Correct Answer: C. \(\frac{22000}{3}\)km

Correct Answer: D. 3.6258

Explanation

log(0.65)² = 2log(0.65)

but log0.65 = 1.8129

2 x 1.8129 = 3.6258

Explanation

Let. X = Prob. that a good mango is selected = 18/24= 3/4

Y Prob. that a bad mango is selected = 6/24 1/4

Using the binomial probability distribution, we have:

(X+Y)\(^6\) = X\(^6\) + \(^6{C}_1 {X}^5 Y\) + \(^6{C}_2 {X}^4 {Y}^2\) + \(^6{C}_3 {X}^3 {Y}^3\) + \(^6{C}_4 {X}^2 {Y}^4\) + \(^6{C}_5 {X} {Y}^5\) + \(^6{C}_6 {Y}^6\)

Probability that not more than 3 are bad is

= \(^6{C}_1 {X}^5 Y\) + \(^6{C}_2 {X}^4 {Y}^2\) + \(^6{C}_3 {X}^3 {Y}^3\)

= 6(3/4)\(^5\) (1/4) + 15(3/4)\(^4\) (1/4)\(^2\) + 20(3/4)\(^3\) (1/4)\(^3\)

= 6(243/1024)(1/4) + 15(81/256)(1/16) + 20(27/64)(1/64)

= 0.36 + 0.30 + 0.13

≈ 0.79

Correct Answer: D. (3y - 4x)(3y + 4x)

Correct Answer: B. -1, 2

Explanation

(a)P(Bad Quality) = \(\frac{30}{10}\) = 0.3 and P(Good Quality) = 1 - 0.3 = 0.7

(a)(i) P(exactly 6) = \(^{12}C_6\) x (0.3)\(^6\) x (0.7)\(^6\) = 0.0792

(a)(ii) P(at least 4) = 1 - [P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3)]

Where P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) = \(^{12}C_0\)(0.3)\(^6\)(0.7)\(^{12}\) + \(^{12}C_1\)(0.3)\(^1\)(0.7)\(^{11}\) + \(^{12}C_2\)(0.3)\(^2\)(0.7)\(^{10}\) + \(^{12}C_3\)(0.3)\(^3\)(0.7)\(^{9}\)

= 0.01384 + 0.071184 + 0.16779 + 0.2397

Therefore; P(at least 4) = 1 - 0.4925 = 0.5075

(a)(iii) P(no bad one) = \(^{12}C_0\)(0.3)\(^o\)(0.7)\(^{12}\) = 0.0138

(b) To find P(4 boys and 3 girls) = \(\frac{^8C_4 \times ^5C_3}{^{13}C_7}\)

= \(\frac{70 \times 10}{1716}\)

= \(\frac{175}{429}\)

= 0.4079

Correct Answer: B. \(\frac{2}{5}\)

Explanation

Let x represent the entire farmland

then, \(\frac{2}{5}\)x + \(\frac{1}{3}\)[x - \(\frac{2}{3}x\)] + M = x

Where M represents the part of the farmland used for growing maize, continuing

\(\frac{2}{5}\)x + \(\frac{1}{3}\)x [1 - \(\frac{2}{3}x\)] + M = x

\(\frac{2}{5}x + \frac{1}{3}\)x [\(\frac{3}{5}\)] + M = x

\(\frac{2}{5}\)x + \(\frac{1x}{5}\) + M = x

\(\frac{3x}{5} + M = x\)

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

= x[1 - \(\frac{3}{5}\)]

= x[\(\frac{2}{5}\)] = \(\frac{2x}{5}\)

Hence the part of the land used for growing maize is

\(\frac{2}{5}\)

Correct Answer: A. (x + b)

Correct Answer: C. \(2\sqrt{21}\)

Explanation

Given lines \(OA\) and \(OB\) inclined at angle \(\theta\), the line \(AB\) is gotten using cosine rule.

\(|AB|^{2} = |OA|^{2} + |OB|^{2} - 2|OA||OB|\cos \theta\)

\(|AB|^{2} = 4^{2} + 5^{2} - 2(4)(5)\cos 60\)

= \(16 + 25 - 20\)

\(|AB|^{2} = 21 \implies |AB| = \sqrt{21}\)

\(\implies \text{The two particles are} \sqrt{21} m \text{apart in 1 sec}\)

In two seconds, the particles will be \(2\sqrt{21} m\) apart.

Correct Answer: D. -4

Explanation

\(\begin{vmatrix} 4 & x \\ 5 & 3 \end{vmatrix} = 12 - 5x = 32\)

\(5x = 12 - 32 = -20\)

\(x = -4\)