when \(\theta \neq - 1\).
  3. Find the general solution when \(\theta = - 1\).
  4. Show that if \(\theta = - 1\) and \(\phi \neq - 1\) then \(\mathbf { A } \mathbf { x } = \left( \begin{array} { r } - 1 \\ 0 \\ \phi \end{array} \right)\) has no solution.
CAIE FP1 2019 November Q11 EITHER
10 marks Challenging +1.8
It is given that \(w = \cos y\) and $$\tan y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } + 2 \tan y \frac { \mathrm {~d} y } { \mathrm {~d} x } = 1 + \mathrm { e } ^ { - 2 x } \sec y$$
  1. Show that $$\frac { \mathrm { d } ^ { 2 } w } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} w } { \mathrm {~d} x } + w = - \mathrm { e } ^ { - 2 x }$$
  2. Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0 , y = \frac { 1 } { 3 } \pi\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 3 } }\). [10]
CAIE FP1 2019 November Q11 OR
Challenging +1.2
The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations, for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), as follows: $$\begin{aligned} & C _ { 1 } : r = 2 \left( \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } \right) , \\ & C _ { 2 } : r = \mathrm { e } ^ { 2 \theta } - \mathrm { e } ^ { - 2 \theta } \end{aligned}$$ The curves intersect at the point \(P\) where \(\theta = \alpha\).
  1. Show that \(\mathrm { e } ^ { 2 \alpha } - 2 \mathrm { e } ^ { \alpha } - 1 = 0\). Hence find the exact value of \(\alpha\) and show that the value of \(r\) at \(P\) is \(4 \sqrt { } 2\).
  2. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
  3. Find the area of the region enclosed by \(C _ { 1 } , C _ { 2 }\) and the initial line, giving your answer correct to 3 significant figures.
    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 2017 Specimen Q1
4 marks Standard +0.8
1 The curve \(C\) is defined parametrically by $$x = 2 \cos ^ { 3 } t \quad \text { and } \quad y = 2 \sin ^ { 3 } t , \quad \text { for } 0 < t < \frac { 1 } { 2 } \pi$$ Show that, at the point with parameter \(t\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 1 } { 6 } \sec ^ { 4 } t \operatorname { cosec } t$$
CAIE FP1 2017 Specimen Q2
6 marks Standard +0.8
2 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 4 x = 7 - 2 t ^ { 2 }$$
CAIE FP1 2017 Specimen Q3
6 marks Challenging +1.2
3 Given that \(a\) is a constant, prove by mathematical induction that, for every positive integer \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( x \mathrm { e } ^ { a x } \right) = n a ^ { n - 1 } \mathrm { e } ^ { a x } + a ^ { n } x \mathrm { e } ^ { a x }$$
CAIE FP1 2017 Specimen Q4
7 marks Challenging +1.2
4 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that, for all positive integers \(n\), $$a _ { n } = \frac { n + 5 } { \sqrt { } \left( n ^ { 2 } - n + 1 \right) } - \frac { n + 6 } { \sqrt { } \left( n ^ { 2 } + n + 1 \right) }$$ The sum \(\sum _ { n = 1 } ^ { N } a _ { n }\) is denoted by \(S _ { N }\).
  1. Find the value of \(S _ { 30 }\) correct to 3 decimal places.
  2. Find the least value of \(N\) for which \(S _ { N } > 4.9\).
CAIE FP1 2017 Specimen Q5
8 marks Standard +0.8
5 The cubic equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers, has roots \(\alpha , \beta\) and \(\gamma\), such that $$\begin{aligned} \alpha + \beta + \gamma & = 15 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 83 \end{aligned}$$
  1. Write down the value of \(p\) and find the value of \(q\).
  2. Given that \(\alpha , \beta\) and \(\gamma\) are all real and that \(\alpha \beta + \alpha \gamma = 36\), find \(\alpha\) and hence find the value of \(r\). [5]
CAIE FP1 2017 Specimen Q6
10 marks Standard +0.3
6 The matrix A, where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 0 & 0 \\ 10 & - 7 & 10 \\ 7 & - 5 & 8 \end{array} \right)$$ has eigenvalues 1 and 3 .
  1. Find corresponding eigenvectors.
    It is given that \(\left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\).
  2. Find the corresponding eigenvalue.
  3. Find a diagonal matrix \(\mathbf { D }\) and matrices \(\mathbf { P }\) and \(\mathbf { P } ^ { - 1 }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\).
CAIE FP1 2017 Specimen Q7
10 marks Challenging +1.2
7 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & - 3 & 1 \\ 3 & - 5 & - 7 & 7 \\ 5 & - 9 & - 13 & 9 \\ 7 & - 13 & - 19 & 11 \end{array} \right)$$
  1. Find the rank of \(\mathbf { M }\) and a basis for the null space of T .
  2. The vector \(\left( \begin{array} { l } 1 \\ 2 \\ 3 \\ 4 \end{array} \right)\) is denoted by \(\mathbf { e }\). Show that there is a solution of the equation \(\mathbf { M x } = \mathbf { M e }\) of the form \(\mathbf { x } = \left( \begin{array} { c } a \\ b \\ - 1 \\ - 1 \end{array} \right)\), where the constants \(a\) and \(b\) are to be found.
CAIE FP1 2017 Specimen Q8
11 marks Standard +0.8
8 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } + k x } { x + 1 }\), where \(k\) is a constant.
  1. Find the set of values of \(k\) for which \(C\) has no stationary points.
  2. For the case \(k = 4\), find the equations of the asymptotes of \(C\) and sketch \(C\), indicating the coordinates of the points where \(C\) intersects the coordinate axes.
CAIE FP1 2017 Specimen Q9
12 marks Challenging +1.3
9 It is given that \(I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x\) for \(n \geqslant 0\).
  1. Show that $$I _ { n } = ( n - 1 ) \left[ I _ { n - 2 } - I _ { n - 1 } \right] \text { for } n \geqslant 2 .$$
  2. Hence find, in an exact form, the mean value of \(( \ln x ) ^ { 3 }\) with respect to \(x\) over the interval \(1 \leqslant x \leqslant \mathrm { e }\). [6]
CAIE FP1 2017 Specimen Q10
12 marks Challenging +1.3
10
  1. Using de Moivre's theorem, show that $$\tan 5 \theta = \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta } .$$
  2. Hence show that the equation \(x ^ { 2 } - 10 x + 5 = 0\) has roots \(\tan ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\tan ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\).
  3. Deduce a quadratic equation, with integer coefficients, having roots \(\sec ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\sec ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\). [3]