2014 - JAMB Mathematics Past Questions and Answers - page 4

If y = 4x³ - 2x² + x, then;

\(\frac{\delta y}{\delta x}\) = 3(4x²) - 2(2x) + 1

= 12x² - 4x + 1

y = cos 3x

Let u = 3x so that y = cos u

Now, \(\frac{\delta y}{\delta x} = 3\),

\(\frac{\delta y}{\delta x} = -sin u\)

By the chain rule,

\(\frac{\delta y}{\delta x} = \frac{\delta y}{\delta u} \times \frac{\delta u}{\delta x}\)

\(\frac{\delta y}{\delta x} = (-\sin u) (3)\)

\(\frac{\delta y}{\delta x} = -3 \sin u\)

\(\frac{\delta y}{\delta x} = -3 \sin 3x\)

y = x² - 2x - 3,

Then \(\frac{\delta y}{\delta x} = 2x - 2\)

But at minimum point,\(\frac{\delta y}{\delta x} = 0\),

Which means 2x - 2 = 0

2x = 2

x = 1.

Hence the minimum value of y = x² - 2x - 3 is;

y_min = (1)² - 2(1) - 3

y_min = 1 - 2 - 3

y_min = -4

\(\int \sin 2x dx = \frac{1}{2} (-\cos 2x) + k\)

\(- \frac{1}{2} \cos 2x + k\)

\(\int (2x + 3)^{\frac{1}{2}} \delta x\)

let u = 2x + 3, \(\frac{\delta y}{\delta x} = 2\)

\(\delta x = \frac{\delta u}{2}\)

Now \(\int (2x + 3)^{\frac{1}{2}} \delta x = \int u^{\frac{1}{2}}.{\frac{\delta x}{2}}\)

\( = \frac{1}{2} \int u^{\frac{1}{2}} \delta u\)

\( = \frac{1}{2} u^{\frac{3}{2}} \times \frac{2}{3} + k\)

\( = \frac{1}{3} u^{\frac{3}{2}} + k\)

\( = \frac{1}{3} (2x + 3)^{\frac{3}{2}} + k\)

Mean x = \(\frac{\sum x}{n}\)

= [(2 - t) + (4 + t) + (3 - 2t) + (2 + t) + (t - 1) \(\div\)] 5

= [11 - 1 + 3t - 3t] \(\div\) 5

= 10 \(\div\) 5

= 2

First arrange the numbers in order of magnitude;
1,2,3,3,4,5,5,5,5,6,7,8,9,9,10

Hence the median = 5

Mean x = \(\frac{\sum x}{n}\)

\( = \frac{5 + 4 + 3 + 2 + 1}{5}\)

\( = \frac{15}{5}\)

= 3

\(\begin{array}{c|c}
x & d = x - 3 & d^2 \\ \hline
5 & 2 & 4 \\ 4 & 1 & 1 \\ 3 & 0 & 0 \\ 2 & -1 & 1 \\ 1 & -2 & 4 \\ \hline
& & \sum d^2 + 10
\end{array}\)

Hence, standard deviation;

\( = \sqrt{\frac{\sum d^2}{n}} = \sqrt{\frac{10}{5}}\)


\( = \sqrt{2}\)

A team of 2 girls can be selected from 7 girls in \(^7C_3\)

\( = \frac{7!}{(7 - 3)! 3!}\)

\( = \frac{7!}{4! 3!} ways\)