CSEC Additional Mathematics Paper 3 May/June 2014 - Free CXC Past Paper with Answers

The student subdivides the interval $[0, 1]$ into $n$ equal subintervals. Calculate the width, $\Delta x$, of each subinterval.

Let the points of subdivision be $x_0 = 0$, $x_1$, $x_2$, ..., $x_{n-1}$, $x_n = 1$ as shown in Figure 1.

Find the values of $x_1$, $x_2$, ..., $x_n$ in terms of .

Determine the heights $h_1$, $h_2$, ..., $h_n$ of the circumscribed rectangles over each of the respective $n$ subintervals.

Determine the area $A_1$, $A_2$, $A_3$, ..., $A_n$ of the respective circumscribed rectangles.

Show that the sum, $S_n$, of the areas of these circumscribed rectangles is given by

$S_n = \frac{n+1}{2n}$.

(Hint: You will need to evaluate the sum of a series. State any theorem used.)

Compute $S(n)$ for $n = 10, 20, 50$ and $100$, giving your answers to three decimal places.

What number does $S(n)$ approach as $n$ gets larger?

The variables $x$ and $y$ are related by a law of the form $y = ax^n$, where $a$ and $n$ are integers. The approximate values for $y$, corresponding to the given values of $x$ are shown in Table 1.

Table 1

Use logarithms to reduce this relation to a linear form, giving your values of $\lg x$ and $\lg y$ correct to two decimal places where appropriate.

Using the graph paper provided and a scale of 2 cm to represent 0.1 units on the $x$-axis, and 1 cm to represent 0.2 units on the $y$-axis, plot a suitable straight line graph of $\lg y$ against $\lg x$.

Use your straight line graph to estimate the value of the constant $a$ and the value of the constant $n$.