Questions — CAIE FP1 (549 questions)

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CAIE FP1 2014 June Q11 OR
With respect to an origin \(O\), the point \(A\) has position vector \(4 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }\) and the plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = ( 4 + \lambda + 3 \mu ) \mathbf { i } + ( - 2 + 7 \lambda + \mu ) \mathbf { j } + ( 2 + \lambda - \mu ) \mathbf { k } ,$$ where \(\lambda\) and \(\mu\) are real. The point \(L\) is such that \(\overrightarrow { O L } = 3 \overrightarrow { O A }\) and \(\Pi _ { 2 }\) is the plane through \(L\) which is parallel to \(\Pi _ { 1 }\). The point \(M\) is such that \(\overrightarrow { A M } = 3 \overrightarrow { M L }\).
  1. Show that \(A\) is in \(\Pi _ { 1 }\).
  2. Find a vector perpendicular to \(\Pi _ { 2 }\).
  3. Find the position vector of the point \(N\) in \(\Pi _ { 2 }\) such that \(O N\) is perpendicular to \(\Pi _ { 2 }\).
  4. Show that the position vector of \(M\) is \(10 \mathbf { i } - 5 \mathbf { j } + 5 \mathbf { k }\) and find the perpendicular distance of \(M\) from the line through \(O\) and \(N\), giving your answer correct to 3 significant figures.
CAIE FP1 2015 June Q1
1 Use the List of Formulae (MF10) to show that \(\sum _ { r = 1 } ^ { 13 } \left( 3 r ^ { 2 } - 5 r + 1 \right)\) and \(\sum _ { r = 0 } ^ { 9 } \left( r ^ { 3 } - 1 \right)\) have the same numerical value.
CAIE FP1 2015 June Q2
2 Find the value of the constant \(k\) for which the system of equations $$\begin{aligned} 2 x - 3 y + 4 z & = 1
3 x - y & = 2
x + 2 y + k z & = 1 \end{aligned}$$ does not have a unique solution. For this value of \(k\), solve the system of equations.
CAIE FP1 2015 June Q3
3 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that \(a _ { 1 } > 5\) and \(a _ { n + 1 } = \frac { 4 a _ { n } } { 5 } + \frac { 5 } { a _ { n } }\) for every positive integer \(n\).
Prove by mathematical induction that \(a _ { n } > 5\) for every positive integer \(n\). Prove also that \(a _ { n } > a _ { n + 1 }\) for every positive integer \(n\).
CAIE FP1 2015 June Q4
4 The roots of the cubic equation \(x ^ { 3 } - 7 x ^ { 2 } + 2 x - 3 = 0\) are \(\alpha , \beta\) and \(\gamma\). Find the values of
  1. \(\frac { 1 } { ( \alpha \beta ) ( \beta \gamma ) ( \gamma \alpha ) }\),
  2. \(\frac { 1 } { \alpha \beta } + \frac { 1 } { \beta \gamma } + \frac { 1 } { \gamma \alpha }\),
  3. \(\frac { 1 } { \alpha ^ { 2 } \beta \gamma } + \frac { 1 } { \alpha \beta ^ { 2 } \gamma } + \frac { 1 } { \alpha \beta \gamma ^ { 2 } }\). Deduce a cubic equation, with integer coefficients, having roots \(\frac { 1 } { \alpha \beta } , \frac { 1 } { \beta \gamma }\) and \(\frac { 1 } { \gamma \alpha }\).
CAIE FP1 2015 June Q5
5 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$\begin{array} { l l } C _ { 1 } : & r = \frac { 1 } { \sqrt { 2 } } , \quad \text { for } 0 \leqslant \theta < 2 \pi
C _ { 2 } : & r = \sqrt { } \left( \sin \frac { 1 } { 2 } \theta \right) , \quad \text { for } 0 \leqslant \theta \leqslant \pi \end{array}$$ Find the polar coordinates of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\). Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram. Find the exact value of the area of the region enclosed by \(C _ { 1 } , C _ { 2 }\) and the half-line \(\theta = 0\).
CAIE FP1 2015 June Q6
6 A curve has equation \(x ^ { 2 } - 6 x y + 25 y ^ { 2 } = 16\). Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) at the point \(( 3,1 )\). By finding the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(( 3,1 )\), determine the nature of this turning point.
CAIE FP1 2015 June Q7
7 Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x\), where \(n\) is a non-negative integer. Show that $$I _ { n } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 } , \quad \text { for } n \geqslant 2$$ Find the exact value of \(I _ { 4 }\).
CAIE FP1 2015 June Q8
8 By considering \(\sum _ { r = 1 } ^ { n } z ^ { 2 r - 1 }\), where \(z = \cos \theta + \mathrm { i } \sin \theta\), show that, if \(\sin \theta \neq 0\), $$\sum _ { r = 1 } ^ { n } \sin ( 2 r - 1 ) \theta = \frac { \sin ^ { 2 } n \theta } { \sin \theta }$$ Deduce that $$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) \cos \left[ \frac { ( 2 r - 1 ) \pi } { 2 n } \right] = - \operatorname { cosec } \left( \frac { \pi } { 2 n } \right) \cot \left( \frac { \pi } { 2 n } \right)$$
CAIE FP1 2015 June Q9
9 The curve \(C\) has parametric equations $$x = 4 t + 2 t ^ { \frac { 3 } { 2 } } , \quad y = 4 t - 2 t ^ { \frac { 3 } { 2 } } , \quad \text { for } 0 \leqslant t \leqslant 4$$ Find the arc length of \(C\), giving your answer correct to 3 significant figures. Find the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant 32\).
CAIE FP1 2015 June Q10
10 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 2 & - 3
2 & 2 & 3
- 3 & 3 & 3 \end{array} \right)$$ The matrix \(\mathbf { A }\) has an eigenvector \(\left( \begin{array} { r } 1
- 1
1 \end{array} \right)\). Find the corresponding eigenvalue. The matrix \(\mathbf { A }\) also has eigenvalues 4 and 6. Find corresponding eigenvectors. Hence find a matrix \(\mathbf { P }\) such that \(\mathbf { A } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\), where \(\mathbf { D }\) is a diagonal matrix which is to be determined. The matrix \(\mathbf { B }\) is such that \(\mathbf { B } = \mathbf { Q A Q } ^ { - 1 }\), where $$\mathbf { Q } = \left( \begin{array} { r r r } 4 & 11 & 5
1 & 4 & 2
1 & 2 & 1 \end{array} \right)$$ By using the expression \(\mathbf { P D P } ^ { - 1 }\) for \(\mathbf { A }\), find the set of eigenvalues and a corresponding set of eigenvectors for \(\mathbf { B }\).
[0pt] [Question 11 is printed on the next page.]
CAIE FP1 2015 June Q11 EITHER
Show that the substitution \(v = \frac { 1 } { y }\) reduces the differential equation $$\frac { 2 } { y ^ { 3 } } \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - \frac { 1 } { y ^ { 2 } } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - \frac { 2 } { y ^ { 2 } } \frac { \mathrm {~d} y } { \mathrm {~d} x } + \frac { 5 } { y } = 17 + 6 x - 5 x ^ { 2 }$$ to the differential equation $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 5 v = 17 + 6 x - 5 x ^ { 2 }$$ Hence find \(y\) in terms of \(x\), given that when \(x = 0 , y = \frac { 1 } { 2 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 1\).
CAIE FP1 2015 June Q11 OR
The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = 8 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } )\) and \(\mathbf { r } = 5 \mathbf { i } + 3 \mathbf { j } - 14 \mathbf { k } + \mu ( 2 \mathbf { j } - 3 \mathbf { k } )\) respectively. 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 \(l _ { 2 }\). Find the position vector of the point \(P\) and the position vector of the point \(Q\). The points with position vectors \(8 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) and \(5 \mathbf { i } + 3 \mathbf { j } - 14 \mathbf { k }\) are denoted by \(A\) and \(B\) respectively. Find
  1. \(\overrightarrow { A P } \times \overrightarrow { A Q }\) and hence the area of the triangle \(A P Q\),
  2. the volume of the tetrahedron \(A P Q B\). (You are given that the volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × perpendicular height.) \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.
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CAIE FP1 2016 June Q1
1 The roots of the cubic equation \(2 x ^ { 3 } + x ^ { 2 } - 7 = 0\) are \(\alpha , \beta\) and \(\gamma\). Using the substitution \(y = 1 + \frac { 1 } { x }\), or otherwise, find the cubic equation whose roots are \(1 + \frac { 1 } { \alpha } , 1 + \frac { 1 } { \beta }\) and \(1 + \frac { 1 } { \gamma }\), giving your answer in the form \(a y ^ { 3 } + b y ^ { 2 } + c y + d = 0\), where \(a , b , c\) and \(d\) are constants to be found.
CAIE FP1 2016 June Q2
2 Express \(\frac { 4 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions and hence find \(\sum _ { r = 1 } ^ { n } \frac { 4 } { r ( r + 1 ) ( r + 2 ) }\). Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 4 } { r ( r + 1 ) ( r + 2 ) }\).
CAIE FP1 2016 June Q3
3 Prove by mathematical induction that, for all positive integers \(n , 10 ^ { n } + 3 \times 4 ^ { n + 2 } + 5\) is divisible by 9 .
CAIE FP1 2016 June Q4
4 A curve \(C\) has polar equation \(r ^ { 2 } = 8 \operatorname { cosec } 2 \theta\) for \(0 < \theta < \frac { 1 } { 2 } \pi\). Find a cartesian equation of \(C\). Sketch \(C\). Determine the exact area of the sector bounded by the arc of \(C\) between \(\theta = \frac { 1 } { 6 } \pi\) and \(\theta = \frac { 1 } { 3 } \pi\), the half-line \(\theta = \frac { 1 } { 6 } \pi\) and the half-line \(\theta = \frac { 1 } { 3 } \pi\).
[0pt] [It is given that \(\int \operatorname { cosec } x \mathrm {~d} x = \ln \left| \tan \frac { 1 } { 2 } x \right| + c\).]
CAIE FP1 2016 June Q5
5 Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \sin ^ { 2 } x \mathrm {~d} x\), for \(n \geqslant 0\). By differentiating \(\cos ^ { n - 1 } x \sin ^ { 3 } x\) with respect to \(x\), prove that $$( n + 2 ) I _ { n } = ( n - 1 ) I _ { n - 2 } \quad \text { for } n \geqslant 2$$ Hence find the exact value of \(I _ { 4 }\).
CAIE FP1 2016 June Q6
6 Use de Moivre's theorem to express \(\cot 7 \theta\) in terms of \(\cot \theta\). Use the equation \(\cot 7 \theta = 0\) to show that the roots of the equation $$x ^ { 6 } - 21 x ^ { 4 } + 35 x ^ { 2 } - 7 = 0$$ are \(\cot \left( \frac { 1 } { 14 } k \pi \right)\) for \(k = 1,3,5,9,11,13\), and deduce that $$\cot ^ { 2 } \left( \frac { 1 } { 14 } \pi \right) \cot ^ { 2 } \left( \frac { 3 } { 14 } \pi \right) \cot ^ { 2 } \left( \frac { 5 } { 14 } \pi \right) = 7$$
CAIE FP1 2016 June Q7
7 A curve \(C\) has equation \(y = \frac { x ^ { 2 } } { x - 2 }\). Find the equations of the asymptotes of \(C\). Show that there are no points on \(C\) for which \(0 < y < 8\). Sketch \(C\), giving the coordinates of the turning points.
CAIE FP1 2016 June Q8
8 Find a cartesian equation of the plane \(\Pi _ { 1 }\) passing through the points with coordinates \(( 2 , - 1,3 )\), \(( 4,2 , - 5 )\) and \(( - 1,3 , - 2 )\). The plane \(\Pi _ { 2 }\) has cartesian equation \(3 x - y + 2 z = 5\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find a vector equation of the line of intersection of the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
CAIE FP1 2016 June Q9
9 Find the value of the constant \(k\) such that \(y = k x ^ { 2 } \mathrm { e } ^ { 2 x }\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 4 \mathrm { e } ^ { 2 x }$$ Hence find the general solution of ( \(*\) ). Find the particular solution of ( \(*\) ) such that \(y = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2\) when \(x = 0\).
CAIE FP1 2016 June Q10
10 Write down the eigenvalues of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 1 & - 1
0 & - 1 & 2
0 & 0 & 1 \end{array} \right)$$ and find corresponding eigenvectors. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\), and hence find the matrix \(\mathbf { A } ^ { n }\), where \(n\) is a positive integer.
[0pt] [Question 11 is printed on the next page.]
CAIE FP1 2016 June Q11 EITHER
A curve \(C\) has parametric equations $$x = \mathrm { e } ^ { 2 t } \cos 2 t , \quad y = \mathrm { e } ^ { 2 t } \sin 2 t , \quad \text { for } - \frac { 1 } { 2 } \pi \leqslant t \leqslant \frac { 1 } { 2 } \pi .$$ 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 2016 June Q11 OR
The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & 3 & - 4
2 & - 4 & 7 & - 9
4 & - 8 & 14 & - 18
5 & - 10 & 17 & - 22 \end{array} \right)$$ Find the rank of \(\mathbf { M }\). Obtain a basis for the null space \(K\) of T . Evaluate $$\mathbf { M } \left( \begin{array} { r } 1
- 2
2
- 1 \end{array} \right)$$ and hence show that any solution of $$\mathbf { M x } = \left( \begin{array} { l }