Questions — Pre-U Pre-U 9795/1 (179 questions)

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Pre-U Pre-U 9795/1 2010 June Q1
4 marks Standard +0.8
1 The equation \(x ^ { 3 } + 2 x ^ { 2 } + x - 7 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). Use the substitution \(y = 1 + x ^ { 2 }\) to find an equation, with integer coefficients, whose roots are \(1 + \alpha ^ { 2 } , 1 + \beta ^ { 2 }\) and \(1 + \gamma ^ { 2 }\).
Pre-U Pre-U 9795/1 2010 June Q2
5 marks Standard +0.3
2 Use the method of differences to express \(\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 }\) in terms of \(n\), and hence deduce the sum of the infinite series $$\frac { 1 } { 3 } + \frac { 1 } { 15 } + \frac { 1 } { 35 } + \ldots + \frac { 1 } { 4 n ^ { 2 } - 1 } + \ldots$$
Pre-U Pre-U 9795/1 2010 June Q3
4 marks Moderate -0.8
3 The points \(A ( 1,3 ) , B ( 4,36 )\) and \(C ( 9,151 )\) lie on the curve with equation \(y = p + q x + r x ^ { 2 }\).
  1. Using this information, write down three simultaneous equations in \(p , q\) and \(r\).
  2. Re-write this system of equations in the matrix form \(\mathbf { C x } = \mathbf { a }\), where \(\mathbf { C }\) is a \(3 \times 3\) matrix, \(\mathbf { x }\) is an unknown vector, and \(\mathbf { a }\) is a fixed vector.
  3. By finding \(\mathbf { C } ^ { - 1 }\), determine the values of \(p , q\) and \(r\).
Pre-U Pre-U 9795/1 2010 June Q4
5 marks Standard +0.3
4
  1. Using the definitions of sinh and cosh in terms of exponentials, prove that $$\cosh A \cosh B + \sinh A \sinh B \equiv \cosh ( A + B )$$
  2. Solve the equation \(5 \cosh x + 3 \sinh x = 12\), giving your answers in the form \(\ln ( p \pm q \sqrt { 2 } )\) for rational numbers \(p\) and \(q\) to be determined.
Pre-U Pre-U 9795/1 2010 June Q5
8 marks Standard +0.8
5 A curve has equation \(y = \frac { x ^ { 2 } + 5 x - 6 } { x + 3 }\) for \(x \neq - 3\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 1\) at all points on the curve.
  2. Sketch the curve, justifying all significant features.
Pre-U Pre-U 9795/1 2010 June Q6
8 marks Challenging +1.2
6
  1. The set \(S\) consists of all \(2 \times 2\) matrices of the form \(\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)\), where \(n \in \mathbb { Z }\).
    1. Show that \(S\), under the operation of matrix multiplication, forms a group \(G\). [You may assume that matrix multiplication is associative.]
    2. State, giving a reason, whether \(G\) is abelian.
    3. The group \(H\) is the set \(\mathbb { Z }\) together with the operation of addition. Explain why \(G\) is isomorphic to \(H\).
    4. The plane transformation \(T\) is given by the matrix \(\left( \begin{array} { l l } 1 & n \\ 0 & 1 \end{array} \right)\), where \(n\) is a non-zero integer. Describe \(T\) fully.
Pre-U Pre-U 9795/1 2010 June Q7
9 marks Challenging +1.2
7 A curve \(C\) has polar equation \(r = 2 + \cos \theta\) for \(- \pi < \theta \leqslant \pi\).
  1. The point \(P\) on \(C\) corresponds to \(\theta = \alpha\), and the point \(Q\) on \(C\) is such that \(P O Q\) is a straight line, where \(O\) is the pole. Show that the length \(P Q\) is independent of \(\alpha\).
  2. Find, in an exact form, the area of the region enclosed by \(C\).
  3. Show that \(\left( x ^ { 2 } + y ^ { 2 } - x \right) ^ { 2 } = 4 \left( x ^ { 2 } + y ^ { 2 } \right)\) is a cartesian equation for \(C\). Identify the coordinates of the point which is included in this cartesian equation but is not on \(C\).
Pre-U Pre-U 9795/1 2010 June Q8
10 marks Challenging +1.8
8 For the differential equation \(t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 4 t \frac { \mathrm {~d} x } { \mathrm {~d} t } + \left( 6 - 4 t ^ { 2 } \right) x = 0\), use the substitution \(x = t ^ { 2 } u\) to find a differential equation involving \(t\) and \(u\) only. Hence solve the above differential equation, given that \(x = \mathrm { e } - 1\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 4 \mathrm { e }\) when \(t = 1\).
Pre-U Pre-U 9795/1 2010 June Q9
10 marks Challenging +1.2
9 Three non-collinear points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to the origin \(O\). The plane through \(A , B\) and \(C\) is denoted by \(\Pi\).
  1. (a) Prove that the area of triangle \(A B C\) is \(\frac { 1 } { 2 } | \mathbf { a } \times \mathbf { b } + \mathbf { b } \times \mathbf { c } + \mathbf { c } \times \mathbf { a } |\).
    (b) Describe the significance of the vector \(\mathbf { a } \times \mathbf { b } + \mathbf { b } \times \mathbf { c } + \mathbf { c } \times \mathbf { a }\) in relation to \(\Pi\).
  2. (a) In the case when \(\mathbf { a } = a \mathbf { i } , \mathbf { b } = b \mathbf { j }\) and \(\mathbf { c } = c \mathbf { k }\), where \(a , b\) and \(c\) are positive scalar constants, determine the equation of \(\Pi\) in the form r.n \(= d\), where the components of \(\mathbf { n }\) and the value of the scalar constant \(d\) are to be given in terms of \(a , b\) and \(c\).
    (b) Deduce the shortest distance from the origin \(O\) to \(\Pi\) in this case.
Pre-U Pre-U 9795/1 2010 June Q10
11 marks Challenging +1.2
10 One root of the equation \(z ^ { 5 } - 1 = 0\) is the complex number \(\omega = \mathrm { e } ^ { \frac { 2 } { 5 } \pi \mathrm { i } }\).
  1. Show that
    1. \(\quad \omega ^ { 5 } = 1\),
    2. \(\quad \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = - 1\),
    3. \(\quad \omega + \omega ^ { 4 } = 2 \cos \frac { 2 } { 5 } \pi\), and write down a similar expression for \(\omega ^ { 2 } + \omega ^ { 3 }\).
    4. Using these results, find the values of \(\cos \frac { 2 } { 5 } \pi + \cos \frac { 4 } { 5 } \pi\) and \(\cos \frac { 2 } { 5 } \pi \times \cos \frac { 4 } { 5 } \pi\), and deduce a quadratic equation, with integer coefficients, which has roots $$\cos \frac { 2 } { 5 } \pi \quad \text { and } \quad \cos \frac { 4 } { 5 } \pi$$
Pre-U Pre-U 9795/1 2010 June Q11
18 marks Challenging +1.8
11
  1. At all points \(( x , y )\) on the curve \(C , \frac { \mathrm {~d} y } { \mathrm {~d} x } + x y = 0\).
    1. Prove by induction that, for all integers \(n \geqslant 1\), $$\frac { \mathrm { d } ^ { n + 1 } y } { \mathrm {~d} x ^ { n + 1 } } + x \frac { \mathrm {~d} ^ { n } y } { \mathrm {~d} x ^ { n } } + n \frac { \mathrm {~d} ^ { n - 1 } y } { \mathrm {~d} x ^ { n - 1 } } = 0$$ where \(\frac { \mathrm { d } ^ { 0 } y } { \mathrm {~d} x ^ { 0 } } = y\).
    2. Given that \(y = 1\) when \(x = 0\), determine the Maclaurin expansion of \(y\) in ascending powers of \(x\), up to and including the term in \(x ^ { 6 }\).
    3. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } + x y = 0\) given that \(y = 1\) when \(x = 0\).
    4. Given that \(Z \sim \mathrm {~N} ( 0,1 )\), use your answers to parts (i) and (ii) to find an approximation, to 4 decimal places, to the probability \(\mathrm { P } ( Z \leqslant 1 )\).
      [0pt] [Note that the probability density function of the standard normal distribution is \(\mathrm { f } ( z ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - \frac { 1 } { 2 } z ^ { 2 } }\).]
Pre-U Pre-U 9795/1 2010 June Q12
22 marks Challenging +1.8
12
  1. Let \(I _ { n } = \int \frac { x ^ { n } } { \sqrt { x ^ { 2 } + 1 } } \mathrm {~d} x\), for integers \(n \geqslant 0\).
    By writing \(\frac { x ^ { n } } { \sqrt { x ^ { 2 } + 1 } }\) as \(x ^ { n - 1 } \times \frac { x } { \sqrt { x ^ { 2 } + 1 } }\), or otherwise, show that, for \(n \geqslant 2\), $$n I _ { n } = x ^ { n - 1 } \sqrt { x ^ { 2 } + 1 } - ( n - 1 ) I _ { n - 2 } .$$
  2. The diagram shows a sketch of the hyperbola \(H\) with equation \(\frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 16 } = 1\). \includegraphics[max width=\textwidth, alt={}, center]{32ed7cc8-3456-4cf0-952a-ee04eada1298-6_593_666_776_776}
    1. Find the coordinates of the points where \(H\) crosses the \(x\)-axis.
    2. The curve \(J\) has parametric equations \(x = 2 \cosh \theta , y = 4 \sinh \theta\), for \(\theta \geqslant 0\). Show that these parametric equations satisfy the cartesian equation of \(H\), and indicate on a copy of the above diagram which part of \(H\) is \(J\).
    3. The arc of the curve \(J\) between the points where \(x = 2\) and \(x = 34\) is rotated once completely about the \(x\)-axis to form a surface of revolution with area \(S\). Show that $$S = 16 \pi \int _ { \alpha } ^ { \beta } \sinh \theta \sqrt { 5 \cosh ^ { 2 } \theta - 1 } \mathrm {~d} \theta$$ for suitable constants \(\alpha\) and \(\beta\).
    4. Use the substitution \(u ^ { 2 } = 5 \cosh ^ { 2 } \theta - 1\) to show that $$S = \frac { 8 \pi } { \sqrt { 5 } } ( 644 \sqrt { 5 } - \ln ( 9 + 4 \sqrt { 5 } ) )$$
Pre-U Pre-U 9795/1 2012 June Q1
4 marks Moderate -0.8
1 Using any standard results given in the List of Formulae (MF20), show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } - r + 1 \right) = \frac { 1 } { 3 } n \left( n ^ { 2 } + 2 \right)$$ for all positive integers \(n\).
Pre-U Pre-U 9795/1 2012 June Q2
4 marks Standard +0.3
2 Find the area enclosed by the curve with polar equation \(r = \sin \theta + \cos \theta , 0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
Pre-U Pre-U 9795/1 2012 June Q3
3 marks Standard +0.8
3
  1. Given that \(y = \sqrt { \sinh x }\) for \(x \geqslant 0\), express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(y\) only.
  2. Find \(\int \frac { 2 t } { \sqrt { 1 + t ^ { 4 } } } \mathrm {~d} t\).
Pre-U Pre-U 9795/1 2012 June Q4
9 marks Standard +0.8
4 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).
  1. By considering a suitable quadratic equation in \(x\), find the set of possible values of \(y\) for points on \(C\).
  2. Deduce the coordinates of the turning points on \(C\).
Pre-U Pre-U 9795/1 2012 June Q5
6 marks Standard +0.8
5
  1. Write down the \(2 \times 2\) matrices which represent the following plane transformations:
    1. an anticlockwise rotation about the origin through an angle \(\alpha\);
    2. a reflection in the line \(y = x \tan \left( \frac { 1 } { 2 } \beta \right)\).
    3. A reflection in the \(x - y\) plane in the line \(y = x \tan \left( \frac { 1 } { 2 } \theta \right)\) is followed by a reflection in the line \(y = x \tan \left( \frac { 1 } { 2 } \phi \right)\). Show that the composition of these two reflections (in this order) is a rotation and describe this rotation fully.
Pre-U Pre-U 9795/1 2012 June Q6
7 marks Challenging +1.8
6 A group \(G\) has order 12.
  1. State, with a reason, the possible orders of the elements of \(G\). The identity element of \(G\) is \(e\), and \(x\) and \(y\) are distinct, non-identity elements of \(G\) satisfying the three conditions
    (1) \(\quad x\) has order 6 ,
    (2) \(x ^ { 3 } = y ^ { 2 }\),
    (3) \(x y x = y\).
  2. Prove that \(y x ^ { 2 } y = x\).
  3. Prove that \(G\) is not a cyclic group.
Pre-U Pre-U 9795/1 2012 June Q7
9 marks Challenging +1.2
7
  1. Use de Moivre's theorem to show that \(\tan 4 \theta = \frac { 4 t \left( 1 - t ^ { 2 } \right) } { 1 - 6 t ^ { 2 } + t ^ { 4 } }\), where \(t = \tan \theta\).
  2. Given that \(\theta\) is the acute angle such that \(\tan \theta = \frac { 1 } { 5 }\), express \(\tan 4 \theta\) as a rational number in its simplest form, and verify that $$\frac { 1 } { 4 } \pi + \tan ^ { - 1 } \left( \frac { 1 } { 239 } \right) = 4 \tan ^ { - 1 } \left( \frac { 1 } { 5 } \right)$$
Pre-U Pre-U 9795/1 2012 June Q8
11 marks Standard +0.8
8 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 \(\mathrm { f } ( 1.1 )\).
Pre-U Pre-U 9795/1 2012 June Q9
9 marks Challenging +1.2
9
  1. Show that the substitution \(u = \frac { 1 } { y ^ { 3 } }\) transforms the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } + y = 3 x y ^ { 4 }\) into $$\frac { \mathrm { d } u } { \mathrm {~d} x } - 3 u = - 9 x$$
  2. Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } + y = 3 x y ^ { 4 }\), given that \(y = \frac { 1 } { 2 }\) when \(x = 0\). Give your answer in the form \(y ^ { 3 } = \mathrm { f } ( x )\).
Pre-U Pre-U 9795/1 2012 June Q10
2 marks Standard +0.3
10 The line \(L\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { l } 3 \\ 4 \\ 6 \end{array} \right)\) and the plane \(\Pi\) has equation \(\mathbf { r } \cdot \left( \begin{array} { r } 2 \\ - 6 \\ 3 \end{array} \right) = k\).
  1. Given that \(L\) lies in \(\Pi\), determine the value of \(k\).
  2. Find the coordinates of the point, \(Q\), in \(\Pi\) which is closest to \(P ( 10,2 , - 43 )\). Deduce the shortest distance from \(P\) to \(\Pi\).
  3. Find, in the form \(a x + b y + c z = d\), where \(a , b , c\) and \(d\) are integers, an equation for the plane which contains both \(L\) and \(P\).
Pre-U Pre-U 9795/1 2012 June Q11
11 marks Standard +0.8
11 The complex number \(w = ( \sqrt { 3 } - 1 ) + \mathrm { i } ( \sqrt { 3 } + 1 )\).
  1. Determine, showing full working, the exact values of \(| w |\) and \(\arg w\).
    [0pt] [You may use the result that \(\tan \left( \frac { 5 } { 12 } \pi \right) = 2 + \sqrt { 3 }\).]
  2. (a) Find, in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), the three roots, \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\), of the equation \(z ^ { 3 } = w\).
    (b) Determine \(z _ { 1 } z _ { 2 } z _ { 3 }\) in the form \(a + \mathrm { i } b\).
    (c) Mark the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) on a sketch of the Argand diagram. Show that they form an equilateral triangle, \(\Delta _ { 1 }\), and determine the side-length of \(\Delta _ { 1 }\).
    (d) The points representing \(k z _ { 1 } , k z _ { 2 }\) and \(k z _ { 3 }\) form \(\Delta _ { 2 }\), an equilateral triangle which is congruent to \(\Delta _ { 1 }\), and one of whose vertices lies on the positive real axis. Write down a suitable value for the complex constant \(k\).
Pre-U Pre-U 9795/1 2012 June Q12
15 marks Challenging +1.8
12
  1. Let \(I _ { n } = \int _ { 0 } ^ { 3 } x ^ { n } \sqrt { 16 + x ^ { 2 } } \mathrm {~d} x\), 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 2012 June Q13
6 marks Challenging +1.8
13 Define the repunit number, \(R _ { n }\), to be the positive integer which consists of a string of \(n 1\) 's. Thus, $$R _ { 1 } = 1 , \quad R _ { 2 } = 11 , \quad R _ { 3 } = 111 , \quad \ldots , \quad R _ { 7 } = 1111111 , \quad \ldots , \text { etc. }$$ Use induction to prove that, for all integers \(n \geqslant 5\), the number $$13579 \times R _ { n }$$ contains a string of ( \(n - 4\) ) consecutive 7's.