A) Illustrate the information provided in a Venn Diagram.

Substitute the expressions for $e$ and $f$ into $d+e+f=14$:

Now find $e$ and $f$:

So, $d=6$, $e=3$, $f=5$.

The sum of all regions must equal the total number of people:

Substitute the known values:

This means the number of people who speak one language only is 26.

Given:
$N(Y) = m^2+3$
$N(H) = 5m-3$
$N(I) = 3m+1$
From the Venn diagram regions:
$N(Y) = a+d+e+g = a+6+3+4 = a+13$
$N(H) = b+d+f+g = b+6+5+4 = b+15$
$N(I) = c+e+f+g = c+3+5+4 = c+12$
Equating these:
$a+13 = m^2+3 \implies a = m^2-10$
$b+15 = 5m-3 \implies b = 5m-18$
$c+12 = 3m+1 \implies c = 3m-11$

The values for the regions are:

• Yoruba and Hausa and Igbo ($g$): 4
• Yoruba and Hausa only ($d$): 6
• Yoruba and Igbo only ($e$): 3
• Hausa and Igbo only ($f$): 5
• None of the languages ($h$): 6 The values for $a, b, c$ depend on $m$, which will be found in part (b)(i). For the diagram, we will use the final values of $a, b, c$ after finding $m$.

b) Find the:

(i) value of $m$;

(ii) number of people who speak one language only.

(i) Value of $m$.

Using the quadratic formula $m = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$:

Two possible values for $m$:

The number of people in any region cannot be negative.
If $m = 5$:
$a = 5^2 - 10 = 25 - 10 = 15$
$b = 5(5) - 18 = 25 - 18 = 7$
$c = 3(5) - 11 = 15 - 11 = 4$
All these values are positive, so $m=5$ is a valid solution.

If $