Explanation

(a) S.P = \(C.P + \frac{% profit}{100} \times C.P\)

= \(300 + (\frac{x}{100} \times 300)\)

S.P = N(300 + 3x)

Therefore, Bola bought it at N(300 + 3x).

James paid \(N(6x + \frac{3}{4})\) extra from what Lamin paid, therefore Bola's S.P = \(N(300 + 6x + \frac{3}{4})\)

= N(300.75 + 6x).

Profit for Bola = \(N(300.75 + 6x - (300 + 3x)) = N(0.75 + 3x)\)

\(\frac{x}{100} \times N(300 + 3x) = N(0.75 + 3x)\)

\(300x + 3x^{2} = 75 + 300x\)

\(\implies 3x^{2} = 75\)

\(x^{2} = 25 \therefore x = 5\)

(b) \(3x - 2 < 10 + x < 2 + 5x\)

\(3x - 2 < 10 + x \implies 3x - x < 10 + 2\)

\(2x < 12 \implies x < 6\)

\(10 + x < 2 + 5x\)

\(x - 5x < 2 - 10\)

\(-4x < -8 \implies x > 2\)

The range = \(2 < x < 6\)

Correct Answer: A. 17/6 units (sq)

Explanation

(a)

(i) Nobody takes Economics and Chemistry \(n(E \cap C) = \null \).

(ii) 4 pupils take Economics and Government; \(n(E \cap G) = 4\)

(b) (i) Suppose \(n(C \cap G) = x\)

\(22 = 8 + 4 + 8 - 4 - x + x + 7 - x\)

\(22 = 23 - x \implies x = 1\)

Only 1 person does Chemistry and Government.

(ii) Number taking only Government = \(8 - (4 + x)\)

\(8 - (4 + 1) = 3\)

Explanation

(a) \(x + 8 < 4x - 15 \implies 4x - 15 > x + 8\)

\(4x - x > 8 + 15 \implies 3x > 23\)

\(x > \frac{23}{3}\)

\(x > 7\frac{2}{3}\).

(b) Extra income = 5k of N200,000

= \(\frac{5}{100} \times N200,000\)

= \(N10,000\)

Total income = Monthly income + commission

= N(2,500 + 10,000)

= N12,500.

(c) Area of semi- circle = \(\frac{\pi r^{2}}{2}\)

= \(\frac{1}{2} \times \frac{22}{7} \times 35 \times 35\)

= \(1925 cm^{2}\)

Length of rectangular part = 50 cm

Breadth = Diameter of semi- circle = 2(35 cm) = 70 cm

Area of rectangular part = \(length \times \breadth\)

= \(50 \times 70\)

= \(3500 cm^{2}\)

Total area of window = \((3500 + 1925) cm^{2}\)

= \(5425 cm^{2}\)

(a) \(x + 8 < 4x - 15 \implies 4x - 15 > x + 8\)

\(4x - x > 8 + 15 \implies 3x > 23\)

\(x > \frac{23}{3}\)

\(x > 7\frac{2}{3}\).

(b) Extra income = 5k of N200,000

= \(\frac{5}{100} \times N200,000\)

= \(N10,000\)

Total income = Monthly income + commission

= N(2,500 + 10,000)

= N12,500.

(c) Area of semi- circle = \(\frac{\pi r^{2}}{2}\)

= \(\frac{1}{2} \times \frac{22}{7} \times 35 \times 35\)

= \(1925 cm^{2}\)

Length of rectangular part = 50 cm

Breadth = Diameter of semi- circle = 2(35 cm) = 70 cm

Area of rectangular part = \(length \times \breadth\)

= \(50 \times 70\)

= \(3500 cm^{2}\)

Total area of window = \((3500 + 1925) cm^{2}\)

= \(5425 cm^{2}\)

Correct Answer: D. 7.15m

Explanation

tan\(\alpha\) = \(\frac{h}{6}\)

h = 6tan\(\alpha\)

= 6tan 50^o

= 6 x 1.1918

= 7.1508

= 7.15m (2d.p)

Correct Answer: A. \(\frac{9}{25}\)

Explanation

Prob. (team scored 4 goals) = Prob. (team scored 7 goals) = \(\frac{3}{25}\)

Hence, probability that a team selected at random scored either 4 or 7 goals;

= \(\frac{6}{25} + \frac{3}{25}\)

= \(\frac{9}{25}\)

Explanation

(a) \(\int _{0} ^{1} \frac{3}{1 + x^{2}} \mathrm {d} x\)

h = 0.25 ;

\(y_{1} = 3, y_{2} = 2.82 , y_{3} = 2.40 , y_{4} = 1.92 , y_{5} = 1.5\)

\(y_{1} + y_{5} = 3 + 1.5 = 4.5\)

\(y_{2} + y_{3} + y_{4} = 2.82 + 2.40 + 1.92 = 7.14\)

\(2(y_{2} + y_{3} + y_{4}) = 14.28\)

= \(3[\frac{1}{2} h ( y_{1} + y_{5} + 2(y_{2} + y_{3} + y_{4})]\)

= \(3[\frac{1}{2} (0.25) (4.5 + 14.28)]\)

= \(3(0.125)(18.78) = 2.3475\)

\(\approxeq 2.348\) (4 sig. figures).

(b) \(A = \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}\)

\(A^{2} + A + 2I = \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} + \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} + 2 \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)

= \(\begin{pmatrix} 4 + 3 & 2 + 2 \\ 6 + 6 & 3 + 4 \end{pmatrix} + \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} + \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\)

= \(\begin{pmatrix} 7 & 4 \\ 12 & 7 \end{pmatrix} + \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} + \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\)

= \(\begin{pmatrix} 11 & 5 \\ 15 & 11\end{pmatrix}\)

\(T : \begin{pmatrix} 1 \\ 2 \end{pmatrix} \to \begin{pmatrix} 11 & 5 \\ 15 & 11 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \end{pmatrix}\)

=\(\begin{pmatrix} 11 + 10 \\ 15 + 22 \end{pmatrix} = \begin{pmatrix} 21 \\ 37 \end{pmatrix}\)

The image of (1, 2) = (21, 37).

Correct Answer: C. 8.8 x 10^-3

Correct Answer: D. 90°

Explanation

Ae is a bisector of BAC => EAC = 45^o

CE is the bisector of ACD => ACE = 45^o

AEC = 180^o - [EAC + ACE]

    = 180^o - (45^o + 45^o) = 180^o - 90^o = 90^o

Correct Answer: D. y = \(x^2 - x - 6\)

Explanation

Since the curve cuts the x-axis at x = -2 and x = 3,

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

\(x^2 - 3x + 2x - 6\) = 0

\(x^2 - x - 6\) = 0

Hence, the equation of the curve is

y = \(x^2 - x - 6\)