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Pre-U Pre-U 9795/2 2014 June Q7
8 marks Challenging +1.2
\includegraphics{figure_7} A light inextensible string of length 8 m is threaded through a smooth fixed ring, \(R\), and carries a particle at each end. One particle, \(P\), of mass 0.5 kg is at rest at a distance 3 m below \(R\). The other particle, \(Q\), is rotating in a horizontal circle whose centre coincides with the position of \(P\) (see diagram). Find the angular speed and the mass of \(Q\). [8]
Pre-U Pre-U 9795/2 2014 June Q8
9 marks Challenging +1.8
\includegraphics{figure_8} A smooth sphere with centre \(A\) and of mass 2 kg, moving at 13 m s\(^{-1}\) on a smooth horizontal plane, strikes a smooth sphere with centre \(B\) and of mass 3 kg moving at 5 m s\(^{-1}\) on the same smooth horizontal plane. The spheres have equal radii. The directions of motion immediately before impact are at angles \(\tan^{-1}\left(\frac{2}{13}\right)\) to \(\overrightarrow{AB}\) and \(\tan^{-1}\left(\frac{4}{3}\right)\) to \(\overrightarrow{BA}\) respectively (see diagram). Given that the coefficient of restitution is \(\frac{2}{3}\), find the speeds of the spheres after impact. [9]
Pre-U Pre-U 9795/2 2014 June Q9
11 marks Challenging +1.2
An engine is travelling along a straight horizontal track against negligible resistances. In travelling a distance of 750 m its speed increases from 5 m s\(^{-1}\) to 15 m s\(^{-1}\). Find the time taken if the engine was
  1. exerting a constant tractive force, [2]
  2. working at constant power. [9]
Pre-U Pre-U 9795/2 2014 June Q10
12 marks Standard +0.3
One end of a light spring of length 0.5 m is attached to a fixed point \(F\). A particle \(P\) of mass 2.5 kg is attached to the other end of the spring and hangs in equilibrium 0.55 m below \(F\). Another particle \(Q\), of mass 1.5 kg, is attached to \(P\), without moving it, and both particles are then released.
  1. Show that the modulus of elasticity of the spring is 250 N. [2]
  2. Prove that the motion is simple harmonic. [4]
  3. Find
    1. the amplitude of the motion, [1]
    2. the greatest speed of the particles, [1]
    3. the period of the motion, [1]
    4. the time taken for the spring to be extended by 0.1 m for the first time. [3]
Pre-U Pre-U 9795/2 2014 June Q11
10 marks Challenging +1.8
It is given that the trajectory of a projectile which is launched with speed \(V\) at an angle \(\alpha\) above the horizontal has equation $$y = x\tan\alpha - \frac{gx^2}{2V^2}(1 + \tan^2\alpha),$$ where the point of projection is the origin, and the \(x\)- and \(y\)-axes are horizontal and vertically upwards respectively.
  1. Express the above equation as a quadratic equation in \(\tan\alpha\) and deduce that the boundary of all accessible points for this projectile has equation $$y = \frac{1}{2gV^2}(V^4 - g^2x^2).$$ [4]
  2. A stone is thrown with speed \(\sqrt{gh}\) from the top of a vertical tower, of height \(h\), which stands on horizontal ground. Find
    1. the maximum distance, from the foot of the tower, at which the stone can land, [3]
    2. the angle of elevation at which the stone must be thrown to achieve this maximum distance. [3]
Pre-U Pre-U 9795/2 2014 June Q12
10 marks Challenging +1.2
A cyclist, when travelling due west at 15 km h\(^{-1}\), finds that the wind appears to be blowing from a bearing of 150°. When the cyclist is travelling due west at 10 km h\(^{-1}\), the wind appears to be blowing from a bearing of 135°. Find the velocity of the wind. [10]
Pre-U Pre-U 9794/3 2014 June Q1
5 marks Easy -1.8
The masses, in kilograms, of 100 chickens on sale in a large supermarket were recorded as follows.
Mass (\(x\) kg)\(1.6 \leq x < 1.8\)\(1.8 \leq x < 2.0\)\(2.0 \leq x < 2.2\)\(2.2 \leq x < 2.4\)\(2.4 \leq x < 2.6\)
Number of chickens1627281811
Calculate estimates of the mean and standard deviation of the masses of these chickens. [5]
Pre-U Pre-U 9794/3 2014 June Q7
5 marks Moderate -0.8
A stone is projected vertically upwards from ground level at a speed of \(30 \mathrm{~m} \mathrm{~s}^{-1}\). It is assumed that there is no wind or air resistance. Find the maximum height it reaches and the total time it takes from its projection to its return to ground level. [5]
Pre-U Pre-U 9794/3 2014 June Q10
10 marks Moderate -0.3
A particle \(P\) is free to move along a straight line \(Ox\). It starts from rest at \(O\) and after \(t\) seconds its acceleration \(a \mathrm{~m} \mathrm{~s}^{-2}\) is given by \(a = 12 - 6t\).
  1. Find an expression in terms of \(t\) for its velocity \(v \mathrm{~m} \mathrm{~s}^{-1}\). Hence find the velocity of \(P\) when \(t = 4\). [4]
  2. Find the displacement of \(P\) from \(O\) when \(t = 4\). [3]
  3. Find the velocity of \(P\) when it returns to \(O\). [3]
Pre-U Pre-U 9794/3 2014 June Q4
6 marks Moderate -0.8
In a certain country 40\% of the population have brown eyes. A random sample of 20 people is chosen from that population.
  1. Find the expected number of people in the sample who have brown eyes. [1]
  2. Find the probability that there are exactly 8 people with brown eyes in the sample. [3]
  3. Find the probability that there are at least 8 people with brown eyes in the sample. [2]
Pre-U Pre-U 9794/3 2014 June Q6
11 marks Standard +0.3
A machine is being used to manufacture ball bearings. The diameters of the ball bearings are normally distributed with mean 8.3 mm and standard deviation 0.20 mm.
  1. Find the probability that the diameter of a randomly chosen ball bearing lies between 8.1 mm and 8.5 mm. [5]
  2. Following an overhaul of the machine, it is now found that the diameters of 88\% of ball bearings are less than 8.5 mm while 10\% are less than 8.1 mm. Estimate the new mean and standard deviation of the diameters. [6]
Pre-U Pre-U 9794/3 2014 June Q7
5 marks Moderate -0.8
A stone is projected vertically upwards from ground level at a speed of \(30 \mathrm{m} \mathrm{s}^{-1}\). It is assumed that there is no wind or air resistance. Find the maximum height it reaches and the total time it takes from its projection to its return to ground level. [5]
Pre-U Pre-U 9794/3 2014 June Q10
10 marks Moderate -0.8
A particle \(P\) is free to move along a straight line \(Ox\). It starts from rest at \(O\) and after \(t\) seconds its acceleration \(a \mathrm{m} \mathrm{s}^{-2}\) is given by \(a = 12 - 6t\).
  1. Find an expression in terms of \(t\) for its velocity \(v \mathrm{m} \mathrm{s}^{-1}\). Hence find the velocity of \(P\) when \(t = 4\). [4]
  2. Find the displacement of \(P\) from \(O\) when \(t = 4\). [3]
  3. Find the velocity of \(P\) when it returns to \(O\). [3]
Pre-U Pre-U 9795/1 2015 June Q1
3 marks Moderate -0.5
Determine the volume of tetrahedron \(OABC\), where \(O\) is the origin and \(A\), \(B\) and \(C\) are, respectively, the points \((2, 3, -2)\), \((2, 0, 4)\) and \((6, 1, 7)\). [3]
Pre-U Pre-U 9795/1 2015 June Q2
3 marks Standard +0.8
The Taylor series expansion, about \(x = 1\), of the function \(y\) is $$y = 1 + \sum_{n=1}^{\infty} \frac{(-2)^{n-1}(x-1)^n}{1 \times 3 \times 5 \times \ldots \times (2n-1)}.$$ Find the value of \(\frac{\text{d}^4 y}{\text{d}x^4}\) when \(x = 1\). [3]
Pre-U Pre-U 9795/1 2015 June Q3
6 marks Challenging +1.2
\(\mathbf{M}\) is the matrix \(\begin{pmatrix} 1 & -2 & 2 \\ 2 & -1 & 2 \\ 2 & -2 & 3 \end{pmatrix}\). Use induction to prove that, for all positive integers \(n\), $$\mathbf{M}^n \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 2n + 1 \\ 2n^2 + 2n \\ 2n^2 + 2n + 1 \end{pmatrix}.$$ [6]
Pre-U Pre-U 9795/1 2015 June Q4
7 marks Challenging +1.2
A curve has polar equation \(r = \sin \frac{1}{4}\theta\) for \(0 \leqslant \theta < 2\pi\).
  1. Sketch the curve. [3]
  2. Determine the area of the region enclosed by the curve. [4]
Pre-U Pre-U 9795/1 2015 June Q5
11 marks Standard +0.8
A curve has equation \(y = \frac{2x^2 + 5x - 25}{x - 3}\).
  1. Determine the equations of the asymptotes. [3]
  2. Find the coordinates of the turning points. [5]
  3. Sketch the curve. [3]
Pre-U Pre-U 9795/1 2015 June Q6
9 marks Challenging +1.2
  1. Given the complex number \(z = \cos \theta + \text{i} \sin \theta\), show that \(z^n + \frac{1}{z^n} = 2 \cos n\theta\). [1]
  2. Deduce the identity \(16 \cos^5 \theta \equiv \cos 5\theta + 5 \cos 3\theta + 10 \cos \theta\). [4]
  3. For \(0 < \theta < 2\pi\), solve the equation \(\cos 5\theta + 5 \cos 3\theta + 9 \cos \theta = 0\). [4]
Pre-U Pre-U 9795/1 2015 June Q7
7 marks Standard +0.8
  1. On an Argand diagram, shade the region whose points satisfy $$|z - 20 + 15\text{i}| \leqslant 7.$$ [3]
  2. The complex number \(z_1\) represents that value of \(z\) in the region described in part (i) for which \(\arg(z)\) is least. Mark \(z_1\) on your Argand diagram and determine \(\arg(z_1)\) correct to 3 decimal places. [4]
Pre-U Pre-U 9795/1 2015 June Q8
9 marks Challenging +1.8
The group \(G\), of order 8, consists of the elements \(\{e, a, b, c, ab, bc, ca, abc\}\), together with a multiplicative binary operation, where \(e\) is the identity and $$a^2 = b^2 = c^2 = e, \quad ab = ba, \quad bc = cb \quad \text{and} \quad ca = ac.$$
  1. Construct the group table of \(G\). [You are not required to show how individual elements of the table are determined.] [4]
  2. List all the proper subgroups of \(G\). [5]
Pre-U Pre-U 9795/1 2015 June Q9
9 marks Challenging +1.2
The differential equation \((\star)\) is $$\frac{\text{d}^2 u}{\text{d}x^2} + 4u = 8x + 1.$$
  1. Find the general solution of \((\star)\). [5]
  2. The differential equation \((\star \star)\) is $$x \frac{\text{d}^2 v}{\text{d}x^2} + 2 \frac{\text{d}v}{\text{d}x} + 4xv = 8x + 1.$$ By using the substitution \(u = xv\), show that \((\star)\) becomes \((\star \star)\) and deduce the general solution of \((\star \star)\). [4]
Pre-U Pre-U 9795/1 2015 June Q10
11 marks Standard +0.8
  1. Find a vector equation for the line of intersection of the planes with cartesian equations $$x + 7y - 6z = -10 \quad \text{and} \quad 3x - 5y + 8z = 48.$$ [5]
  2. Determine the value of \(k\) for which the system of equations \begin{align} x + 7y - 6z &= -10
    3x - 5y + 8z &= 48
    kx + 2y + 3z &= 16 \end{align} does not have a unique solution and show that, for this value of \(k\), the system of equations is inconsistent. [6]
Pre-U Pre-U 9795/1 2015 June Q11
13 marks Challenging +1.3
  1. The cubic equation \(x^3 + 2x^2 + 3x - 4 = 0\) has roots \(p\), \(q\) and \(r\). A second cubic equation has roots \(qr\), \(rp\) and \(pq\). Show how the substitution \(y = \frac{4}{x}\) can be used to determine this second equation. Hence, or otherwise, find this equation in the form \(y^3 + ay^2 + by + c = 0\). [6]
  2. The cubic equation \(x^3 - 4x^2 + 5x - 4 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\). You are given that \(\alpha\) is real and positive, and that \(\beta\) and \(\gamma\) are complex.
    1. Describe the relationship between \(\beta\) and \(\gamma\). [1]
    2. Explain why \(|\beta| = \frac{2}{\sqrt{\alpha}}\). [2]
    3. Verify that \(\alpha = 2.70\) correct to 3 significant figures, and deduce that \(\text{Re}(\beta) = 0.65\) correct to 2 significant figures. [4]
Pre-U Pre-U 9795/1 2015 June Q12
22 marks Challenging +1.8
Let \(I_n = \int_0^2 x^n \sqrt{1 + 2x^2} \, \text{d}x\) for \(n = 0, 1, 2, 3, \ldots\).
    1. Evaluate \(I_1\). [3]
    2. Prove that, for \(n \geqslant 2\), $$(2n + 4)I_n = 27 \times 2^{n-1} - (n - 1)I_{n-2}.$$ [6]
    3. Using a suitable substitution, or otherwise, show that $$I_0 = 3 + \frac{1}{\sqrt{2}} \ln(1 + \sqrt{2}).$$ [8]
  2. The curve \(y = \frac{1}{\sqrt{2}} x^2\), between \(x = 0\) and \(x = 2\), is rotated through \(2\pi\) radians about the \(x\)-axis to form a surface with area \(S\). Find the exact value of \(S\). [5]