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Pre-U Pre-U 9795/1 Specimen Q1
4 marks Moderate -0.8
1 Using standard results given in MF20, show that $$\sum _ { r = 1 } ^ { n } \left( 4 r ^ { 3 } + 2 r ^ { 2 } + 5 \right) = \frac { 1 } { 3 } n \left( n ^ { 2 } + 2 \right) ( 3 n + 8 )$$
Pre-U Pre-U 9795/1 Specimen Q2
5 marks Standard +0.3
2 The equation \(x ^ { 3 } - 14 x ^ { 2 } + 16 x + 21 = 0\) has roots \(\alpha , \beta , \gamma\). Determine the values of \(\alpha + \beta + \gamma\), \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) and \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
Pre-U Pre-U 9795/1 Specimen Q3
3 marks Standard +0.3
3
  1. Evaluate, in terms of \(k\), the determinant of the matrix \(\left( \begin{array} { r r r } 1 & 2 & 1 \\ - 3 & 5 & 8 \\ 6 & 12 & k \end{array} \right)\). Three planes have equations \(x + 2 y + z = 4 , - 3 x + 5 y + 8 z = 21\) and \(6 x + 12 y + k z = 31\).
  2. State the value of \(k\) for which these three planes do not meet at a single point.
  3. Find the coordinates of the point of intersection of the three planes when \(k = 7\).
Pre-U Pre-U 9795/1 Specimen Q4
6 marks Challenging +1.2
4 Two skew lines have equations \(\mathbf { r } = \left( \begin{array} { r } - 4 \\ 2 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ 0 \\ - 1 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { l } 6 \\ 5 \\ 2 \end{array} \right) + \mu \left( \begin{array} { l } 5 \\ 8 \\ 3 \end{array} \right)\). Find a vector which is perpendicular to both lines and determine the shortest distance between the two lines.
Pre-U Pre-U 9795/1 Specimen Q5
7 marks Challenging +1.2
5 The variables \(y\) and \(x\) are related by the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } , \quad - 2 < x < 2 .$$ By writing \(u = \frac { \mathrm { d } y } { \mathrm {~d} x }\), determine \(y\) explicitly in terms of \(x\), given that \(y = \frac { 1 } { 2 }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 4 }\) when \(x = 0\).
Pre-U Pre-U 9795/1 Specimen Q6
9 marks Standard +0.8
6 The set \(S\) consists of all real numbers except 1. The binary operation * is defined for all \(a , b\) in \(S\) by $$a * b = a + b - a b$$
  1. By considering the identity \(a + b - a b \equiv 1 - ( a - 1 ) ( b - 1 )\), or otherwise, show that \(S\) is closed under *.
  2. Show that * is associative on \(S\).
  3. Find the identity of \(S\) under \(*\), and the inverse of \(x\) for all \(x \in S\).
  4. The set \(S\), together with the binary operation *, forms a group \(G\). Find a subgroup of \(G\) of order 2 .
Pre-U Pre-U 9795/1 Specimen Q7
6 marks Standard +0.8
7 A curve has equation \(y = \frac { 4 x + 11 } { ( x + 3 ) ^ { 2 } }\).
  1. Show that the curve meets the line \(y = k\) if and only if \(k \leq 4\), and deduce the coordinates of the turning point on the curve.
  2. Sketch the curve, stating the coordinates of the points where it cuts the axes, and showing clearly its asymptotes and the turning point.
Pre-U Pre-U 9795/1 Specimen Q8
7 marks Standard +0.8
8 The curve \(C\) has polar equation \(r = \theta ^ { 2 } + 2 \theta\) for \(0 \leq \theta \leq 3\).
  1. Find the area of the region enclosed by \(C\) and the half-lines \(\theta = 0\) and \(\theta = 3\).
  2. Determine the length of \(C\).
Pre-U Pre-U 9795/1 Specimen Q9
16 marks Standard +0.8
9
  1. (a) Given that \(y = \tanh ^ { - 1 } x , - 1 < x < 1\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\).
    (b) Show that \(y = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\).
  2. Show that \(\int _ { 0 } ^ { 1 / \sqrt { 3 } } \frac { 2 } { 1 - x ^ { 4 } } \mathrm {~d} x = \ln ( 1 + \sqrt { 3 } ) - \frac { 1 } { 2 } \ln 2 + \frac { 1 } { 6 } \pi\).
Pre-U Pre-U 9795/1 Specimen Q10
24 marks Challenging +1.8
10
  1. Use de Moivre's theorem to prove that \(\sin 5 \theta \equiv s \left( 16 s ^ { 4 } - 20 s ^ { 2 } + 5 \right)\), where \(s = \sin \theta\), and deduce that \(\sin \frac { 2 \pi } { 5 } = \sqrt { \frac { 5 + \sqrt { 5 } } { 8 } }\). The complex number \(\omega = 16 ( - 1 + \mathrm { i } \sqrt { 3 } )\).
  2. State the value of \(| \omega |\) and find \(\arg \omega\) as a rational multiple of \(\pi\).
  3. (a) Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leq \pi\).
    (b) These five roots are represented in the complex plane by the points \(A , B , C , D\) and \(E\). Show these points on an Argand diagram, and find the area of the pentagon \(A B C D E\) in an exact surd form.
Pre-U Pre-U 9795/1 Specimen Q11
17 marks Challenging +1.2
11
    1. Write down the matrix which represents a rotation through an angle \(\alpha\) anticlockwise about the origin.
    2. Show that the plane transformation given by the matrix $$\left( \begin{array} { c c } \cos \theta + \sin \theta & - ( \sin \theta - \cos \theta ) \\ \sin \theta - \cos \theta & \cos \theta + \sin \theta \end{array} \right)$$ is the composition of a rotation, \(R\), and a second transformation, \(S\). Describe both \(R\) and \(S\) fully.
    1. Write down the matrix which represents a reflection in the line \(y = x \tan \frac { 1 } { 2 } \beta\). For \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\), the plane transformation \(T\) is given by the matrix $$\left( \begin{array} { c c } 1 + \cos 2 \theta & \sin 2 \theta \\ \sin 2 \theta & - 1 - \cos 2 \theta \end{array} \right)$$
    2. Show that \(T\) is the composition of a reflection and an enlargement, and describe these transformations in full.
    3. Find also the values of \(\theta\) for which \(T\) is an area-preserving transformation.
Pre-U Pre-U 9795/1 Specimen Q12
13 marks Challenging +1.2
12
  1. The sequence \(\left\{ u _ { n } \right\}\) is defined for all integers \(n \geq 0\) by $$u _ { 0 } = 1 \quad \text { and } \quad u _ { n } = n u _ { n - 1 } + 1 , \quad n \geq 1 .$$ Prove by induction that \(u _ { n } = n ! \sum _ { r = 0 } ^ { n } \frac { 1 } { r ! }\).
  2. (a) Given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { - x } \mathrm {~d} x\) for \(n \geq 0\), show that, for \(n \geq 1\), $$I _ { n } = n I _ { n - 1 } - \frac { 1 } { \mathrm { e } }$$ (b) Evaluate \(I _ { 0 }\) exactly and deduce the value of \(I _ { 1 }\).
    (c) Show that \(I _ { n } = n ! - \frac { u _ { n } } { \mathrm { e } }\) for all integers \(n \geq 1\).
Pre-U Pre-U 9795 Specimen Q1
Standard +0.3
1 The \(n\)th triangular number, \(T _ { n }\), is given by the formula \(T _ { n } = \frac { 1 } { 2 } n ( n + 1 )\).
  1. Express \(\frac { 1 } { T _ { n } }\) in terms of partial fractions.
  2. Hence, using the method of differences, show that \(\sum _ { n = 1 } ^ { N } \left( \frac { 1 } { T _ { n } } \right) = \frac { 2 N } { N + 1 }\).
Pre-U Pre-U 9795 Specimen Q2
Moderate -0.3
2
  1. On a single Argand diagram, sketch and clearly label each of the following loci:
    1. \(| z | = 4\),
    2. \(\quad \arg ( z + 4 \mathrm { i } ) = \frac { 1 } { 4 } \pi\).
    3. On the same Argand diagram, shade the region \(R\) defined by the inequalities $$| z | \leqslant 4 \quad \text { and } \quad 0 \leqslant \arg ( z + 4 i ) \leqslant \frac { 1 } { 4 } \pi$$
Pre-U Pre-U 9795 Specimen Q3
Standard +0.3
3 Solve the equation $$5 \cosh x - \sinh x = 7$$ giving your answers in an exact logarithmic form.
Pre-U Pre-U 9795 Specimen Q4
Standard +0.8
4 Let \(z = \cos \theta + \mathrm { i } \sin \theta\).
  1. Use de Moivre's theorem to prove that \(z ^ { n } + z ^ { - n } = 2 \cos n \theta\).
  2. Deduce the identity \(\cos ^ { 5 } \theta = \frac { 1 } { 16 } ( \cos 5 \theta + 5 \cos 3 \theta + 10 \cos \theta )\).
Pre-U Pre-U 9795 Specimen Q5
Standard +0.3
5 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 9 y = 72 \mathrm { e } ^ { 3 x }$$
Pre-U Pre-U 9795 Specimen Q6
Challenging +1.2
6
  1. Given that \(y = \cos ( \ln ( 1 + x ) )\), prove that
    1. \(\quad ( 1 + x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = - \sin ( \ln ( 1 + x ) )\),
    2. \(( 1 + x ) ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + ( 1 + x ) \frac { \mathrm { d } y } { \mathrm {~d} x } + y = 0\).
    3. Obtain an equation relating \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } , \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
    4. Hence find the Maclaurin series for \(y\), up to and including the term in \(x ^ { 3 }\).
Pre-U Pre-U 9795 Specimen Q7
Standard +0.3
7 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 2 x - 3 } { x + 2 } .$$
  1. Find the equations of the asymptotes of \(C\).
  2. Sketch \(C\), indicating clearly the asymptotes and any points where \(C\) meets the coordinate axes.
Pre-U Pre-U 9795 Specimen Q8
Challenging +1.2
8 The equation \(8 x ^ { 3 } + 12 x ^ { 2 } + 4 x - 1 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. By considering a suitable substitution, or otherwise, show that the equation whose roots are \(2 \alpha + 1,2 \beta + 1,2 \gamma + 1\) can be written in the form $$y ^ { 3 } - y - 1 = 0 .$$
  2. The sum \(( 2 \alpha + 1 ) ^ { n } + ( 2 \beta + 1 ) ^ { n } + ( 2 \gamma + 1 ) ^ { n }\) is denoted by \(S _ { n }\). Evaluate \(S _ { 3 }\) and \(S _ { - 2 }\).
Pre-U Pre-U 9795 Specimen Q9
Standard +0.8
9
  1. Find the inverse of the matrix \(\left( \begin{array} { r r r } 1 & 3 & 4 \\ 2 & 5 & - 1 \\ 3 & 8 & 2 \end{array} \right)\), and hence solve the set of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 10 \\ 3 x + 8 y + 2 z & = 8 \end{aligned}$$
  2. Find the value of \(k\) for which the set of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 15 \\ 3 x + 8 y + 3 z & = k \end{aligned}$$ is consistent. Find the solution in this case and interpret it geometrically.
Pre-U Pre-U 9795 Specimen Q10
Challenging +1.8
10 A group \(G\) has distinct elements \(e , a , b , c , \ldots\), where \(e\) is the identity element and \(\circ\) is the binary operation.
  1. Prove that if \(a \circ a = b\) and \(b \circ b = a\), then the set of elements \(\{ e , a , b \}\) forms a subgroup of \(G\).
  2. Prove that if \(a \circ a = b , b \circ b = c\) and \(c \circ c = a\), then the set of elements \(\{ e , a , b , c \}\) does not form a subgroup of \(G\).
Pre-U Pre-U 9795 Specimen Q11
Challenging +1.2
11 With respect to an origin \(O\), the points \(A , B , C\) and \(D\) have position vectors $$\mathbf { a } = 2 \mathbf { i } - \mathbf { j } + \mathbf { k } , \quad \mathbf { b } = \mathbf { i } - 2 \mathbf { k } , \quad \mathbf { c } = - \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } , \quad \mathbf { d } = - \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ,$$ respectively. Find
  1. a vector perpendicular to the plane \(O A B\),
  2. the acute angle between the planes \(O A B\) and \(O C D\), correct to the nearest \(0.1 ^ { \circ }\),
  3. the shortest distance between the line \(A B\) and the line \(C D\),
  4. the perpendicular distance from the point \(A\) to the line \(C D\).
Pre-U Pre-U 9795 Specimen Q12
Challenging +1.8
12 \includegraphics[max width=\textwidth, alt={}, center]{0f5edc87-cb14-4583-a54d-badec47741d1-08_414_659_804_744} The diagram shows a sketch of the curve \(C\) with polar equation \(r = 4 \cos ^ { 2 } \theta\) for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  1. Explain briefly how you can tell from this form of the equation that \(C\) is symmetrical about the line \(\theta = 0\) and that the tangent to \(C\) at the pole \(O\) is perpendicular to the line \(\theta = 0\).
  2. The equation of \(C\) may be expressed in the form \(r = k ( 1 + \cos 2 \theta )\). State the value of \(k\) and use this form to show that the area of the region enclosed by \(C\) is given by $$\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } ( 3 + 4 \cos 2 \theta + \cos 4 \theta ) d \theta ,$$ and hence find this area.
  3. The length of \(C\) is denoted by \(L\). Show that $$L = 8 \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos \theta \sqrt { 1 + 3 \sin ^ { 2 } \theta } \mathrm {~d} \theta$$ and use the substitution \(\sinh x = \sqrt { 3 } \sin \theta\) to determine \(L\) in an exact form.
Pre-U Pre-U 9795 Specimen Q13
Challenging +1.8
13 Let \(I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x\), where \(n\) is a positive integer.
  1. By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x ( \ln x ) ^ { n } \right)\), or otherwise, show that \(I _ { n } = \mathrm { e } - n I _ { n - 1 }\).
  2. Let \(J _ { n } = \frac { I _ { n } } { n ! }\). Prove by induction that $$\sum _ { r = 2 } ^ { n } \frac { ( - 1 ) ^ { r } } { r ! } = \frac { 1 } { \mathrm { e } } \left( 1 + ( - 1 ) ^ { n } J _ { n } \right)$$ for all positive integers \(n \geqslant 2\).