Explanation

\(< PRQ = 90°\)

Gradient of PR \(\times\) Gradient of QR = -1.

\(\frac{5 - 3}{k + 1} \times \frac{5 - 4}{k + 4} = -1\)

\(\frac{2}{(k + 1)(k + 4)} = -1\)

\(-2 = k^{2} + 5k + 4 \implies k^{2} + 5k + 4 + 2 = 0\)

\(k^{2} + 5k + 6 = 0\)

\(k^{2} + 3k + 2k + 6 = 0\)

\(k(k + 3) + 2(k + 3) = 0 \implies (k + 3)(k + 2) = 0\)

\(\implies \text{k} = -2 or -3\)

Explanation

At y-intercept, 3(0) — 2y + 6 = 0 and solving for y, y = 3.

Therefore, the coordinates of y — intercept is (0, 3).

Then, r\(^2\) = (0 — 2)\(^2\) + (3 — 3)\(^2\) = 4. With the centre at (2, 3). the equation of the circle was (x — 2)\(^2\) + (y — 3)\(^2\) = 4 and which simplified to \(x^2\) - y\(^2\) — 4x — 6y + 9 = 0

Correct Answer: B. \(3^{\frac{1}{2}}\)

Explanation

\(\log_{3} x = \log_{9} 3 \implies \log_{3} x = \log_{9} 9^{\frac{1}{2}} = \frac{1}{2}\log_{9} 9\)

\(\log_{3} x = \frac{1}{2} \)

\(\therefore x = 3^{\frac{1}{2}}\)

Correct Answer: A. 2 -√3

Correct Answer: D. 0.2007

Explanation

f:x → x\(^2\) + 2 and g:x → \(\frac{1}{x+2}\).

(g?f)\(^{-1}\)

(g?f)\(^{x^2+2}\)

g\(^{x^2+2}\) = \(\frac{1}{(x^2+2)+2}\)

(g?f) = \(\frac{1}{(x^2+4}\)

let y = \(\frac{1}{(x^2+4}\)

y\(((x^2+4)\) = 1

\(yx^2+4y = 1\)

x\(^2\) = \(\frac{1-4y}{y}\)

x = \(\sqrt \frac{1-4y}{y}\)

(g?f)\(^{-1}\) = \(\sqrt \frac{1-4x}{x}\)

Correct Answer: D. 3

Correct Answer: A. 30.4

Explanation

Another word for mean is average. To find the average of this set of numbers, add them up, then divide by the amount of numbers in the set. In this case there are 5 numbers in the set, meaning we will divide our answer by 5. (6+91+12+31+125=152)/5=30.4

Correct Answer: C. 12

Explanation

If logx 64 = 3, then \(64 = x^3\)

\(4^3 = x^3\)

Since the indices are equal,

x = 4

Hence, x log\(_2\)8

= 4(3.log\(_2\)2)

= where log\(_2\)2 = 1(1)

= 12 X 1

= 12

Correct Answer: (b) 15