Questions FP1 (1385 questions)

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CAIE FP1 2013 June Q10
13 marks Challenging +1.2
10 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 }\). State the equations of the asymptotes of \(C\). Show that \(y \leqslant \frac { 25 } { 12 }\) at all points of \(C\). Find the coordinates of any stationary points of \(C\). Sketch \(C\), stating the coordinates of any intersections of \(C\) with the coordinate axes and the asymptotes.
CAIE FP1 2013 June Q11 EITHER
Challenging +1.2
The curve \(C\) has equation \(y = 2 \sec x\), for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). Show that the arc length \(s\) of \(C\) is given by $$S = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 2 \sec ^ { 2 } x - 1 \right) d x$$ Find the exact value of \(s\). The surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(S\). Show that
  1. \(S = 4 \pi \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 2 \sec ^ { 3 } x - \sec x \right) \mathrm { d } x\),
  2. \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x \tan x ) = 2 \sec ^ { 3 } x - \sec x\). Hence find the exact value of \(S\).
CAIE FP1 2013 June Q11 OR
Standard +0.8
The points \(A , B , C\) and \(D\) have coordinates as follows: $$A ( 2,1 , - 2 ) , \quad B ( 4,1 , - 1 ) , \quad C ( 3 , - 2 , - 1 ) \quad \text { and } \quad D ( 3,6,2 ) .$$ The plane \(\Pi _ { 1 }\) passes through the points \(A , B\) and \(C\). Find a cartesian equation of \(\Pi _ { 1 }\). Find the area of triangle \(A B C\) and hence, or otherwise, find the volume of the tetrahedron \(A B C D\).
[0pt] [The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × perpendicular height.]
The plane \(\Pi _ { 2 }\) passes through the points \(A , B\) and \(D\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE FP1 2013 June Q1
5 marks Challenging +1.2
1 Let \(\mathrm { f } ( r ) = r ! ( r - 1 )\). Simplify \(\mathrm { f } ( r + 1 ) - \mathrm { f } ( r )\) and hence find \(\sum _ { r = n + 1 } ^ { 2 n } r ! \left( r ^ { 2 } + 1 \right)\).
CAIE FP1 2013 June Q2
6 marks Challenging +1.2
2 The roots of the equation \(x ^ { 4 } - 4 x ^ { 2 } + 3 x - 2 = 0\) are \(\alpha , \beta , \gamma\) and \(\delta\); the sum \(\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } + \delta ^ { n }\) is denoted by \(S _ { n }\). By using the relation \(y = x ^ { 2 }\), or otherwise, show that \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }\) and \(\delta ^ { 2 }\) are the roots of the equation $$y ^ { 4 } - 8 y ^ { 3 } + 12 y ^ { 2 } + 7 y + 4 = 0$$ State the value of \(S _ { 2 }\) and hence show that $$S _ { 8 } = 8 S _ { 6 } - 12 S _ { 4 } - 72 .$$
CAIE FP1 2013 June Q3
7 marks Challenging +1.2
3 Prove by mathematical induction that, for every positive integer \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x } \sin x \right) = ( \sqrt { } 2 ) ^ { n } \mathrm { e } ^ { x } \sin \left( x + \frac { 1 } { 4 } n \pi \right)$$
CAIE FP1 2013 June Q4
8 marks Standard +0.3
4 Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 4 } { 3 }\) at the point \(A ( 1 , - 2 )\) on the curve with equation $$y ^ { 3 } - 3 x ^ { 2 } y + 2 = 0$$ and find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).
CAIE FP1 2013 June Q5
8 marks Challenging +1.2
5 Show that \(\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } - \frac { 1 } { 2 \mathrm { e } }\). Let \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x ^ { 2 } } \mathrm {~d} x\). Show that \(I _ { 2 n + 1 } = n I _ { 2 n - 1 } - \frac { 1 } { 2 \mathrm { e } }\) for \(n \geqslant 1\). Find the exact value of \(I _ { 7 }\).
CAIE FP1 2013 June Q6
8 marks Challenging +1.2
6 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } - 2 & 5 & 3 & - 1 \\ 0 & 1 & - 4 & - 2 \\ 6 & - 14 & - 13 & 1 \\ \alpha & \alpha & - 2 \alpha & - 11 \alpha \end{array} \right)$$ and \(\alpha\) is a constant. The null space of T is denoted by \(K _ { 1 }\) when \(\alpha \neq 0\), and by \(K _ { 2 }\) when \(\alpha = 0\). Find a basis for \(K _ { 1 }\) and a basis for \(K _ { 2 }\).
CAIE FP1 2013 June Q7
10 marks Challenging +1.2
7 Find the value of the constant \(\lambda\) such that \(\lambda x \mathrm { e } ^ { - x }\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 6 \mathrm { e } ^ { - x }$$ Find the solution of the differential equation for which \(y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) when \(x = 0\).
CAIE FP1 2013 June Q8
11 marks Challenging +1.2
8 The curve \(C\) has parametric equations \(x = \frac { 3 } { 2 } t ^ { 2 } , y = t ^ { 3 }\), for \(0 \leqslant t \leqslant 2\). Find the arc length of \(C\). Find the coordinates of the centroid of the region enclosed by \(C\), the \(x\)-axis and the line \(x = 6\).
CAIE FP1 2013 June Q9
11 marks Standard +0.8
9 The square matrix \(\mathbf { A }\) has an eigenvalue \(\lambda\) with corresponding eigenvector \(\mathbf { e }\). The non-singular matrix \(\mathbf { M }\) is of the same order as \(\mathbf { A }\). Show that \(\mathbf { M e }\) is an eigenvector of the matrix \(\mathbf { B }\), where \(\mathbf { B } = \mathbf { M } \mathbf { A } \mathbf { M } ^ { - 1 }\), and that \(\lambda\) is the corresponding eigenvalue. Let $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & 2 & 1 \\ 0 & 1 & 4 \\ 0 & 0 & 2 \end{array} \right)$$ Write down the eigenvalues of \(\mathbf { A }\) and obtain corresponding eigenvectors. Given that $$\mathbf { M } = \left( \begin{array} { l l l } 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$$ find the eigenvalues and corresponding eigenvectors of \(\mathbf { B }\).
CAIE FP1 2013 June Q10
12 marks Challenging +1.3
10 Use the identity \(2 \sin P \cos Q \equiv \sin ( P + Q ) + \sin ( P - Q )\) to show that $$2 \sin \theta \cos \left( \theta - \frac { 1 } { 4 } \pi \right) \equiv \cos \left( 2 \theta - \frac { 3 } { 4 } \pi \right) + \frac { 1 } { \sqrt { } 2 }$$ A curve has polar equation \(r = 2 \sin \theta \cos \left( \theta - \frac { 1 } { 4 } \pi \right)\), for \(0 \leqslant \theta \leqslant \frac { 3 } { 4 } \pi\). Sketch the curve and state the polar equation of its line of symmetry, justifying your answer. Show that the area of the region enclosed by the curve is \(\frac { 3 } { 8 } ( \pi + 1 )\).
CAIE FP1 2013 June Q11 EITHER
Challenging +1.2
The line \(l _ { 1 }\) passes through the point \(A\) whose position vector is \(4 \mathbf { i } + 7 \mathbf { j } - \mathbf { k }\) and is parallel to the vector \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\). The line \(l _ { 2 }\) passes through the point \(B\) whose position vector is \(\mathbf { i } + 7 \mathbf { j } + 11 \mathbf { k }\) and is parallel to the vector \(\mathbf { i } - 6 \mathbf { j } - 2 \mathbf { k }\). The points \(P\) on \(l _ { 1 }\) and \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\). Find the position vectors of \(P\) and \(Q\). Find the shortest distance between the line through \(A\) and \(B\) and the line through \(P\) and \(Q\), giving your answer correct to 3 significant figures.
CAIE FP1 2013 June Q11 OR
Standard +0.8
Show the cube roots of 1 on an Argand diagram. Show that the two non-real cube roots can be expressed in the form \(\omega\) and \(\omega ^ { 2 }\), and find these cube roots in exact cartesian form \(x + i y\). Evaluate the determinant $$\left| \begin{array} { c c c } 1 & 3 \omega & 2 \omega ^ { 2 } \\ 3 \omega ^ { 2 } & 2 & \omega \\ 2 \omega & \omega ^ { 2 } & 3 \end{array} \right|$$ It is given that \(z = ( 4 \sqrt { } 3 ) \left( \cos \frac { 4 } { 3 } \pi + i \sin \frac { 4 } { 3 } \pi \right) - 4 \left( \cos \frac { 11 } { 6 } \pi + i \sin \frac { 11 } { 6 } \pi \right)\). Express \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), giving exact values for \(r\) and \(\theta\). Hence find the cube roots of \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
CAIE FP1 2014 June Q1
5 marks Standard +0.8
1 The equation \(x ^ { 3 } + p x + q = 0\), where \(p\) and \(q\) are constants, with \(q \neq 0\), has one root which is the reciprocal of another root. Prove that \(p + q ^ { 2 } = 1\).
CAIE FP1 2014 June Q2
5 marks Standard +0.8
2 Expand and simplify \(( r + 1 ) ^ { 4 } - r ^ { 4 }\). Use the method of differences together with the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$
CAIE FP1 2014 June Q3
6 marks Standard +0.8
3 Prove by mathematical induction that, for all non-negative integers \(n\), $$11 ^ { 2 n } + 25 ^ { n } + 22$$ is divisible by 24 .
CAIE FP1 2014 June Q4
6 marks Standard +0.8
4 Obtain the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 25 x = 195 \sin 2 t$$
CAIE FP1 2014 June Q5
6 marks Standard +0.8
5 The curve \(C\) has polar equation \(r = a ( 1 + \sin \theta )\), where \(a\) is a positive constant and \(0 \leqslant \theta < 2 \pi\). Draw a sketch of \(C\). Find the exact value of the area of the region enclosed by \(C\) and the half-lines \(\theta = \frac { 1 } { 3 } \pi\) and \(\theta = \frac { 2 } { 3 } \pi\).
CAIE FP1 2014 June Q6
8 marks Challenging +1.2
6 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 2 & - 1 & 1 & 3 \\ 2 & 0 & 0 & 5 \\ 6 & - 2 & 2 & 11 \\ 10 & - 3 & 3 & 19 \end{array} \right)$$
  1. Find the rank of \(\mathbf { M }\) and state a basis for the range space of T .
  2. Obtain a basis for the null space of T .
CAIE FP1 2014 June Q7
9 marks Challenging +1.2
7 Use de Moivre's theorem to show that $$\tan 5 \theta = \frac { 5 t - 10 t ^ { 3 } + t ^ { 5 } } { 1 - 10 t ^ { 2 } + 5 t ^ { 4 } }$$ where \(t = \tan \theta\). Deduce that the roots of the equation \(t ^ { 4 } - 10 t ^ { 2 } + 5 = 0\) are \(\pm \tan \frac { 1 } { 5 } \pi\) and \(\pm \tan \frac { 2 } { 5 } \pi\). Hence show that \(\tan \frac { 1 } { 5 } \pi \tan \frac { 2 } { 5 } \pi = \sqrt { } 5\).
CAIE FP1 2014 June Q8
10 marks Challenging +1.2
8 The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find
  1. the arc length of \(C\),
  2. the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
CAIE FP1 2014 June Q9
10 marks Standard +0.3
9 The matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r } - 2 & 2 & 2 \\ 2 & 1 & 2 \\ - 3 & - 6 & - 7 \end{array} \right)$$ has an eigenvector \(\left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right)\). Find the corresponding eigenvalue. It is given that if the eigenvalues of a general \(3 \times 3\) matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { l l l } a & b & c \\ d & e & f \\ g & h & i \end{array} \right)$$ are \(\lambda _ { 1 } , \lambda _ { 2 }\) and \(\lambda _ { 3 }\) then $$\lambda _ { 1 } + \lambda _ { 2 } + \lambda _ { 3 } = a + e + i$$ and the determinant of \(\mathbf { A }\) has the value \(\lambda _ { 1 } \lambda _ { 2 } \lambda _ { 3 }\). Use these results to find the other two eigenvalues of the matrix \(\mathbf { M }\), and find corresponding eigenvectors.
CAIE FP1 2014 June Q10
10 marks Challenging +1.8
10 It is given that \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 2 n } x } { \cos x } \mathrm {~d} x\), where \(n \geqslant 0\). Show that $$I _ { n } - I _ { n + 1 } = \frac { 2 ^ { - \left( n + \frac { 1 } { 2 } \right) } } { 2 n + 1 }$$ Hence show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 6 } x } { \cos x } \mathrm {~d} x = \ln ( 1 + \sqrt { } 2 ) - \frac { 73 } { 120 } \sqrt { } 2\).