WAEC Further Mathematics Past Questions & Answers - Page 114

$^{n}C_{r} = \frac{n!}{(n-r)! r!}$

$^{n}C_{r - 1} = \frac{n!}{(n - (r - 1))! (r - 1)!}$

$^{n}C_{r} ÷ ^{n}C_{r - 1} = \frac{n!}{(n - r)! r!} ÷ \frac{n!}{(n-(r-1))!(r-1)!}$

= $\frac{n!}{(n-r)! r!} \times \frac{(n-(r-1)! (r-1)!}{n!}$

= $\frac{(n + 1 - r)! (r - 1)!}{(n - r)! r!}$

= $\frac{(n+1-r)(n-r)! (r-1)!}{(n-r)! r (r - 1)!}$

= $\frac{n + 1 - r}{r}$

Let the sum of the 8 numbers = y.

$\frac{y}{8} = 20 \implies y = 20 \times 8 = 160$

$160 - 17 = 143 (\text{the sum of the other 7 numbers})$

$mean = \frac{143 + 25}{8} = \frac{168}{8}$

= 21

Let $p(head) = p = \frac{1}{2}$ and $p(tail) = q = \frac{1}{2}$

$(p + q)^{4} = p^{4} + 4p^{3}q + 6p^{2}q^{2} + 4pq^{3} + q^{4}$

The probability of equal heads and tails = $6p^{2}q^{2} = 6(\frac{1}{2}^{2})(\frac{1}{2}^{2})$

= $\frac{6}{16} = \frac{3}{8}$.

$y = 4 - 9x$

$\frac{\mathrm d y}{\mathrm d x} = \frac{\Delta y}{\Delta x} = -9$

$\frac{\Delta y}{0.1} = -9 \implies \Delta y = -9 \times 0.1 = -0.9$ 

Given, $a(t) =  (3t - 2) ms^{-2}$, the first integration of a(t), with respect to t, gives v(t) (the velocity). The second integration of a(t) or first integration of v(t) gives s(t).

$v(t) = \int (3t - 2) \mathrm {d} t = \frac{3}{2}t^{2} - 2t$

$s(t) = \int (\frac{3}{2}t^{2} - 2t) \mathrm {d} t$

= $\frac{t^{3}}{2} - t^{2}$

When t = 3s,

$s(3) = \frac{3^{3}}{2} - 3^{2} = \frac{27}{2} - 9$

= $\frac{9}{2}$