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CAIE FP1 2019 November Q1
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
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
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
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 \(T _ { N } = \sum _ { r = 1 } ^ { N } \frac { 1 } { ( 5 r + 1 ) ( 5 r + 6 ) }\).
  4. Use standard results from the List of Formulae (MF10) to show that $$S _ { N } = \frac { 1 } { 3 } N \left( 25 N ^ { 2 } + 90 N + 83 \right)$$
  5. Use the method of differences to express \(T _ { N }\) in terms of \(N\).
  6. Find \(\lim _ { N \rightarrow \infty } \left( N ^ { - 3 } S _ { N } T _ { N } \right)\).
CAIE FP1 2019 November Q6
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
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
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
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
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
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
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 2019 November Q1
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
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
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
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 \(T _ { N } = \sum _ { r = 1 } ^ { N } \frac { 1 } { ( 5 r + 1 ) ( 5 r + 6 ) }\).
  4. Use standard results from the List of Formulae (MF10) to show that $$S _ { N } = \frac { 1 } { 3 } N \left( 25 N ^ { 2 } + 90 N + 83 \right)$$
  5. Use the method of differences to express \(T _ { N }\) in terms of \(N\).
  6. Find \(\lim _ { N \rightarrow \infty } \left( N ^ { - 3 } S _ { N } T _ { N } \right)\).
CAIE FP1 2019 November Q6
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
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
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
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
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
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
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 P2 2013 June Q1
1 Solve the equation \(\left| 2 ^ { x } - 7 \right| = 1\), giving answers correct to 2 decimal places where appropriate.
CAIE P2 2013 June Q2
2 Solve the equation \(\ln ( 3 - 2 x ) - 2 \ln x = \ln 5\).
CAIE P2 2013 June Q3
3
  1. Show that \(12 \sin ^ { 2 } x \cos ^ { 2 } x \equiv \frac { 3 } { 2 } ( 1 - \cos 4 x )\).
  2. Hence show that $$\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } 12 \sin ^ { 2 } x \cos ^ { 2 } x d x = \frac { \pi } { 8 } + \frac { 3 \sqrt { } 3 } { 16 }$$