Correct Answer: C. \(\frac{16}{3}\)

Explanation

\(\int_{0}^{2} (8x - 4x^{2}) \mathrm {d} x = (\frac{8x^{1 + 1}}{2} - \frac{4x^{2+1}}{3})|_{0}^{2}\)

= \((4x^{2} - \frac{4x^{3}}{3}) |_{0}^{2}\)

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

= \(16 - \frac{32}{3} = \frac{16}{3}\)

Explanation

(a)

(b) The estimate of the modal height of the seedlings is 1.08

Correct Answer: A. \(4(6 + 5\pi)\)

Explanation

The angle subtended by the minor arc = \(\frac{\pi}{3} radians\)

The angle subtended by the major arc = \(2\pi - \frac{\pi}{3} = \frac{5\pi}{3}\)

Perimeter of the major arc = \(r\theta + 2r\)

= \(12 \times \frac{5\pi}{3} + 2(12) = 20\pi + 24\)

= \(4(5\pi + 6)\)

Correct Answer: D. 0

Explanation

\(\frac{1}{x^2 - 4} = \frac{P}{(x + 2)} + \frac{Q}{(x - 2)}\)

I = p(x - 2) + Q(x + 2)

Let x = 2

I = P(2 - 2) + Q(2+ 2)

I = -4Q

Q = \(\frac{1}{4}\)

Let x = -2

I = P(-2 - 2) + Q(-2 + 2)

I = -4p

P = \(\frac{1}{-4}\)

PQQ = - \(\frac{1}{4} + \frac{1}{4}\)

= 0

Correct Answer: C. 36m

Explanation

from the equation of motion

u = 5m/s, a = 2m/s\(^2\), t = 4s

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

s = 5*4 + \(\frac{1}{2} 2*4^2\)

s = 20 + 16

s = 36m

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

Explanation

\(f(x) = \frac{4}{x} - 1\). Let y = f(x)

\(y = \frac{4 - x}{x} \implies xy + x = 4\)

\(x(y + 1) = 4 \therefore x = \frac{4}{y + 1}\)

\(f^{-1}(7) = \frac{4}{7 + 1} = \frac{1}{2}\)

Correct Answer: C. 41 \(ms ^{-2}\)

Explanation

\(S = 5t^3 - \frac{19}{2} t^2 + 6t - 4\)

\(v(t)=\frac{dS}{dt}=15t^2-19t+6\)

\(a(t)=\frac{dv}{dt}=30t-19\)

∴a(2)=30(2)-19=60-19=41 \(ms ^{-2}\)

Correct Answer: D. 13

Explanation

\(\lim\limits_{x \to 3} \frac{2x^{2} + x - 21}{x - 3}\)

\(2x^{2} + x - 21 = 2x^{2} - 6x + 7x - 21 \) (by factorizing)

= \((2x + 7)(x - 3)\)

\(\therefore \lim\limits_{x \to 3} \frac{2x^{2} + x - 21}{x - 3} \equiv \lim\limits_{x \to 3} \frac{(2x+7)(x-3)}{x-3}\)

\(\lim\limits_{x \to 3} (2x + 7) = 2(3) + 7 = 13\)

Correct Answer: D. \(\cos^{2} \theta\)

Explanation

\((1 + \sin \theta)(1 - \sin \theta) = 1 - \sin \theta + \sin \theta - \sin^{2} \theta\)

\(= 1 - \sin^{2} \theta\)

Recall, \(\cos^{2} \theta + \sin^{2} \theta = 1\)

\(\therefore 1 - \sin^{2} \theta = \cos^{2} \theta\).

Explanation

Spearman's rank correlation coefficient = \(1 - \frac{6 \sum d^{2}}{n(n^{2} - 1)}\)

= \(1 - \frac{6(10)}{5(5^{2} - 1)}\)

= \(1 - \frac{60}{120}\)

= \(0.5\)