Questions Pre-U 9795/1 (179 questions)

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Pre-U Pre-U 9795/1 2020 Specimen Q7
2 marks Standard +0.8
7 The function f satisfies the differential equation $$x ^ { 2 } \mathrm { f } ^ { \prime \prime } ( x ) + ( 2 x - 1 ) \mathrm { f } ^ { \prime } ( x ) - 2 \mathrm { f } ( x ) = 3 \mathrm { e } ^ { x - 1 } + 1$$ and the conditions \(f ( 1 ) = 2 , f ^ { \prime } ( 1 ) = 3\).
  1. Determine \(f ^ { \prime \prime } ( 1 )\).
  2. Differentiate ( \(*\) ) with respect to \(x\) and hence evaluate \(\mathrm { f } ^ { \prime \prime \prime } ( 1 )\).
  3. Hence determine the Taylor series approximation for \(\mathrm { f } ( x )\) about \(x = 1\), up to and including the term in \(( x - 1 ) ^ { 3 }\).
  4. Deduce, to 3 decimal places, an approximation for f(1.1).
Pre-U Pre-U 9795/1 2020 Specimen Q8
5 marks Challenging +1.8
8 Consider the set \(S\) of all matrices of the form \(\left( \begin{array} { l l } p & p \\ p & p \end{array} \right)\), where p is a non-zero rational number.
  1. Show that \(S\), under the operation of matrix multiplication, forms a group, \(G\). (You may assume that matrix multiplication is associative.)
  2. Find a subgroup of \(G\) of order 2 and show that \(G\) contains no subgroups of order 3.
Pre-U Pre-U 9795/1 2020 Specimen Q11
8 marks Challenging +1.8
11
  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 \(\pi\).
    1. Determine the five roots of the equation \(z ^ { 5 } = \omega\), giving your answers in the form \(( \mathrm { r } , \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    2. 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 2020 Specimen Q12
6 marks Challenging +1.8
12
  1. Let \(I _ { \mathrm { n } } = \int _ { 0 } ^ { 3 } x ^ { n } \sqrt { 16 + x ^ { 2 } } \mathrm {~d} x\), for \(n \geqslant 0\). Show that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = 125 \times 3 ^ { n - 1 } - 16 ( n - 1 ) I _ { n - 2 }$$
  2. A curve has polar equation \(r = \frac { 1 } { 4 } \theta ^ { 4 }\) for \(0 \leqslant \theta \leqslant 3\).
    1. Sketch this curve.
    2. Find the exact length of the curve.
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 9795/1 Specimen Q2
7 marks Standard +0.3
2
  1. Verify that, for all positive values of \(n\), $$\frac { 1 } { ( n + 2 ) ( 2 n + 3 ) } - \frac { 1 } { ( n + 3 ) ( 2 n + 5 ) } = \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) }$$ For the series $$\sum _ { n = 1 } ^ { N } \frac { 4 n + 9 } { ( n + 2 ) ( n + 3 ) ( 2 n + 3 ) ( 2 n + 5 ) }$$ find
  2. the sum to \(N\) terms,
  3. the sum to infinity.
Pre-U Pre-U 9795/1 Specimen Q3
5 marks Challenging +1.2
3 A curve has equation $$y = \frac { 1 } { 3 } x ^ { 3 } + 1$$ The length of the arc of the curve joining the point where \(x = 0\) to the point where \(x = 1\) is denoted by \(s\).
  1. Show that $$s = \int _ { 0 } ^ { 1 } \sqrt { 1 + x ^ { 4 } } \mathrm {~d} x$$ The surface area generated when this arc is rotated through one complete revolution about the \(x\)-axis is denoted by \(S\).
  2. Show that $$S = \frac { 1 } { 9 } \pi ( 18 s + 2 \sqrt { 2 } - 1 )$$ [Do not attempt to evaluate \(s\) or \(S\).]
Pre-U Pre-U 9795/1 Specimen Q4
14 marks Standard +0.3
4
  1. Draw a sketch of the curve \(C\) whose polar equation is \(r = \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  2. On the same diagram draw the line \(\theta = \alpha\), where \(0 < \alpha < \frac { 1 } { 2 } \pi\). The region bounded by \(C\) and the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(R\).
  3. Find the exact value of \(\alpha\) for which the line \(\theta = \alpha\) divides \(R\) into two regions of equal area.
Pre-U Pre-U 9795/1 Specimen Q5
6 marks Challenging +1.2
5 Let $$I _ { n } = \int _ { 0 } ^ { 1 } t ^ { n } \mathrm { e } ^ { - t } \mathrm {~d} t$$ where \(n \geqslant 0\).
  1. Show that, for all \(n \geqslant 1\), $$I _ { n } = n I _ { n - 1 } - \mathrm { e } ^ { - 1 } .$$
  2. Hence prove by induction that, for all positive integers \(n\), $$I _ { n } < n ! .$$
Pre-U Pre-U 9795/1 Specimen Q6
9 marks Standard +0.8
6
  1. Find the general solution of the differential equation $$4 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 65 y = 65 x ^ { 2 } + 8 x + 73 .$$
  2. Show that, whatever the initial conditions, \(\frac { y } { x ^ { 2 } } \rightarrow 1\) as \(x \rightarrow \infty\).
Pre-U Pre-U 9795/1 Specimen Q7
9 marks Challenging +1.8
7 The multiplicative group \(G\) has eight elements \(e , a , b , c , a b , a c , b c , a b c\), where \(e\) is the identity. The group is commutative, and the order of each of the elements \(a , b , c\) is 2 .
  1. Find four subgroups of \(G\) of order 4.
  2. Give a reason why no group of order 8 can have a subgroup of order 3 . The group \(H\) has elements \(0,1,2 , \ldots , 7\) with group operation addition modulo 8 .
  3. Find the order of each element of \(H\).
  4. Determine whether \(G\) and \(H\) are isomorphic and justify your conclusion.
Pre-U Pre-U 9795/1 Specimen Q8
10 marks Standard +0.8
8
  1. Show that if \(a \neq 3\) then the system of equations $$\begin{aligned} x + 3 y + 4 z & = - 5 \\ 2 x + 5 y - z & = 5 a \\ 3 x + 8 y + a z & = b \end{aligned}$$ has a unique solution.
  2. By use of the inverse matrix of a suitable \(3 \times 3\) matrix, find the unique solution in the case \(a = 1\) and \(b = 2\).
  3. Given that \(a = 3\), find the value of \(b\) for which the equations are consistent.
Pre-U Pre-U 9795/1 Specimen Q9
13 marks Standard +0.8
9 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { x + \lambda }$$ where \(\lambda\) is a non-zero constant.
  1. Obtain the equation of each of the asymptotes of \(C\).
  2. Find the coordinates of the turning points of \(C\).
  3. In separate diagrams, sketch \(C\) for the cases \(\lambda > 0\) and \(\lambda < 0\).
Pre-U Pre-U 9795/1 Specimen Q10
14 marks Challenging +1.8
10 The line \(l _ { 1 }\) is parallel to the vector \(4 \mathbf { j } - \mathbf { k }\) and passes through the point \(A\) whose position vector is \(2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\). The variable line \(l _ { 2 }\) is parallel to the vector \(\mathbf { i } - ( 2 \sin t ) \mathbf { j }\), where \(0 \leqslant t < 2 \pi\), and passes through the point \(B\) whose position vector is \(\mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\). The points \(P\) and \(Q\) are on \(l _ { 1 }\) and \(l _ { 2 }\) respectively, and \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).
  1. Find the length of \(P Q\) in terms of \(t\).
  2. Hence find the values of \(t\) for which \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
  3. For the case \(t = \frac { 1 } { 4 } \pi\), find the perpendicular distance from \(A\) to the plane \(B P Q\), giving your answer correct to 3 decimal places.