Explanation

W = 8i + 3J

X = 6i - 5J

Y = 2i + 3J

w + x - 2y = 8i + 3j + 6i - 5j - 2(2i + 3j)

= 10i - 8j

Z = \(\frac{10i - 8j}{|Z|}\)

= \(\frac{10i - 8j}{4}\)

= \(\frac{10i}{4}\)

\(\frac{8j}{4}\)

= \(\frac{5}{2}\)i - 2j

Correct Answer: B. 250 m

Explanation

\(s = ut + \frac{1}{2} at^{2}\)

\(u = 0, t = 10 secs, a = 5 ms^{-2}\)

\(s = 0 + \frac{1}{2} 5 \times 10^{2}\)

\(s = 250 m\)

Explanation

(a) Resultant of the forces acting on the plane = (6i + 2j) + (5i + 4j) + (4i - j) = 15i + 5j

\(F = \frac{m \Delta v}{t}\)

Resultant force = \(\frac{mv}{t} \)

\(15i + 5j = \frac{5v}{t}\)

\(v = \frac{(15i + 5j). t}{5} = 3t i + t j\)

(b) velocity after 4 seconds = \(3(4) i + 4 j = 12 i + 4 j\)

\(\therefore |12i + 4j| = \sqrt{12^{2} + 4^{2}} = \sqrt{144 + 16}\)

= \(\sqrt{160} \)

= \(4\sqrt{10} m/s\)

Correct Answer: D. -22

Explanation

Taking the LCM of the right hand side of the equation, we have

\(\frac{4(2x-3) - 2(x+5)}{(x+5)(2x-3)} = \frac{6x+m}{2x^{2}+7x-15}\)

Comparing the numerators, we have

\(4(2x-3) - 2(x+5) = 6x+m\)

\(8x-12-2x-10 = 6x -22 = 6x + m\)

\(\implies m = -22\)

Explanation

The probability of success (p) = \(\frac{3}{5}\) and the probability of failure (q) = 1 - \(\frac{3}{5} = \frac{2}{5}\)

The probability that at least two of them failed is

P(x \(\geq\) 2) = 1 - p(x < 2) = 1 - [P(x = 0)] + [P(x = 1)]= 1 - [(\(^{10}_{0}\)) (\(\frac{2}{5}\))\(^0\) (\(\frac{3}{5}\))\(^{10}\) + (\(^{10}_1\)) (\(\frac{2}{5}\))\(^1\) (\(\frac{3}{5}\))\(^9\)]

Which simplified to 1 - 0.046357401 = 0.9536 correct to four decimal places

(b) The probability that exactly half of them passes is P(x = 6) = (\(^{10}_{5}\)) (\(\frac{3}{5}\))\(^5\) (\(\frac{2}{5}\))\(^{5}\)

Simplified to give; 252 x 0.7776 x 0.01024 = 0.2007 correct to four decimal places

(c) The probability that at most two of them failed is

P(x \(\leq\) 2) = P(x = 0) + P(x = 1) + P(x = 2). In this case p = \(\frac{2}{5}\) q = \(\frac{3}{5}\) so that

p(x \(\leq\) 2) = (\(^{10}_{0}\)) (\(\frac{2}{5}\))\(^0\) (\(\frac{3}{5}\))\(^{10}\) + (\(^{10}_1\)) (\(\frac{2}{5}\))\(^1\) (\(\frac{3}{5}\))\(^9\) + (\(^{10}_{2}\)) (\(\frac{2}{5}\))\(^2\) (\(\frac{3}{5}\))\(^{8}\)

Which simplies to 0.1673 correct to four decimal places.

Correct Answer: C. \(\frac{1}{2(x - 2)} + \frac{5x}{2(x- 2)^{2}}\)

Explanation

\(\frac{3x - 1}{(x - 2)^{2}} = \frac{A}{(x - 2)} + \frac{Bx}{(x - 2)^{2}}\)

\(\frac{3x - 1}{(x - 2)^{2}} = \frac{A(x - 2) + Bx}{(x - 2)^{2}}\)

Comparing, we have

\(3x - 1 = Ax - 2A + Bx \implies -2A = -1; A + B = 3\)

\(\therefore A = \frac{1}{2}; B = \frac{5}{2}\)

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

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

Explanation

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

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

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

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

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

Correct Answer: C. \(\sqrt{65}\) units

Explanation

distance = \(\sqrt{(3 - (-5)^2 + (-1 - (-2)^2)}\)

= \(\sqrt{8^2 + 1^2}\)

= \(\sqrt{65}\) units

Correct Answer: B. \(\begin{pmatrix} -1 & -6 \\ 1 & 18 \end{pmatrix}\)

Explanation

3M = 3 \(\begin{pmatrix} 3 & 2 \\ -1 & 4 \end{pmatrix}\)

= \(\begin{pmatrix} 9 & 6 \\ -3 & 12 \end{pmatrix}\)

2N = 2 \(\begin{pmatrix} 5 & 6 \\ -2 & -3 \end{pmatrix}\)

= \(\begin{pmatrix} 10 & 12 \\ -4 & -6 \end{pmatrix}\)

(3M - 2N) = \(\begin{pmatrix} 9 & 6 \\ -3 & 12 \end{pmatrix}\) - \(\begin{pmatrix} 10 & 12 \\ -4 & -6 \end{pmatrix}\)

Correct Answer: C. \(\frac{1}{2}\)

Explanation

\(log_{y}\frac{1}{8} = 3 \implies y^{3} = \frac{1}{8}\) (Laws of logarithm)

\(y^{3} = \frac{1}{2^{3}} = (\frac{1}{2})^{3}\)

Equating both sides, we have

\(y = \frac{1}{2}\)