OCR C3 (Core Mathematics 3)

1. Show that

$$\int _ { 1 } ^ { 7 } \frac { 2 } { 4 x - 1 } \mathrm {~d} x = \ln 3$$

1. Find the set of values of \(x\) such that

$$| 3 x + 1 | \leq | x - 2 |$$

1. Find all values of \(\theta\) in the interval \(- 180 < \theta < 180\) for which

$$\tan ^ { 2 } \theta ^ { \circ } + \sec \theta ^ { \circ } = 1$$

1. Solve each equation, giving your answers in exact form.
   1. \(\mathrm { e } ^ { 4 x - 3 } = 2\)
   2. \(\quad \ln ( 2 y - 1 ) = 1 + \ln ( 3 - y )\)
   3. (i) Prove, by counter-example, that the statement
      "cosec \(\theta - \sin \theta > 0\) for all values of \(\theta\) in the interval \(0 < \theta < \pi\) " is false.
   4. Find the values of \(\theta\) in the interval \(0 < \theta < \pi\) such that
   $$\operatorname { cosec } \theta - \sin \theta = 2$$ giving your answers to 2 decimal places.

6. 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 \(C\).

7.
\includegraphics[max width=\textwidth, alt={}, center]{6cdbc2bc-8863-4003-a218-44552d75d137-2_556_777_246_468} The diagram shows the curve \(y = \mathrm { f } ( x )\) which has a maximum point at ( \(- 45,7\) ) and a minimum point at \(( 135 , - 1 )\).

1. Showing the coordinates of any stationary points, sketch the curve with equation \(y = 1 + 2 \mathrm { f } ( x )\). Given that $$f ( x ) = A + 2 \sqrt { 2 } \cos x ^ { \circ } - 2 \sqrt { 2 } \sin x ^ { \circ } , \quad x \in \mathbb { R } , \quad - 180 \leq x \leq 180 ,$$ where \(A\) is a constant,
2. show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = A + R \cos ( x + \alpha ) ^ { \circ }$$ where \(R > 0\) and \(0 < \alpha < 90\),
3. state the value of \(A\),
4. find, to 1 decimal place, the \(x\)-coordinates of the points where the curve \(y = \mathrm { f } ( x )\) crosses the \(x\)-axis.

8. The function f is defined by $$\mathrm { f } ( x ) \equiv 3 - x ^ { 2 } , \quad x \in \mathbb { R } , \quad x \geq 0 .$$

1. State the range of f .
2. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram.
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. The function g is defined by $$\mathrm { g } ( x ) \equiv \frac { 8 } { 3 - x } , \quad x \in \mathbb { R } , \quad x \neq 3$$
4. Evaluate fg(-3).
5. Solve the equation $$\mathrm { f } ^ { - 1 } ( x ) = \mathrm { g } ( x )$$

1. A curve has the equation \(y = ( 2 x + 3 ) \mathrm { e } ^ { - x }\).
   1. Find the exact coordinates of the stationary point of the curve.
   The curve crosses the \(y\)-axis at the point \(P\).
2. Find an equation for the normal to the curve at \(P\). The normal to the curve at \(P\) meets the curve again at \(Q\).
3. Show that the \(x\)-coordinate of \(Q\) lies between - 2 and - 1 .
4. Use the iterative formula $$x _ { n + 1 } = \frac { 3 - 3 \mathrm { e } ^ { x _ { n } } } { \mathrm { e } ^ { x _ { n } } - 2 }$$ with \(x _ { 0 } = - 1\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Give the value of \(x _ { 4 }\) to 2 decimal places.
5. Show that your value for \(x _ { 4 }\) is the \(x\)-coordinate of \(Q\) correct to 2 decimal places.