Pre-U Pre-U 9794/1 (Pre-U Mathematics Paper 1) 2017 June

1 The equation of a circle is given by \(( x - 3 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = r ^ { 2 }\).

1. Write down the coordinates of the centre of the circle.
2. The circle passes through the point \(( 0,2 )\). Find the length of the diameter.

2 Express each of the following as a single logarithm.

1. \(\log 3 + \log 4 - \log 2\)
2. \(2 \log x - 3 \log y + 2 \log z\)

3 A triangle \(A B C\) has sides \(A B , B C\) and \(C A\) of lengths \(7 \mathrm {~cm} , 6 \mathrm {~cm}\) and 8 cm respectively.

1. Show that \(\cos A B C = \frac { 1 } { 4 }\).
2. Find the area of triangle \(A B C\).

4 Solve the equation \(\sin 2 x = \sqrt { 3 } \cos x\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).

5 Solve \(| x - \sqrt { 3 } | < | x + 2 \sqrt { 3 } |\) giving the answer in exact form.

6

1. Expand \(( 1 + x ) ^ { \frac { 1 } { 2 } }\), for \(| x | < 1\), in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying the coefficients.
2. In the expansion of \(( 2 + k x ) ( 1 + x ) ^ { \frac { 1 } { 2 } }\) the coefficient of \(x ^ { 3 }\) is 1 . Find the value of \(k\).

7

1. Describe the transformation which maps the graph of \(y = \ln x\) onto the graph of \(y = \ln ( 1 + x )\).
2. By sketching the curves \(y = \ln ( 1 + x )\) and \(y = 4 - x\) on a single diagram, show that the equation $$\ln ( 1 + x ) = 4 - x$$ has exactly one root.
3. Use the Newton-Raphson method with \(x _ { 0 } = 2\) to find the root of the equation \(\ln ( 1 + x ) = 4 - x\) correct to 3 decimal places. Show the result of each iteration.

8 The curve \(C\) has equation \(y ^ { 3 } + 6 y ^ { 2 } - 2 y = 3 x ^ { 2 } + 2 x\). Show that the equation of the normal to \(C\) at the point \(( 1,1 )\) can be written in the form \(8 y + 13 x - 21 = 0\).

9 Solve the equation \(z ^ { 3 } + 6 z - 20 = 0\). Find the modulus and argument of each root and illustrate the roots on an Argand diagram.

10 \includegraphics[max width=\textwidth, alt={}, center]{a3cad2ad-e06b-4aa4-a3a9-a2840cd54893-3_529_527_264_810} The diagram shows the region \(R\) in the first quadrant bounded by the curves \(y = \frac { 1 } { 3 } \left( 9 - x ^ { 2 } \right)\) and \(y = \frac { 1 } { 5 } \left( 9 - x ^ { 2 } \right)\). \(R\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. Calculate the volume of the solid formed.

11 The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }\) and \(3 \mathbf { i } - 2 \mathbf { j } - \mathbf { k }\) respectively, relative to the origin \(O\). The point \(P\) lies on \(O A\) extended so that \(\overrightarrow { O P } = 3 \overrightarrow { O A }\) and the point \(Q\) lies on \(O B\) extended so that \(\overrightarrow { O Q } = 2 \overrightarrow { O B }\).

1. Find the coordinates of the point of intersection of the lines \(A Q\) and \(B P\).
2. Find the acute angle between the lines \(A Q\) and \(B P\).

12 Boyle's Law states that when a gas is kept at a constant temperature, its pressure \(P\) pascals is inversely proportional to its volume \(V \mathrm {~m} ^ { 3 }\). When the volume of a certain gas is \(80 \mathrm {~m} ^ { 3 }\), its pressure is 5 pascals and the rate at which the volume is increasing is \(10 \mathrm {~m} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate at which the pressure is decreasing at this volume.