Correct Answer: A. \(\frac{5\pi}{12}\)

Explanation

\(180° = \pi rads\)

\(1° = \frac{\pi}{180}\)

\(\therefore 75° = \frac{\pi}{180} \times 75 \)

= \(\frac{5\pi}{12}\)

Correct Answer: B. -3

Explanation

\(P = begin{pmatrix} y - 2 & y - 1 \\ y - 4 & y + 2 \end{pmatrix}\)

\(|P| = (y - 2)(y + 2) - (y - 1)(y - 4) = (y^{2} - 4) - (y^{2} - 5y + 4) = -23\)

\(5y - 8 = -23 \implies 5y = -23 + 8 = -15\)

\(y = \frac{-15}{5} = -3\)

Correct Answer: B. 5cm

Explanation

Volume of water after 9 seconds = \(5\pi \times 9 = 45\pi cm^3\)

Volume of cylinder = \(\pi r^2 h\)

\(\therefore \pi r^2 h = 45\pi\)

\(\pi \times 3^2 \times h = 45\pi\)

\(\implies 9h = 45 \)

\(h = 5 cm\)

(where h = height of the water after 9 secs)

Correct Answer: C. 121°

Explanation

Joining SQ. In \(\bigtriangleup\) SPQ,

(22^o + a) + 55^o + (41^o + b) = 180^o

121^o + a + b = 180^o

a + b = 180 - 121

a + b = 59^o.....(1)

In \(\bigtriangleup\) SRPQ; R + a + b = 180^o

R + 59^o = 180^o

(in (1), a + b = 59^o)

R = 180 - 59

R = 121^o

Correct Answer: B. 36cm

Explanation

Area = \(\frac{1}{2} \times base \times height\)

\(height : base = 1 : 3\)

\(\implies base = 3 \times height\)

Let height = h;

Area = \(\frac{1}{2} \times 3h \times h = 216\)

\(3h^2 = 216 \times 2 = 432\)

\(h^2 = \frac{432}{3} = 144\)

\(h = \sqrt{144} = 12.0 cm\)

\(\therefore base = 3 \times 12 = 36 cm\)

Explanation

No. of red = 10, No of green = 8

Total = 18

P(RG) = P(R) x P(G)

= \(\frac{10}{18} \times \frac{8}{17} = \frac{40}{152}\)

P(GR) = P(G) x P(R)

= \(\frac{8}{18} \times \frac{10}{17} = \frac{40}{153}\)

P(one red and one green)

= \(\frac{40}{153} + \frac{40}{153} = \frac{80}{153}\)

(b) P = 20% = \(\frac{20}{100}\) = 0.2

q = 1 - P = 1 - 0.2 = 0.8

n = 12

P(2 \(\leq x \leq 5\)) = P(5) - P(2)

= \(^{12}C_5(0.5)^5 (0.8)^7 - ^{12}C_2(0.2)^2(0.8)^{10}\)

= 792 x (0.2)\(^5\) (0.8)\(^7\) - 65(0.2)\(^2\)(0.8)\(^10\)

= 0.04892 - 0.2791

= 0.2302

Correct Answer: A. 2

Explanation

4\(^{3x}\) = 4\(^{2(x+1)}\)

3x = 2x + 2

3x - 2x = 2

x = 2

Correct Answer: C. 36

Explanation

Sample size = 6 x 6 = 36.

Correct Answer: A. 21x² - 19x - 12 = 0

Explanation

Let x = \(\frac{4}{3}\), x = -\(\frac{3}{7}\)

Then 3x = 4, 7x = -3

3x - 4 = 0, 7x + 3 = 0

(3x - 4)(7x + 3) = 0

21x² + 9x - 28x - 12 = 0

21x² - 19x - 12 = 0

Correct Answer: A. 29\(\sqrt{5}\)

Explanation

3 \(\sqrt{45} - 12\sqrt{5} + 16\sqrt{20}\)

= 3 x \(\sqrt{9 \times 5} - 12 \times \sqrt{5} + 16 \times \sqrt{4 \times 5}\)

= 3 x 3 x \(\sqrt{5} - 12 \times \sqrt{5} + 16 \times 2 \times \sqrt{5}\)

= 9\(\sqrt{5} - 2 \sqrt{5} + 32 \sqrt{5}\)

= 9\(\sqrt{5} + 32\sqrt{5} - 12\sqrt{5}\)

= 29\(\sqrt{5}\)