CAIE P3 (Pure Mathematics 3) 2014 June

Find the set of values of \(x\) satisfying the inequality $$|x + 2a| > 3|x - a|,$$ where \(a\) is a positive constant. [4]

Solve the equation $$2\ln(5 - e^{-2x}) = 1,$$ giving your answer correct to 3 significant figures. [4]

Solve the equation $$\cos(x + 30°) = 2\cos x,$$ giving all solutions in the interval \(-180° < x < 180°\). [5]

The parametric equations of a curve are $$x = t - \tan t, \quad y = \ln(\cos t),$$ for \(-\frac{1}{4}\pi < t < \frac{1}{4}\pi\).

1. Show that \(\frac{dy}{dx} = \cot t\). [5]
2. Hence find the \(x\)-coordinate of the point on the curve at which the gradient is equal to 2. Give your answer correct to 3 significant figures. [2]

1. The polynomial \(f(x)\) is of the form \((x - 2)^2g(x)\), where \(g(x)\) is another polynomial. Show that \((x - 2)\) is a factor of \(f'(x)\). [2]
2. The polynomial \(x^5 + ax^4 + 3x^3 + bx^2 + a\), where \(a\) and \(b\) are constants, has a factor \((x - 2)^2\). Using the factor theorem and the result of part (i), or otherwise, find the values of \(a\) and \(b\). [5]

\includegraphics{figure_6} In the diagram, \(A\) is a point on the circumference of a circle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the circumference at \(B\) and \(C\). The angle \(\angle OAB\) is equal to \(x\) radians. The shaded region is bounded by \(AB\), \(AC\) and the circular arc with centre \(A\) joining \(B\) and \(C\). The perimeter of the shaded region is equal to half the circumference of the circle.

1. Show that \(x = \cos^{-1}\left(\frac{\pi}{4 + 4x}\right)\). [3]
2. Verify by calculation that \(x\) lies between 1 and 1.5. [2]
3. Use the iterative formula $$x_{n+1} = \cos^{-1}\left(\frac{\pi}{4 + 4x_n}\right)$$ to determine the value of \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places. [3]

1. It is given that \(-1 + (\sqrt{5})i\) is a root of the equation \(z^3 + 2z + a = 0\), where \(a\) is real. Showing your working, find the value of \(a\), and write down the other complex root of this equation. [4]
2. The complex number \(w\) has modulus 1 and argument \(2\theta\) radians. Show that \(\frac{w - 1}{w + 1} = i\tan\theta\). [4]

\includegraphics{figure_8} The diagram shows the curve \(y = x\cos\frac{1}{2}x\) for \(0 \leqslant x \leqslant \pi\).

1. Find \(\frac{dy}{dx}\) and show that \(4\frac{d^2y}{dx^2} + y + 4\sin\frac{1}{2}x = 0\). [5]
2. Find the exact value of the area of the region enclosed by this part of the curve and the \(x\)-axis. [5]

The population of a country at time \(t\) years is \(N\) millions. At any time, \(N\) is assumed to increase at a rate proportional to the product of \(N\) and \((1 - 0.01N)\). When \(t = 0\), \(N = 20\) and \(\frac{dN}{dt} = 0.32\).

1. Treating \(N\) and \(t\) as continuous variables, show that they satisfy the differential equation $$\frac{dN}{dt} = 0.02N(1 - 0.01N).$$ [1]
2. Solve the differential equation, obtaining an expression for \(t\) in terms of \(N\). [8]
3. Find the time at which the population will be double its value at \(t = 0\). [1]

Referred to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors given by $$\overrightarrow{OA} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}, \quad \overrightarrow{OB} = 2\mathbf{i} + 4\mathbf{j} + \mathbf{k} \quad \text{and} \quad \overrightarrow{OC} = 3\mathbf{i} + 5\mathbf{j} - 3\mathbf{k}.$$

1. Find the exact value of the cosine of angle \(BAC\). [4]
2. Hence find the exact value of the area of triangle \(ABC\). [3]
3. Find the equation of the plane which is parallel to the \(y\)-axis and contains the line through \(B\) and \(C\). Give your answer in the form \(ax + by + cz = d\). [5]