OCR C3 (Core Mathematics 3) 2006 January

1 Show that \(\int _ { 2 } ^ { 8 } \frac { 3 } { x } \mathrm {~d} x = \ln 64\).

2 Solve, for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), the equation \(\sec ^ { 2 } \theta = 4 \tan \theta - 2\).

3

1. Differentiate \(x ^ { 2 } ( x + 1 ) ^ { 6 }\) with respect to \(x\).
2. Find the gradient of the curve \(y = \frac { x ^ { 2 } + 3 } { x ^ { 2 } - 3 }\) at the point where \(x = 1\).

4
\includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-2_529_737_900_701} The function f is defined by \(\mathrm { f } ( x ) = 2 - \sqrt { x }\) for \(x \geqslant 0\). The graph of \(y = \mathrm { f } ( x )\) is shown above.

1. State the range of f.
2. Find the value of \(\mathrm { ff } ( 4 )\).
3. Given that the equation \(| \mathrm { f } ( x ) | = k\) has two distinct roots, determine the possible values of the constant \(k\).

5
\includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-2_486_746_1978_696} The diagram shows the curves \(y = ( 1 - 2 x ) ^ { 5 }\) and \(y = \mathrm { e } ^ { 2 x - 1 } - 1\). The curves meet at the point \(\left( \frac { 1 } { 2 } , 0 \right)\). Find the exact area of the region (shaded in the diagram) bounded by the \(y\)-axis and by part of each curve.

6

1. \(t\)	0	10	20
   \(X\)	275	440

   The quantity \(X\) is increasing exponentially with respect to time \(t\). The table above shows values of \(X\) for different values of \(t\). Find the value of \(X\) when \(t = 20\).
2. The quantity \(Y\) is decreasing exponentially with respect to time \(t\) where $$Y = 80 \mathrm { e } ^ { - 0.02 t }$$
   1. Find the value of \(t\) for which \(Y = 20\), giving your answer correct to 2 significant figures.
   2. Find by differentiation the rate at which \(Y\) is decreasing when \(t = 30\), giving your answer correct to 2 significant figures.

7
\includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-3_465_748_1133_717} The diagram shows the curve with equation \(y = \cos ^ { - 1 } x\).

1. Sketch the curve with equation \(y = 3 \cos ^ { - 1 } ( x - 1 )\), showing the coordinates of the points where the curve meets the axes.
2. By drawing an appropriate straight line on your sketch in part (i), show that the equation \(3 \cos ^ { - 1 } ( x - 1 ) = x\) has exactly one root.
3. Show by calculation that the root of the equation \(3 \cos ^ { - 1 } ( x - 1 ) = x\) lies between 1.8 and 1.9 .
4. The sequence defined by $$x _ { 1 } = 2 , \quad x _ { n + 1 } = 1 + \cos \left( \frac { 1 } { 3 } x _ { n } \right)$$ converges to a number \(\alpha\). Find the value of \(\alpha\) correct to 2 decimal places and explain why \(\alpha\) is the root of the equation \(3 \cos ^ { - 1 } ( x - 1 ) = x\).

8
\includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-4_787_742_276_719} The diagram shows part of the curve \(y = \ln \left( 5 - x ^ { 2 } \right)\) which meets the \(x\)-axis at the point \(P\) with coordinates \(( 2,0 )\). The tangent to the curve at \(P\) meets the \(y\)-axis at the point \(Q\). The region \(A\) is bounded by the curve and the lines \(x = 0\) and \(y = 0\). The region \(B\) is bounded by the curve and the lines \(P Q\) and \(x = 0\).

1. Find the equation of the tangent to the curve at \(P\).
2. Use Simpson's Rule with four strips to find an approximation to the area of the region \(A\), giving your answer correct to 3 significant figures.
3. Deduce an approximation to the area of the region \(B\).