CAIE P1 (Pure Mathematics 1) 2017 June

The coefficients of \(x\) and \(x^2\) in the expansion of \((2 + ax)^7\) are equal. Find the value of the non-zero constant \(a\). [3]

The common ratio of a geometric progression is \(r\). The first term of the progression is \((r^2 - 3r + 2)\) and the sum to infinity is \(S\).

1. Show that \(S = 2 - r\). [2]
2. Find the set of possible values that \(S\) can take. [2]

Find the coordinates of the points of intersection of the curve \(y = x^{\frac{2}{3}} - 1\) with the curve \(y = x^{\frac{1}{3}} + 1\). [4]

Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow{OA} = \begin{pmatrix} 5 \\ 1 \\ 3 \end{pmatrix} \text{ and } \overrightarrow{OB} = \begin{pmatrix} 5 \\ 4 \\ -3 \end{pmatrix}$$ The point \(P\) lies on \(AB\) and is such that \(\overrightarrow{AP} = \frac{3}{4}\overrightarrow{AB}\).

1. Find the position vector of \(P\). [3]
2. Find the distance \(OP\). [1]
3. Determine whether \(OP\) is perpendicular to \(AB\). Justify your answer. [2]

1. Show that the equation \(\frac{2 \sin \theta + \cos \theta}{\sin \theta + \cos \theta} = 2 \tan \theta\) may be expressed as \(\cos^2 \theta = 2 \sin^2 \theta\). [3]
2. Hence solve the equation \(\frac{2 \sin \theta + \cos \theta}{\sin \theta + \cos \theta} = 2 \tan \theta\) for \(0° < \theta < 180°\). [3]

The line \(3y + x = 25\) is a normal to the curve \(y = x^2 - 5x + k\). Find the value of the constant \(k\). [6]

\includegraphics{figure_7} The diagram shows two circles with centres \(A\) and \(B\) having radii 8 cm and 10 cm respectively. The two circles intersect at \(C\) and \(D\) where \(CAD\) is a straight line and \(AB\) is perpendicular to \(CD\).

1. Find angle \(ABC\) in radians. [1]
2. Find the area of the shaded region. [6]

\(A(-1, 1)\) and \(P(a, b)\) are two points, where \(a\) and \(b\) are constants. The gradient of \(AP\) is 2.

1. Find an expression for \(b\) in terms of \(a\). [2]
2. \(B(10, -1)\) is a third point such that \(AP = AB\). Calculate the coordinates of the possible positions of \(P\). [6]

1. Express \(9x^2 - 6x + 6\) in the form \((ax + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants. [3]

The function f is defined by \(\text{f}(x) = 9x^2 - 6x + 6\) for \(x \geqslant p\), where \(p\) is a constant.

1. State the smallest value of \(p\) for which f is a one-one function. [1]
2. For this value of \(p\), obtain an expression for \(\text{f}^{-1}(x)\), and state the domain of \(\text{f}^{-1}\). [4]
3. State the set of values of \(q\) for which the equation \(\text{f}(x) = q\) has no solution. [1]

1. \includegraphics{figure_1} Fig. 1 shows part of the curve \(y = x^2 - 1\) and the line \(y = h\), where \(h\) is a constant.
   1. The shaded region is rotated through \(360°\) about the \(y\)-axis. Show that the volume of revolution, \(V\), is given by \(V = \pi\left(\frac{1}{2}h^2 + h\right)\). [3]
   2. Find, showing all necessary working, the area of the shaded region when \(h = 3\). [4]
2. \includegraphics{figure_2} Fig. 2 shows a cross-section of a bowl containing water. When the height of the water level is \(h\) cm, the volume, \(V\) cm\(^3\), of water is given by \(V = \pi\left(\frac{1}{4}h^2 + h\right)\). Water is poured into the bowl at a constant rate of 2 cm\(^3\) s\(^{-1}\). Find the rate, in cm s\(^{-1}\), at which the height of the water level is increasing when the height of the water level is 3 cm. [4]

The function f is defined for \(x \geqslant 0\). It is given that f has a minimum value when \(x = 2\) and that \(\text{f}''(x) = (4x + 1)^{-\frac{1}{2}}\).

1. Find \(\text{f}'(x)\). [3]

It is now given that \(\text{f}''(0)\), \(\text{f}'(0)\) and \(\text{f}(0)\) are the first three terms respectively of an arithmetic progression.

1. Find the value of \(\text{f}(0)\). [3]
2. Find \(\text{f}(x)\), and hence find the minimum value of f. [5]