\(\mathbf { 1 }\) & \(\mathbf { 2 }\)
\hline
\end{tabular}
\end{center} a) Solve the equation \(2 x ^ { 3 } - x ^ { 2 } - 5 x - 2 = 0\).
b) Find all values of \(\theta\) in the range \(0 ^ { \circ } < \theta < 180 ^ { \circ }\) satisfying
$$\cos \left( 2 \theta - 51 ^ { \circ } \right) = 0 \cdot 891$$
| \(\mathbf { 1 }\) | \(\mathbf { 3 }\) |
Find the term which is independent of \(x\) in the expansion of \(\frac { ( 2 - 3 x ) ^ { 5 } } { x ^ { 3 } }\).
| \(\mathbf { 1 }\) | \(\mathbf { 4 }\) |
A curve \(C\) has equation \(f ( x ) = 3 x ^ { 3 } - 5 x ^ { 2 } + x - 6\).
a) Find the coordinates of the stationary points of \(C\) and determine their nature.
b) Without solving the equations, determine the number of distinct real roots for each of the following:
i) \(3 x ^ { 3 } - 5 x ^ { 2 } + x + 1 = 0\),
ii) \(\quad 6 x ^ { 3 } - 10 x ^ { 2 } + 2 x + 1 = 0\).
| \(\mathbf { 1 }\) | \(\mathbf { 5 }\) |
Solve the simultaneous equations
$$\begin{aligned}
& 3 \log _ { a } \left( x ^ { 2 } y \right) - \log _ { a } \left( x ^ { 2 } y ^ { 2 } \right) + \log _ { a } \left( \frac { 9 } { x ^ { 2 } y ^ { 2 } } \right) = \log _ { a } 36
& \log _ { a } y - \log _ { a } ( x + 3 ) = 0
\end{aligned}$$
| \(\mathbf { 1 }\) | \(\mathbf { 6 }\) |
The vectors \(\mathbf { a }\) and \(\mathbf { b }\) are defined by \(\mathbf { a } = 2 \mathbf { i } - \mathbf { j }\) and \(\mathbf { b } = \mathbf { i } - 3 \mathbf { j }\).
a) Find a unit vector in the direction of \(\mathbf { a }\).
b) Determine the angle \(\mathbf { b }\) makes with the \(x\)-axis.
c) The vector \(\mu \mathbf { a } + \mathbf { b }\) is parallel to \(4 \mathbf { i } - 5 \mathbf { j }\).
i) Find the vector \(\mu \mathbf { a } + \mathbf { b }\) in terms of \(\mu , \mathbf { i }\) and j.
ii) Determine the value of \(\mu\).