A uniform rod AB, of weight W and length 2a, rests with the end A on a rough horizontal plane - CIE - A-Level Further Maths - Question 10 - 2011 - Paper 1

Given h = 3a, we can substitute this value into the previously derived equation for μ. Then we find:

By substituting h back into the equation: $k = \frac{\mu(h + 2a \sin \theta)}{2a \cos \theta}$ This requires previously determined values of the resultant forces, thus confirming its numerical evaluation.

The horizontal component of the force on point P can be found using the tension in the string BC. Knowing that T = W at equilibrium and given its angle:

Let T be the tension, and thus we can analyze:

$T_{horizontal} = W \sin(\theta)$ Consequently, the horizontal component exerted on P will align to this force.

Using the data provided, we can derive sums involving the values of x and y from the regression line: Using the normal equations of regression, where: $\sum x = \frac{10}{5} \sum y.$ Summing the respective values gives the combined expression for p + q equating to 19 upon substitution.

Identifying p and q from the regression equations yields: We can define p and q through substituting into the normal equations, allowing explicit resolution of these variables from established regression properties.

To find the product moment correlation coefficient (r), we will use the established formula:

$r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$ This will involve calculating averages of x and y, conducting total moments, arriving at an r value shaped by the linear relationship indicated by our regression line.

When the values of y are adjusted by a factor of 10, the new regression line must accommodate this scale alteration:

Thus, the new regression line reformed becomes: $y = 0.25x - 0.15$ This alteration reflects the remix of products applied to the original line.

The correlation coefficient r remains unchanged in scale adjustments, meaning it preserves its original calculated value despite y being scaled down by a divisor. Hence, the final outcome yields the same r value as before.