CAIE P3 (Pure Mathematics 3) 2018 June

The coefficient of \(x^2\) in the expansion of \(\left(2 + \frac{x}{2}\right)^6 + (a + x)^5\) is 330. Find the value of the constant \(a\). [5]

The equation of a curve is \(y = x^2 - 6x + k\), where \(k\) is a constant.

1. Find the set of values of \(k\) for which the whole of the curve lies above the \(x\)-axis. [2]
2. Find the value of \(k\) for which the line \(y + 2x = 7\) is a tangent to the curve. [3]

A company producing salt from sea water changed to a new process. The amount of salt obtained each week increased by 2% of the amount obtained in the preceding week. It is given that in the first week after the change the company obtained 8000 kg of salt.

1. Find the amount of salt obtained in the 12th week after the change. [3]
2. Find the total amount of salt obtained in the first 12 weeks after the change. [2]

The function f is such that \(\mathrm{f}(x) = a + b \cos x\) for \(0 \leqslant x \leqslant 2\pi\). It is given that \(\mathrm{f}\left(\frac{1}{3}\pi\right) = 5\) and \(\mathrm{f}(\pi) = 11\).

1. Find the values of the constants \(a\) and \(b\). [3]
2. Find the set of values of \(k\) for which the equation \(\mathrm{f}(x) = k\) has no solution. [3]

\includegraphics{figure_5} The diagram shows a three-dimensional shape. The base \(OAB\) is a horizontal triangle in which angle \(AOB\) is 90°. The side \(OBCD\) is a rectangle and the side \(OAD\) lies in a vertical plane. Unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(OA\) and \(OB\) respectively and the unit vector \(\mathbf{k}\) is vertical. The position vectors of \(A\), \(B\) and \(D\) are given by \(\overrightarrow{OA} = 8\mathbf{i}\), \(\overrightarrow{OB} = 5\mathbf{j}\) and \(\overrightarrow{OD} = 2\mathbf{i} + 4\mathbf{k}\).

1. Express each of the vectors \(\overrightarrow{DA}\) and \(\overrightarrow{CA}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [2]
2. Use a scalar product to find angle \(CAD\). [4]

\includegraphics{figure_6} The diagram shows points \(A\) and \(B\) on a circle with centre \(O\) and radius \(r\). The tangents to the circle at \(A\) and \(B\) meet at \(T\). The shaded region is bounded by the minor arc \(AB\) and the lines \(AT\) and \(BT\). Angle \(AOB\) is \(2\theta\) radians.

1. In the case where the area of the sector \(AOB\) is the same as the area of the shaded region, show that \(\tan \theta = 2\theta\). [3]
2. In the case where \(r = 8\) cm and the length of the minor arc \(AB\) is 19.2 cm, find the area of the shaded region. [3]

The function f is defined by \(\mathrm{f} : x \mapsto 7 - 2x^2 - 12x\) for \(x \in \mathbb{R}\).

1. Express \(7 - 2x^2 - 12x\) in the form \(a - 2(x + b)^2\), where \(a\) and \(b\) are constants. [2]
2. State the coordinates of the stationary point on the curve \(y = \mathrm{f}(x)\). [1]

The function g is defined by \(\mathrm{g} : x \mapsto 7 - 2x^2 - 12x\) for \(x \geqslant k\).

1. State the smallest value of \(k\) for which g has an inverse. [1]
2. For this value of \(k\), find \(\mathrm{g}^{-1}(x)\). [3]

Points \(A\) and \(B\) have coordinates \((h, h)\) and \((4h + 6, 5h)\) respectively. The equation of the perpendicular bisector of \(AB\) is \(3x + 2y = k\). Find the values of the constants \(h\) and \(k\). [7]

A curve is such that \(\frac{\mathrm{d}y}{\mathrm{d}x} = \sqrt{(4x + 1)}\) and \((2, 5)\) is a point on the curve.

1. Find the equation of the curve. [4]
2. A point \(P\) moves along the curve in such a way that the \(y\)-coordinate is increasing at a constant rate of 0.06 units per second. Find the rate of change of the \(x\)-coordinate when \(P\) passes through \((2, 5)\). [2]
3. Show that \(\frac{\mathrm{d}^2y}{\mathrm{d}x^2} \times \frac{\mathrm{d}y}{\mathrm{d}x}\) is constant. [2]

1. Solve the equation \(2 \cos x + 3 \sin x = 0\), for \(0° \leqslant x \leqslant 360°\). [3]
2. Sketch, on the same diagram, the graphs of \(y = 2 \cos x\) and \(y = -3 \sin x\) for \(0° \leqslant x \leqslant 360°\). [3]
3. Use your answers to parts (i) and (ii) to find the set of values of \(x\) for \(0° \leqslant x \leqslant 360°\) for which \(2 \cos x + 3 \sin x > 0\). [2]

\includegraphics{figure_11} The diagram shows part of the curve \(y = \frac{x}{2} + \frac{6}{x}\). The line \(y = 4\) intersects the curve at the points \(P\) and \(Q\).

1. Show that the tangents to the curve at \(P\) and \(Q\) meet at a point on the line \(y = x\). [6]
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through 360° about the \(x\)-axis. Give your answer in terms of \(\pi\). [6]