Correct Answer: D. 48cm

Correct Answer: D. 7

Explanation

2p - m = 6 ... (i)

2p + 4m = 1 ... (ii)

From (i), m = 2p - 6.

2p + 4(2p - 6) = 1

2p + 8p - 24 = 1

10p = 1 + 24 = 25

p = 2.5

m = 2(2.5) - 6

= 5 - 6

= -1

\(\therefore\) 4p + 3m = 4(2.5) + 3(-1)

= 10 - 3

= 7

Explanation

(a) \(x = (-2i + 5j) ; y = (i - 7j)\)

(i) \((3x + 2y) = 3(-2i + 5j) + 2(i - 7j)\)

= \((-6i + 15j) + (2i - 14j)\)

= \(-4i + j\)

(ii) \((y - 2x) = (i - 7j) - 2(-2i + 5j)\)

\((i - 7j) - (-4i + 10j) = (5i - 17j)\)

\(|y - 2x| = |-3i - 17j| = \sqrt{(5^{2}) + (-17 ^{2})}\)

= \(\sqrt{25 + 289} = \sqrt{314}\)

(iii) Let the angle between x and y be \(\theta\).

\(x . y = |x||y| \cos \theta\)

\((-2i + 5j) . (i - 7j) = |-2i + 5j||i - 7j| \cos \theta\)

\(-2 - 35 = (\sqrt{(-2)^{2} + (5)^{2}})(\sqrt{(1)^{2} + (-7)^{2}}) \cos \theta\)

\(-37 = (\sqrt{29})(\sqrt{50}) \cos \theta \implies -37 = (\sqrt{1450})\cos \theta\)

\(\cos \theta = \frac{-37}{\sqrt{1450}} = -0.9717\)

\(\theta = \cos^{-1} (-0.9717) = 166.34°\)

(iv) \(x + y = (-2i + 5j) + (i - 7j) = -i - 2j\)

\(|x + y| = \sqrt{(-1)^{2} + (-2)^{2}} = \sqrt{5}\)

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

(b)

\(m_{1} = 0.084kg ; m_{2} = 20 kg ; m = 20.084 kg\)

\(u_{1} = ? ; u_{2} = 0 m/s ; v = 0.24 m/s\)

Momentum before collision = \(0.084 u_{1} kg m/s\)

Momentum after collision = \(20.084 \times 0.24 = 4.82016 kg m/s\)

\(\therefore 0.084 u_{1} = 4.82016 \)

\(u_{1} = \frac{4.82016}{0.084} = 57.3829 m/s\)

The initial velocity of bullet is 57.38 ms\(^{-1}\).

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

Explanation

No. of teams that scored at most 3 goals = 3 + 1 + 6 = 10

Hence, probability that a team selected at random scored at most 3 goals

= \(\frac{10}{25}\) = \(\frac{2}{5}\)

Correct Answer: A. 0.18

Explanation

(a) 1km = 100,000 cm.

Area = \(\frac{85 \times 100,000 \times 100,000}{20,000 \times 20,000}\)

= \(85 \times 5 \times 5\)

= \(2125 cm^{2}\)

(b)

Perimeter of path = \(2(length + width)\)

= \(2(30 + (12 + a)) = 2(42 + a)\)

= \(84 + 2a\)

Area of field = \(length \times width\)

= \(18 \times a\)

= 18a

\(\therefore 18a = 84 + 2a\)

\(18a - 2a = 84 \implies 16a = 84\)

\(a = \frac{84}{16}\)

= \(5.25 cm\)

(c) Area of major sector = \(\frac{360 - x}{360} \times \pi r^{2}\)

\(27.5 = \frac{360 - x}{360} \times \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2}\)

\(27.5 = \frac{360 - x}{360} \times \frac{77}{2}\)

\(360 - x = \frac{27.5 \times 360 \times 2}{77}\)

\(360 - x = 257.14\)

\(x = 360° - 257.14° = 102.86°\)

\(x \approxeq 103°\) (to the nearest degree).

Correct Answer: B. 1.5

Explanation

3^2x = 27

3^2x = 3³

2x = 3

x = 1.5

Correct Answer: D. \(\frac{x - 4}{2x - 3}\)

Explanation

\(\frac{2x^2 - 5x - 12}{4x^2 - 9}\) = \(\frac{3x^2 - 8x + 3x - 12}{(2x)^2 - 3^2}\)

= \(\frac{32(x - 4) + 3(x - 4)}{(2x - 3)(2x + 3)} - \frac{(x - 4) + (2x + 3)}{(2x - 3) (2x + 3)}\)

= \(\frac{x - 4}{2x - 3}\)

Correct Answer: (a) 9

Correct Answer: D. N76,800.00