Questions — CAIE FP1 (594 questions)

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CAIE FP1 2015 June Q1
4 marks Moderate -0.8
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
6 marks Standard +0.8
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
7 marks Challenging +1.2
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
8 marks Challenging +1.2
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
9 marks Standard +0.8
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
9 marks Standard +0.3
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
9 marks Challenging +1.2
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
11 marks Challenging +1.8
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
11 marks Challenging +1.2
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
12 marks Challenging +1.2
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
Challenging +1.3
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
Challenging +1.8
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.) {www.cie.org.uk} after the live examination series.
    }
CAIE FP1 2016 June Q1
4 marks Standard +0.8
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
6 marks Standard +0.3
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
6 marks Standard +0.3
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
8 marks Standard +0.8
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
9 marks Challenging +1.3
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
9 marks Challenging +1.8
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
10 marks Standard +0.8
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
11 marks Standard +0.8
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
11 marks Challenging +1.2
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
12 marks Standard +0.8
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
Challenging +1.8
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
Challenging +1.2
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 $$\mathbf { M x } = \left( \begin{array} { l } 15 \\ 33 \\ 66 \\ 81 \end{array} \right)$$
CAIE FP1 2016 June Q15
Challenging +1.2
has the form \(\left( \begin{array} { r } 1 \\ - 2 \\ 2 \\ - 1 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 }\), where \(\lambda\) and \(\mu\) are scalars and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\). Hence obtain a solution \(\mathbf { x } ^ { \prime }\) of ( \(*\) ) such that the sum of the components \(\mathbf { x } ^ { \prime }\) is 6 and the sum of the squares of the components of \(\mathbf { x } ^ { \prime }\) is 26 . {www.cie.org.uk} after the live examination series. }