Explanation

\((x + 1)(x + 2) = k(3x + 7)\)

\(x^{2} + 3x + 2 = 3kx + 7k \)

\(x^{2} + 3x - 3kx + 2 - 7k = 0\)

\(x^{2} + (3 - 3k)x + 2 - 7k = 0\)

For equal roots, \(b^{2} - 4ac = 0, a \neq 0\)

\(\therefore (3 - 3k)^{2} - 4(1)(2 - 7k) = 0\)

\(9 - 18k + 9k^{2} - 8 + 28k = 0\)

\(9k^{2} + 10k + 1 = 0 \)

\(9k(k + 1) + 1(k + 1) = 0 \implies (9k + 1)(k + 1) = 0\)

\(\implies k = -\frac{1}{9} ; k = -1\)

Explanation

(a)

Difference in longitude = 47° + 25° = 72°

Distance /XY/ = \(\frac{\theta}{360°} \times 2\pi r\)

Where \(r = R \cos 32°\)

\(\therefore /XY/ = \frac{72}{360} \times 2 \times \frac{22}{7} \times 6400 \times \cos 32\)

= \(2 \times \frac{22}{7} \times 1280 \times 0.8480\)

= \(6822.765km\)

\(Speed = \frac{Distance}{Time}\)

= \(\frac{6822.766}{11} = 620.25 km/h\)

(b)\

\( <TQP = 180° - 45° = 135°\) (angles on a straight line)

\(< QTP = 180° - (135° + 35°) = 10°\) (sum of angles in a triangle)

\(\frac{TQ}{\sin 35°} = \frac{15}{\sin 10°}\)

\(\therefore TQ = \frac{15 \sin 35°}{\sin 10°}\)

= \(49.546m\)

\(\therefore \sin 45° = \frac{TA}{TQ}\)

\(\sin 45° = \frac{TA}{49.546}\)

\(TA = 49.546 \times 0.7071 = 35.03m\)

\(\therefore\) The height of the kite = 35m (correct to the nearest whole number)

Correct Answer: A. (-3, -2)

Correct Answer: B. 040000111906.

Explanation

\(U = 25\)

\(\therefore (2x + 1) + x + (x - 2)^{2} = 25\)

\(x^{2} - 4x + 4 + x + 2x + 1 = 25\)

\(x^{2} - x + 5 - 25 = 0\)

\(x^{2} - x - 20 = 0\)

\(\implies x^{2} - 5x + 4x - 20 = 0\)

\(x(x - 5) + 4(x - 5) = 0\)

\(\implies \text{x = -4 or 5}\)

But x cannot be negative in this case, hence, x = 5.

(b) Number of guests that selected just one meal = \(2x + 1 + x^{2} - 4x + 4\)

\(x^{2} - 2x + 5 = 5^{2} - 2(5) + 5 = 25 - 10 + 5 = 20\)

Probability of selecting a person who picked one meal = \(\frac{20}{25} = \frac{4}{5}\).

Correct Answer: D. 115°

Explanation

In triangle RXY, < RXY = < RYX (base angles of an isosceles triangle)

\(\implies\) 180° - 62° = 2 < RYX

118° = 2 < RYX \(\implies\) < RYX = 59°

In triangle RYZ, < RZY = 62° (base angles of an isosceles triangle)

\(\therefore\) < RYZ = 180° - (62° + 62°)

= 180° - 124° = 56°

\(\therefore\) < XYZ = 56° + 59°

= 115°

Correct Answer: A. 1.2

Explanation

Average performance rating of Team B = \(\frac{7+9+1+9+6}{5} = \frac{32}{5}\) = 6.4

Average performance rating of Team A = \(\frac{5+3+6+10+2}{5} = \frac{26}{5}\) = 5.2

∴ The difference in the average performance ratings between Team B and Team A = 6.4 - 5.2 = 1.2

Correct Answer: A. 2

Correct Answer: B. 7.2 units

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

Explanation

\(F = F_{1} + F_{2}\)

\((2i - 5j) + (-3i + 4j) = (-i - j)\)

\(F = ma \implies (-1, -1) = 5a\)

\(a = (-\frac{1}{5}, -\frac{1}{5})\)

\(|a| = \sqrt{(\frac{-1}{5})^2 + (\frac{-1}{5})^2} = \sqrt{2}{25}\)

\(|a| = \frac{\sqrt{2}}{5} ms^{-2}\)