Correct Answer: B. \(x^{2} - 6x + 6 = 0\)

Explanation

\((x - \alpha)(x - \beta) = 0\)

\((x - (3 - \sqrt{3}))(x - (3 + \sqrt{3})) = 0\)

\((x^{2} - (3 - \sqrt{3})x - (3 + \sqrt{3})x + (9 + 3\sqrt{3} - 3\sqrt{3} - 3) = 0\)

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

\(x^{2} - 6x + 6 = 0\)

Explanation

(a) \(^{18}C_{r} = ^{18}C_{r + 2}\)

\(\frac{18!}{r! (18 - r)!} = \frac{18!}{(r + 2)! (18 - r - 2)!}\)

\(r! (18 - r)! = (r + 2)(r + 1)r! (18 - r - 2)!\)

\((18 - r)! = (r + 2)(r + 1)(18 - r - 2)!\)

\((18 - r)(18 - r - 1)(18 - r - 2)! = (r + 2)(r + 1)(18 - r - 2)!\)

\((18 - r)(17 - r ) = r^{2} + 3r + 2\)

\(306 - 35r + r^{2} = r^{2} + 3r + 2\)

\(306 -2 = r^{2} - r^{2} + 3r + 35r\)

\(304 = 38r \implies r = 8\)

\(\therefore ^{8}C_{5} = \frac{8!}{5! (8 - 5)!}\)

\(\frac{8 \times 7 \times 6}{3 \times 2} = 56\)

(b) p(H) = 10% = 0.1 = a; p(H') = 0.9 = b

6 persons tested. We use the binomial probability distribution.

\((a + b)^{6} = a^{6} + 6a^{5}b + 15a^{4}b^{2} +20a^{3}b^{3} + 15a^{2}b^{4} + 6ab^{5} + b^{6}\)

(i) p(exactly 5 tested positive to the virus) = \(6a^{5}b\)

= \(6 \times (0.1)^{5} \times (0.9) = 0.000054 \approxeq 0.0001\) (to 4 d.p)

(ii) p(none tested positive) = \(b^{6}\)

= \((0.9)^{6} = 0.531441 \approxeq 0.5314\) (to 4 d.p)

(iii) p(at most 2) = p(none) + p(one) + p(two)

= \(b^{6} + 6ab^{5} + 15a^{2}b^{4}\)

= \((0.9)^{6} + 6(0.1)(0.9)^{5} + 15(0.1^{2})(0.9^{4})\)

= \(0.531441 + 0.354294 + 0.098415 = 0.98415\)

\(\approxeq 0.9842\) (to 4 d.p)

Correct Answer: D. 15.45%

Explanation

Original population = 1,230

New population = 1,040

Decrease in population = 1,230 – 1,040 = 190

Percentage decrease in population =decrease in population x 100%

original population

= \(\frac {190}{1,230}\) x 100 = 15.45%

Correct Answer: D. 224

Correct Answer: B. 21.5

Explanation

To find the median, first you arrange the numbers in order from least to greatest. Then you count how many numbers you have and divide that number by two. In this case (12,15,16,21,22,32,93,10)/8= 8 numbers. So 8/2=4 Then starting from the least side of the numbers count 4 numbers till you reach the median number of 21 Then starting from the greatest side count 4 numbers until you reach the other median number of 22 Finally find the mean of the two numbers by adding them together and dividing them by two (21+22)/2=43/2 to find the median number of 21.5.

Correct Answer: B. 8%

Explanation

From \(\Delta\) BCD,

\(\sin \theta = \frac{50}{56}\)

\(\theta = \sin^{-1} (0.8929)\)

\( < BCD = 63.23°\)

\(\therefore < ACB = 63.23° \times 2 = 126.46°\)

\(< ACB (reflex) = 360° - 126.46°\)

= \(233.54°\)

Perimeter of the arc = \(\frac{233.54}{360} \times 2 \times \frac{22}{7} \times 56\)

= \(228.35 m\)

Perimeter of the railway tunnel = Perimeter of the arc + |AB|

= 228.35 m + 100 m

= 328.35 m.

Correct Answer: D. (2x -15(x - 3)

Explanation

2x\(^2\) - 21x + 45

= 2x\(^2\) - 15x - 6x + 45

= x(2x - 15) - 3(2x - 15)

= (x - 3)(2x - 15)

Correct Answer: A. \(\frac{25}{4} - m\)

Explanation

\(2x^{2} - 5x + m = 0\)

\(a = 2, b = -5, c = m\)

\(\alpha + \beta = \frac{-b}{a} = \frac{5}{2}\)

\(\alpha \beta = \frac{c}{a} = \frac{m}{2}\)

\(\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2\alpha\beta\)

= \((\frac{5}{2})^{2} - 2(\frac{m}{2}) \)

= \(\frac{25}{4} - m\)

Correct Answer: A. forbidden