Correct Answer: B. tan θ + k

Correct Answer: D. 17.75

Correct Answer: A. 31

Explanation

First, write out the individual values found within this stem and leaf plot to figure out how many values there are. Values:20,21,24,24,30,31,31,32,32,35,37,41,45 Now, finding the mean is the same as finding the average. Recall how to find the average: Mean/Average=Sum of all valuesNumber of values Thus, the mean number of baseball cards owned is found by the following: Mean=(20+21+24+24+30+31+31+32+32+35+37+41+45)/13

Correct Answer: B. neither commutative nor closed on the set of ppositive integers

Explanation

(a) For P, Q and R to lie on a straight line,the gradient of any two of PQ, PR and QR must be equal.

Gradient of PR : \(\frac{6 - 1}{2 - 11} = -\frac{5}{9}\)

Gradient of QR : \(\frac{6 - \frac{13}{3}}{2 - 5} = \frac{\frac{5}{3}}{-3}\)

= \(- \frac{5}{9}\)

The gradients are the same therefore P, Q and R are on the same line.

(b) \(\overrightarrow{PQ} = (5i + \frac{13}{3}j) - (11i + j)\)

= \(-6i + \frac{10}{3}j\)

\(|\overrightarrow{PQ}| = \sqrt{(-6)^{2} + (\frac{10}{3})^{2}}\)

= \(\sqrt{36 + \frac{100}{9}}\)

= \(\sqrt{\frac{424}{9}}\)

\(\overrightarrow{QR} = (2i + 6j) - (5i + \frac{13}{3}j)\)

= \(-3i + \frac{5}{3}j\)

\(|\overrightarrow{QR}| = \sqrt{(-3)^{2} + (\frac{5}{3})^{2}}\)

= \(\sqrt{9 + \frac{25}{9}}\)

= \(\sqrt{\frac{106}{9}}\)

\(\frac{|\overrightarrow{PQ}|}{|\overrightarrow{QR}|} = \frac{\sqrt{\frac{424}{9}}}{\sqrt{\frac{106}{9}}}\)

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

\(\therefore |\overrightarrow{PQ}| : |\overrightarrow{QR}| = 4 : 1\)

Explanation

(a) If Musa's age = x and Manya's age = y, then x = y + 3.... (1) and (x — 7) = 2(y— 7) .......(2)

Solving both equations will yield Musa's age = 13 years and Manya's age = 10 years.

(b) Let n be the year when the sum of their ages will be 45,

so that (10 + n) + (13 + n) = 45

and solving will give n = 11 years.

Correct Answer: C. 125

Correct Answer: A. -21

Explanation

\(y = x^{2} + 6x - 12\)

\(\frac{\mathrm d y}{\mathrm d x} = 2x + 6 = 0\)

\(2x = -6 \implies x = -3\)

\(y(-3) = (-3)^{2} + 6(-3) - 12 = 9 - 18 - 12 = -21\)

Correct Answer: D. 5/12

Correct Answer: B. p = -1, q = 2