Questions FP1 (1385 questions)

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CAIE FP1 2018 November Q10
10
  1. Find the particular solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 10 x = 37 \sin 3 t$$ given that \(x = 3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\) when \(t = 0\).
  2. Show that, for large positive values of \(t\) and for any initial conditions, $$x \approx \sqrt { } ( 37 ) \sin ( 3 t - \phi ) ,$$ where the constant \(\phi\) is such that \(\tan \phi = 6\).
CAIE FP1 2018 November Q11 EITHER
  1. By considering \(( 2 r + 1 ) ^ { 2 } - ( 2 r - 1 ) ^ { 2 }\), use the method of differences to prove that $$\sum _ { r = 1 } ^ { n } r = \frac { 1 } { 2 } n ( n + 1 )$$
  2. By considering \(( 2 r + 1 ) ^ { 4 } - ( 2 r - 1 ) ^ { 4 }\), use the method of differences and the result given in part (i) to prove that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$ The sums \(S\) and \(T\) are defined as follows: $$\begin{aligned} & S = 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } + 4 ^ { 3 } + \ldots + ( 2 N ) ^ { 3 } + ( 2 N + 1 ) ^ { 3 } ,
    & T = 1 ^ { 3 } + 3 ^ { 3 } + 5 ^ { 3 } + 7 ^ { 3 } + \ldots + ( 2 N - 1 ) ^ { 3 } + ( 2 N + 1 ) ^ { 3 } . \end{aligned}$$
  3. Use the result given in part (ii) to show that \(S = ( 2 N + 1 ) ^ { 2 } ( N + 1 ) ^ { 2 }\).
  4. Hence, or otherwise, find an expression in terms of \(N\) for \(T\), factorising your answer as far as possible.
  5. Deduce the value of \(\frac { S } { T }\) as \(N \rightarrow \infty\).
CAIE FP1 2018 November Q11 OR
The curve \(C\) has equation $$x ^ { 2 } + 2 x y = y ^ { 3 } - 2$$
  1. Show that \(A ( - 1,1 )\) is the only point on \(C\) with \(x\)-coordinate equal to - 1 .
    For \(n \geqslant 1\), let \(A _ { n }\) denote the value of \(\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } }\) at the point \(A ( - 1,1 )\).
  2. Show that \(A _ { 1 } = 0\).
  3. Show that \(A _ { 2 } = \frac { 2 } { 5 }\).
    Let \(I _ { n } = \int _ { - 1 } ^ { 0 } x ^ { n } \frac { \mathrm {~d} ^ { n } y } { \mathrm {~d} x ^ { n } } \mathrm {~d} x\).
  4. Show that for \(n \geqslant 2\), $$I _ { n } = ( - 1 ) ^ { n + 1 } A _ { n - 1 } - n I _ { n - 1 } .$$
  5. Deduce the value of \(I _ { 3 }\) in terms of \(I _ { 1 }\).
    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 2018 November Q1
1 The roots of the cubic equation $$x ^ { 3 } - 5 x ^ { 2 } + 13 x - 4 = 0$$ are \(\alpha , \beta , \gamma\).
  1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
  2. Find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
CAIE FP1 2018 November Q2
2 It is given that $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 3 & 1
0 & - 2 & 1
0 & 0 & 1 \end{array} \right)$$
  1. Find the eigenvalue of \(\mathbf { A }\) corresponding to the eigenvector \(\left( \begin{array} { l } 1
    0
    0 \end{array} \right)\).
  2. Write down the negative eigenvalue of \(\mathbf { A }\) and find a corresponding eigenvector.
  3. Find an eigenvalue and a corresponding eigenvector of the matrix \(\mathbf { A } + \mathbf { A } ^ { 6 }\).
CAIE FP1 2018 November Q3
3 The curve \(C\) has polar equation \(r = a \cos 3 \theta\), for \(- \frac { 1 } { 6 } \pi \leqslant \theta \leqslant \frac { 1 } { 6 } \pi\), where \(a\) is a positive constant.
  1. Sketch \(C\).
  2. Find the area of the region enclosed by \(C\), showing full working.
  3. Using the identity \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\), find a cartesian equation of \(C\).
CAIE FP1 2018 November Q4
4
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = 4 \sin t$$
  2. State an approximate solution for large positive values of \(t\).
CAIE FP1 2018 November Q5
5 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 3 & 2 & 0 & 1
6 & 5 & - 1 & 3
9 & 8 & - 2 & 5
- 3 & - 2 & 0 & - 1 \end{array} \right)$$
  1. Find the rank of \(\mathbf { M }\).
    Let \(K\) be the null space of T .
  2. Find a basis for \(K\).
  3. Find the general solution of $$\mathbf { M } \mathbf { x } = \left( \begin{array} { r } 2
    5
    8
    - 2 \end{array} \right) .$$
CAIE FP1 2018 November Q6
6 It is given that \(y = \mathrm { e } ^ { x } u\), where \(u\) is a function of \(x\). The \(r\) th derivatives \(\frac { \mathrm { d } ^ { r } y } { \mathrm {~d} x ^ { r } }\) and \(\frac { \mathrm { d } ^ { r } u } { \mathrm {~d} x ^ { r } }\) are denoted by \(y ^ { ( r ) }\) and \(u ^ { ( r ) }\) respectively. Prove by mathematical induction that, for all positive integers \(n\), $$y ^ { ( n ) } = \mathrm { e } ^ { x } \left( \binom { n } { 0 } u + \binom { n } { 1 } u ^ { ( 1 ) } + \binom { n } { 2 } u ^ { ( 2 ) } + \ldots + \binom { n } { r } u ^ { ( r ) } + \ldots + \binom { n } { n } u ^ { ( n ) } \right)$$ [You may use without proof the result \(\binom { k } { r } + \binom { k } { r - 1 } = \binom { k + 1 } { r }\).]
CAIE FP1 2018 November Q8
8
- 2 \end{array} \right) .$$ 6 It is given that \(y = \mathrm { e } ^ { x } u\), where \(u\) is a function of \(x\). The \(r\) th derivatives \(\frac { \mathrm { d } ^ { r } y } { \mathrm {~d} x ^ { r } }\) and \(\frac { \mathrm { d } ^ { r } u } { \mathrm {~d} x ^ { r } }\) are denoted by \(y ^ { ( r ) }\) and \(u ^ { ( r ) }\) respectively. Prove by mathematical induction that, for all positive integers \(n\), $$y ^ { ( n ) } = \mathrm { e } ^ { x } \left( \binom { n } { 0 } u + \binom { n } { 1 } u ^ { ( 1 ) } + \binom { n } { 2 } u ^ { ( 2 ) } + \ldots + \binom { n } { r } u ^ { ( r ) } + \ldots + \binom { n } { n } u ^ { ( n ) } \right)$$ [You may use without proof the result \(\binom { k } { r } + \binom { k } { r - 1 } = \binom { k + 1 } { r }\).]
7 Let $$S _ { N } = \sum _ { r = 1 } ^ { N } ( 3 r + 1 ) ( 3 r + 4 ) \quad \text { and } \quad T _ { N } = \sum _ { r = 1 } ^ { N } \frac { 1 } { ( 3 r + 1 ) ( 3 r + 4 ) } .$$
  1. Use standard results from the List of Formulae (MF10) to show that $$S _ { N } = N \left( 3 N ^ { 2 } + 12 N + 13 \right)$$
  2. Use the method of differences to show that $$T _ { N } = \frac { 1 } { 12 } - \frac { 1 } { 3 ( 3 N + 4 ) } .$$
  3. Deduce that \(\frac { S _ { N } } { T _ { N } }\) is an integer.
  4. Find \(\lim _ { N \rightarrow \infty } \frac { S _ { N } } { N ^ { 3 } T _ { N } }\).
    8
  5. By considering the binomial expansion of \(\left( z + \frac { 1 } { z } \right) ^ { 6 }\), where \(z = \cos \theta + i \sin \theta\), express \(\cos ^ { 6 } \theta\) in the form $$\frac { 1 } { 32 } ( p + q \cos 2 \theta + r \cos 4 \theta + s \cos 6 \theta ) ,$$ where \(p , q , r\) and \(s\) are integers to be determined.
  6. Hence find the exact value of $$\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 6 } \left( \frac { 1 } { 2 } x \right) \mathrm { d } x$$
CAIE FP1 2018 November Q9
9 The curve \(C\) has equation $$y = \frac { 5 x ^ { 2 } + 5 x + 1 } { x ^ { 2 } + x + 1 } .$$
  1. Find the equation of the asymptote of \(C\).
  2. Show that, for all real values of \(x , - \frac { 1 } { 3 } \leqslant y < 5\).
  3. Find the coordinates of any stationary points of \(C\).
  4. Sketch \(C\), stating the coordinates of any intersections with the \(y\)-axis.
CAIE FP1 2018 November Q10
10 The position vectors of the points \(A , B , C , D\) are $$\mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad 3 \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k } , \quad - \mathbf { i } + 3 \mathbf { k } , \quad m \mathbf { j } + 4 \mathbf { k } ,$$ respectively, where \(m\) is a constant.
  1. Show that the lines \(A B\) and \(C D\) are parallel when \(m = \frac { 3 } { 2 }\).
  2. Given that \(m \neq \frac { 3 } { 2 }\), find the shortest distance between the lines \(A B\) and \(C D\).
  3. When \(m = 2\), find the acute angle between the planes \(A B C\) and \(A B D\), giving your answer in degrees.
CAIE FP1 2018 November Q11 EITHER
The curve \(C\) is defined parametrically by $$x = 18 t - t ^ { 2 } \quad \text { and } \quad y = 8 t ^ { \frac { 3 } { 2 } }$$ where \(0 < t \leqslant 4\).
  1. Show that at all points of \(C\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 3 ( 9 + t ) } { 2 t ^ { \frac { 1 } { 2 } } ( 9 - t ) ^ { 3 } }$$
  2. Show that the mean value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) with respect to \(x\) over the interval \(0 < x \leqslant 56\) is \(\frac { 3 } { 70 }\).
  3. Find the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis, showing full working.
CAIE FP1 2018 November Q11 OR
Let \(I _ { n } = \int _ { 1 } ^ { \sqrt { } 2 } \left( x ^ { 2 } - 1 \right) ^ { n } \mathrm {~d} x\).
  1. Show that, for \(n \geqslant 1\), $$( 2 n + 1 ) I _ { n } = \sqrt { } 2 - 2 n I _ { n - 1 } .$$
  2. Using the substitution \(x = \sec \theta\), show that $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \tan ^ { 2 n + 1 } \theta \sec \theta \mathrm {~d} \theta$$
  3. Deduce the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 7 } \theta } { \cos ^ { 8 } \theta } \mathrm {~d} \theta$$ 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 \(T _ { N } = \sum _ { r = 1 } ^ { N } \frac { 1 } { ( 5 r + 1 ) ( 5 r + 6 ) }\).
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.