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SPS SPS FM Mechanics 2022 January Q3
9 marks Standard +0.3
A car of mass 800 kg is driven with its engine generating a power of 15 kW.
  1. The car is first driven along a straight horizontal road and accelerates from rest. Assuming that there is no resistance to motion, find the speed of the car after 6 seconds. [2]
  2. The car is next driven at constant speed up a straight road inclined at an angle \(\theta\) to the horizontal. The resistance to motion is now modelled as being constant with magnitude of 150 N. Given that \(\sin \theta = \frac{1}{20}\), find the speed of the car. [3]
  3. The car is now driven at a constant speed of 30 ms\(^{-1}\) along the horizontal road pulling a trailer of mass 150 kg which is attached by means of a light rigid horizontal towbar. Assuming the resistance to motion of the car is three times the resistance to motion of the trailer. Find:
    1. the resistance to motion of the car,
    2. the magnitude of the tension in the towbar
    [4]
SPS SPS FM Mechanics 2022 January Q4
9 marks Challenging +1.3
\includegraphics{figure_4} Two uniform smooth spheres A and B of equal radius are moving on a horizontal surface when they collide. A has mass 0.1 kg and B has mass 0.4 kg. Immediately before the collision A is moving with speed 2.8 ms\(^{-1}\) along the line of centres, and B is moving with speed 1 ms\(^{-1}\) at an angle \(\theta\) to the line of centres, where \(\cos \theta = 0.8\) (see diagram). Immediately after the collision A is stationary. Find:
  1. the coefficient of restitution between A and B, [5]
  2. the angle turned through by the direction of motion of B as a result of the collision. [4]
SPS SPS FM Mechanics 2022 January Q5
9 marks Challenging +1.2
A right circular cone C of height 4 m and base radius 3 m has its base fixed to a horizontal plane. One end of a light elastic string of natural length 2 m and modulus of elasticity 32 N is fixed to the vertex of C. The other end of the string is attached to a particle P of mass 2.5 kg. P moves in a horizontal circle with constant speed and in contact with the smooth curved surface of C. The extension of the string is 1.5 m.
  1. Find the tension in the string. [2]
  2. Find the speed of P. [7]
SPS SPS FM Mechanics 2022 January Q6
8 marks Challenging +1.8
A uniform rod, PQ, of length \(2a\), rests with one end, P, on rough horizontal ground and a point T resting on a rough fixed prism of semi-circular cross-section of radius \(a\), as shown in the diagram. The rod is in a vertical plane which is parallel to the prism's cross-section. The coefficient of friction at both P and T is \(\mu\). \includegraphics{figure_6} The rod is on the point of slipping when it is inclined at an angle of 30\(^0\) to the horizontal. Find the value of \(\mu\). [8]
SPS SPS FM Mechanics 2022 January Q7
14 marks Challenging +1.2
The diagram shows the cross-section through the centre of mass of a uniform solid prism. The cross-section is a trapezium ABCD with AB and CD perpendicular to AD. The lengths of AB and AD are each 5 cm and the length of CD is \((a + 5)\) cm. \includegraphics{figure_7}
  1. Show the distance of the centre of mass of the prism from AD is $$\frac{a^2 + 15a + 75}{3(a + 10)} \text{ cm.}$$ [5]
The prism is placed with the face containing AB in contact with a horizontal surface.
  1. Find the greatest value of \(a\) for which the prism does not topple. [3]
The prism is now placed on an inclined plane which makes an angle \(\theta^o\) with the horizontal. AB lies along a line of greatest slope with B higher than A.
  1. Using the value for \(a\) found in part (ii), and assuming the prism does not slip down the plane, find the great value of \(\theta\) for which the prism does not topple. [6]
SPS SPS FM 2021 November Q1
3 marks Moderate -0.3
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. The roots of the equation $$x^3 - 8x^2 + 28x - 32 = 0$$ are \(\alpha\), \(\beta\) and \(\gamma\). Without solving the equation, find the value of $$(\alpha + 2)(\beta + 2)(\gamma + 2).$$ [3 marks]
SPS SPS FM 2021 November Q2
3 marks Standard +0.3
The equation of a curve in polar coordinates is $$r = 11 + 9 \sec \theta.$$ Show that a cartesian equation of the curve is $$(x - 9)\sqrt{x^2 + y^2} = 11x.$$ [3 marks]
SPS SPS FM 2021 November Q3
6 marks Standard +0.3
The point \(P\) represents a complex number \(z\) on an Argand diagram such that $$|z - 6i| = 2|z - 3|.$$ Show that, as \(z\) varies, the locus of \(P\) is a circle, stating the radius and the coordinates of the centre of this circle. [6 marks]
SPS SPS FM 2021 November Q4
4 marks Moderate -0.3
Prove that $$\sum_{r=1}^{n} 18(r^2 - 4) = n(6n^2 + 9n - 69).$$ [4 marks]
SPS SPS FM 2021 November Q5
4 marks Standard +0.8
Use a trigonometrical substitution to show that $$\int_0^2 \frac{1}{(16 - x^2)^{\frac{3}{2}}} dx = \frac{1}{16\sqrt{3}}$$ [4 marks]
SPS SPS FM 2021 November Q6
7 marks Challenging +1.8
In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Find $$\int_1^{\infty} \frac{1}{\cosh u} du,$$ giving your answer in an exact form. [7 marks]
SPS SPS FM 2021 November Q7
7 marks Challenging +1.3
The curve with equation $$y = -x + \tanh(36x), \quad x \geq 0,$$ has a maximum turning point \(A\).
  1. Find, in exact logarithmic form, the \(x\)-coordinate of \(A\). [4 marks]
  2. Show that the \(y\)-coordinate of \(A\) is $$\frac{\sqrt{35}}{6} - \frac{1}{36}\ln(6 + \sqrt{35}).$$ [3 marks]
SPS SPS FM 2021 November Q8
11 marks Standard +0.3
In this question you must show all stages of your working. The function \(f\) is defined by \(f(x) = (1 + 2x)^{\frac{1}{2}}\).
  1. Find \(f'''(x)\) (i.e. the third derivative of \(f\)) showing all your intermediate steps. Hence, find the Maclaurin series for \(f(x)\) up to and including the \(x^3\) term. [8 marks]
  2. Use the expansion of \(e^x\) together with the result in part (a) to show that, up to and including the \(x^3\) term, $$e^x(1 + 2x)^{\frac{1}{2}} = 1 + 2x + x^2 + kx^3,$$ where \(k\) is a rational number to be found. [3 marks]
SPS SPS FM 2021 November Q9
7 marks Standard +0.3
  1. Show that $$\frac{1}{9r - 4} - \frac{1}{9r + 5} = \frac{9}{(9r - 4)(9r + 5)}$$ [2 marks]
  2. Hence use the method of differences to find $$\sum_{r=1}^{n} \frac{1}{(9r - 4)(9r + 5)}.$$ [5 marks]
SPS SPS FM 2021 November Q10
13 marks Challenging +1.8
\includegraphics{figure_1} Figure 1 shows a closed curve \(C\) with equation $$r = 3\sqrt{\cos(2\theta)}, \quad \text{where } -\frac{\pi}{4} < \theta \leq \frac{\pi}{4}, \quad \frac{3\pi}{4} < \theta \leq \frac{5\pi}{4}$$ The lines \(PQ\), \(SR\), \(PS\) and \(QR\) are tangents to \(C\), where \(PQ\) and \(SR\) are parallel to the initial line and \(PS\) and \(QR\) are perpendicular to the initial line. The point \(O\) is the pole.
  1. Find the total area enclosed by the curve \(C\), shown unshaded inside the rectangle in Figure 1. [4 marks]
  2. Find the total area of the region bounded by the curve \(C\) and the four tangents, shown shaded in Figure 1. [9 marks]
SPS SPS SM 2021 November Q1
8 marks Moderate -0.8
Find \(\frac{dy}{dx}\) for the following functions, simplifying your answers as far as possible.
  1. \(y = \cos x - 2 \sin 2x\) [2]
  2. \(y = \frac{1}{2}x^4 + 2x^4 \ln x\) [3]
  3. \(y = \frac{2e^{3x} - 1}{3e^{3x} - 1}\) [3]
SPS SPS SM 2021 November Q2
6 marks Moderate -0.3
  1. Express \(\frac{5x+7}{(x+3)(x+1)^2}\) in partial fractions. In this question you must show all of your algebraic steps clearly. [3] The function \(f(x) = \frac{2-6x+5x^2}{x^2(1-2x)}\) can be written in the form; $$f(x) = \frac{-2}{x} + \frac{2}{x^2} + \frac{1}{1-2x}$$
  2. Hence find the exact value of \(\int_2^3 \frac{2-6x+5x^2}{x^2(1-2x)} dx\) [3]
SPS SPS SM 2021 November Q3
5 marks Standard +0.3
In this question you must show detailed algebraic reasoning. Find the coordinates of any stationary points on the curve below. $$y = (1 - 3x)(3 - x)^3$$ [5]
SPS SPS SM 2021 November Q4
5 marks Standard +0.3
Find the equation of the normal to the curve \(y = 4 \ln(2x - 3)\) at the point where the curve crosses the \(x\) axis. Give your answer in the form \(ax + by + k = 0\) where \(a > 0\). [5]
SPS SPS SM 2021 November Q5
4 marks Moderate -0.3
  1. Write \(\log_{16} y - \log_{16} x\) as a single logarithm. [1]
  2. Solve the simultaneous equations, giving your answers in an exact form. $$\log_3 y = \log_3(9 - 6x) + 1$$ $$\log_{16} y - \log_{16} x = \frac{1}{4}$$ [3]
SPS SPS SM 2021 November Q6
7 marks Challenging +1.2
  1. Prove the following trigonometric identities. You must show all of your algebraic steps clearly. $$(\cos x + \sin x)(\cos x - \sec x) \equiv 2 \cot 2x$$ [3]
  2. Solve the following equation for \(x\) in the interval \(0 \leq x \leq \pi\). Giving your answers in terms of \(\pi\). $$\sin\left(2x + \frac{\pi}{6}\right) = \frac{1}{2}\sin\left(2x - \frac{\pi}{6}\right)$$ [4]
SPS SPS SM 2021 November Q7
5 marks Standard +0.3
The diagram below represents the graph of the function \(y = (2x - 5)^4 - 1\) \includegraphics{figure_7}
  1. Find the intersections of this graph with the \(x\) axis. [1]
  2. Hence find the exact value of the area bounded by the curve and the \(x\) axis. [4]
SPS SPS SM 2021 November Q8
11 marks Standard +0.3
  1. Express \(2\sqrt{3} \cos 2x - 6 \sin 2x\) in the form \(R\cos(2x + \alpha)\) where \(R > 0\) and \(0 < \alpha < \frac{\pi}{2}\) [3]
  2. Hence
    1. Solve the equation \(2\sqrt{3} \cos 2x - 6 \sin 2x = 6\) for \(0 \leq x \leq 2\pi\) Giving your answers in terms of \(\pi\). [3]
  3. It can be shown that \(y = 9 \sin 2x + 4 \cos 2x\) can be written as \(y = \sqrt{97} \sin(2x + 24.0°)\)
    1. State the transformations in the order of occurrence which transform the curve \(y = 9 \sin 2x + 4 \cos 2x\) to the curve \(y = \sin x\) [3]
    2. Find the exact maximum and minimum values of the function; $$f(x) = \frac{1}{11 - 9 \sin 2x - 4 \cos 2x}$$ [2]
SPS SPS SM 2021 November Q9
7 marks Moderate -0.3
    1. Show that \(\cos^2 x \equiv \frac{1}{2} + \frac{1}{2}\cos 2x\) [1]
    2. Hence find \(\int 2\cos^2 4x \, dx\) [3]
  2. Find \(\int \sin^3 x \, dx\) [3]
SPS SPS SM 2021 November Q10
7 marks Standard +0.3
  1. The parametric equations of a curve are \(x = \theta \cos \theta\) and \(y = \sin \theta\) Find the gradient of the curve at the point for which \(\theta = \pi\) [3]
  2. A curve is defined parametrically by the equations; $$x = \cos \theta \qquad y = \left(\frac{\sin \theta}{2}\right)\left(\sin \frac{\theta}{2}\right)$$ Show that the cartesian equation of the curve can be written as \(y^2 = \frac{1}{8}(1-x)^2(1+x)\) [4]