Edexcel C3 (Core Mathematics 3)

1. (a) Simplify

$$\frac { x ^ { 2 } + 7 x + 12 } { 2 x ^ { 2 } + 9 x + 4 }$$ (b) Solve the equation $$\ln \left( x ^ { 2 } + 7 x + 12 \right) - 1 = \ln \left( 2 x ^ { 2 } + 9 x + 4 \right)$$ giving your answer in terms of e.

2. A curve has the equation \(y = \sqrt { 3 x + 11 }\). The point \(P\) on the curve has \(x\)-coordinate 3 .

1. Show that the tangent to the curve at \(P\) has the equation $$3 x - 4 \sqrt { 5 } y + 31 = 0$$ The normal to the curve at \(P\) crosses the \(y\)-axis at \(Q\).
2. Find the \(y\)-coordinate of \(Q\) in the form \(k \sqrt { 5 }\).

3. (a) Use the identities for \(\sin ( A + B )\) and \(\sin ( A - B )\) to prove that $$\sin P + \sin Q \equiv 2 \sin \frac { P + Q } { 2 } \cos \frac { P - Q } { 2 } \text {. }$$ (b) Find, in terms of \(\pi\), the solutions of the equation $$\sin 5 x + \sin x = 0$$ for \(x\) in the interval \(0 \leq x < \pi\).

4. The curve with equation \(y = x ^ { \frac { 5 } { 2 } } \ln \frac { x } { 4 } , x > 0\) crosses the \(x\)-axis at the point \(P\).

1. Write down the coordinates of \(P\). The normal to the curve at \(P\) crosses the \(y\)-axis at the point \(Q\).
2. Find the area of triangle \(O P Q\) where \(O\) is the origin. The curve has a stationary point at \(R\).
3. Find the \(x\)-coordinate of \(R\) in exact form.

5. $$\mathrm { f } ( x ) \equiv 2 x ^ { 2 } + 4 x + 2 , \quad x \in \mathbb { R } , \quad x \geq - 1 .$$

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. Describe fully two transformations that would map the graph of \(y = x ^ { 2 } , x \geq 0\) onto the graph of \(y = \mathrm { f } ( x )\).
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
4. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram and state the relationship between them.

6. $$\mathrm { f } ( x ) = \mathrm { e } ^ { 3 x + 1 } - 2 , \quad x \in \mathbb { R } .$$

1. State the range of f . The curve \(y = \mathrm { f } ( x )\) meets the \(y\)-axis at the point \(P\) and the \(x\)-axis at the point \(Q\).
2. Find the exact coordinates of \(P\) and \(Q\).
3. Show that the tangent to the curve at \(P\) has the equation $$y = 3 \mathrm { e } x + \mathrm { e } - 2 .$$
4. Find to 3 significant figures the \(x\)-coordinate of the point where the tangent to the curve at \(P\) meets the tangent to the curve at \(Q\).

7. (a) Solve the equation $$\pi - 3 \arccos \theta = 0$$ (b) Sketch on the same diagram the curves \(y = \arccos ( x - 1 ) , 0 \leq x \leq 2\) and \(y = \sqrt { x + 2 } , x \geq - 2\). Given that \(\alpha\) is the root of the equation $$\arccos ( x - 1 ) = \sqrt { x + 2 }$$ (c) show that \(0 < \alpha < 1\),
(d) use the iterative formula $$x _ { n + 1 } = 1 + \cos \sqrt { x _ { n } + 2 }$$ with \(x _ { 0 } = 1\) to find \(\alpha\) correct to 3 decimal places. END