Questions — CAIE FP1 (549 questions)

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CAIE FP1 2015 June Q3
3 Prove by mathematical induction that, for all positive integers \(n , \sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r ) ^ { 2 } - 1 } = \frac { n } { 2 n + 1 }\). State the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r ) ^ { 2 } - 1 }\).
CAIE FP1 2015 June Q4
4 Use the formula for \(\tan ( A - B )\) in the List of Formulae (MF10) to show that $$\tan ^ { - 1 } ( x + 1 ) - \tan ^ { - 1 } ( x - 1 ) = \tan ^ { - 1 } \left( \frac { 2 } { x ^ { 2 } } \right)$$ Deduce the sum to \(n\) terms of the series $$\tan ^ { - 1 } \left( \frac { 2 } { 1 ^ { 2 } } \right) + \tan ^ { - 1 } \left( \frac { 2 } { 2 ^ { 2 } } \right) + \tan ^ { - 1 } \left( \frac { 2 } { 3 ^ { 2 } } \right) + \ldots .$$
CAIE FP1 2015 June Q5
5 Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { \sin 2 n \theta } { \cos \theta } \mathrm {~d} \theta\), where \(n\) is a non-negative integer.
  1. Use the identity \(\sin P + \sin Q \equiv 2 \sin \frac { 1 } { 2 } ( P + Q ) \cos \frac { 1 } { 2 } ( P - Q )\) to show that $$I _ { n } + I _ { n - 1 } = \frac { 2 } { 2 n - 1 } , \text { for all positive integers } n$$
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \frac { \sin 8 \theta } { \cos \theta } d \theta\).
CAIE FP1 2015 June Q6
6 Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Use the binomial expansion of \(( 1 + z ) ^ { n }\), where \(n\) is a positive integer, to show that $$\binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta + \ldots + \binom { n } { n } \cos n \theta = 2 ^ { n } \cos ^ { n } \left( \frac { 1 } { 2 } \theta \right) \cos \left( \frac { 1 } { 2 } n \theta \right) - 1$$ Find $$\binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \ldots + \binom { n } { n } \sin n \theta$$
CAIE FP1 2015 June Q7
7 The curve \(C\) has equation \(x ^ { 2 } + 2 x y - 4 y ^ { 2 } + 20 = 0\). Show that if the tangent to \(C\) at the point \(( x , y )\) is parallel to the \(x\)-axis then \(x + y = 0\). Hence find the coordinates of the stationary points on \(C\), and determine their nature.
CAIE FP1 2015 June Q8
8 A line, passing through the point \(A ( 3,0,2 )\), has vector equation \(\mathbf { r } = 3 \mathbf { i } + 2 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )\). It meets the plane \(\Pi\), which has equation \(\mathbf { r } \cdot ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) = 3\), at the point \(P\). Find the coordinates of \(P\). Write down a vector \(\mathbf { n }\) which is perpendicular to \(\Pi\), and calculate the vector \(\mathbf { w }\), where $$\mathbf { w } = \mathbf { n } \times ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )$$ The point \(Q\) lies in \(\Pi\) and is the foot of the perpendicular from \(A\) to \(\Pi\). Use the vector \(\mathbf { w }\) to determine an equation of the line \(P Q\) in the form \(\mathbf { r } = \mathbf { u } + \mu \mathbf { v }\).
CAIE FP1 2015 June Q9
9 Find the particular solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 3 \frac { \mathrm {~d} x } { \mathrm {~d} t } - 10 x = 2 \sin t - 3 \cos t$$ given that, when \(t = 0 , x = 3.3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.9\).
CAIE FP1 2015 June Q10
10 The curve \(C\) has equation \(y = \frac { 4 x ^ { 2 } - 3 x } { x ^ { 2 } + 1 }\). Verify that the equation of \(C\) may be written in the form \(y = - \frac { 1 } { 2 } + \frac { ( 3 x - 1 ) ^ { 2 } } { 2 \left( x ^ { 2 } + 1 \right) }\) and also in the form \(y = \frac { 9 } { 2 } - \frac { ( x + 3 ) ^ { 2 } } { 2 \left( x ^ { 2 } + 1 \right) }\). Hence show that \(- \frac { 1 } { 2 } \leqslant y \leqslant \frac { 9 } { 2 }\). Without differentiating, write down the coordinates of the turning points of \(C\). State the equation of the asymptote of \(C\). Sketch the graph of \(C\), stating the coordinates of the intersections with the coordinate axes and the asymptote.
CAIE FP1 2015 June Q11 EITHER
The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & 2 & 3 & 4
1 & - 1 & 2 & 3
1 & - 3 & 3 & 5
1 & 4 & 2 & 2 \end{array} \right)$$ The range space of T is denoted by \(V\).
  1. Determine the dimension of \(V\).
  2. Show that the vectors \(\left( \begin{array} { l } 1
    1
    1
    1 \end{array} \right) , \left( \begin{array} { r } 2
    - 1
    - 3
    4 \end{array} \right) , \left( \begin{array} { l } 3
    2
    3
    2 \end{array} \right)\) are a basis of \(V\). The set of elements of \(\mathbb { R } ^ { 4 }\) which do not belong to \(V\) is denoted by \(W\).
  3. State, with a reason, whether \(W\) is a vector space.
  4. Show that if the vector \(\left( \begin{array} { l } x
    y
    z
    t \end{array} \right)\) belongs to \(W\) then \(x + y \neq z + t\).
CAIE FP1 2015 June Q11 OR
One of the eigenvalues of the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r } 3 & - 4 & 2
- 4 & \alpha & 6
2 & 6 & - 2 \end{array} \right)$$ is - 9 . Find the value of \(\alpha\). Find
  1. the other two eigenvalues, \(\lambda _ { 1 }\) and \(\lambda _ { 2 }\), of \(\mathbf { M }\), where \(\lambda _ { 1 } > \lambda _ { 2 }\),
  2. corresponding eigenvectors for all three eigenvalues of \(\mathbf { M }\). It is given that \(\mathbf { x } = a \mathbf { e } _ { 1 } + b \mathbf { e } _ { 2 }\), where \(\mathbf { e } _ { 1 }\) and \(\mathbf { e } _ { 2 }\) are eigenvectors of \(\mathbf { M }\) corresponding to the eigenvalues \(\lambda _ { 1 }\) and \(\lambda _ { 2 }\) respectively, and \(a\) and \(b\) are scalar constants. Show that \(\mathbf { M x } = p \mathbf { e } _ { 1 } + q \mathbf { e } _ { 2 }\), expressing \(p\) and \(q\) in terms of \(a\) and \(b\). \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. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. 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 2007 November Q1
1 A curve is defined parametrically by $$x = a t ^ { 2 } , \quad y = a t$$ where \(a\) is a positive constant. The part of the curve joining the point where \(t = 0\) to the point where \(t = \sqrt { } 2\) is rotated through one complete revolution about the \(x\)-axis. Show that the area of the surface obtained is \(\frac { 13 } { 3 } \pi a ^ { 2 }\).
CAIE FP1 2007 November Q2
2 Express $$\frac { 2 n + 3 } { n ( n + 1 ) }$$ in partial fractions and hence use the method of differences to find $$\sum _ { n = 1 } ^ { N } \frac { 2 n + 3 } { n ( n + 1 ) } \left( \frac { 1 } { 3 } \right) ^ { n + 1 }$$ in terms of \(N\). Deduce the value of $$\sum _ { n = 1 } ^ { \infty } \frac { 2 n + 3 } { n ( n + 1 ) } \left( \frac { 1 } { 3 } \right) ^ { n + 1 }$$
CAIE FP1 2007 November Q3
3 Prove by induction that, for all \(n \geqslant 1\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \mathrm { e } ^ { x ^ { 2 } } \right) = \mathrm { P } _ { n } ( x ) \mathrm { e } ^ { x ^ { 2 } } ,$$ where \(\mathrm { P } _ { n } ( x )\) is a polynomial in \(x\) of degree \(n\) with the coefficient of \(x ^ { n }\) equal to \(2 ^ { n }\).
CAIE FP1 2007 November Q4
4 The roots of the equation $$x ^ { 3 } - 8 x ^ { 2 } + 5 = 0$$ are \(\alpha , \beta , \gamma\). Show that $$\alpha ^ { 2 } = \frac { 5 } { \beta + \gamma } .$$ It is given that the roots are all real. Without reference to a graph, show that one of the roots is negative and the other two roots are positive.
CAIE FP1 2007 November Q5
5 The positive variables \(x\) and \(y\) are related by $$y = x ^ { 2 } + 2 \ln ( x y )$$ Find the values of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when both \(x\) and \(y\) are equal to 1 .
CAIE FP1 2007 November Q6
6 The points \(A , B\) and \(C\) have position vectors \(2 \mathbf { i } , 3 \mathbf { j }\) and \(4 \mathbf { k }\) respectively. Find a vector which is perpendicular to the plane \(\Pi _ { 1 }\) containing \(A , B\) and \(C\). The plane \(\Pi _ { 2 }\) has equation $$\mathbf { r } = \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } ) + \mu ( \mathbf { j } - \mathbf { k } ) .$$ Find the acute angle between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
CAIE FP1 2007 November Q7
7 The curve \(C\) has polar equation $$r = \theta \sin \theta ,$$ where \(0 \leqslant \theta \leqslant \pi\). Draw a sketch of \(C\). Find the area of the region enclosed by \(C\), leaving your answer in terms of \(\pi\).
CAIE FP1 2007 November Q8
8 Let \(I _ { n } = \int _ { 0 } ^ { \ln 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n } \mathrm {~d} x\).
  1. Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ \left( \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } \right) \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n - 1 } \right] = n \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n } - 4 ( n - 1 ) \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { n - 2 } .$$
  2. Hence show that $$n I _ { n } = 4 ( n - 1 ) I _ { n - 2 } + \frac { 3 } { 2 } \left( \frac { 5 } { 2 } \right) ^ { n - 1 } .$$
  3. Use the result in part (ii) to find the \(y\)-coordinate of the centroid of the region bounded by the axes, the line \(x = \ln 2\) and the curve $$y = \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right) ^ { 2 } .$$ Give your answer correct to 3 decimal places.
CAIE FP1 2007 November Q9
9 Write down, in any form, all the roots of the equation \(z ^ { 5 } - 1 = 0\). Hence find all the roots of the equation $$( w - 1 ) ^ { 4 } + ( w - 1 ) ^ { 3 } + ( w - 1 ) ^ { 2 } + w = 0$$ and deduce that none of them is real. Find the arguments of the two roots which have the smaller modulus.
CAIE FP1 2007 November Q10
10 The vectors \(\mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 3 } , \mathbf { b } _ { 4 }\) are defined as follows: $$\mathbf { b } _ { 1 } = \left( \begin{array} { c } 1
0
0
0 \end{array} \right) , \quad \mathbf { b } _ { 2 } = \left( \begin{array} { c } 1
1
0
0 \end{array} \right) , \quad \mathbf { b } _ { 3 } = \left( \begin{array} { c } 1
1
1
0 \end{array} \right) , \quad \mathbf { b } _ { 4 } = \left( \begin{array} { c } 1
1
1
1 \end{array} \right) .$$ The linear space spanned by \(\mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 3 }\) is denoted by \(V _ { 1 }\) and the linear space spanned by \(\mathbf { b } _ { 1 } , \mathbf { b } _ { 2 } , \mathbf { b } _ { 4 }\) is denoted by \(V _ { 2 }\).
  1. Give a reason why \(V _ { 1 } \cup V _ { 2 }\) is not a linear space.
  2. State the dimension of the linear space \(V _ { 1 } \cap V _ { 2 }\) and write down a basis. Consider now the set \(V _ { 3 }\) of all vectors of the form \(q \mathbf { b } _ { 2 } + r \mathbf { b } _ { 3 } + s \mathbf { b } _ { 4 }\), where \(q , r , s\) are real numbers. Show that \(V _ { 3 }\) is a linear space, and show also that it has dimension 3 . Determine whether each of the vectors $$\left( \begin{array} { l } 4
    4
    2
    5 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { l } 5
    4
    2
    5 \end{array} \right)$$ belongs to \(V _ { 3 }\) and justify your conclusions.
CAIE FP1 2007 November Q11
11 Find the eigenvalues of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & 1 & 4
1 & 1 & - 1
2 & 1 & 1 \end{array} \right)$$ and corresponding eigenvectors. The matrix \(\mathbf { B }\) is defined by $$\mathbf { B } = \mathbf { A } - k \mathbf { I } ,$$ where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix and \(k\) is a real number. Find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { B } ^ { 3 } = \mathbf { P D } \mathbf { P } ^ { - 1 } .$$
CAIE FP1 2007 November Q12 EITHER
The curve \(C\) has equation $$y = \frac { a x ^ { 2 } + b x + c } { x + 4 }$$ where \(a\), \(b\) and \(c\) are constants. It is given that \(y = 2 x - 5\) is an asymptote of \(C\).
  1. Find the values of \(a\) and \(b\).
  2. Given also that \(C\) has a turning point at \(x = - 1\), find the value of \(c\).
  3. Find the set of values of \(y\) for which there are no points on \(C\).
  4. Draw a sketch of the curve with equation $$y = \frac { 2 ( x - 7 ) ^ { 2 } + 3 ( x - 7 ) - 2 } { x - 3 }$$ [You should state the equations of the asymptotes and the coordinates of the turning points.]
CAIE FP1 2007 November Q12 OR
Show that the substitution \(y = \frac { 1 } { w }\) reduces the differential equation $$y \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 y \frac { \mathrm {~d} y } { \mathrm {~d} x } - 2 \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - 5 y ^ { 2 } = \left( 5 x ^ { 2 } + 4 x + 2 \right) y ^ { 3 }$$ to $$\frac { \mathrm { d } ^ { 2 } w } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} w } { \mathrm {~d} x } + 5 w = - 5 x ^ { 2 } - 4 x - 2$$ Find the general solution for \(w\) in terms of \(x\). Find a function f such that \(\lim _ { x \rightarrow \infty } \left( \frac { y } { \mathrm { f } ( x ) } \right) = 1\).
CAIE FP1 2011 November Q1
1 The equation \(x ^ { 3 } + p x + q = 0\) has a repeated root. Prove that \(4 p ^ { 3 } + 27 q ^ { 2 } = 0\).
CAIE FP1 2011 November Q2
2 The position vectors of points \(A , B , C\), relative to the origin \(O\), are \(\mathbf { a } , \mathbf { b } , \mathbf { c }\), where $$\mathbf { a } = 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \quad \mathbf { b } = 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } , \quad \mathbf { c } = 3 \mathbf { i } - \mathbf { j } - \mathbf { k }$$ Find \(\mathbf { a } \times \mathbf { b }\) and deduce the area of the triangle \(O A B\). Hence find the volume of the tetrahedron \(O A B C\), given that the volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × perpendicular height.