Questions FP1 (1491 questions)

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CAIE FP1 2010 June Q6
8 marks Standard +0.8
6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 3 x - 7 } { x + 1 }$$
  1. Obtain the equations of the asymptotes of \(C\).
  2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 1\) at all points of \(C\).
  3. Draw a sketch of \(C\).
CAIE FP1 2010 June Q7
8 marks Standard +0.8
7 It is given that $$x = t ^ { 2 } \mathrm { e } ^ { - t ^ { 2 } } \quad \text { and } \quad y = t \mathrm { e } ^ { - t ^ { 2 } }$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 - 2 t ^ { 2 } } { 2 t - 2 t ^ { 3 } }$$
  2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(t\).
CAIE FP1 2010 June Q8
9 marks Challenging +1.2
8 Obtain the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 10 \sin 3 x - 20 \cos 3 x$$ Show that, for large positive \(x\) and independently of the initial conditions, $$y \approx R \sin ( 3 x + \phi )$$ where the constants \(R\) and \(\phi\), such that \(R > 0\) and \(0 < \phi < 2 \pi\), are to be determined correct to 2 decimal places.
CAIE FP1 2010 June Q9
10 marks Challenging +1.2
9 Let $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } \theta \mathrm {~d} \theta$$ where \(n\) is a non-negative integer. Show that \(I _ { n + 2 } = \frac { n + 1 } { n + 2 } I _ { n }\). The region \(R\) of the \(x - y\) plane is bounded by the \(x\)-axis, the line \(x = \frac { \pi } { 2 m }\) and the curve whose equation is \(y = \sin ^ { 4 } m x\), where \(m > 0\). Find the \(y\)-coordinate of the centroid of \(R\).
CAIE FP1 2010 June Q10
10 marks Challenging +1.2
10 The equation $$x ^ { 4 } + x ^ { 3 } + c x ^ { 2 } + 4 x - 2 = 0$$ where \(c\) is a constant, has roots \(\alpha , \beta , \gamma , \delta\).
  1. Use the substitution \(y = \frac { 1 } { x }\) to find an equation which has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta } , \frac { 1 } { \gamma } , \frac { 1 } { \delta }\).
  2. Find, in terms of \(c\), the values of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }\) and \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }\).
  3. Hence find $$\left( \alpha - \frac { 1 } { \alpha } \right) ^ { 2 } + \left( \beta - \frac { 1 } { \beta } \right) ^ { 2 } + \left( \gamma - \frac { 1 } { \gamma } \right) ^ { 2 } + \left( \delta - \frac { 1 } { \delta } \right) ^ { 2 }$$ in terms of \(c\).
  4. Deduce that when \(c = - 3\) the roots of the given equation are not all real.
CAIE FP1 2010 June Q11
12 marks Challenging +1.3
11 The curve \(C\) has polar equation $$r = \frac { a } { 1 + \theta }$$ where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  1. Show that \(r\) decreases as \(\theta\) increases.
  2. The point \(P\) of \(C\) is further from the initial line than any other point of \(C\). Show that, at \(P\), $$\tan \theta = 1 + \theta$$ and verify that this equation has a root between 1.1 and 1.2.
  3. Draw a sketch of \(C\).
  4. Find the area of the region bounded by the initial line, the line \(\theta = \frac { 1 } { 2 } \pi\) and \(C\), leaving your answer in terms of \(\pi\) and \(a\).
CAIE FP1 2010 June Q12 EITHER
Challenging +1.8
The line \(l _ { 1 }\) passes through the point \(A\) whose position vector is \(3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\) and is parallel to the vector \(\mathbf { i } + \mathbf { j }\). The line \(l _ { 2 }\) passes through the point \(B\) whose position vector is \(- \mathbf { i } - \mathbf { k }\) and is parallel to the vector \(\mathbf { j } + 2 \mathbf { k }\). The point \(P\) is on \(l _ { 1 }\) and the point \(Q\) is on \(l _ { 2 }\) and \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).
  1. Find the length of \(P Q\).
  2. Find the position vector of \(Q\).
  3. Show that the perpendicular distance from \(Q\) to the plane containing \(A B\) and the line \(l _ { 1 }\) is \(\sqrt { } 3\).
CAIE FP1 2010 June Q12 OR
Challenging +1.2
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,
    1. show that the dimension of \(R\), the range space of T , is equal to 2 ,
    2. obtain a basis for \(R\).
    3. Show that the vector \(\left( \begin{array} { r } 1 \\ - 15 \\ - 17 \\ - 6 \end{array} \right)\) belongs to \(R\).
    4. 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 $$\mathbf { M x } = \left( \begin{array} { r } 1 \\ - 15 \\ - 17 \\ - 6 \end{array} \right)$$
    5. Hence find a solution of \(( * )\) of the form \(\left( \begin{array} { c } \alpha \\ 0 \\ \gamma \\ \delta \end{array} \right)\).
CAIE FP1 2011 June Q1
5 marks Standard +0.3
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
6 marks Standard +0.8
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
6 marks Challenging +1.2
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
6 marks Standard +0.3
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
8 marks Standard +0.8
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
9 marks Challenging +1.8
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
11 marks Challenging +1.8
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
11 marks Challenging +1.2
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
12 marks Standard +0.8
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
12 marks Challenging +1.2
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
Challenging +1.3
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
Challenging +1.2
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
5 marks Standard +0.3
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
5 marks Standard +0.3
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
6 marks Standard +0.3
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
8 marks Standard +0.8
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
8 marks Standard +0.8
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 \(I _ { 8 } = \frac { 1 } { 7 } - \frac { 1 } { 5 } + \frac { 1 } { 3 } - 1 + \frac { 1 } { 4 } \pi\).