Edexcel C3 (Core Mathematics 3)

2. \begin{figure}[h] \end{figure} Figure 1 shows the curves \(y = 3 + 2 \mathrm { e } ^ { x }\) and \(y = \mathrm { e } ^ { x + 2 }\) which cross the \(y\)-axis at the points \(A\) and \(B\) respectively.

1. Find the exact length \(A B\). The two curves intersect at the point \(C\).
2. Find an expression for the \(x\)-coordinate of \(C\) and show that the \(y\)-coordinate of \(C\) is \(\frac { 3 \mathrm { e } ^ { 2 } } { \mathrm { e } ^ { 2 } - 2 }\).

3. $$f ( x ) = \frac { x ^ { 2 } + 3 } { 4 x + 1 } , \quad x \in \mathbb { R } , \quad x \neq - \frac { 1 } { 4 }$$

1. Find and simplify an expression for \(\mathrm { f } ^ { \prime } ( x )\).
2. Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.

4. The curve \(C\) has the equation \(y = x ^ { 2 } - 5 x + 2 \ln \frac { x } { 3 } , x > 0\).

1. Show that the normal to \(C\) at the point where \(x = 3\) has the equation $$3 x + 5 y + 21 = 0$$
2. Find the \(x\)-coordinates of the stationary points of \(C\).

5. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) \equiv 6 x - 1 , \quad x \in \mathbb { R } ,
& \mathrm {~g} ( x ) \equiv \log _ { 2 } ( 3 x + 1 ) , \quad x \in \mathbb { R } , \quad x > - \frac { 1 } { 3 } \end{aligned}$$

1. Evaluate \(\operatorname { gf } ( 1 )\).
2. Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\).
3. Find, in terms of natural logarithms, the solution of the equation $$\mathrm { fg } ^ { - 1 } ( x ) = 2$$

1. (a) Use the identities for \(\cos ( A + B )\) and \(\cos ( A - B )\) to prove that

$$\cos P - \cos Q \equiv - 2 \sin \frac { P + Q } { 2 } \sin \frac { P - Q } { 2 }$$ (b) Hence find all solutions in the interval \(0 \leq x < 180\) to the equation $$\cos 5 x ^ { \circ } + \sin 3 x ^ { \circ } - \cos x ^ { \circ } = 0$$ Turn over

7. The function f is defined by $$\mathrm { f } ( x ) \equiv x ^ { 2 } - 2 a x , \quad x \in \mathbb { R } ,$$ where \(a\) is a positive constant.

1. Showing the coordinates of any points where each graph meets the axes, sketch on separate diagrams the graphs of
   1. \(\quad y = | \mathrm { f } ( x ) |\),
   2. \(y = \mathrm { f } ( | x | )\). The function g is defined by $$\mathrm { g } ( x ) \equiv 3 a x , \quad x \in \mathbb { R } .$$
2. Find fg(a) in terms of \(a\).
3. Solve the equation $$\operatorname { gf } ( x ) = 9 a ^ { 3 }$$

8. $$f ( x ) = 2 x + \sin x - 3 \cos x$$

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root in the interval [0.7, 0.8].
2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where it crosses the \(y\)-axis.
3. Find the values of the constants \(a , b\) and \(c\), where \(b > 0\) and \(0 < c < \frac { \pi } { 2 }\), such that $$f ^ { \prime } ( x ) = a + b \cos ( x - c )$$
4. Hence find the \(x\)-coordinates of the stationary points of the curve \(y = \mathrm { f } ( x )\) in the interval \(0 \leq x \leq 2 \pi\), giving your answers to 2 decimal places.