Correct Answer: C. 107°

Explanation

QPS - QRK = 31^o

QRK + RKQ + KQR = 180

31 + 42 + KQR = 180^o

KQR = 180 - 73 = 107^o

Correct Answer: B. (a - c)(x - y)

Explanation

x(a - c) + y(c - a)

= x(a - c) - y(a - c)

= (x - y)(a - c)

Correct Answer: D. -1

Correct Answer: d. 5000

Explanation

If d = deceleration, then retarding force = \(md\).

This is also the force down the plane \(mg \sin \theta\)

\(\therefore md = mg\sin \theta \implies d = g \sin \theta\)

= \(10 \sin 30° = 10 \times \frac{1}{2} = 5 ms^{-2}\)

(i) \(v^{2} = u^{2} + 2as\)

At the highest point, v = 0 ms\(^{-1}\).

\(0^{2} = u^{2} - 2(5)(15) \)

\(u^{2} = 150 \implies u = \sqrt{150} = 12.247 ms^{-1}\)

(ii) \(v = u - dt\)

\(0 = 12.247 - 5t \implies 5t = 12.247\)

\(t = \frac{12.247}{5} = 2.4494 \approxeq 2.45 s\)

(b)(i) Resultant force

\(F = \begin{pmatrix} 10 \cos 0° \\ 10 \sin 0° \end{pmatrix} + \begin{pmatrix} 5\sqrt{3} \cos 30° \\ 5\sqrt{3} \sin 30° \end{pmatrix} + \begin{pmatrix} 5 \cos 30° \\ 5 \sin 30° \end{pmatrix} + \begin{pmatrix} 5 \cos 300° \\ 5 \sin 300° \end{pmatrix} + \begin{pmatrix} 5\sqrt{3} \cos 330° \\ 5 \sqrt{3} \sin 330° \end{pmatrix}\)

= \(\begin{pmatrix} 10 \times 1 \\ 10 \times 0 \end{pmatrix} + \begin{pmatrix} 5\sqrt{3} \times \frac{\sqrt{3}}{2} \\ 5 \sqrt{3} \times \frac{1}{2} \end{pmatrix} + \begin{pmatrix} 5 \times \frac{1}{2} \\ 5 \times \frac{\sqrt{3}}{2} \end{pmatrix} + \begin{pmatrix} 5 \times \frac{1}{2} \\ -5 \times \frac{\sqrt{3}}{2} \end{pmatrix} + \begin{pmatrix} 5\sqrt{3} \times \frac{\sqrt{3}}{2} \\ -5\sqrt{3} \times \frac{1}{2} \end{pmatrix}\)

= \(\begin{pmatrix} 10 \\ 0 \end{pmatrix} + \begin{pmatrix} 7.5 \\ 2.5 \sqrt{3} \end{pmatrix} + \begin{pmatrix} 2.5 \\ 2.5\sqrt{3} \end{pmatrix} + \begin{pmatrix} 2.5 \\ -2.5 \sqrt{3} \end{pmatrix} + \begin{pmatrix} 7.5 \\ -2.5 \sqrt{3} \end{pmatrix}\)

= \(\begin{pmatrix} 30 \\ 0 \end{pmatrix}\)

Magnitude of resultant = \(\sqrt{30^{2}} = 30N\)

(ii) \(F = ma \implies a = \frac{F}{m}\)

= \(\frac{30}{5} = 6 ms^{-2}\)

Correct Answer: B. -5 or +1

Explanation

\(\frac{6x - 1}{x^2 + 4x - 5}\)

The expression is undefined when \(x^2 + 4x - 5 = 0\)

\(x^2 + 5x - x - 5 = 0\)

\(x(x + 5) - 1(x + 5) = 0\)

\((x - 1)(x + 5) = 0\)

The expression is undefined when x = 1 or -5.

Correct Answer: D. log a²¹⁰

Correct Answer: A. 432

Explanation

\((3x + 4)^{4} = ^{4}C_{0}(3x)^{0}(4)^{4} + ^{4}C_{1}(3x)^{1}(4)^{3} + ^{4}C_{2}(3x)^{2}(4)^{2} + ^{4}C_{3}(3x)^{3}(4)^{1} + ^{4}C_{4}(3x)^{4}(4)^{0}\)

\(x^{3} = ^{4}C_{3}(3x)^{3}(4) = \frac{4!}{3!1!} \times 3^{3} \times 4\)

= \(432x^{3}\)

Correct Answer: A. 3

Explanation

Volume of pyramid = \(\frac{1}{3}\)lbh

90 = \(\frac{1}{3} \times x \times 6 \times 15\)

x = \(\frac{90 \times 33}{6 \times 15}\)

= 3

Correct Answer: C. x = 0