OCR C3 (Core Mathematics 3)

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

$$| 2 x - 3 | > | x + 2 |$$

1. Find, to 2 decimal places, the solutions of the equation

$$3 \cot ^ { 2 } x - 4 \operatorname { cosec } x + \operatorname { cosec } ^ { 2 } x = 0$$ in the interval \(0 \leq x \leq 2 \pi\).

3. A curve has the equation \(x = y ^ { 2 } - 3 \ln 2 y\).

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { 2 y ^ { 2 } - 3 }$$
2. Find an equation for the tangent to the curve at the point where \(y = \frac { 1 } { 2 }\). Give your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

4. (i) Use Simpson's rule with four intervals, each of width 0.25 , to estimate the value of the integral $$\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { 2 x } \mathrm {~d} x$$ (ii) Find the exact value of the integral $$\int _ { \frac { 1 } { 2 } } ^ { 1 } e ^ { 1 - 2 x } d x$$

5. The diagram shows the curve with equation \(y = \frac { 1 } { \sqrt { 3 x + 1 } }\).
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 5\).

1. Find the area of the shaded region. The shaded region is rotated through four right angles about the \(x\)-axis.
2. Find the volume of the solid formed, giving your answer in the form \(k \pi \ln 2\).

6.
\includegraphics[max width=\textwidth, alt={}, center]{14a2477a-c40e-4b4b-bc39-7100d1df9b4d-2_397_488_1299_632} The diagram shows a vertical cross-section through a vase.
The inside of the vase is in the shape of a right-circular cone with the angle between the sides in the cross-section being \(60 ^ { \circ }\). When the depth of water in the vase is \(h \mathrm {~cm}\), the volume of water in the vase is \(V \mathrm {~cm} ^ { 3 }\).

1. Show that \(V = \frac { 1 } { 9 } \pi h ^ { 3 }\). The vase is initially empty and water is poured in at a constant rate of \(120 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\).
2. Find, to 2 decimal places, the rate at which \(h\) is increasing
   1. when \(h = 6\),
   2. after water has been poured in for 8 seconds.

7. (i) Prove that, for \(\cos x \neq 0\), $$\sin 2 x - \tan x \equiv \tan x \cos 2 x$$ (ii) Hence, or otherwise, solve the equation $$\sin 2 x - \tan x = 2 \cos 2 x$$ for \(x\) in the interval \(0 \leq x \leq 180 ^ { \circ }\).

8. A rock contains a radioactive substance which is decaying. The mass of the rock, \(m\) grams, at time \(t\) years after initial observation is given by $$m = 400 + 80 \mathrm { e } ^ { - k t }$$ where \(k\) is a positive constant.
Given that the mass of the rock decreases by \(0.2 \%\) in the first 10 years, find

1. the value of \(k\),
2. the value of \(t\) when \(m = 475\),
3. the rate at which the mass of the rock is decreasing when \(t = 100\).

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

1. State the range of f .
2. Find the exact value of \(\mathrm { ff } ( 0 )\).
3. Define the inverse function \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain. Given that \(\alpha\) is the solution of the equation \(\mathrm { f } ( x ) = \mathrm { f } ^ { - 1 } ( x )\),
4. explain why \(\alpha\) satisfies the equation $$x = \mathrm { f } ^ { - 1 } ( x )$$
5. use the iterative formula $$x _ { n + 1 } = \mathrm { f } ^ { - 1 } \left( x _ { n } \right)$$ with \(x _ { 0 } = 0.5\) to find \(\alpha\) correct to 3 significant figures.