Questions — CAIE FP1 (549 questions)

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CAIE FP1 2010 June Q12 OR
The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M } = \left( \begin{array} { r r r r } 1 & 1 & 5 & 7
3 & 9 & 17 & 25
1 & 7 & 7 & 11
3 & 6 & 16 & 23 \end{array} \right)\).
  1. In either order,
    (a) show that the dimension of \(R\), the range space of T , is equal to 2 ,
    (b) obtain a basis for \(R\).
  2. Show that the vector \(\left( \begin{array} { r } 1
    - 15
    - 17
    - 6 \end{array} \right)\) belongs to \(R\).
  3. It is given that \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for the null space of T , where \(\mathbf { e } _ { 1 } = \left( \begin{array} { r } 14
    1
    - 3
    0 \end{array} \right)\) and \(\mathbf { e } _ { 2 } = \left( \begin{array} { r } 19
    2
    0
    - 3 \end{array} \right)\). Show that, for all \(\lambda\) and \(\mu\), $$\mathbf { x } = \left( \begin{array} { r } 4
    - 3
    0
    0 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 }$$ is a solution of $$\mathbf { M x } = \left( \begin{array} { r } 1
    - 15
    - 17
    - 6 \end{array} \right)$$
  4. Hence find a solution of \(( * )\) of the form \(\left( \begin{array} { c } \alpha
    0
    \gamma
    \delta \end{array} \right)\).
CAIE FP1 2011 June Q1
1 Express \(\frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }\) in partial fractions and hence use the method of differences to find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }$$ Deduce the value of $$\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }$$
CAIE FP1 2011 June Q2
2 The roots of the equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ are \(\frac { \beta } { k } , \beta , k \beta\), where \(p , q , r , k\) and \(\beta\) are non-zero real constants. Show that \(\beta = - \frac { q } { p }\). Deduce that \(r p ^ { 3 } = q ^ { 3 }\).
CAIE FP1 2011 June Q3
3 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M } = \left( \begin{array} { r r r r } 1 & 3 & - 2 & 4
5 & 15 & - 9 & 19
- 2 & - 6 & 3 & - 7
3 & 9 & - 5 & 11 \end{array} \right)\).
  1. Find the rank of \(\mathbf { M }\).
  2. Obtain a basis for the null space of T .
CAIE FP1 2011 June Q4
4 It is given that \(\mathrm { f } ( n ) = 3 ^ { 3 n } + 6 ^ { n - 1 }\).
  1. Show that \(\mathrm { f } ( n + 1 ) + \mathrm { f } ( n ) = 28 \left( 3 ^ { 3 n } \right) + 7 \left( 6 ^ { n - 1 } \right)\).
  2. Hence, or otherwise, prove by mathematical induction that \(\mathrm { f } ( n )\) is divisible by 7 for every positive integer \(n\).
CAIE FP1 2011 June Q5
5 The curve \(C\) has polar equation \(r = 2 \cos 2 \theta\). Sketch the curve for \(0 \leqslant \theta < 2 \pi\). Find the exact area of one loop of the curve.
CAIE FP1 2011 June Q6
6 The line \(l _ { 1 }\) passes through the point with position vector \(8 \mathbf { i } + 8 \mathbf { j } - 7 \mathbf { k }\) and is parallel to the vector \(4 \mathbf { i } + 3 \mathbf { j }\). The line \(l _ { 2 }\) passes through the point with position vector \(7 \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k }\) and is parallel to the vector \(4 \mathbf { i } - \mathbf { k }\). 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 \(l _ { 2 }\). In either order,
  1. show that \(P Q = 13\),
  2. find the position vectors of \(P\) and \(Q\).
CAIE FP1 2011 June Q7
7 The variables \(x\) and \(y\) are related by the differential equation $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - 5 y ^ { 3 } = 8 \mathrm { e } ^ { - x }$$ Given that \(v = y ^ { 3 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} v } { \mathrm {~d} x } - 15 v = 24 \mathrm { e } ^ { - x }$$ Hence find the general solution for \(y\) in terms of \(x\).
CAIE FP1 2011 June Q8
8 Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf { A } = \left( \begin{array} { r r r } 4 & - 1 & 1
- 1 & 0 & - 3
1 & - 3 & 0 \end{array} \right)\). Find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
CAIE FP1 2011 June Q9
9 The curve \(C\) has equation \(y = x ^ { \frac { 3 } { 2 } }\). Find the coordinates of the centroid of the region bounded by \(C\), the lines \(x = 1 , x = 4\) and the \(x\)-axis. Show that the length of the arc of \(C\) from the point where \(x = 5\) to the point where \(x = 28\) is 139 .
CAIE FP1 2011 June Q10
10 Let $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \mathrm {~d} x$$ where \(n \geqslant 0\). Show that, for all \(n \geqslant 2\), $$I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$$ A curve has parametric equations \(x = a \sin ^ { 3 } t\) and \(y = a \cos ^ { 3 } t\), where \(a\) is a constant and \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). Show that the mean value \(m\) of \(y\) over the interval \(0 \leqslant x \leqslant a\) is given by $$m = 3 a \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( \cos ^ { 4 } t - \cos ^ { 6 } t \right) \mathrm { d } t$$ Find the exact value of \(m\), in terms of \(a\).
CAIE FP1 2011 June Q11 EITHER
Use de Moivre's theorem to prove that $$\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }$$ State the exact values of \(\theta\), between 0 and \(\pi\), that satisfy \(\tan 3 \theta = 1\). Express each root of the equation \(t ^ { 3 } - 3 t ^ { 2 } - 3 t + 1 = 0\) in the form \(\tan ( k \pi )\), where \(k\) is a positive rational number. For each of these values of \(k\), find the exact value of \(\tan ( k \pi )\).
CAIE FP1 2011 June Q11 OR
The curve \(C\) has equation $$y = \frac { x ^ { 2 } + \lambda x - 6 \lambda ^ { 2 } } { x + 3 }$$ where \(\lambda\) is a constant such that \(\lambda \neq 1\) and \(\lambda \neq - \frac { 3 } { 2 }\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and deduce that if \(C\) has two stationary points then \(- \frac { 3 } { 2 } < \lambda < 1\).
  2. Find the equations of the asymptotes of \(C\).
  3. Draw a sketch of \(C\) for the case \(0 < \lambda < 1\).
  4. Draw a sketch of \(C\) for the case \(\lambda > 3\).
CAIE FP1 2011 June Q1
1 Find \(2 ^ { 2 } + 4 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }\). Hence find \(1 ^ { 2 } - 2 ^ { 2 } + 3 ^ { 2 } - 4 ^ { 2 } + \ldots - ( 2 n ) ^ { 2 }\), simplifying your answer.
CAIE FP1 2011 June Q2
2 Let \(\mathbf { A } = \left( \begin{array} { l l } 2 & 3
0 & 1 \end{array} \right)\). Prove by mathematical induction that, for every positive integer \(n\), $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 3 \left( 2 ^ { n } - 1 \right)
0 & 1 \end{array} \right)$$
CAIE FP1 2011 June Q3
3 Find a cubic equation with roots \(\alpha , \beta\) and \(\gamma\), given that $$\alpha + \beta + \gamma = - 6 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 38 , \quad \alpha \beta \gamma = 30 .$$ Hence find the numerical values of the roots.
CAIE FP1 2011 June Q4
4 The curve \(C\) has equation $$2 x y ^ { 2 } + 3 x ^ { 2 } y = 1$$ Show that, at the point \(A ( - 1,1 )\) on \(C , \frac { \mathrm {~d} y } { \mathrm {~d} x } = - 4\). Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
CAIE FP1 2011 June Q5
5 Let $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \tan ^ { n } x \mathrm {~d} x$$ where \(n \geqslant 0\). Use the fact that \(\tan ^ { 2 } x = \sec ^ { 2 } x - 1\) to show that, for \(n \geqslant 2\), $$I _ { n } = \frac { 1 } { n - 1 } - I _ { n - 2 }$$ Show that \(I _ { 8 } = \frac { 1 } { 7 } - \frac { 1 } { 5 } + \frac { 1 } { 3 } - 1 + \frac { 1 } { 4 } \pi\).
CAIE FP1 2011 June Q6
6 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$\begin{array} { l l } C _ { 1 } : & r = a
C _ { 2 } : & r = 2 a \cos 2 \theta , \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi \end{array}$$ where \(a\) is a positive constant. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram. The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the point with polar coordinates ( \(a , \beta\) ). State the value of \(\beta\). Show that the area of the region bounded by the initial line, the arc of \(C _ { 1 }\) from \(\theta = 0\) to \(\theta = \beta\), and the arc of \(C _ { 2 }\) from \(\theta = \beta\) to \(\theta = \frac { 1 } { 4 } \pi\) is $$a ^ { 2 } \left( \frac { 1 } { 6 } \pi - \frac { 1 } { 8 } \sqrt { } 3 \right)$$
CAIE FP1 2011 June Q7
7 A curve \(C\) has parametric equations \(x = \mathrm { e } ^ { t } \cos t , y = \mathrm { e } ^ { t } \sin t\), for \(0 \leqslant t \leqslant \pi\). Find the arc length of \(C\). Find the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
CAIE FP1 2011 June Q8
8 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 10 \sin t$$ Find the particular solution, given that \(x = 5\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 2\) when \(t = 0\). State an approximate solution for large positive values of \(t\).
CAIE FP1 2011 June Q9
9 The curve \(C\) with equation $$y = \frac { a x ^ { 2 } + b x + c } { x - 1 }$$ where \(a , b\) and \(c\) are constants, has two asymptotes. It is given that \(y = 2 x - 5\) is one of these asymptotes.
  1. State the equation of the other asymptote.
  2. Find the value of \(a\) and show that \(b = - 7\).
  3. Given also that \(C\) has a turning point when \(x = 2\), find the value of \(c\).
  4. Find the set of values of \(k\) for which the line \(y = k\) does not intersect \(C\).
CAIE FP1 2011 June Q10
10 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$l _ { 1 } : \mathbf { r } = 6 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad l _ { 2 } : \mathbf { r } = 6 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } + \mu ( 4 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } ) .$$ Find a cartesian equation of the plane \(\Pi\) containing \(l _ { 1 }\) and \(l _ { 2 }\). Find the position vector of the foot of the perpendicular from the point with position vector \(\mathbf { i } + 10 \mathbf { j } + 3 \mathbf { k }\) to \(\Pi\). The line \(l _ { 3 }\) has equation \(\mathbf { r } = \mathbf { i } + 10 \mathbf { j } + 3 \mathbf { k } + v ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )\). Find the shortest distance between \(l _ { 1 }\) and \(l _ { 3 }\).
CAIE FP1 2011 June Q11 EITHER
A \(3 \times 3\) matrix \(\mathbf { A }\) has eigenvalues \(- 1,1,2\), with corresponding eigenvectors $$\left( \begin{array} { r } 0
1
- 1 \end{array} \right) , \quad \left( \begin{array} { r } - 1
0
1 \end{array} \right) , \quad \left( \begin{array} { l } 1
1
0 \end{array} \right) ,$$ respectively. Find
  1. the matrix \(\mathbf { A }\),
  2. \(\mathbf { A } ^ { 2 n }\), where \(n\) is a positive integer.
CAIE FP1 2011 June Q11 OR
Determine the rank of the matrix $$\mathbf { A } = \left( \begin{array} { l l l l } 1 & - 1 & - 1 & 1
2 & - 1 & - 4 & 3
3 & - 3 & - 2 & 2
5 & - 4 & - 6 & 5 \end{array} \right)$$ Show that if $$\mathbf { A x } = p \left( \begin{array} { l } 1
2
3
5 \end{array} \right) + q \left( \begin{array} { l } - 1
- 1
- 3
- 4 \end{array} \right) + r \left( \begin{array} { l } - 1
- 4
- 2
- 6 \end{array} \right)$$ where \(p , q\) and \(r\) are given real numbers, then $$\mathbf { x } = \left( \begin{array} { c } p + \lambda
q + \lambda
r + \lambda
\lambda \end{array} \right) ,$$ where \(\lambda\) is real. Find the values of \(p , q\) and \(r\) such that $$p \left( \begin{array} { l } 1
2
3
5 \end{array} \right) + q \left( \begin{array} { l } - 1
- 1
- 3
- 4 \end{array} \right) + r \left( \begin{array} { l } - 1
- 4
- 2
- 6 \end{array} \right) = \left( \begin{array} { r } 3
7
8