3 questions: Q14, Q17, Q21. Solve each one here: tap the answer, press Check, then open the solution.
MCQ · +3 / −1
The number of common terms in the two sequences $18, 24, 30, 36, \ldots, 606$ and $14, 24, 34, 44, \ldots, 604$ is:
A 18 B 19 C 20 D 21 Check
Answer and solution Correct answer: C
Solution
First common term $= 24$ ; common terms repeat every $\text{LCM}(6, 10) = 30$ .$24, 54, 84, \ldots, 594$ (last one $\le 604$ )$= \dfrac{594 - 24}{30} + 1 = 19 + 1 = 20$
MCQ · +3 / −1
If $z$ is a negative number such that $2^{\,z^2 \log_5 7} = 7^{\log_2 5}$ , then $z$ equals:
A $\log_2 \tfrac{1}{7}$ B $\log_2 \tfrac{1}{5}$ C $-\log_2 \tfrac{1}{7}$ D $-\log_2 \tfrac{1}{5}$ Check
Answer and solution Correct answer: B
Solution
Take $\log_2$ of both sides: $z^2 \log_5 7 = \log_2 5 \cdot \log_2 7$ $\log_5 7 = \dfrac{\log_2 7}{\log_2 5}$ , so $z^2 \dfrac{\log_2 7}{\log_2 5} = \log_2 5 \cdot \log_2 7$ $z^2 = (\log_2 5)^2 \Rightarrow z = \pm \log_2 5$ $z$ is negative: $z = -\log_2 5 = \log_2 \tfrac{1}{5}$
TITA (type the answer) · +3 / 0
How many integers $n$ satisfy $(n - 3)(n - 6) - 3(n - 3) \le 0$ ?
Your answer Check
Answer and solution Correct answer: 7
Solution
$(n - 3)\big[(n - 6) - 3\big] = (n - 3)(n - 9) \le 0$ $\Rightarrow 3 \le n \le 9$ $7$
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