Correct Answer: B. x = 1\(\frac{1}{2}\) or x = 2\(\frac{1}{2}\)

Explanation

4x\(^2\) - 16x + 15 = 0

(2x - 3)(2x - 5) = 0

x = 1\(\frac{1}{2}\) or x = 2\(\frac{1}{2}\)

Correct Answer: B. 1

Explanation

\(2\frac{1}{3} \div 2\frac{2}{3} \times 1\frac{1}{7}\)

= \(\frac{7}{3} \div \frac{8}{3} \times \frac{8}{7}\)

= \(\frac{7}{3} \times \frac{3}{8} \times \frac{8}{7}\)

= 1

Correct Answer: C. 6

Explanation

012 + 01 = 01

101\(_2\) + 12\(_y\) = 2.3\(_5\)

1 x 2\(^o\) + 0 x 2\(^o\) + 1 x2\(^2\) + 1 x y\(^o\) + 2 x y\(^1\) = 3 x 5\(^o\) + 3 x 5\(^1\)

1 + 4 + 1 + 2y = 3 + 15

6 + 2y = 18

2y = 18 - 6

\(\frac{2y}{2} = \frac{12}{2}\)

y = 6

Explanation

(a) \(\int_{1} ^{4} \frac{x(3x -2)}{2\sqrt{x}} \mathrm {d} x\)

= \(\int_{1} ^{4} \frac{1}{2} x(x^{-\frac{1}{2}})(3x - 2) \mathrm {d} x\)

= \(\frac{1}{2} \int_{1} ^{4} (3x^{\frac{3}{2}} - 2x^{\frac{1}{2}}) \mathrm {d} x\)

= \(\frac{1}{2} [\frac{3x^{\frac{5}{2}}}{\frac{5}{2}} - \frac{2x^{\frac{3}{2}}}{\frac{3}{2}}]_{1} ^{4}\)

= \(\frac{1}{2} [\frac{6x^{\frac{5}{2}}}{5} - \frac{4x^{\frac{3}{2}}}{3}]_{1} ^{4}\)

= \(\frac{1}{2} [\frac{6(4^\frac{5}{2})}{5} - \frac{4(4^\frac{3}{2})}{3}] - \frac{1}{2} [\frac{6(1^\frac{5}{2})}{5} - \frac{4(1^\frac{3}{2})}{3}]\)

= \(\frac{1}{2} [\frac{6 \times 32}{5} - \frac{4 \times 8}{3}] - \frac{1}{2} [\frac{6}{5} - \frac{4}{3}]\)

= \(\frac{1}{2}[38.4 - 10.67] - \frac{1}{2} [1.2 - 1.33]\)

= \(\frac{1}{2} [27.73 + 0.13]\)

= \(13.93\)

(b) \(2x^{2} + 2y^{2} - 8x + 5y - 10 = 0\)

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

\(x^{2} - 4x + [\frac{1}{2} (-4)]^{2} + y^{2} + \frac{5}{2}y + [\frac{1}{2} (\frac{5}{2})]^{2} = 5 + 4 + \frac{25}{16} = \frac{169}{16}\)

\((x - 2)^{2} + (y + \frac{5}{2})^{2} = \frac{169}{16}\)

(i) coordinates of the centre = \((2, -\frac{5}{4})\)

(ii) Radius = \(\sqrt{\frac{169}{16}} = \frac{13}{4}\)

Correct Answer: B. 20

Explanation

mean age = \(\frac{\text{sum of ages}}{\text{no. of men}}\)

50 = \9\frac{sum}{R}\)

sum = 50R.....(1)

Sum of ages of the men that left = 55 + 63 = 188

remaining sum = 50R - 118

remaining no. of men = R - 2

now mean age = 50 - 1 = 49 years

49 = \(\frac{50R - 118}{R - 2}\)

49(R - 2) = 50R - 118

49R - 50R = -188 - 98

-R = -20

R = 20

Correct Answer: D. S70° W

Explanation

250° = S 70° W

Correct Answer: C. P = 4 and Q = -2

Explanation

\(\frac{2x}{(x + 6)(x + 3)} = \frac{P}{x + 6} + \frac{Q}{x + 3}\)

\(\frac{2x}{(x + 6)(x + 3)} = \frac{P(x + 3) + Q(x + 6)}{(x + 6)(x + 3)}\)

Comparing equations, we have

\(2x = Px + 3P + Qx + 6Q\)

\(\implies 3P + 6Q = 0 ... (1) ; P + Q = 2 .... (2)\)

From equation (1), \(3P = -6Q \implies P = -2Q\)

\(\therefore -2Q + Q = -Q = 2 \)

\(Q = -2\)

\(P = -2Q = -2(-2) = 4\)

\(P = 4, Q = -2\)

Explanation

(i) Let the length of the longitude be \(\theta\).

Longitude difference = \(50 + \theta\)

Distance along the parallel of latitude (KL) = \(\frac{50 + \theta}{360} \times 2\pi R \cos 60\)

\(3520 = \frac{50 + \theta}{360} \times 2 \times \frac{22}{7} \times 6400 \cos 60\)

\(3520 = \frac{50 + \theta}{360} \times 2 \times \frac{22}{7} \times 3200\)

\(50 + \theta = \frac{3520 \times 360 \times 7}{22 times 2 \times 3200}\)

\(50 + \theta = 63°\)

\(\theta = 63° - 50° = 13°\)

(ii) Let the latitude of N be \(\alpha\).

The distance along the parallel of longitude (KN) = \(\frac{60 + \alpha}{360} \times 2 \times \frac{22}{7} \times 6400\)

\(10951 = \frac{60 + \alpha}{360} \times 2 \times \frac{22}{7} \times 6400\)

\(60 + \alpha = \frac{10951 \times 360 \times 7}{2 \times 22 \times 6400}\)

\(60 + \alpha = 98°\)

\(\alpha = 98° - 60° = 38°\)

The latitude of N = 38°S.

(b) Let the man's income tax be x.

80% of x = \(\frac{1200}{0.25} = N4800\)

\(0.8x = N4,800\)

\(x = \frac{4800}{0.8} = N6000\)

Correct Answer: A. 2.9.seconds

Explanation

s = \(\frac{1}{2}gt^2\)

40 = \(\frac{1}{2} \times 9.8 \times t^2\)

\(\frac{80}{9.8} = t^2\)

t = \(\sqrt{8.16326}\)

t≈ 2.9 secs

Correct Answer: A. 3 9/17%

Explanation

Error = 4.25 - 4.10 = 0.15

% error = \(\frac{0.15}{4.25} \times 100%\)

= \(\frac{15}{\frac{17}{4}} = \frac{15 \times 4}{17}\)

= \(3\frac{9}{17} %\)