Correct Answer: B. 35₆

Correct Answer: D. \(\frac{3}{4} \hspace{1mm}or \hspace{1mm}-3\)

Explanation

The equation \(\frac{3x - 2}{4x^2 + 9x - 9}\) is undefined when the denominator = 0.

\(4x^2 + 9x - 9 = 0\)

\(4x^2 + 12x - 3x - 9 = 0\)

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

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

x = \(\frac{3}{4}\) or x = -3.

Correct Answer: C. 180cm

Explanation

d \(\alpha\) t²

d = t² k

where k is a constant. d = 45cm, when t = 3s; thus 45 = 3² x t

k = \(\frac{45}{9}\) = 5

thus equation connecting d and t is d = 5t²

when t = 6s, d = 5 x 6²

= 5 x 36

= 180cm

Correct Answer: B. \(\frac{\sqrt{2}}{4}(\sqrt{3} - 1)\)

Explanation

\(\cos(a + b) = \cos a\cos b - \sin a\sin b\)

\(\cos75° = \cos(30 + 45) = (\cos30)(\cos45) - (\sin30)(\sin45)\)

= \((\frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2}) - (\frac{1}{2} \times \frac{\sqrt{2}}{2})\)

= \(\frac{\sqrt{6} - \sqrt{2}}{4}\)

= \(\frac{\sqrt{2}(\sqrt{3} - 1)}{4}\)

Correct Answer: C. 9

Explanation

\(log_3 x + log_3 (x - 8) = 2\)

\(log_3 x(x - 8) = log_39\) since 2 = \(log_39\)

\(log_3\) cancels out

⇒ x(x - 8) = 9

⇒ \(x^2 - 8x = 9\)

⇒ \(x^2 - 8x - 9 = 0\)

⇒ \(x^2 - 9x + x - 9 = 0\)

⇒ x(x - 9) + 1(x - 9) = 0

⇒ (x - 9)(x + 1) = 0

⇒ x = 9 or x = -1

Since we can't have a log of negative numbers,

∴ x = 9.

Correct Answer: C. 59.5°

Explanation

\(\overrightarrow{r} . \overrightarrow{t} = |\overrightarrow{r}||\overrightarrow{t}|\cos \theta\)

\(\overrightarrow{r} . \overrightarrow{t} = (3i + 4j) . (-5i + 12j) = -15 + 48 = 33\)

\(|\overrightarrow{r}| = \sqrt{3^{2} + 4^{2}} = \sqrt{25} = 5\)

\(|\overrightarrow{t}| = \sqrt{(-5)^{2} + 12^{2}| = \sqrt{169} = 13\)

\(\cos \theta = \frac{\overrightarrow{r} . \overrightarrow{t}}{|\overrightarrow{r}||\overrightarrow{t}|}\)

\(\cos \theta = \frac{33}{5 \times 13} = \frac{33}{65}\)

\(\theta = \cos^{-1} {\frac{33}{65}} \approxeq 59.5°\)

Correct Answer: A. 5y - 4x - 7z

Explanation

(4y - x - 2z) - (-y + 3x + 5z)

= 4y + y - x - 3x - 2z - 5z

= 5y - 4x - 7z

Correct Answer: D. \(-\frac{11}{3}\)

Correct Answer: C. 68°

Correct Answer: A. 36