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Pre-U Pre-U 9795/1 2017 June Q5
8 marks Standard +0.3
5
  1. Use the definition \(\tanh y = \frac { \mathrm { e } ^ { 2 y } - 1 } { \mathrm { e } ^ { 2 y } + 1 }\) to show that \(\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\) for \(| x | < 1\).
  2. Solve the equation \(\tanh x + \operatorname { coth } x = 4\), giving your answer in the form \(p \ln m\), where \(p\) is a positive rational number and \(m\) is a positive integer.
Pre-U Pre-U 9795/1 2017 June Q6
7 marks Standard +0.3
6 The curve \(S\) has equation \(y = \frac { x ^ { 2 } + 1 } { ( x + 1 ) ^ { 2 } }\).
  1. Write down the equations of the asymptotes of \(S\).
  2. Determine \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and hence find the coordinates of any turning points of \(S\).
  3. Sketch \(S\).
Pre-U Pre-U 9795/1 2017 June Q7
11 marks Standard +0.8
7
  1. Find the value of the constant \(k\) for which \(y = k x \sin 2 x\) is a particular integral of the differential equation \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = 8 \cos 2 x\).
  2. Solve \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 y = 8 \cos 2 x\), given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\) when \(x = 0\).
Pre-U Pre-U 9795/1 2017 June Q8
11 marks Standard +0.3
8 The line \(l\) has equation \(\mathbf { r } = \lambda \mathbf { d }\) and the plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } . \mathbf { n } = 35\), where $$\mathbf { d } = \left( \begin{array} { r } 2 \\ - 1 \\ 2 \end{array} \right) \quad \text { and } \quad \mathbf { n } = \left( \begin{array} { r } 6 \\ - 2 \\ 3 \end{array} \right) .$$
  1. (a) Determine the exact value of \(\cos \theta\), where \(\theta\) is the angle between \(\mathbf { d }\) and \(\mathbf { n }\).
    (b) Determine the position vector of the point of intersection of \(l\) and \(\Pi _ { 1 }\).
    (c) Determine the shortest distance from \(O\) to \(\Pi _ { 1 }\).
  2. The plane \(\Pi _ { 2 }\) has cartesian equation \(12 x - 4 y + 6 z + 21 = 0\). Determine the distance between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
Pre-U Pre-U 9795/1 2017 June Q9
11 marks Challenging +1.2
9
  1. Given that \(x \geqslant 1\), use the substitution \(x = \cosh \theta\) to show that $$\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x = \frac { \sqrt { x ^ { 2 } - 1 } } { x } + C$$ where \(C\) is an arbitrary constant.
  2. By differentiating sec \(y = x\) implicitly, show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sec ^ { - 1 } x \right) = \frac { 1 } { x \sqrt { x ^ { 2 } - 1 } }\) for \(x \geqslant 1\).
  3. Use integration by parts to determine \(\int \frac { \sec ^ { - 1 } x } { x ^ { 2 } } \mathrm {~d} x\) for \(x \geqslant 1\).
Pre-U Pre-U 9795/1 2017 June Q10
10 marks Challenging +1.8
10
  1. Express \(\frac { 1 } { ( k - 1 ) k ( k + 1 ) }\) in partial fractions.
  2. Let \(S _ { n } = \sum _ { k = 3 } ^ { n } \frac { 1 } { ( k - 1 ) k ( k + 1 ) }\) for \(n \geqslant 3\). Use the method of differences to show that $$S _ { n } = \frac { 1 } { 12 } - \frac { 1 } { 2 n ( n + 1 ) }$$ and write down the limit of \(S _ { n }\) as \(n \rightarrow \infty\).
  3. Given that \(k\) is a positive integer greater than 1 , explain why \(\frac { 1 } { k ^ { 3 } } < \frac { 1 } { ( k - 1 ) k ( k + 1 ) }\).
  4. Show that \(\frac { 27 } { 24 } < \sum _ { k = 1 } ^ { \infty } \frac { 1 } { k ^ { 3 } } < \frac { 29 } { 24 }\).
Pre-U Pre-U 9795/1 2017 June Q11
13 marks Standard +0.3
11
  1. (a) Given \(\mathbf { A } = \left( \begin{array} { l l } a & b \\ c & d \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } e & f \\ g & h \end{array} \right)\), work out the matrix \(\mathbf { A B }\) and write down expressions for \(\operatorname { det } \mathbf { A }\) and \(\operatorname { det } \mathbf { B }\).
    (b) Verify, by direct calculation, that \(\operatorname { det } ( \mathbf { A B } ) = \operatorname { det } \mathbf { A } \times \operatorname { det } \mathbf { B }\). Let \(S\) be the set of all \(2 \times 2\) matrices with determinant equal to 1 .
  2. Show that \(\left( S , \times _ { \mathrm { M } } \right)\) forms a group, \(G\), where \(\times _ { \mathrm { M } }\) is the operation of matrix multiplication. [You may assume that \(\mathrm { X } _ { \mathrm { M } }\) is associative.]
  3. (a) Show that \(\mathbf { K } = \left( \begin{array} { l l } 1 & \mathrm { i } \\ \mathrm { i } & 0 \end{array} \right)\) is an element of \(G\). Let \(H\) be the smallest subgroup of \(G\) that contains \(\mathbf { K }\) and let \(n\) be the order of \(H\).
    (b) Determine the value of \(n\).
    (c) Give a second subgroup of \(G\), also of order \(n\), which is isomorphic to \(H\).
Pre-U Pre-U 9795/1 2017 June Q12
Challenging +1.8
12 For each positive integer \(n\), the function \(\mathrm { F } _ { n }\) is defined for all real angles \(\theta\) by $$\mathrm { F } _ { n } ( \theta ) = c ^ { 2 n } + s ^ { 2 n }$$ where \(c = \cos \theta\) and \(s = \sin \theta\).
  1. Prove the identity $$\mathrm { F } _ { n + 2 } ( \theta ) - \frac { 1 } { 4 } \sin ^ { 2 } 2 \theta \times \mathrm { F } _ { n + 1 } ( \theta ) \equiv \mathrm { F } _ { n + 3 } ( \theta )$$ Let \(z\) denote the complex number \(c + \mathrm { i } s\).
  2. Using de Moivre's theorem,
    1. express \(z + z ^ { - 1 }\) and \(z - z ^ { - 1 }\) in terms of \(c\) and \(s\) respectively,
    2. prove the identity \(8 \left( c ^ { 6 } + s ^ { 6 } \right) \equiv 3 \cos 4 \theta + 5\) and deduce that $$c ^ { 6 } + s ^ { 6 } \equiv \cos ^ { 2 } 2 \theta + \frac { 1 } { 4 } \sin ^ { 2 } 2 \theta$$
    3. Prove by induction that, for all positive integers \(n\), $$c ^ { 2 n + 4 } + s ^ { 2 n + 4 } \leqslant \cos ^ { 2 } 2 \theta + \frac { 1 } { 2 ^ { n + 1 } } \sin ^ { 2 } 2 \theta$$ [You are given that the range of the function \(\mathrm { F } _ { n }\) is \(\frac { 1 } { 2 ^ { n - 1 } } \leqslant \mathrm {~F} _ { n } ( \theta ) \leqslant 1\).] {www.cie.org.uk} after the live examination series. }
Pre-U Pre-U 9794/1 2017 June Q1
3 marks Easy -1.8
1 The equation of a circle is given by \(( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = r ^ { 2 }\).
  1. Write down the coordinates of the centre of the circle.
  2. The circle passes through the point \(( 0,2 )\). Find the length of the diameter.
Pre-U Pre-U 9794/1 2017 June Q2
5 marks Easy -1.3
2 Express each of the following as a single logarithm.
  1. \(\log 3 + \log 4 - \log 2\)
  2. \(2 \log x - 3 \log y + 2 \log z\)
Pre-U Pre-U 9794/1 2017 June Q3
6 marks Moderate -0.8
3 A triangle \(A B C\) has sides \(A B , B C\) and \(C A\) of lengths \(7 \mathrm {~cm} , 6 \mathrm {~cm}\) and 8 cm respectively.
  1. Show that \(\cos A B C = \frac { 1 } { 4 }\).
  2. Find the area of triangle \(A B C\).
Pre-U Pre-U 9794/1 2017 June Q4
4 marks Moderate -0.3
4 Solve the equation \(\sin 2 x = \sqrt { 3 } \cos x\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
Pre-U Pre-U 9794/1 2017 June Q5
5 marks Standard +0.3
5 Solve \(| x - \sqrt { 3 } | < | x + 2 \sqrt { 3 } |\) giving the answer in exact form.
Pre-U Pre-U 9794/1 2017 June Q6
7 marks Standard +0.3
6
  1. Expand \(( 1 + x ) ^ { \frac { 1 } { 2 } }\), for \(| x | < 1\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
  2. In the expansion of \(( 2 + k x ) ( 1 + x ) ^ { \frac { 1 } { 2 } }\) the coefficient of \(x ^ { 3 }\) is 1 . Find the value of \(k\).
Pre-U Pre-U 9794/1 2017 June Q7
5 marks Standard +0.3
7
  1. Describe the transformation which maps the graph of \(y = \ln x\) onto the graph of \(y = \ln ( 1 + x )\).
  2. By sketching the curves \(y = \ln ( 1 + x )\) and \(y = 4 - x\) on a single diagram, show that the equation $$\ln ( 1 + x ) = 4 - x$$ has exactly one root.
  3. Use the Newton-Raphson method with \(x _ { 0 } = 2\) to find the root of the equation \(\ln ( 1 + x ) = 4 - x\) correct to 3 decimal places. Show the result of each iteration.
Pre-U Pre-U 9794/1 2017 June Q8
7 marks Standard +0.3
8 The curve \(C\) has equation \(y ^ { 3 } + 6 y ^ { 2 } - 2 y = 3 x ^ { 2 } + 2 x\). Show that the equation of the normal to \(C\) at the point \(( 1,1 )\) can be written in the form \(8 y + 13 x - 21 = 0\).
Pre-U Pre-U 9794/1 2017 June Q9
9 marks Standard +0.3
9 Solve the equation \(z ^ { 3 } + 6 z - 20 = 0\). Find the modulus and argument of each root and illustrate the roots on an Argand diagram.
Pre-U Pre-U 9794/1 2017 June Q10
7 marks Challenging +1.2
10 \includegraphics[max width=\textwidth, alt={}, center]{a3cad2ad-e06b-4aa4-a3a9-a2840cd54893-3_529_527_264_810} The diagram shows the region \(R\) in the first quadrant bounded by the curves \(y = \frac { 1 } { 3 } \left( 9 - x ^ { 2 } \right)\) and \(y = \frac { 1 } { 5 } \left( 9 - x ^ { 2 } \right)\). \(R\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Calculate the volume of the solid formed.
Pre-U Pre-U 9794/1 2017 June Q11
10 marks Standard +0.3
11 The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }\) and \(3 \mathbf { i } - 2 \mathbf { j } - \mathbf { k }\) respectively, relative to the origin \(O\). The point \(P\) lies on \(O A\) extended so that \(\overrightarrow { O P } = 3 \overrightarrow { O A }\) and the point \(Q\) lies on \(O B\) extended so that \(\overrightarrow { O Q } = 2 \overrightarrow { O B }\).
  1. Find the coordinates of the point of intersection of the lines \(A Q\) and \(B P\).
  2. Find the acute angle between the lines \(A Q\) and \(B P\).
Pre-U Pre-U 9794/1 2017 June Q12
8 marks Standard +0.3
12 Boyle's Law states that when a gas is kept at a constant temperature, its pressure \(P\) pascals is inversely proportional to its volume \(V \mathrm {~m} ^ { 3 }\). When the volume of a certain gas is \(80 \mathrm {~m} ^ { 3 }\), its pressure is 5 pascals and the rate at which the volume is increasing is \(10 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate at which the pressure is decreasing at this volume.
Pre-U Pre-U 9794/2 2017 June Q1
4 marks Easy -1.8
1 Find the equation of the line which passes through the points \(( 2,5 )\) and \(( 8 , - 1 )\). Show that this line also passes through the point \(( - 2,9 )\).
Pre-U Pre-U 9794/2 2017 June Q2
6 marks Moderate -0.8
2
    1. Find the value of the discriminant of \(x ^ { 2 } + 3 x + 5\).
    2. Use your value from part (i) to determine the number of real roots of the equation \(x ^ { 2 } + 3 x + 5 = 0\).
  2. Find the non-zero value of \(k\) for which the equation \(k x ^ { 2 } + 3 x + 5 = 0\) has only one distinct real root.
Pre-U Pre-U 9794/2 2017 June Q3
4 marks Moderate -0.8
3 Solve the equation \(\tan \left( \theta + 10 ^ { \circ } \right) = 0.1\) in the range \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
Pre-U Pre-U 9794/2 2017 June Q4
4 marks Moderate -0.3
4 A sequence of complex numbers is defined by $$u _ { 1 } = 1 + \mathrm { i } \quad \text { and } \quad u _ { n + 1 } = \mathrm { i } u _ { n } ( n = 1,2,3 , \ldots )$$
  1. Find \(u _ { 2 } , u _ { 3 } , u _ { 4 } , u _ { 5 }\) and \(u _ { 6 }\).
  2. Describe the behaviour of the sequence.
  3. Hence evaluate \(\sum _ { n = 1 } ^ { 73 } u _ { n }\).
Pre-U Pre-U 9794/2 2017 June Q5
7 marks Moderate -0.3
5
  1. Differentiate \(\frac { x } { \sqrt { 1 + x ^ { 2 } } }\) with respect to \(x\).
  2. Hence show that \(\frac { x } { \sqrt { 1 + x ^ { 2 } } }\) is increasing for all \(x\).