OCR H240/03 (Pure Mathematics and Mechanics) 2022 June

Solve the equation \(|2x - 3| = 9\). [3]

1. Give full details of the single transformation that transforms the graph of \(y = x^3\) to the graph of \(y = x^3 - 8\). [2]

The function f is defined by \(\mathrm{f}(x) = x^3 - 8\).

1. Find an expression for \(\mathrm{f}^{-1}(x)\). [2]
2. State how the graphs of \(y = \mathrm{f}(x)\) and \(y = \mathrm{f}^{-1}(x)\) are related geometrically. [1]

The points \(P\) and \(Q\) have coordinates \((2, -5)\) and \((3, 1)\) respectively. Determine the equation of the circle that has \(PQ\) as a diameter. Give your answer in the form \(x^2 + y^2 + ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. [4]

The positive integers \(x\), \(y\) and \(z\) are the first, second and third terms, respectively, of an arithmetic progression with common difference \(-4\). Also, \(x\), \(\frac{15}{y}\) and \(z\) are the first, second and third terms, respectively, of a geometric progression.

1. Show that \(y\) satisfies the equation \(y^4 - 16y^2 - 225 = 0\). [4]
2. Hence determine the sum to infinity of the geometric progression. [4]

In this question you must show detailed reasoning. \includegraphics{figure_5} The diagram shows the curve with equation \(y = \frac{2x - 3}{4x^2 + 1}\). The tangent to the curve at the point \(P\) has gradient 2.

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation $$4x^3 + 3x - 3 = 0.$$ [5]
2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 0.5 and 1. [2]
3. Show that the iteration $$x_{n+1} = \frac{3 - 4x_n^3}{3}$$ cannot converge to the \(x\)-coordinate of \(P\) whatever starting value is used. [2]
4. Use the Newton-Raphson method, with initial value 0.5, to determine the coordinates of \(P\) correct to 5 decimal places. [5]

In this question you must show detailed reasoning. \includegraphics{figure_6} The diagram shows the curves \(y = \sqrt{2x + 9}\) and \(y = 4\mathrm{e}^{-2x} - 1\) which intersect on the \(y\)-axis. The shaded region is bounded by the curves and the \(x\)-axis. Determine the area of the shaded region, giving your answer in the form \(p + q \ln 2\) where \(p\) and \(q\) are constants to be determined. [8]

In this question you must show detailed reasoning.

1. Show that the equation \(m \sec \theta + 3 \cos \theta = 4 \sin \theta\) can be expressed in the form $$m \tan^2 \theta - 4 \tan \theta + (m + 3) = 0.$$ [3]
2. It is given that there is only one value of \(\theta\), for \(0 < \theta < \pi\), satisfying the equation \(m \sec \theta + 3 \cos \theta = 4 \sin \theta\). Given also that \(m\) is a negative integer, find this value of \(\theta\), correct to 3 significant figures. [5]

\includegraphics{figure_8} A child attempts to drag a sledge along horizontal ground by means of a rope attached to the sledge. The rope makes an angle of \(15°\) with the horizontal (see diagram). Given that the sledge remains at rest and that the frictional force acting on the sledge is 60 N, find the tension in the rope. [2]

\includegraphics{figure_9} The diagram shows a velocity-time graph representing the motion of two cars \(A\) and \(B\) which are both travelling along a horizontal straight road. At time \(t = 0\), car \(B\), which is travelling with constant speed \(12 \mathrm{m s}^{-1}\), is overtaken by car \(A\) which has initial speed \(20 \mathrm{m s}^{-1}\). From \(t = 0\) car \(A\) travels with constant deceleration for 30 seconds. When \(t = 30\) the speed of car \(A\) is \(8 \mathrm{m s}^{-1}\) and the car maintains this speed in subsequent motion.

1. Calculate the deceleration of car \(A\). [2]
2. Determine the value of \(t\) when \(B\) overtakes \(A\). [4]

\includegraphics{figure_10} A rectangular block \(B\) is at rest on a horizontal surface. A particle \(P\) of mass 2.5 kg is placed on the upper surface of \(B\). The particle \(P\) is attached to one end of a light inextensible string which passes over a smooth fixed pulley. A particle \(Q\) of mass 3 kg is attached to the other end of the string and hangs freely below the pulley. The part of the string between \(P\) and the pulley is horizontal (see diagram). The particles are released from rest with the string taut. It is given that \(B\) remains in equilibrium while \(P\) moves on the upper surface of \(B\). The tension in the string while \(P\) moves on \(B\) is 16.8 N.

1. Find the acceleration of \(Q\) while \(P\) and \(B\) are in contact. [2]
2. Determine the coefficient of friction between \(P\) and \(B\). [3]
3. Given that the coefficient of friction between \(B\) and the horizontal surface is \(\frac{5}{49}\), determine the least possible value for the mass of \(B\). [3]

\includegraphics{figure_11} A uniform rod \(AB\) of mass 4 kg and length 3 m rests in a vertical plane with \(A\) on rough horizontal ground. A particle of mass 1 kg is attached to the rod at \(B\). The rod makes an angle of \(60°\) with the horizontal and is held in limiting equilibrium by a light inextensible string \(CD\). \(D\) is a fixed point vertically above \(A\) and \(CD\) makes an angle of \(60°\) with the vertical. The distance \(AC\) is \(x\) m (see diagram).

1. Find, in terms of \(g\) and \(x\), the tension in the string. [3]

The coefficient of friction between the rod and the ground is \(\frac{9\sqrt{3}}{35}\).

1. Determine the value of \(x\). [4]

In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in the directions east and north respectively. A particle \(P\) is moving on a smooth horizontal surface under the action of a single force \(\mathbf{F}\) N. At time \(t\) seconds, where \(t \geq 0\), the velocity \(\mathbf{v} \mathrm{m s}^{-1}\) of \(P\), relative to a fixed origin \(O\), is given by $$\mathbf{v} = (1 - 2t)\mathbf{i} + (2t^2 + t - 13)\mathbf{j}.$$

1. Show that \(P\) is never stationary. [2]
2. Find, in terms of \(\mathbf{i}\) and \(\mathbf{j}\), the acceleration of \(P\) at time \(t\). [1]

The mass of \(P\) is 0.5 kg.

1. Determine the magnitude of \(\mathbf{F}\) when \(P\) is moving in the direction of the vector \(-2\mathbf{i} + \mathbf{j}\). Give your answer correct to 3 significant figures. [5]

When \(t = 1\), \(P\) is at the point with position vector \(\frac{1}{6}\mathbf{j}\).

1. Determine the bearing of \(P\) from \(O\) at time \(t = 1.5\). [5]

A small ball \(B\) moves in the plane of a fixed horizontal axis \(Ox\), which lies on horizontal ground, and a fixed vertically upwards axis \(Oy\). \(B\) is projected from \(O\) with a velocity whose components along \(Ox\) and \(Oy\) are \(U \mathrm{m s}^{-1}\) and \(V \mathrm{m s}^{-1}\), respectively. The units of \(x\) and \(y\) are metres. \(B\) is modelled as a particle moving freely under gravity.

1. Show that the path of \(B\) has equation \(2U^2 y = 2UVx - gx^2\). [3]

During its motion, \(B\) just clears a vertical wall of height \(\frac{1}{3}a\) m at a horizontal distance \(a\) m from \(O\). \(B\) strikes the ground at a horizontal distance \(3a\) m beyond the wall.

1. Determine the angle of projection of \(B\). Give your answer in degrees correct to 3 significant figures. [5]
2. Given that the speed of projection of \(B\) is \(54.6 \mathrm{m s}^{-1}\), determine the value of \(a\). [2]
3. Hence find the maximum height of \(B\) above the ground during its motion. [3]
4. State one refinement of the model, other than including air resistance, that would make it more realistic. [1]