OCR C3 (Core Mathematics 3)

1. Evaluate

$$\int _ { 2 } ^ { 6 } \sqrt { 3 x - 2 } \mathrm {~d} x$$

1. Differentiate each of the following with respect to \(x\) and simplify your answers.
   1. \(\frac { 6 } { \sqrt { 2 x - 7 } }\)
   2. \(x ^ { 2 } \mathrm { e } ^ { - x }\)
   3. (i) Prove the identity
   $$\sqrt { 2 } \cos ( x + 45 ) ^ { \circ } + 2 \cos ( x - 30 ) ^ { \circ } \equiv ( 1 + \sqrt { 3 } ) \cos x ^ { \circ }$$
2. Hence, find the exact value of \(\cos 75 ^ { \circ }\) in terms of surds.

4. \(\mathrm { f } ( x ) = x ^ { 2 } + 5 x - 2\) sec \(x , \quad x \in \mathbb { R } , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).

1. Show that the equation \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), such that \(1 < \alpha < 1.5\)
2. Show that a suitable rearrangement of the equation \(\mathrm { f } ( x ) = 0\) leads to the iterative formula $$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 2 } { x _ { n } ^ { 2 } + 5 x _ { n } } \right)$$
3. Use the iterative formula in part (ii) with a starting value of 1.25 to find \(\alpha\) correct to 3 decimal places. You should show the result of each iteration.

5. The function \(f\) is defined by $$f ( x ) \equiv 2 + \ln ( 3 x - 2 ) , \quad x \in \mathbb { R } , \quad x > \frac { 2 } { 3 }$$

1. Find the exact value of \(\mathrm { ff } ( 1 )\).
2. Find an equation for the tangent to the curve \(y = \mathrm { f } ( x )\) at the point where \(x = 1\).
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

6. (i) Sketch on the same diagram the graphs of \(y = | x | - a\) and \(y = | 3 x + 5 a |\), where \(a\) is a positive constant. Show on your diagram the coordinates of any points where each graph meets the coordinate axes.
(ii) Solve the equation $$| x | - a = | 3 x + 5 a |$$

7. The diagram shows the curve with equation \(y = 2 x - \mathrm { e } ^ { \frac { 1 } { 2 } x }\).
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 2\) and \(x = 4\).

1. Find the area of the shaded region, giving your answer in terms of e. The shaded region is rotated through four right angles about the \(x\)-axis.
2. Using Simpson's rule with two strips, estimate the volume of the solid formed.

8. (i) Sketch on the same diagram the graphs of $$y = \sin ^ { - 1 } x , - 1 \leq x \leq 1$$ and $$y = \cos ^ { - 1 } ( 2 x ) , \quad - \frac { 1 } { 2 } \leq x \leq \frac { 1 } { 2 }$$ Given that the graphs intersect at the point with coordinates \(( a , b )\),
(ii) show that \(\tan b = \frac { 1 } { 2 }\),
(iii) find the value of \(a\) in the form \(k \sqrt { 5 }\).

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

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 $$y = 3 e x + e - 2$$
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 \(Q\).