CAIE P1 (Pure Mathematics 1) 2014 November

\includegraphics{figure_1} The diagram shows part of the curve \(y = x^2 + 1\). Find the volume obtained when the shaded region is rotated through \(360°\) about the \(y\)-axis. [4]

\includegraphics{figure_2} The diagram shows a triangle \(AOB\) in which \(OA\) is 12 cm, \(OB\) is 5 cm and angle \(AOB\) is a right angle. Point \(P\) lies on \(AB\) and \(OP\) is an arc of a circle with centre \(A\). Point \(Q\) lies on \(AB\) and \(OQ\) is an arc of a circle with centre \(B\).

1. Show that angle \(BAO\) is 0.3948 radians, correct to 4 decimal places. [1]
2. Calculate the area of the shaded region. [5]

1. Find the first 3 terms, in ascending powers of \(x\), in the expansion of \((1 + x)^5\). [2]

The coefficient of \(x^2\) in the expansion of \((1 + (px + x^2))^5\) is 95.

1. Use the answer to part (i) to find the value of the positive constant \(p\). [3]

A curve has equation \(y = \frac{12}{5 - 2x}\).

1. Find \(\frac{dy}{dx}\). [2]

A point moves along this curve. As the point passes through \(A\), the \(x\)-coordinate is increasing at a rate of 0.15 units per second and the \(y\)-coordinate is increasing at a rate of 0.4 units per second.

1. Find the possible \(x\)-coordinates of \(A\). [4]

1. Show that the equation \(1 + \sin x \tan x = 5 \cos x\) can be expressed as $$6 \cos^2 x - \cos x - 1 = 0.$$ [3]
2. Hence solve the equation \(1 + \sin x \tan x = 5 \cos x\) for \(0° \leqslant x \leqslant 180°\). [3]

The equation of a curve is \(y = x^3 + ax^2 + bx\), where \(a\) and \(b\) are constants.

1. In the case where the curve has no stationary point, show that \(a^2 < 3b\). [3]
2. In the case where \(a = -6\) and \(b = 9\), find the set of values of \(x\) for which \(y\) is a decreasing function of \(x\). [3]

\includegraphics{figure_7} The diagram shows a pyramid \(OABCX\). The horizontal square base \(OABC\) has side 8 units and the centre of the base is \(D\). The top of the pyramid, \(X\), is vertically above \(D\) and \(XD = 10\) units. The mid-point of \(OX\) is \(M\). The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(\overrightarrow{OA}\) and \(\overrightarrow{OC}\) respectively and the unit vector \(\mathbf{k}\) is vertically upwards.

1. Express the vectors \(\overrightarrow{AM}\) and \(\overrightarrow{AC}\) in terms of \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\). [3]
2. Use a scalar product to find angle \(MAC\). [4]

1. The sum, \(S_n\), of the first \(n\) terms of an arithmetic progression is given by \(S_n = 32n - n^2\). Find the first term and the common difference. [3]
2. A geometric progression in which all the terms are positive has sum to infinity 20. The sum of the first two terms is 12.8. Find the first term of the progression. [5]

\includegraphics{figure_9} The diagram shows a trapezium \(ABCD\) in which \(AB\) is parallel to \(DC\) and angle \(BAD\) is \(90°\). The coordinates of \(A\), \(B\) and \(C\) are \((2, 6)\), \((5, -3)\) and \((8, 3)\) respectively.

1. Find the equation of \(AD\). [3]
2. Find, by calculation, the coordinates of \(D\). [3]

The point \(E\) is such that \(ABCE\) is a parallelogram.

1. Find the length of \(BE\). [2]

A curve is such that \(\frac{d^2y}{dx^2} = \frac{24}{x^3} - 4\). The curve has a stationary point at \(P\) where \(x = 2\).

1. State, with a reason, the nature of this stationary point. [1]
2. Find an expression for \(\frac{dy}{dx}\). [4]
3. Given that the curve passes through the point \((1, 13)\), find the coordinates of the stationary point \(P\). [4]

The function \(f : x \mapsto 6 - 4\cos(\frac{1}{2}x)\) is defined for \(0 \leqslant x \leqslant 2\pi\).

1. Find the exact value of \(x\) for which \(f(x) = 4\). [3]
2. State the range of \(f\). [2]
3. Sketch the graph of \(y = f(x)\). [2]
4. Find an expression for \(f^{-1}(x)\). [3]