Correct Answer: B. 15.00

Explanation

Note that \(\frac{10\sqrt{3}}{\sqrt{5}} = \frac{10\sqrt{3}}{\sqrt{5}} \times - \frac{\sqrt{5}}{\sqrt{5}}\)

= \(\frac{10\sqrt{15}}{\sqrt{5}} = 2\sqrt{15}\)

hence, (\(\frac{10\sqrt{3}}{\sqrt{5}} - \sqrt{15}\))² = (\(2\sqrt{15} - \sqrt{15}\))²

= (\(2\sqrt{15} - \sqrt{15}\))(\(2\sqrt{15} - \sqrt{15}\))

= 4\(\sqrt{15 \times 15} - 2\sqrt{15 \times 15} - 2\sqrt{15 x 15} + \sqrt{15 \times 15}\)

= 4 x 15 - 2 x 15 - 2 x 15 + 15

= 60 - 30 - 30 + 15

= 15

Correct Answer: D. 1.8 years

Explanation

Mean (\(\bar{x}\)) = \(\frac{5 + 6 + 6 + 8 + 10}{5} \)

= \(\frac{35}{5} = 7\)

\(SD = \sqrt{\frac{\sum (x - \bar{x})^{2}}{n}}\)

= \(\sqrt{\frac{16}{5}} \)

= \(\sqrt{3.2}\)

\(\approxeq 1.8 years\)

Correct Answer: C. 1.95

Explanation

(ai) \(p=\frac{4}{16}=\frac{1}{4}\)

\(∴q=1-\frac{1}{4}=\frac{3}{4}\)

P(The probability that he did not pick agreenball) = \(^nC_r p^rq^{n - r}\)

Where n = 5 and r = 0

\(=^5C_0(\frac{1}{4})^o(\frac{3}{4})^{5 - 0}\)

\(=^5C_0(\frac{1}{4})^o(\frac{3}{4})^5\)

\(=1\times1\times\frac{243}{1024}\)

\(=\frac{243}{1024}\)

(aii) Pr(at least three) = Pr(for 3 green balls) + Pr (for 4 green balls) + Pr (for 5 green balls)

\(^5C_3(\frac{1}{4})^3(\frac{3}{4})^{5 - 3}+^5C_4(\frac{1}{4})^4(\frac{3}{4})^{5 - 4}+^5C_5(\frac{1}{4})^5(\frac{3}{4})^{5 - 5}\)

\(=^5C_3(\frac{1}{4})^3(\frac{3}{4})^2+^5C_4(\frac{1}{4})^4(\frac{3}{4})^1+^5C_5(\frac{1}{4})^5(\frac{3}{4})^0\)

\(=10\times\frac{1}{6}4\times\frac{9}{16}+5\times\frac{1}{256}\times\frac{3}{4}+1\times\frac{1}{1024}\times1\)

\(=\frac{45}{512}+\frac{15}{1024}+\frac{1}{1024}\)

\(=\frac{53}{512}\)

(b) The sum of deviations from the mean is always equal to 0. This is a fundamental property of deviations and the definition of the mean.

\(= -2 + (m - 1) + (m^2 + 1) + (-1) + 2 + (2m - 1) + (-2) = 0\)

\(= -2 + m - 1 + m^2 + 1 - 1 + 2 + 2m - 1 - 2 = 0\)

\(= m^2 + 3m - 4 = 0\)

\(= m^2 + 4m - m - 4 = 0\)

= m(m + 4) - 1(m + 4) = 0

= (m + 4)(m - 1) = 0

This gives us two possible values for m: -4 and 1

So, the possible values of m are -4 and 1.

Correct Answer: D. It would stay the same

Explanation

The median is the middle number of a set. If the same amount of values are used and the actual middle number is not changed, the median of the set will not change. Changing the actual values of the two extremes did not change them position wise in the set when placed in value order.

Correct Answer: B. \(x \leq 2\)

Explanation

\(f : x \to \sqrt{4 -2x}\) defined on the set of real numbers,R, which has range from \((-\infty, \infty)\) but because of the root sign, it is defined from \([0, \infty)\).

This is because the root of numbers only has real number values from 0 and upwards.

\(\sqrt{4-2x} \geq 0 \implies 4-2x \geq 0\)

\(-2x \geq -4; x \leq 2\)

Correct Answer: A. 5850

Correct Answer: D. 2n - 1

Correct Answer: C. 34.45

Explanation

\(\frac{12 + 47 + 49 + 15 + 43 + 41 + 13 + 39 + 43 + 41 + 36}{11}\)

= \(\frac{379}{11}\)

= 34.45

Correct Answer: D. 210