Questions — Pre-U (885 questions)

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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]
Pre-U Pre-U 9795/1 2015 June Q13
10 marks Challenging +1.2
  1. By sketching a suitable triangle, show that \(\tan^{-1} a + \tan^{-1} \left(\frac{1}{a}\right) = \frac{1}{4}\pi\), for \(a > 0\). [1]
  2. Given that \(a\) and \(b\) are positive and less than 1, express \(\tan(\tan^{-1} a \pm \tan^{-1} b)\) in terms of \(a\) and \(b\). [2]
  3. By letting \(a = \frac{1}{n-1}\) and \(b = \frac{1}{n+1}\), use the method of differences to prove that $$\sum_{n=1}^{\infty} \tan^{-1} \left(\frac{2}{n^2}\right) = \frac{3}{4}\pi.$$ [7]
Pre-U Pre-U 9794/2 2016 June Q1
3 marks Easy -1.3
  1. Find the remainder when \(x^3 + 2x\) is divided by \(x + 2\). [2]
  2. Write down the value of \(k\) for which \(x + 2\) is a factor of \(x^3 + 2x + k\). [1]
Pre-U Pre-U 9794/2 2016 June Q2
4 marks Easy -1.2
Solve the equation \(4 \times 3^x = 5\), giving the solution in an exact form. [4]
Pre-U Pre-U 9794/2 2016 June Q3
4 marks Moderate -0.8
The graph of \(\log_{10} y\) against \(x\) is a straight line with gradient 2 and the intercept on the vertical axis at 4. Write down an equation for this straight line and show that \(y = 10000 \times 100^x\). [4]
Pre-U Pre-U 9794/2 2016 June Q4
6 marks Moderate -0.8
The complex numbers \(z_1\) and \(z_2\) are given by \(z_1 = 2 + i\) and \(z_2 = 3 + 4i\).
  1. Verify that \(|z_1| + |z_2| > |z_1 + z_2|\). [4]
  2. Sketch on an Argand diagram the locus \(|z - z_1| = 2\). [2]
Pre-U Pre-U 9794/2 2016 June Q5
7 marks Moderate -0.3
  1. Show that \(\frac{3}{x+2} + \frac{1}{x+1} \equiv \frac{4x+5}{x^2+3x+2}\). [2]
  2. Differentiate \(\frac{4x+5}{x^2+3x+2}\) with respect to \(x\). [3]
  3. Hence show that the function given by $$f(x) = \frac{4x+5}{x^2+3x+2}, \quad x \neq -1, x \neq -2,$$ is a decreasing function. [2]
Pre-U Pre-U 9794/2 2016 June Q6
7 marks Moderate -0.8
The points \(A\) and \(B\) are at \((2, 3, 5)\) and \((8, 2, 4)\) with respect to the origin \(O\).
  1. Find the size of angle \(AOB\). [4]
  2. Show that triangle \(AOB\) is isosceles. [3]
Pre-U Pre-U 9794/2 2016 June Q7
11 marks Moderate -0.3
  1. Use a change of sign to verify that the equation \(\cos x - x = 0\) has a root \(\alpha\) between \(x = 0.7\) and \(x = 0.8\). [2]
  2. Sketch, on a single diagram, the curve \(y = \cos x\) and the line \(y = x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), giving the coordinates of all points of intersection with the coordinate axes. [2]
An iteration of the form \(x_{n+1} = \cos(x_n)\) is to be used to find \(\alpha\).
  1. By considering the gradient of \(y = \cos x\), show that this iteration will converge. [3]
  2. On a copy of your sketch from part (ii), illustrate how this iteration converges to \(\alpha\). [2]
  3. Use a change of sign to verify that \(\alpha = 0.7391\) to 4 decimal places. [2]
Pre-U Pre-U 9794/2 2016 June Q8
5 marks Standard +0.3
\(P\) and \(Q\) are points on the circumference of a circle with centre \(O\) and radius \(r\). The angle \(POQ\) is \(\theta\) radians. Given that the chord \(PQ\) has length 4, find an expression for the length of the arc \(PQ\) in terms of \(\theta\) of only. [5]
Pre-U Pre-U 9794/2 2016 June Q9
11 marks Challenging +1.2
  1. Show that \(\frac{\sin x}{1 + \sin x} \equiv \sec x \tan x - \sec^2 x + 1\). [5]
  2. Hence show that \(\int_0^{\frac{\pi}{4}} \frac{\sin x}{1 + \sin x} \, dx = \frac{1}{4}\pi + \sqrt{2} - 2\). [6]