Questions FP1 (1491 questions)

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CAIE FP1 2011 June Q6
8 marks Challenging +1.2
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
11 marks Challenging +1.2
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
11 marks Standard +0.8
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
11 marks Standard +0.8
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
13 marks Standard +0.8
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
Challenging +1.2
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
Challenging +1.8
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 $$p \left( \begin{array} { l } 1 \\ 2 \\ 3 \\ 5 \end{array} \right) + q \left( \begin{array} { l } - 1 \\ - 1 \\ - 3 \\ - 4 \end{array} \right) + r \left( \begin{array} { l } - 1 \\ - 4 \\ - 2 \\ - 6 \end{array} \right) = \left( \begin{array} { r } 3 \\ 7 \\ 8 \\ 15 \end{array} \right) .$$ Find the solution \(\mathbf { x } = \left( \begin{array} { l } \alpha \\ \beta \\ \gamma \\ \delta \end{array} \right)\) of the equation \(\mathbf { A } \mathbf { x } = \left( \begin{array} { r } 3 \\ 7 \\ 8 \\ 15 \end{array} \right)\) for which \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = \frac { 11 } { 4 }\).
CAIE FP1 2012 June Q1
5 marks Standard +0.8
1 The roots of the cubic equation \(x ^ { 3 } - 7 x ^ { 2 } + 2 x - 3 = 0\) are \(\alpha , \beta , \gamma\). Find the values of
  1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\),
  2. \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
CAIE FP1 2012 June Q2
5 marks Standard +0.3
2 Prove, by mathematical induction, that, for integers \(n \geqslant 2\), $$4 ^ { n } > 2 ^ { n } + 3 ^ { n }$$
CAIE FP1 2012 June Q3
6 marks Standard +0.3
3 Given that \(\mathrm { f } ( r ) = \frac { 1 } { ( r + 1 ) ( r + 2 ) }\), show that $$\mathrm { f } ( r - 1 ) - \mathrm { f } ( r ) = \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$ Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\). Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).
CAIE FP1 2012 June Q4
9 marks Standard +0.8
4 The curve \(C\) has polar equation \(r = 2 + 2 \cos \theta\), for \(0 \leqslant \theta \leqslant \pi\). Sketch the graph of \(C\). Find the area of the region \(R\) enclosed by \(C\) and the initial line. The half-line \(\theta = \frac { 1 } { 5 } \pi\) divides \(R\) into two parts. Find the area of each part, correct to 3 decimal places.
CAIE FP1 2012 June Q5
9 marks Standard +0.3
5 A matrix \(\mathbf { A }\) has eigenvalues \(- 1,1\) and 2 , with corresponding eigenvectors $$\left( \begin{array} { r } 0 \\ 1 \\ - 2 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ - 1 \\ 3 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } 2 \\ - 3 \\ 5 \end{array} \right) ,$$ respectively. Find \(\mathbf { A }\).
CAIE FP1 2012 June Q6
9 marks Challenging +1.2
6 Write down the values of \(\theta\), in the interval \(0 \leqslant \theta < 2 \pi\), for which \(\cos \theta + \mathrm { i } \sin \theta\) is a fifth root of unity. By writing the equation \(( z + 1 ) ^ { 5 } = z ^ { 5 }\) in the form $$\left( \frac { z + 1 } { z } \right) ^ { 5 } = 1$$ show that its roots are $$- \frac { 1 } { 2 } \left\{ 1 + \mathrm { i } \cot \left( \frac { k \pi } { 5 } \right) \right\} , \quad k = 1,2,3,4$$
CAIE FP1 2012 June Q7
10 marks Challenging +1.2
7 The linear transformations \(\mathrm { T } _ { 1 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) and \(\mathrm { T } _ { 2 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) are represented by the matrices $$\mathbf { M } _ { 1 } = \left( \begin{array} { r r r r } 1 & 1 & 1 & 4 \\ 2 & 1 & 4 & 11 \\ 3 & 4 & 1 & 9 \\ 4 & - 3 & 18 & 37 \end{array} \right) \quad \text { and } \quad \mathbf { M } _ { 2 } = \left( \begin{array} { r r r r } 1 & 1 & 1 & - 1 \\ 2 & 3 & 0 & 1 \\ 3 & 4 & 1 & 0 \\ 4 & 5 & 2 & 0 \end{array} \right)$$ respectively. The null space of \(\mathrm { T } _ { 1 }\) is denoted by \(K _ { 1 }\) and the null space of \(\mathrm { T } _ { 2 }\) is denoted by \(K _ { 2 }\). Show that the dimension of \(K _ { 1 }\) is 2 and that the dimension of \(K _ { 2 }\) is 1 . Find the basis of \(K _ { 1 }\) which has the form \(\left\{ \left( \begin{array} { c } p \\ q \\ 1 \\ 0 \end{array} \right) , \left( \begin{array} { c } r \\ s \\ 0 \\ 1 \end{array} \right) \right\}\) and show that \(K _ { 2 }\) is a subspace of \(K _ { 1 }\).
CAIE FP1 2012 June Q8
11 marks Standard +0.8
8 Find the particular solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 10 \mathrm { e } ^ { - 2 x }$$ given that \(y = 5\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 0\).
CAIE FP1 2012 June Q9
11 marks Standard +0.3
9 The curve \(C\) has equation $$y = \frac { 2 x ^ { 2 } + 2 x + 3 } { x ^ { 2 } + 2 }$$ Show that, for all \(x , 1 \leqslant y \leqslant \frac { 5 } { 2 }\). Find the coordinates of the turning points on \(C\). Find the equation of the asymptote of \(C\). Sketch the graph of \(C\), stating the coordinates of any intersections with the \(y\)-axis and the asymptote.
CAIE FP1 2012 June Q10
11 marks Challenging +1.2
10 The curve \(C\) has equation $$y = 2 \left( \frac { x } { 3 } \right) ^ { \frac { 3 } { 2 } }$$ where \(0 \leqslant x \leqslant 3\). Show that the arc length of \(C\) is \(2 ( 2 \sqrt { 2 } - 1 )\). Find the coordinates of the centroid of the region enclosed by \(C\), the \(x\)-axis and the line \(x = 3\).
CAIE FP1 2012 June Q11 EITHER
Challenging +1.8
Show that $$\int _ { 0 } ^ { \pi } \mathrm { e } ^ { x } \sin x \mathrm {~d} x = \frac { 1 + \mathrm { e } ^ { \pi } } { 2 }$$ Given that $$I _ { n } = \int _ { 0 } ^ { \pi } \mathrm { e } ^ { x } \sin ^ { n } x \mathrm {~d} x$$ show that, for \(n \geqslant 2\), $$I _ { n } = n ( n - 1 ) \int _ { 0 } ^ { \pi } \mathrm { e } ^ { x } \cos ^ { 2 } x \sin ^ { n - 2 } x \mathrm {~d} x - n I _ { n }$$ and deduce that $$\left( n ^ { 2 } + 1 \right) I _ { n } = n ( n - 1 ) I _ { n - 2 } .$$ A curve has equation \(y = \mathrm { e } ^ { x } \sin ^ { 5 } x\). Find, in an exact form, the mean value of \(y\) over the interval \(0 \leqslant x \leqslant \pi\).
CAIE FP1 2012 June Q11 OR
Challenging +1.2
The position vectors of the points \(A , B , C , D\) are $$2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } , \quad - 2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + 5 \mathbf { j } + m \mathbf { k }$$ respectively, where \(m\) is an integer. It is given that the shortest distance between the line through \(A\) and \(B\) and the line through \(C\) and \(D\) is 3 . Show that the only possible value of \(m\) is 2 . Find the shortest distance of \(D\) from the line through \(A\) and \(C\). Show that the acute angle between the planes \(A C D\) and \(B C D\) is \(\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { } 3 } \right)\).
CAIE FP1 2012 June Q7
10 marks Challenging +1.2
7 The linear transformations \(\mathrm { T } _ { 1 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) and \(\mathrm { T } _ { 2 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) are represented by the matrices $$\mathbf { M } _ { 1 } = \left( \begin{array} { r r r r } 1 & 1 & 1 & 4 \\ 2 & 1 & 4 & 11 \\ 3 & 4 & 1 & 9 \\ 4 & - 3 & 18 & 37 \end{array} \right) \quad \text { and } \quad \mathbf { M } _ { 2 } = \left( \begin{array} { r r r r } 1 & 1 & 1 & - 1 \\ 2 & 3 & 0 & 1 \\ 3 & 4 & 1 & 0 \\ 4 & 5 & 2 & 0 \end{array} \right)$$ respectively. The null space of \(\mathrm { T } _ { 1 }\) is denoted by \(K _ { 1 }\) and the null space of \(\mathrm { T } _ { 2 }\) is denoted by \(K _ { 2 }\). Show that the dimension of \(K _ { 1 }\) is 2 and that the dimension of \(K _ { 2 }\) is 1 . Find the basis of \(K _ { 1 }\) which has the form \(\left\{ \left( \begin{array} { c } p \\ q \\ 1 \\ 0 \end{array} \right) , \left( \begin{array} { l } r \\ s \\ 0 \\ 1 \end{array} \right) \right\}\) and show that \(K _ { 2 }\) is a subspace of \(K _ { 1 }\).
CAIE FP1 2012 June Q11 OR
Challenging +1.2
The position vectors of the points \(A , B , C , D\) are $$2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } , \quad - 2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + 5 \mathbf { j } + m \mathbf { k } ,$$ respectively, where \(m\) is an integer. It is given that the shortest distance between the line through \(A\) and \(B\) and the line through \(C\) and \(D\) is 3 . Show that the only possible value of \(m\) is 2 . Find the shortest distance of \(D\) from the line through \(A\) and \(C\). Show that the acute angle between the planes \(A C D\) and \(B C D\) is \(\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { } 3 } \right)\).
CAIE FP1 2012 June Q1
5 marks Standard +0.8
1 Find the sum of the first \(n\) terms of the series $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 2 \times 4 } + \frac { 1 } { 3 \times 5 } + \ldots$$ and deduce the sum to infinity.
CAIE FP1 2012 June Q2
5 marks Standard +0.3
2 For the sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), it is given that \(u _ { 1 } = 1\) and \(u _ { r + 1 } = \frac { 3 u _ { r } - 2 } { 4 }\) for all \(r\). Prove by mathematical induction that \(u _ { n } = 4 \left( \frac { 3 } { 4 } \right) ^ { n } - 2\), for all positive integers \(n\).
CAIE FP1 2012 June Q3
8 marks Standard +0.3
3 The curve \(C\) has equation $$x y + ( x + y ) ^ { 3 } = 1$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 } { 4 }\) at the point \(A ( 1,0 )\) on \(C\). Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
CAIE FP1 2012 June Q4
8 marks Challenging +1.2
4 Let $$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } x ^ { 2 } ( \ln x ) ^ { n } \mathrm {~d} x$$ for \(n \geqslant 0\). Show that, for all \(n \geqslant 1\), $$I _ { n } = \frac { 1 } { 3 } \mathrm { e } ^ { 3 } - \frac { 1 } { 3 } n I _ { n - 1 }$$ Find the exact value of \(I _ { 3 }\).