CAIE P1 (Pure Mathematics 1) 2014 November

In the expansion of \((2 + ax)^6\), the coefficient of \(x^2\) is equal to the coefficient of \(x^3\). Find the value of the non-zero constant \(a\). [4]

\includegraphics{figure_2} In the diagram, \(OADC\) is a sector of a circle with centre \(O\) and radius 3 cm. \(AB\) and \(CB\) are tangents to the circle and angle \(ABC = \frac{1}{4}\pi\) radians. Find, giving your answer in terms of \(\sqrt{3}\) and \(\pi\),

1. the perimeter of the shaded region, [3]
2. the area of the shaded region. [3]

1. Express \(9x^2 - 12x + 5\) in the form \((ax + b)^2 + c\). [3]
2. Determine whether \(3x^3 - 6x^2 + 5x - 12\) is an increasing function, a decreasing function or neither. [3]

Three geometric progressions, \(P\), \(Q\) and \(R\), are such that their sums to infinity are the first three terms respectively of an arithmetic progression. Progression \(P\) is \(2, 1, \frac{1}{2}, \frac{1}{4}, \ldots\) Progression \(Q\) is \(3, 1, \frac{1}{3}, \frac{1}{9}, \ldots\)

1. Find the sum to infinity of progression \(R\). [3]
2. Given that the first term of \(R\) is 4, find the sum of the first three terms of \(R\). [3]

1. Show that \(\sin^2 \theta - \cos^4 \theta = 2 \sin^2 \theta - 1\). [3]
2. Hence solve the equation \(\sin^2 \theta - \cos^4 \theta = \frac{1}{2}\) for \(0° \leq \theta \leq 360°\). [4]

\(A\) is the point \((a, 2a - 1)\) and \(B\) is the point \((2a + 4, 3a + 9)\), where \(a\) is a constant.

1. Find, in terms of \(a\), the gradient of a line perpendicular to \(AB\). [3]
2. Given that the distance \(AB\) is \(\sqrt{260}\), find the possible values of \(a\). [4]

Three points, \(O\), \(A\) and \(B\), are such that \(\overrightarrow{OA} = \mathbf{i} + 3\mathbf{j} + p\mathbf{k}\) and \(\overrightarrow{OB} = -7\mathbf{i} + (1 - p)\mathbf{j} + p\mathbf{k}\), where \(p\) is a constant.

1. Find the values of \(p\) for which \(\overrightarrow{OA}\) is perpendicular to \(\overrightarrow{OB}\). [3]
2. The magnitudes of \(\overrightarrow{OA}\) and \(\overrightarrow{OB}\) are \(a\) and \(b\) respectively. Find the value of \(p\) for which \(b^2 = 2a^2\). [2]
3. Find the unit vector in the direction of \(\overrightarrow{AB}\) when \(p = -8\). [3]

A curve \(y = f(x)\) has a stationary point at \((3, 7)\) and is such that \(f''(x) = 36x^{-3}\).

1. State, with a reason, whether this stationary point is a maximum or a minimum. [1]
2. Find \(f'(x)\) and \(f(x)\). [7]

\includegraphics{figure_9} The diagram shows parts of the graphs of \(y = x + 2\) and \(y = 3\sqrt{x}\) intersecting at points \(A\) and \(B\).

1. Write down an equation satisfied by the \(x\)-coordinates of \(A\) and \(B\). Solve this equation and hence find the coordinates of \(A\) and \(B\). [4]
2. Find by integration the area of the shaded region. [6]

1. The functions \(f\) and \(g\) are defined for \(x \geq 0\) by $$f : x \mapsto (ax + b)^{\frac{1}{3}}, \text{ where } a \text{ and } b \text{ are positive constants,}$$ $$g : x \mapsto x^2.$$ Given that \(fg(1) = 2\) and \(gf(9) = 16\),
   1. calculate the values of \(a\) and \(b\), [4]
   2. obtain an expression for \(f^{-1}(x)\) and state the domain of \(f^{-1}\). [4]
2. A point \(P\) travels along the curve \(y = (7x^2 + 1)^{\frac{1}{3}}\) in such a way that the \(x\)-coordinate of \(P\) at time \(t\) minutes is increasing at a constant rate of 8 units per minute. Find the rate of increase of the \(y\)-coordinate of \(P\) at the instant when \(P\) is at the point \((3, 4)\). [5]