Questions — Pre-U (885 questions)

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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 9794/2 2016 Specimen Q1
9 marks Easy -1.3
1
  1. Express each of the following as a single logarithm.
    1. \(\log _ { a } 5 + \log _ { a } 3\)
    2. \(5 \log _ { b } 2 - 3 \log _ { b } 4\)
  2. Express \(\left( 9 a ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }\) as an algebraic fraction in its simplest form.
  3. Show that \(\frac { 3 \sqrt { 3 } - 1 } { 2 \sqrt { 3 } - 3 } = \frac { 15 + 7 \sqrt { 3 } } { 3 }\).
Pre-U Pre-U 9794/2 2016 Specimen Q2
5 marks Moderate -0.5
2 \includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-2_399_933_968_561} The diagram shows a triangle \(A B C\) in which angle \(C = 30 ^ { \circ } , B C = x \mathrm {~cm}\) and \(A C = ( x + 2 ) \mathrm { cm }\). Given that the area of triangle \(A B C\) is \(12 \mathrm {~cm} ^ { 2 }\), calculate the value of \(x\).
Pre-U Pre-U 9794/2 2016 Specimen Q3
5 marks Easy -1.2
3
  1. The points \(A\) and \(B\) have coordinates ( \(- 4,4\) ) and ( 8,1 ) respectively. Find the equation of the line \(A B\). Give your answer in the form \(y = m x + c\).
  2. Determine, with a reason, whether the line \(y = 7 - 4 x\) is perpendicular to the line \(A B\).
Pre-U Pre-U 9794/2 2016 Specimen Q4
7 marks Moderate -0.3
4
  1. Show that \(2 x ^ { 2 } - 10 x - 3\) may be expressed in the form \(a ( x + b ) ^ { 2 } + c\) where \(a , b\) and \(c\) are real numbers to be found. Hence write down the co-ordinates of the minimum point on the curve.
  2. Solve the equation \(4 x ^ { 4 } - 13 x ^ { 2 } + 9 = 0\).
Pre-U Pre-U 9794/2 2016 Specimen Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-3_570_736_292_667} The diagram shows a sector of a circle, \(O M N\). The angle \(M O N\) is \(2 x\) radians, the radius of the circle is \(r\) and \(O\) is the centre.
  1. Find expressions, in terms of \(r\) and \(x\), for the area, \(A\), and the perimeter, \(P\), of the sector.
  2. Given that \(P = 20\), show that \(A = \frac { 100 x } { ( 1 + x ) ^ { 2 } }\).
  3. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\), and hence find the value of \(x\) for which the area of the sector is a maximum.
Pre-U Pre-U 9794/2 2016 Specimen Q6
8 marks Moderate -0.3
6 Diane is given an injection that combines two drugs, Antiflu and Coldcure. At time \(t\) hours after the injection, the concentration of Antiflu in Diane's bloodstream is \(3 \mathrm { e } ^ { - 0.02 t }\) units and the concentration of Coldcure is \(5 \mathrm { e } ^ { - 0.07 t }\) units. Each drug becomes ineffective when its concentration falls below 1 unit.
  1. Show that Coldcure becomes ineffective before Antiflu.
  2. Sketch, on the same diagram, the graphs of concentration against time for each drug.
  3. 20 hours after the first injection, Diane is given a second injection. Determine the concentration of Coldcure 10 hours later.
Pre-U Pre-U 9794/2 2016 Specimen Q7
6 marks Standard +0.3
7 Solve the differential equation \(x ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = \sec y\) given that \(y = \frac { \pi } { 6 }\) when \(x = 4\) giving your answer in the form \(y = \mathrm { f } ( x )\).
Pre-U Pre-U 9794/2 2016 Specimen Q8
8 marks Moderate -0.3
8 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } - 5 t , \quad y = \mathrm { e } ^ { 2 t } - 3 t .$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the equation of the tangent to the curve at the point when \(t = 0\), giving your answer in the form \(a y + b x + c = 0\) where \(a , b\) and \(c\) are integers.