Questions Pre-U 9794/1 (194 questions)

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Pre-U Pre-U 9794/1 2016 June Q1
3 marks Easy -1.3
1 Find the equation of the line perpendicular to the line \(y = 5 x + 6\) which passes through the point \(( 1,11 )\). Give your answer in the form \(y = m x + c\).
Pre-U Pre-U 9794/1 2016 June Q2
2 marks Easy -1.8
2 Without using a calculator, simplify the following, giving each answer in the form \(a \sqrt { 5 }\) where \(a\) is an integer. Show all your working.
  1. \(4 \sqrt { 10 } \times \sqrt { 2 }\)
  2. \(\sqrt { 500 } + \sqrt { 125 }\)
Pre-U Pre-U 9794/1 2016 June Q3
4 marks Easy -1.2
3 Solve \(3 x ^ { 2 } + 11 x - 20 > 0\).
Pre-U Pre-U 9794/1 2016 June Q4
3 marks Easy -1.3
4 A sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), is defined by \(u _ { n } = 3 n + 5\).
  1. State the values of \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
  2. Find the value of \(n\) such that \(u _ { n } = 254\).
  3. Evaluate \(\sum _ { n = 1 } ^ { 500 } u _ { n }\).
Pre-U Pre-U 9794/1 2016 June Q5
4 marks Moderate -0.8
5 The circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x - k = 0\) has radius 5 . Find the coordinates of the centre of the circle and the value of \(k\).
Pre-U Pre-U 9794/1 2016 June Q6
9 marks Moderate -0.3
6
  1. Find the coordinates of the stationary points of the curve with equation $$y = 3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 }$$ and determine their nature.
  2. Sketch the graph of \(y = 3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 }\) and hence state the set of values of \(k\) for which the equation \(3 x ^ { 4 } - 20 x ^ { 3 } + 36 x ^ { 2 } = k\) has exactly four distinct real roots.
Pre-U Pre-U 9794/1 2016 June Q7
8 marks Moderate -0.8
7 The functions f and g are defined for all real numbers by $$\mathrm { f } ( x ) = x ^ { 2 } + 2 \quad \text { and } \quad \mathrm { g } ( x ) = 4 x + 3$$
  1. State the range of each of the functions f and g .
  2. Find the values of \(x\) for which \(\mathrm { fg } ( x ) = \mathrm { gf } ( x )\).
  3. The function h , given by \(\mathrm { h } ( x ) = x ^ { 2 } + 2\), has the same range as f but is such that \(\mathrm { h } ^ { - 1 } ( x )\) exists. State a possible domain for h and find an expression for \(\mathrm { h } ^ { - 1 } ( x )\).
Pre-U Pre-U 9794/1 2016 June Q8
4 marks Moderate -0.3
8
  1. Evaluate exactly \(\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { - x } \mathrm {~d} x\).
  2. Find \(\int \frac { x - 1 } { x + 1 } \mathrm {~d} x\).
Pre-U Pre-U 9794/1 2016 June Q9
6 marks Moderate -0.3
9 Determine whether the lines whose equations are $$\mathbf { r } = ( 4 + 2 \mu ) \mathbf { i } + ( 7 + 3 \mu ) \mathbf { j } + ( 3 + 7 \mu ) \mathbf { k } \quad \text { and } \quad \mathbf { r } = ( 35 - 5 \lambda ) \mathbf { i } + ( 6 + 2 \lambda ) \mathbf { j } + ( 14 + 3 \lambda ) \mathbf { k }$$ intersect, are parallel or are skew.
Pre-U Pre-U 9794/1 2016 June Q10
6 marks Standard +0.3
10 The diagram shows the curve with equation $$x = ( y - 4 ) \ln ( 2 y + 3 ) .$$ The curve crosses the \(y\)-axis at \(A\) and \(B\). \includegraphics[max width=\textwidth, alt={}, center]{afc8561d-94ae-42c0-bc6c-e9b091938368-3_588_780_1087_680}
  1. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
  2. Find the exact gradient of the curve at each of the points \(A\) and \(B\).
Pre-U Pre-U 9794/1 2016 June Q11
5 marks Challenging +1.2
11
  1. Prove that $$\sin ^ { 2 } \left( \theta + \frac { 1 } { 3 } \pi \right) + \frac { 1 } { 2 } \sin ^ { 2 } \theta - \frac { 3 } { 4 } = \frac { 1 } { 4 } \sqrt { 3 } \sin 2 \theta .$$
  2. Hence solve the equation $$2 \sin ^ { 2 } \left( \theta + \frac { 1 } { 3 } \pi \right) + \sin ^ { 2 } \theta = 1 \text { for } - \pi \leqslant \theta \leqslant \pi .$$
Pre-U Pre-U 9794/1 2016 June Q12
10 marks Standard +0.3
12 A patch of disease on a leaf is being chemically treated. At time \(t\) days after treatment starts, its length is \(x \mathrm {~cm}\) and the rate of decrease of its length is observed to be inversely proportional to the square root of its length. At time \(t = 3 , x = 4\) and, at this instant, the length is decreasing at 0.05 cm per day. Write down and solve a differential equation to model this situation. Hence find the time it takes for the length to decrease to 0.01 cm .
[0pt] [10]
Pre-U Pre-U 9794/1 2016 Specimen Q1
4 marks Easy -1.8
1 A circle has equation \(( x - 4 ) ^ { 2 } + ( y + 7 ) ^ { 2 } = 64\).
  1. Write down the coordinates of the centre and the radius of the circle. Two points, \(A\) and \(B\), lie on the circle and have coordinates \(( 4,1 )\) and \(( 12 , - 7 )\) respectively.
  2. Find the coordinates of the midpoint of the chord \(A B\).
Pre-U Pre-U 9794/1 2016 Specimen Q2
6 marks Moderate -0.8
2 The equation of a curve is \(y = x ^ { 3 } - 2 x ^ { 2 } - 4 x + 3\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the coordinates of the stationary points on the curve.
Pre-U Pre-U 9794/1 2016 Specimen Q3
7 marks Easy -1.2
3 Let \(\mathrm { f } ( x ) = x ^ { 2 }\) and \(\mathrm { g } ( x ) = 7 x - 2\) for all real values of \(x\).
  1. Give a reason why f has no inverse function.
  2. Write down an expression for \(\operatorname { gf } ( x )\).
  3. Find \(\mathrm { g } ^ { - 1 } ( x )\).
  4. Explain the relationship between the graph of \(y = \mathrm { g } ( x )\) and \(y = \mathrm { g } ^ { - 1 } ( x )\).
Pre-U Pre-U 9794/1 2016 Specimen Q4
6 marks Moderate -0.8
4
  1. Show that \(x = 2\) is a root of the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
  2. Hence solve the equation \(2 x ^ { 3 } - x ^ { 2 } - 15 x + 18 = 0\).
Pre-U Pre-U 9794/1 2016 Specimen Q5
4 marks Moderate -0.3
5 The coefficient of \(x ^ { 3 }\) in the expansion of \(( 2 + a x ) ^ { 5 }\) is 10 times the coefficient of \(x ^ { 2 }\) in \(\left( 1 + \frac { a x } { 3 } \right) ^ { 4 }\). Find \(a\).
Pre-U Pre-U 9794/1 2016 Specimen Q6
6 marks Moderate -0.5
6 Solve the simultaneous equations $$x + y = 1 , \quad x ^ { 2 } - 2 x y + y ^ { 2 } = 9$$
Pre-U Pre-U 9794/1 2016 Specimen Q7
9 marks Moderate -0.3
7
  1. Express \(\frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) }\) in the form \(\frac { A } { 2 x - 1 } + \frac { B } { x + 1 }\) where \(A\) and \(B\) are constants.
  2. Hence show that \(\int _ { 2 } ^ { 5 } \frac { 8 x - 1 } { ( 2 x - 1 ) ( x + 1 ) } \mathrm { d } x = \ln 24\).
Pre-U Pre-U 9794/1 2016 Specimen Q8
4 marks Moderate -0.3
8 Given that the equation \(x ^ { 3 } + 2 x - 7 = 0\) has a root between \(x = 1\) and \(x = 2\), use the Newton-Raphson formula with \(x _ { \mathrm { o } } = 1\) to find this root correct to 3 decimal places.
Pre-U Pre-U 9794/1 2016 Specimen Q9
6 marks Easy -1.3
9 The complex number \(3 - 4 \mathrm { i }\) is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find
  1. \(2 z + z ^ { * }\),
  2. \(\frac { 5 } { z }\).
  3. Show \(z\) and \(z ^ { * }\) on an Argand diagram.
Pre-U Pre-U 9794/1 2016 Specimen Q10
8 marks Standard +0.3
10
  1. Prove that \(\cot \theta + \frac { \sin \theta } { 1 + \cos \theta } = \operatorname { cosec } \theta\).
  2. Hence solve the equation \(\cot \left( \theta + \frac { \pi } { 4 } \right) + \frac { \sin \left( \theta + \frac { \pi } { 4 } \right) } { 1 + \cos \left( \theta + \frac { \pi } { 4 } \right) } = \frac { 5 } { 2 }\) for \(0 \leqslant \theta \leqslant 2 \pi\).
Pre-U Pre-U 9794/1 2016 Specimen Q11
9 marks Standard +0.8
11 An arithmetic progression has first term \(a\) and common difference \(d\). The first, ninth and fourteenth terms are, respectively, the first three terms of a geometric progression with common ratio \(r\), where \(r \neq 1\).
  1. Find \(d\) in terms of \(a\) and show that \(r = \frac { 5 } { 8 }\).
  2. Find the sum to infinity of the geometric progression in terms of \(a\).
Pre-U Pre-U 9794/1 2016 Specimen Q12
11 marks Standard +0.8
12
  1. Use integration by parts to show that \(\int \ln x \mathrm {~d} x = x \ln x - x + c\).
  2. Find
    1. \(\quad \int ( \ln x ) ^ { 2 } \mathrm {~d} x\),
    2. \(\quad \int \frac { \ln ( \ln x ) } { x } \mathrm {~d} x\).
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.