OCR MEI C3 (Core Mathematics 3)

1 Prove that the product of consecutive integers is always even.

2 Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \sqrt { 1 + x ^ { 3 } }\).

3 The graph shows part of the function \(y = a \ln ( b x )\).
\includegraphics[max width=\textwidth, alt={}, center]{2f403099-2813-40d8-a9ae-1f7e64d41f80-2_377_762_900_685} The graph passes through the points \(( 2,0 )\) and \(( 4,1 )\).

1. Show that \(b = \frac { 1 } { 2 }\) and find the exact value of \(a\).
2. Solve the inequality \(| a \ln ( b x ) | < 2\).

4

1. Show that \(y = a x e ^ { - x }\) for \(a > 0\) has only one stationary point for all values of \(x\). Determine whether this stationary value is a maximum or minimum point.
2. Sketch the curve.

5 Find \(\int _ { 2 } ^ { 3 } x \mathrm { e } ^ { 2 x } \mathrm {~d} x\), giving your answer to 1 decimal place.

6 Find \(\frac { \mathrm { d } } { \mathrm { d } x } ( x \ln x )\) and hence or otherwise find the value of \(\int _ { 2 } ^ { 3 } \ln x \mathrm {~d} x\), giving your answer in the form \(\ln a + b\), where \(a\) and \(b\) are to be determined.

7 Two quantities, \(x\) and \(\theta\), vary with time and are related by the equation \(x = 5 \sin \theta - 4 \cos \theta\).

1. Find the value of \(x\) when \(\theta = \frac { \pi } { 2 }\).
2. When \(\theta = \frac { \pi } { 2 }\), its rate of increase (in suitable units) is given by \(\frac { \mathrm { d } \theta } { \mathrm { d } t } = 0.1\). Show that at that moment \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0.4\).

8 You are given that \(\mathrm { f } ( x ) = \frac { x } { x ^ { 2 } + 1 }\) for all real values of \(x\).

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 1 - x ^ { 2 } } { \left( x ^ { 2 } + 1 \right) ^ { 2 } }\).
2. Hence show that there is a stationary value at \(\left( 1 , \frac { 1 } { 2 } \right)\) and find the coordinates of the other stationary point.
3. The graph of the curve is shown in Fig. 8. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2f403099-2813-40d8-a9ae-1f7e64d41f80-3_518_892_1612_705} \captionsetup{labelformat=empty} \caption{Fig. 8}
   \end{figure} State whether the curve is odd or even and prove the result algebraically.
4. Show that \(\int _ { 1 } ^ { 4 } \frac { x } { x ^ { 2 } + 1 } \mathrm {~d} x = \int _ { a } ^ { b } k \frac { 1 } { u + 1 } \mathrm {~d} u\), where the values of \(a , b\) and \(k\) are to be determined.
5. Hence find the area of the shaded region in Fig. 8.

9 The curve in Fig. 9.1 has equation \(\sqrt { x } + \sqrt { y } = 1\). \begin{figure}[h] \end{figure}

1. Show that this is part, but not all of the curve \(y = 1 - 2 \sqrt { x } + x\). Sketch the full curve \(y = 1 - 2 \sqrt { x } + x\).
2. Fig.9.2 shows a star shape made up of four parts, one of which is given in part (i) above. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2f403099-2813-40d8-a9ae-1f7e64d41f80-4_380_681_1197_651} \captionsetup{labelformat=empty} \caption{Fig. 9.2}
   \end{figure} For each of the sections of the shape labelled \(\mathrm { A } , \mathrm { B }\) and C , state the equation of the curve and the domain.
3. The shape shown in Fig.9.2 is made into that in Fig. 10.3 by stretching the part of the figure for which \(y > 0\) by a scale factor of 2 . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2f403099-2813-40d8-a9ae-1f7e64d41f80-4_405_686_1996_605} \captionsetup{labelformat=empty} \caption{Fig. 9.3}
   \end{figure} Find the area of this shape.