Questions — Pre-U Pre-U 9794/2 (176 questions)

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Pre-U Pre-U 9794/2 2012 June Q10
12 marks Standard +0.3
    1. Find \(\int \frac{e^x}{1 + e^x} dx\). [2]
    2. Hence evaluate \(\int_0^{\ln 3} \frac{e^x}{1 + e^x} dx\), giving your answer in the form \(\ln k\), where \(k\) is an integer. [3]
    1. Using the substitution \(u = 1 + e^x\), find \(\int \left(\frac{e^x}{1 + e^x}\right)^2 dx\). [5]
    2. Hence find the exact volume of the solid of revolution generated when the curve given by \(y = \frac{e^x}{1 + e^x}\), between \(x = -\ln 3\) and \(x = \ln 3\), is rotated through \(2\pi\) radians about the \(x\)-axis. [2]
Pre-U Pre-U 9794/2 2012 June Q11
15 marks Challenging +1.2
The function f is defined by \(f : t \mapsto 2 \sin t + \cos 2t\) for \(0 \leqslant t < 2\pi\).
  1. Show that \(\frac{df}{dt} = 2 \cos t(1 - 2 \sin t)\). [2]
  2. Determine the range of f. [5]
A curve \(C\) is given parametrically by \(x = 2 \cos t + \sin 2t\), \(y = f(t)\) for \(0 \leqslant t < 2\pi\).
  1. Show that \(x^2 + y^2 = 5 + 4 \sin 3t\). [3]
  2. Deduce that \(C\) lies between two circles centred at the origin, and touches both. [2]
  3. Find the gradient of the tangent to \(C\) at the point at which \(t = 0\). [3]
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]
Pre-U Pre-U 9794/2 2016 June Q10
10 marks Challenging +1.2
  1. Using the substitution \(u = \frac{1}{x}\), or otherwise, find \(\int \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\). [4]
  2. Evaluate \(\int_{\frac{1}{2\pi}}^{\frac{1}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\) and \(\int_{\frac{1}{2\pi}}^{\frac{1}{\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\). [3]
  3. Show that, when \(n\) is a positive integer, the integral \(\int_{\frac{1}{(n+1)\pi}}^{\frac{1}{n\pi}} \frac{\sin\left(\frac{1}{x}\right)}{x^2} \, dx\) takes one of the two values found in part (ii), distinguishing between the two cases. [3]
Pre-U Pre-U 9794/2 2016 June Q11
12 marks Standard +0.3
The function f is defined by \(f(x) = \sqrt{x}, x > 0\).
  1. Use differentiation from first principles to find an expression for \(f'(x)\). [5]
The lines \(l_1\) and \(l_2\) are the tangents to the curve \(y = f(x)\) at the points \(A\) and \(B\) where \(x = a\) and \(x = b\) respectively, \(a \neq b\).
    1. Show that the tangents intersect at the point \(\left(\sqrt{ab}, \frac{1}{2}(\sqrt{a} + \sqrt{b})\right)\). [5]
    2. Given that \(l_1\) and \(l_2\) intersect at a point with integer coordinates, write down a possible pair of values for \(a\) and \(b\). [2]
Pre-U Pre-U 9794/2 Specimen Q1
4 marks Moderate -0.3
  1. Show that \(\binom{n}{n-2} = \frac{n(n-1)}{2}\), where the positive integer \(n\) satisfies \(n \geqslant 2\). [1]
  2. Solve the equation \(\binom{2n+1}{2n-1} - 2 \times \binom{n}{n-2} = 24\). [3]
Pre-U Pre-U 9794/2 Specimen Q2
4 marks Standard +0.3
Solve the simultaneous equations $$x - 2y = 5,$$ $$\frac{4}{x} - \frac{2}{y} = 5.$$ [4]
Pre-U Pre-U 9794/2 Specimen Q3
5 marks Standard +0.8
The equation of a curve is \(y = x^{\frac{3}{2}} \ln x\). Find the exact coordinates of the stationary point on the curve. [5]
Pre-U Pre-U 9794/2 Specimen Q4
7 marks Standard +0.3
A circle, of radius \(\sqrt{5}\) and centre the origin \(O\), is divided into two segments by the line \(y = 1\).
  1. Determine the area of the smaller segment. [4]
The line is rotated clockwise about \(O\) through \(45^{\circ}\) and then reflected in the \(x\)-axis.
  1. Find the equation of the line in its final position. [3]
Pre-U Pre-U 9794/2 Specimen Q5
9 marks Standard +0.3
  1. Divide the quartic \(2x^4 - 5x^3 + 4x^2 + 2x - 3\) by the quadratic \(x^2 + x - 2\), identifying the quotient and the remainder. [4]
    1. Show that \((x - 1)\) is a factor of \(nx^{n+1} - (n + 1)x^n + 1\), where \(n\) is a positive integer. [1]
    2. Hence, or otherwise, find all the roots of \(3x^4 - 4x^3 + 1 = 0\). [4]
Pre-U Pre-U 9794/2 Specimen Q6
10 marks Standard +0.8
  1. Express \(y^3 - 3y - 2\) in terms of \(x\), where \(x = y + 1\). [1]
  2. Hence express $$\frac{2y + 5}{y^3 - 3y - 2}$$ in partial fractions. [5]
  3. Find the exact value of $$\int_0^1 \frac{2y + 5}{y^3 - 3y - 2} dy.$$ [4]
Pre-U Pre-U 9794/2 Specimen Q7
12 marks Moderate -0.3
  1. Given that \(\cos \theta = \frac{7}{25}\), where \(\frac{3}{2}\pi < \theta < 2\pi\), determine the exact values of
    1. \(\sin \theta\), [3]
    2. \(\sin(\frac{1}{2}\theta)\), [3]
    3. \(\sec(\frac{1}{2}\theta)\). [1]
    1. Express \(4 \cos x - 3 \sin x\) in the form \(A \cos(x + \alpha)\), where \(A > 0\) and \(0 < \alpha < \frac{1}{2}\pi\). [2]
    2. Hence find the greatest and least values of \(4 \cos x - 3 \sin x\) for \(0 \leqslant x \leqslant \pi\). [3]
Pre-U Pre-U 9794/2 Specimen Q8
14 marks Standard +0.8
    1. Find the general solution of the differential equation $$x \frac{dy}{dx} = y(1 + x \cot x),$$ expressing \(y\) in terms of \(x\). [5]
    2. Find the particular solution given that \(y = 1\) when \(x = \frac{1}{2}\pi\). [2]
  2. The real variables \(x\) and \(y\) are related by \(x^2 - y^2 = 2ax - b\), where \(a\) and \(b\) are real constants.
    1. Show that \(\frac{dy}{dx} = 0\) can only be solved for \(x\) and \(y\) if \(b \geqslant a^2\). [5]
    2. Show that \(y \frac{d^2y}{dx^2} = 1 - \left(\frac{dy}{dx}\right)^2\). [2]
Pre-U Pre-U 9794/2 Specimen Q9
15 marks Challenging +1.3
Two curves are defined by \(y = x^k\) and \(y = x^{\frac{1}{k}}\), for \(x \geqslant 0\), where \(k > 0\).
  1. Prove that, except for one value of \(k\), the curves intersect in exactly two points. [4]
The two curves enclose a finite region \(R\).
  1. Find the area, \(A\), of \(R\), giving your answer in the form \(A = f(k)\) and distinguishing clearly between the cases \(k < 1\) and \(k > 1\). [4]
  2. Determine the set of values of \(k\) for which \(A \leqslant 0.5\). [3]
  3. The function \(f\) is given by \(f : x \mapsto A\) with \(k > 1\). Prove that \(f\) is one-one and determine its inverse. [4]
Pre-U Pre-U 9794/2 Specimen Q10
7 marks Moderate -0.3
  1. Determine the impulse of a force of magnitude \(6\) N that acts on a particle of mass \(3\) kg for \(1.5\) seconds. [1]
Particles \(A\) and \(B\), of masses \(0.1\) kg and \(0.2\) kg respectively, can move on a smooth horizontal table. Initially \(A\) is moving with speed \(3\) m s\(^{-1}\) towards \(B\), which is moving with speed \(1\) m s\(^{-1}\) in the same direction as the motion of \(A\). During a collision \(B\) experiences an impulse from \(A\) of magnitude \(0.2\) kg m s\(^{-1}\).
  1. Find the speeds of the particles immediately after the collision. [4]
  2. Determine the coefficient of restitution between the particles. [2]
Pre-U Pre-U 9794/2 Specimen Q11
11 marks Standard +0.3
A particle \(P\) of mass \(1.5\) kg is placed on a smooth horizontal table. The particle is initially at the origin of a \(2\)-dimensional vector system defined by perpendicular unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) in the plane of the table. The particle is subject to three forces of magnitudes \(10\) N, \(12\) N and \(F\) N, acting in the directions of the vectors \(3\mathbf{i} + 4\mathbf{j}\), \(-\mathbf{j}\) and \(-\cos \theta \mathbf{i} + \sin \theta \mathbf{j}\) respectively, and no others.
  1. Given that the system is in equilibrium, determine \(F\) and \(\theta\). [6]
The force of magnitude \(12\) N is replaced by one of magnitude \(4\) N, but in the opposite direction. The particle is initially at rest.
  1. Find the position vector of the particle \(3\) seconds later. [5]
Pre-U Pre-U 9794/2 Specimen Q12
11 marks Standard +0.3
A particle \(P\) of mass \(2\) kg rests on a long rough horizontal table. The coefficient of friction between \(P\) and the table is \(0.2\). A light inextensible string has one end attached to \(P\) and the other end attached to a particle \(Q\) of mass \(4\) kg. The particle \(Q\) is placed on a smooth plane inclined at \(30^{\circ}\) to the horizontal. The string passes over a smooth light pulley fixed at a point in the line of intersection of the table and the plane (see diagram). \includegraphics{figure_12} Initially the system is held in equilibrium with the string taut. The system is released from rest at time \(t = 0\), where \(t\) is measured in seconds. In the subsequent motion \(P\) does not reach the pulley.
  1. Show that the magnitude of the acceleration of the particles is \(\frac{8}{3}\) m s\(^{-2}\). [4]
After the particles have moved a distance of \(12\) m the string is cut.
  1. Find the corresponding value of \(t\) and the speed of the particles at this instant. [4]
  2. Find the value of \(t\) when \(P\) comes to rest. [3]