Verify, factorise, solve with substitution

6 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 6 x ^ { 3 } + a x ^ { 2 } - 4 x - 3$$ where \(a\) is a constant. It is given that \(( x + 3 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. Using this value of \(a\), factorise \(\mathrm { p } ( x )\) completely.
3. Hence solve the equation \(\mathrm { p } ( \operatorname { cosec } \theta ) = 0\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

3 The polynomial \(\mathrm { p } ( x )\) is defined by $$p ( x ) = 6 x ^ { 3 } + a x ^ { 2 } + 3 x - 10$$ where \(a\) is a constant. It is given that \(( 2 x - 1 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\) and hence factorise \(\mathrm { p } ( x )\) completely.
2. Solve the equation \(\mathrm { p } ( \operatorname { cosec } \theta ) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
   \includegraphics[max width=\textwidth, alt={}, center]{7b39a2ab-305d-43c5-a1e7-9442d6c13886-06_442_706_278_667} The diagram shows the curve with equation \(\mathrm { y } = \sqrt { 1 + \mathrm { e } ^ { 0.5 \mathrm { x } } }\). The shaded region is bounded by the curve and the straight lines \(x = 0 , x = 6\) and \(y = 0\).
3. Use the trapezium rule with three intervals to find an approximation to the area of the shaded region. Give your answer correct to 3 significant figures.
4. The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = a x ^ { 3 } - a x ^ { 2 } - 15 x + 18$$ where \(a\) is a constant. It is given that ( \(x + 2\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. Hence factorise \(\mathrm { p } ( x )\) completely.
   
3. Solve the equation \(\mathrm { p } \left( \operatorname { cosec } ^ { 2 } \theta \right) = 0\) for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).

12

1. Prove that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\)
   12
2. Factorise \(\mathrm { p } ( x )\) completely.
   12
3. Prove that there are no real solutions to the equation $$\frac { 30 \sec ^ { 2 } x + 2 \cos x } { 7 } = \sec x + 1$$