OCR C3 (Core Mathematics 3) 2013 January

1 For each of the following curves, find the gradient at the point with \(x\)-coordinate 2 .

1. \(y = \frac { 3 x } { 2 x + 1 }\)
2. \(y = \sqrt { 4 x ^ { 2 } + 9 }\)

2 The acute angle \(A\) is such that \(\tan A = 2\).

1. Find the exact value of \(\operatorname { cosec } A\).
2. The angle \(B\) is such that \(\tan ( A + B ) = 3\). Using an appropriate identity, find the exact value of \(\tan B\).

3

1. Given that \(| t | = 3\), find the possible values of \(| 2 t - 1 |\).
2. Solve the inequality \(| x - \sqrt { 2 } | > | x + 3 \sqrt { 2 } |\).

4 The mass, \(m\) grams, of a substance is increasing exponentially so that the mass at time \(t\) hours is given by $$m = 250 \mathrm { e } ^ { 0.021 t } .$$

1. Find the time taken for the mass to increase to twice its initial value, and deduce the time taken for the mass to increase to 8 times its initial value.
2. Find the rate at which the mass is increasing at the instant when the mass is 400 grams.

5
\includegraphics[max width=\textwidth, alt={}, center]{b8ff33d4-dfe5-4067-855e-86d5765cc249-2_454_770_1628_635} The diagram shows the curve \(y = \frac { 6 } { \sqrt { 3 x + 1 } }\). The shaded region is bounded by the curve and the lines \(x = 2 , x = 9\) and \(y = 0\).

1. Show that the area of the shaded region is \(4 \sqrt { 7 }\) square units.
2. The shaded region is rotated completely about the \(x\)-axis. Show that the volume of the solid produced can be written in the form \(k \ln 2\), where the exact value of the constant \(k\) is to be determined.

7

1. By sketching the curves \(y = \ln x\) and \(y = 8 - 2 x ^ { 2 }\) on a single diagram, show that the equation $$\ln x = 8 - 2 x ^ { 2 }$$ has exactly one real root.
2. Explain how your diagram shows that the root is between 1 and 2 .
3. Use the iterative formula $$x _ { n + 1 } = \sqrt { 4 - \frac { 1 } { 2 } \ln x _ { n } } ,$$ with a suitable starting value, to find the root. Show all your working and give the root correct to 3 decimal places.
4. The curves \(y = \ln x\) and \(y = 8 - 2 x ^ { 2 }\) are each translated by 2 units in the positive \(x\)-direction and then stretched by scale factor 4 in the \(y\)-direction. Find the coordinates of the point where the new curves intersect, giving each coordinate correct to 2 decimal places.
   \includegraphics[max width=\textwidth, alt={}, center]{b8ff33d4-dfe5-4067-855e-86d5765cc249-3_389_917_1117_557} The diagram shows the curve with equation $$x = ( y + 4 ) \ln ( 2 y + 3 ) .$$ The curve crosses the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
5. Find an expression for \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) in terms of \(y\).
6. Find the gradient of the curve at each of the points \(A\) and \(B\), giving each answer correct to 2 decimal places.

8 The functions f and g are defined for all real values of \(x\) by $$\mathrm { f } ( x ) = x ^ { 2 } + 4 a x + a ^ { 2 } \text { and } \mathrm { g } ( x ) = 4 x - 2 a ,$$ where \(a\) is a positive constant.

1. Find the range of f in terms of \(a\).
2. Given that \(\mathrm { fg } ( 3 ) = 69\), find the value of \(a\) and hence find the value of \(x\) such that \(\mathrm { g } ^ { - 1 } ( x ) = x\).

9

1. Prove that $$\cos ^ { 2 } \left( \theta + 45 ^ { \circ } \right) - \frac { 1 } { 2 } ( \cos 2 \theta - \sin 2 \theta ) \equiv \sin ^ { 2 } \theta .$$
2. Hence solve the equation $$6 \cos ^ { 2 } \left( \frac { 1 } { 2 } \theta + 45 ^ { \circ } \right) - 3 ( \cos \theta - \sin \theta ) = 2$$ for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).
3. It is given that there are two values of \(\theta\), where \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\), satisfying the equation $$6 \cos ^ { 2 } \left( \frac { 1 } { 3 } \theta + 45 ^ { \circ } \right) - 3 \left( \cos \frac { 2 } { 3 } \theta - \sin \frac { 2 } { 3 } \theta \right) = k ,$$ where \(k\) is a constant. Find the set of possible values of \(k\).