CAIE P3 (Pure Mathematics 3) 2018 June

Express \(3x^2 - 12x + 7\) in the form \(a(x + b)^2 + c\), where \(a\), \(b\) and \(c\) are constants. [3]

Find the coefficient of \(\frac{1}{x}\) in the expansion of \(\left(x - \frac{2}{x}\right)^5\). [3]

The common ratio of a geometric progression is 0.99. Express the sum of the first 100 terms as a percentage of the sum to infinity, giving your answer correct to 2 significant figures. [5]

A curve with equation \(y = \mathrm{f}(x)\) passes through the point \(A(3, 1)\) and crosses the \(y\)-axis at \(B\). It is given that \(\mathrm{f}'(x) = (3x - 1)^{-\frac{1}{3}}\). Find the \(y\)-coordinate of \(B\). [6]

\includegraphics{figure_5} The diagram shows a triangle \(OAB\) in which angle \(OAB = 90°\) and \(OA = 5\) cm. The arc \(AC\) is part of a circle with centre \(O\). The arc has length 6 cm and it meets \(OB\) at \(C\). Find the area of the shaded region. [5]

The coordinates of points \(A\) and \(B\) are \((-3k - 1, k + 3)\) and \((k + 3, 3k + 5)\) respectively, where \(k\) is a constant \((k \neq -1)\).

1. Find and simplify the gradient of \(AB\), showing that it is independent of \(k\). [2]
2. Find and simplify the equation of the perpendicular bisector of \(AB\). [5]

1. 1. Express \(\frac{\tan^2 \theta - 1}{\tan^2 \theta + 1}\) in the form \(a \sin^2 \theta + b\), where \(a\) and \(b\) are constants to be found. [3]
   2. Hence, or otherwise, and showing all necessary working, solve the equation $$\frac{\tan^2 \theta - 1}{\tan^2 \theta + 1} = \frac{1}{4}$$ for \(-90° \leqslant \theta \leqslant 0°\). [2]
2. \includegraphics{figure_7b} The diagram shows the graphs of \(y = \sin x\) and \(y = 2 \cos x\) for \(-\pi \leqslant x \leqslant \pi\). The graphs intersect at the points \(A\) and \(B\).
   1. Find the \(x\)-coordinate of \(A\). [2]
   2. Find the \(y\)-coordinate of \(B\). [2]

1. The tangent to the curve \(y = x^3 - 9x^2 + 24x - 12\) at a point \(A\) is parallel to the line \(y = 2 - 3x\). Find the equation of the tangent at \(A\). [6]
2. The function f is defined by \(\mathrm{f}(x) = x^3 - 9x^2 + 24x - 12\) for \(x > k\), where \(k\) is a constant. Find the smallest value of \(k\) for f to be an increasing function. [2]

\includegraphics{figure_9} The diagram shows a pyramid \(OABCD\) with a horizontal rectangular base \(OABC\). The sides \(OA\) and \(AB\) have lengths of 8 units and 6 units respectively. The point \(E\) on \(OB\) is such that \(OE = 2\) units. The point \(D\) of the pyramid is 7 units vertically above \(E\). Unit vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\) are parallel to \(OA\), \(OC\) and \(ED\) respectively.

1. Show that \(\overrightarrow{OE} = 1.6\mathbf{i} + 1.2\mathbf{j}\). [2]
2. Use a scalar product to find angle \(BDO\). [7]

The one-one function f is defined by \(\mathrm{f}(x) = (x - 2)^2 + 2\) for \(x \geqslant c\), where \(c\) is a constant.

1. State the smallest possible value of \(c\). [1]

In parts (ii) and (iii) the value of \(c\) is 4.

1. Find an expression for \(\mathrm{f}^{-1}(x)\) and state the domain of \(\mathrm{f}^{-1}\). [3]
2. Solve the equation \(\mathrm{f f}(x) = 51\), giving your answer in the form \(a + \sqrt{b}\). [5]

\includegraphics{figure_11} The diagram shows part of the curve \(y = (x + 1)^2 + (x + 1)^{-1}\) and the line \(x = 1\). The point \(A\) is the minimum point on the curve.

1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(2(x + 1)^3 = 1\) and find the exact value of \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}\) at \(A\). [5]
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis. [6]