Questions — Pre-U (885 questions)

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Pre-U Pre-U 9794/3 2012 June Q10
10 marks Challenging +1.2
10 \includegraphics[max width=\textwidth, alt={}, center]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-4_81_949_1283_598} Three particles \(A , B\) and \(C\), having masses \(1 \mathrm {~kg} , 2 \mathrm {~kg}\) and 5 kg , respectively, are placed 1 metre apart in a straight line on a smooth horizontal plane (see diagram). The particles \(B\) and \(C\) are initially at rest and \(A\) is moving towards \(B\) with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between each pair of particles is 0.5 .
  1. Find the velocity of \(B\) immediately after the first impact and show that \(A\) comes to rest.
  2. Show that \(B\) reverses direction after an impact with \(C\).
  3. Find the distance between \(B\) and \(C\) at the instant that \(B\) collides with \(A\) for the second time.
Pre-U Pre-U 9794/3 2012 June Q11
13 marks Standard +0.3
11 A particle \(P\) of mass 2 kg can move along a line of greatest slope on the smooth surface of a wedge which is fixed to the ground. The sloping face \(O A\) of the wedge has length 10 metres and is inclined at \(30 ^ { \circ }\) to the horizontal (see Fig. 1). \(P\) is fired up the slope from the lowest point \(O\), with an initial speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-5_295_1529_484_310} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Find the time taken for \(P\) to reach \(A\) and show that the speed of \(P\) at \(A\) is \(10 \sqrt { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After \(P\) has reached \(A\) it becomes a projectile (see Fig. 2). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-5_424_1533_1123_306} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
  2. Find the total horizontal distance travelled by \(P\) from \(O\) when it hits the ground.
Pre-U Pre-U 9794/1 2012 Specimen Q1
2 marks Easy -1.8
1 Write down the coordinates of the centre and the radius of the circle with equation $$( x + 5 ) ^ { 2 } + ( y - 3 ) ^ { 2 } = 36 .$$
Pre-U Pre-U 9794/1 2012 Specimen Q2
5 marks Moderate -0.8
2
  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 2012 Specimen Q3
6 marks Easy -1.2
3
  1. In an arithmetic progression, the first term is 7 and the sum of the first 40 terms is 4960. Find the common difference.
  2. A geometric progression has first term 14 and common ratio 0.3. Find the sum to infinity.
Pre-U Pre-U 9794/1 2012 Specimen Q4
7 marks Moderate -0.3
4 A sector \(A O B\) of a circle has radius \(r \mathrm {~cm}\) and the angle \(A O B\) is \(\theta\) radians. The perimeter of the sector is 40 cm and its area is \(100 \mathrm {~cm} ^ { 2 }\).
  1. Write down equations for the perimeter and area of the sector in terms of \(r\) and \(\theta\).
  2. Use your equations to show that \(r ^ { 2 } - 20 r + 100 = 0\) and hence find the value of \(r\) and of \(\theta\).
Pre-U Pre-U 9794/1 2012 Specimen Q5
8 marks Moderate -0.3
5
  1. Find \(\int \left( \frac { 1 } { x - 2 } - \frac { 2 } { 2 x + 3 } \right) \mathrm { d } x\) giving your answer in its simplest form.
  2. Use integration by parts to find \(\int x ^ { 2 } \ln x \mathrm {~d} x\).
Pre-U Pre-U 9794/1 2012 Specimen Q6
7 marks Moderate -0.3
6
  1. Find and simplify the first four terms in the expansion of \(( 1 - 2 x ) ^ { 9 }\) in ascending powers of \(x\).
  2. In the expansion of $$( 2 + a x ) ( 1 - 2 x ) ^ { 9 }$$ the coefficient of \(x ^ { 2 }\) is 66 . Find the value of \(a\).
Pre-U Pre-U 9794/1 2012 Specimen Q7
4 marks Standard +0.3
7 Given that the equation \(x = 2 - \frac { 1 } { ( x + 1 ) ^ { 2 } }\) has a root between \(x = 1\) and \(x = 2\), use the Newton-Raphson formula with \(x _ { 0 } = 2\) to find this root correct to 3 decimal places.
Pre-U Pre-U 9794/1 2012 Specimen Q8
7 marks Moderate -0.8
8 A curve has equation \(y = 2 x ^ { 3 } - 5 x ^ { 2 } - 4 x + 1\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. Hence find the \(x\)-coordinates of the stationary points of the curve.
  3. By using the second derivative, determine whether each of the stationary points is a maximum or a minimum.
Pre-U Pre-U 9794/1 2012 Specimen Q9
10 marks Moderate -0.3
9
  1. On the same axes, sketch the curves \(y = 3 + 2 x - x ^ { 2 }\) and \(y = x + 1\).
  2. Find the exact area of the region contained between the curves \(y = 3 + 2 x - x ^ { 2 }\) and \(y = x + 1\).
Pre-U Pre-U 9794/1 2012 Specimen Q10
6 marks Moderate -0.3
10 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) relative to an origin \(O\), where \(\mathbf { a } = 5 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = - 7 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\).
  1. Find the length of \(A B\).
  2. Use a scalar product to find angle \(O A B\).
Pre-U Pre-U 9794/1 2012 Specimen Q11
6 marks Standard +0.3
11 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 9794/1 2012 Specimen Q12
11 marks Challenging +1.2
12 Calculate the maximum and minimum values of \(\frac { 1 } { 2 + \cos \theta + \sqrt { 2 } \sin \theta }\) and the smallest positive values of \(\theta\) for which they occur.
Pre-U Pre-U 9794/2 2012 Specimen Q1
7 marks Easy -1.8
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.
Pre-U Pre-U 9794/2 2012 Specimen Q2
5 marks Moderate -0.8
2 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 2012 Specimen Q3
6 marks Standard +0.3
3 Solve the simultaneous equations $$x + y = 1 , \quad x ^ { 2 } - x y + y ^ { 2 } = 7 .$$
Pre-U Pre-U 9794/2 2012 Specimen Q4
5 marks Moderate -0.8
4 Find
  1. \(\quad \int ( 2 x + 3 ) ^ { 4 } \mathrm {~d} x\)
  2. \(\quad \int \left( 1 + \tan ^ { 2 } 2 x \right) \mathrm { d } x\)
Pre-U Pre-U 9794/2 2012 Specimen Q5
5 marks Standard +0.3
5 When \(x ^ { 4 } - 4 x ^ { 3 } + 5 x ^ { 2 } + x + a\) is divided by \(x ^ { 2 } - x + 1\), the quotient is \(x ^ { 2 } + b x + 1\) and the remainder is \(c x - 3\). Find the values of the constants \(a , b\) and \(c\).
Pre-U Pre-U 9794/2 2012 Specimen Q6
8 marks Moderate -0.8
6 The complex number \(5 - 3 \mathrm { i }\) is denoted by \(z\). Giving your answers in the form \(x + \mathrm { i } y\), and showing clearly how you obtain them, find
  1. \(\quad 6 z - z ^ { * }\),
  2. \(\quad ( z - \mathrm { i } ) ^ { 2 }\),
  3. \(\frac { 5 } { z }\).
Pre-U Pre-U 9794/2 2012 Specimen Q7
5 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{f8b66d63-96ce-43d2-bd28-c048070feac3-3_456_606_182_735} The diagram shows the region \(R\) bounded by the curve \(y = \frac { 1 } { \sqrt { 5 x + 3 } }\) and the lines \(x = 0\), \(x = 3\) and \(y = 0\). Find the exact volume of the solid formed when the region \(R\) is rotated completely about the \(x\)-axis, simplifying your answer.
Pre-U Pre-U 9794/2 2012 Specimen Q8
9 marks Moderate -0.3
8
  1. Express \(\frac { 3 x + 2 } { ( x - 2 ) ^ { 2 } }\) in the form \(\frac { A } { x - 2 } + \frac { B } { ( x - 2 ) ^ { 2 } }\) where \(A\) and \(B\) are constants.
  2. Hence find the exact value of \(\int _ { 6 } ^ { 10 } \frac { 3 x + 2 } { ( x - 2 ) ^ { 2 } } \mathrm {~d} x\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are integers.
Pre-U Pre-U 9794/2 2012 Specimen Q9
8 marks Standard +0.3
9 The parametric equations of a curve are $$x = \mathrm { e } ^ { 2 t } - 5 t , \quad y = \mathrm { e } ^ { 2 t } - 2 t$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\).
  2. Find the exact value of \(t\) at the point on the curve where the gradient is 2 .
Pre-U Pre-U 9794/2 2012 Specimen Q10
7 marks Standard +0.3
10 Lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have vector equations $$\begin{aligned} & L _ { 1 } = ( 4 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) + s ( 6 \mathbf { i } + 9 \mathbf { j } - 3 \mathbf { k } ) , \\ & L _ { 2 } = ( 2 \mathbf { i } + 3 \mathbf { j } ) + t ( - 3 \mathbf { i } - 8 \mathbf { j } + 6 \mathbf { k } ) , \\ & L _ { 3 } = ( 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } ) + u ( - 2 \mathbf { i } + c \mathbf { j } + \mathbf { k } ) . \end{aligned}$$ In each of the following cases, find the value of \(c\).
  1. \(\quad L _ { 1 }\) and \(L _ { 3 }\) are parallel.
  2. \(\quad L _ { 2 }\) and \(L _ { 3 }\) intersect.
Pre-U Pre-U 9794/2 2012 Specimen Q11
16 marks Standard +0.3
11 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.
  1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\). \includegraphics[max width=\textwidth, alt={}, center]{f8b66d63-96ce-43d2-bd28-c048070feac3-4_631_901_532_571} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled C has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed.
  2. In the first model the equation is $$y = \mathrm { e } ^ { - x } \cos 12 x$$ Show that this model has a maximum point close to \(A\) and a minimum point close to \(B\), and state the coordinates of these maximum and minimum points and also the \(y\) value when \(x = 0.3\).
  3. In an alternative model the equation is $$y = f \cos ( \lambda x ) + g$$ where the constants \(f , \lambda\) and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). Find suitable values for \(f , \lambda\) and \(g\).
  4. Using the alternative model, state the value of \(y\) when \(x = 0.3\) and hence comment on how accurate each model is in fitting the three given points.