OCR C3 (Core Mathematics 3) 2008 June

1 Find the exact solutions of the equation \(| 4 x - 5 | = | 3 x - 5 |\).

2
\includegraphics[max width=\textwidth, alt={}, center]{5c501214-b41c-43a8-b9c6-986758e83e7d-2_529_855_397_646} The diagram shows the graph of \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ( - 3 ) = 0\) and \(\mathrm { f } ( 0 ) = 2\). Sketch, on separate diagrams, the following graphs, indicating in each case the coordinates of the points where the graph crosses the axes:

1. \(y = \mathrm { f } ^ { - 1 } ( x )\),
2. \(y = - 2 \mathrm { f } ( x )\).

3 Find, in the form \(y = m x + c\), the equation of the tangent to the curve $$y = x ^ { 2 } \ln x$$ at the point with \(x\)-coordinate e.

4 The gradient of the curve \(y = \left( 2 x ^ { 2 } + 9 \right) ^ { \frac { 5 } { 2 } }\) at the point \(P\) is 100 .

1. Show that the \(x\)-coordinate of \(P\) satisfies the equation \(x = 10 \left( 2 x ^ { 2 } + 9 \right) ^ { - \frac { 3 } { 2 } }\).
2. Show by calculation that the \(x\)-coordinate of \(P\) lies between 0.3 and 0.4 .
3. Use an iterative formula, based on the equation in part (i), to find the \(x\)-coordinate of \(P\) correct to 4 decimal places. You should show the result of each iteration.

5

1. Express \(\tan 2 \alpha\) in terms of \(\tan \alpha\) and hence solve, for \(0 ^ { \circ } < \alpha < 180 ^ { \circ }\), the equation $$\tan 2 \alpha \tan \alpha = 8 .$$
2. Given that \(\beta\) is the acute angle such that \(\sin \beta = \frac { 6 } { 7 }\), find the exact value of
   1. \(\operatorname { cosec } \beta\),
   2. \(\cot ^ { 2 } \beta\).

6
\includegraphics[max width=\textwidth, alt={}, center]{5c501214-b41c-43a8-b9c6-986758e83e7d-3_586_798_267_676} The diagram shows the curves \(y = \mathrm { e } ^ { 3 x }\) and \(y = ( 2 x - 1 ) ^ { 4 }\). The shaded region is bounded by the two curves and the line \(x = \frac { 1 } { 2 }\). The shaded region is rotated completely about the \(x\)-axis. Find the exact volume of the solid produced.

7 It is claimed that the number of plants of a certain species in a particular locality is doubling every 9 years. The number of plants now is 42 . The number of plants is treated as a continuous variable and is denoted by \(N\). The number of years from now is denoted by \(t\).

1. Two equivalent expressions giving \(N\) in terms of \(t\) are $$N = A \times 2 ^ { k t } \quad \text { and } \quad N = A \mathrm { e } ^ { m t } .$$ Determine the value of each of the constants \(A , k\) and \(m\).
2. Find the value of \(t\) for which \(N = 100\), giving your answer correct to 3 significant figures.
3. Find the rate at which the number of plants will be increasing at a time 35 years from now.

8 The expression \(\mathrm { T } ( \theta )\) is defined for \(\theta\) in degrees by $$\mathrm { T } ( \theta ) = 3 \cos \left( \theta - 60 ^ { \circ } \right) + 2 \cos \left( \theta + 60 ^ { \circ } \right) .$$

1. Express \(\mathrm { T } ( \theta )\) in the form \(A \sin \theta + B \cos \theta\), giving the exact values of the constants \(A\) and \(B\). [3]
2. Hence express \(\mathrm { T } ( \theta )\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
3. Find the smallest positive value of \(\theta\) such that \(\mathrm { T } ( \theta ) + 1 = 0\).

9
\includegraphics[max width=\textwidth, alt={}, center]{5c501214-b41c-43a8-b9c6-986758e83e7d-4_534_935_264_605} The function f is defined for the domain \(x \geqslant 0\) by $$f ( x ) = \frac { 15 x } { x ^ { 2 } + 5 }$$ The diagram shows the curve with equation \(y = \mathrm { f } ( x )\).

1. Find the range of f .
2. The function g is defined for the domain \(x \geqslant k\) by $$\mathrm { g } ( x ) = \frac { 15 x } { x ^ { 2 } + 5 }$$ Given that g is a one-one function, state the least possible value of \(k\).
3. Show that there is no point on the curve \(y = \mathrm { g } ( x )\) at which the gradient is - 1 .