2013 - WAEC Mathematics Past Questions and Answers - page 3

p = (y : 2y \(\geq\) 6)

2y \(\leq\) 6

y \(\leq \frac{6}{2}\)

y = \(\leq\) 3

and Q = (y : y -3 \(\geq\) 4)

y - 3 \(\geq\) 4

y \(\geq\) 4 + 3

y \(\geq\) 7

therefore p = {3, 4, 5, 6, 7} and Q = {7, 6, 5, 4, 3....}

P\(\cap\)Q = {3, 4, 5, 6, 7}

6k² = 5k + 6

6k² - 5k - 6 = 0

6k² - 0k + 4k - 6 = 0

3k(2k - 3) + 2(2k - 3) = 0

(3k + 2)(2k - 3) = 0

3k + 2 = 0 or 2k - 3 = 0

3k = -2 or 2k = 3

k = \(\frac{-2}{3}\) or k = \(\frac{3}{2}\)

k = (\(\frac{-2}{3}\), k = \(\frac{3}{2}\))

y \(\alpha\) \sqrt{x + 1}\), y = k\sqrt{x + 1}\)

6 = k\(\sqrt{3 + 1}\)

6 = k\(\sqrt{4}\)

6 = 2k

k = \(\frac{6}{2}\) = 3

y = \(\sqrt{(x + 1)}\)

9 = 3\(\sqrt{(x + 1)}\)(divide both side by 3)

\(\frac{9}{3}\) = \(\frac{3\sqrt{x + 1}}{3}\)

3 = \(\sqrt{x + 1}\)(square both sides)

9 = x + 1

x = 9 - 1

x = 8

y = x² + 2x + k at point(2,0) x = 2, y = 0

0 = (2)² + 2(20 + k)

0 = 4 + 4 + k

0 = 8 + k

k = -8

Let the number be y, subtract y from 2 i.e 2 - y

2 - y = 4 < \(\frac{1}{5}\) x y, 2 - y = 4 < \(\frac{y}{5}\)

2 - y - 4 < \(\frac{y}{5}\)

2 - 4 - y \(\frac{x}{5} - 4\), multiplying through by 5

5(2 - x) = 5(\(\frac{x}{5}\)) - 5(4)

10 - 5x = x - 20

-5x - x = -20 - 10

-6x = -30

x = \(\frac{-30}{-6}\)

= 5

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

= \(\frac{2x - 4 - x - 3}{x^2 - 2x + 3x - 6}\)

= \(\frac{x -7}{x^2 + x - 6}\)

= \(\frac{x - 7}{x^2 + x - 6}\)

Let the interior angle = x^o

interior angle = 5x^o (sum of int. angle ann exterior)

(angles = angle or straight line)

6x = 180

x = \(\frac{180}{6}\)

x = 30^o

no. of sides = \(\frac{\text{sum of exterior angles}}{\text{exterior angle}}\)

= \(\frac{360}{30}\) = 12

P = x² + 4x - 2, Q = 2x - 1

Q - p = 2, (2x - 1) - (x² + 4x - 2) = 2

2x - 1 - x² - 4x + 2 = 2

-2x - x² + 1

-x² - 2x - 1 = 0

x² + 2x + 1 = 0

x² + x + x + 1 = 0

x(x + 1) + 1(x + 1) = 0

(x + 1)(x + 1) = 0

x + 1 = 0 or x + 1 = 0

x = -1 or x = -1

x = -1

Volume of pyramid = \(\frac{1}{3}\) x base area x height

= \(\frac{1}{3} \times 12^4 \times 8 \times 14\)

= 4 x 8 x 14 = 448m³