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Arjun S.
Math and Chemistry major at Columbia University
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SAT II Mathematics Level 2
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Question:

(From the CollegeBoard) A game has two spinners. For the first spinner, the probability of landing on blue is $$\frac{4}{5}$$. Independently, for the second spinner, the probability of landing on blue is $$\frac{1}{7}$$. What is the probability that the first spinner lands on blue and the second spinner does not land on blue?

Arjun S.
Answer:

The key to this question is understanding that the probability of the first and second event are independent. Since these probabilities are independent, we can simply multiply the probability of the first occurrence by the probability of the second occurrence to get the total probability. P(1st spinner is blue) =$$ \frac{4}{5}$$ P(2nd spinner isn't blue) = 1 - P(2nd spinner is blue) = $$1- \frac{1}{7} = \frac{6}{7}$$ P(total) = P(1st spinner is blue) $$\times$$ P(2nd spinner isn't blue) = $$\frac{4}{5} \times \frac{6}{7} = \frac{24}{35}$$ Our answer is $$\frac{24}{35}$$

Physics (Newtonian Mechanics)
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Question:

A projectile moves in a parabolic path without air resistance. Is there any point at which the acceleration is parallel to the velocity? Perpendicular?

Arjun S.
Answer:

Since the object is traveling in projectile(2-dimensional) motion, the only force acting on the object is acting straight downwards and is the force of gravity. Therefore the acceleration at all points of motion is a vector downward. The object is never traveling straight downward, therefore the velocity is never parallel to acceleration. At the top of the parabola, the velocity is directly to the right, and the acceleration is downward, therefore both quantities are perpendicular the the maximum height of the parabola.

Chemistry
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Question:

Order the following ions from smallest to largest atomic radius: Na+, F-, Li+

Arjun S.
Answer:

When considering this question, we should first look at each ion's electron count on the periodic table. We see that Na+ and F- have 10 electrons and Li+ has two electrons. Since Na+ and F- have an extra energy level compared to Li+, both these ions must be larger than Li+. We now must decide whether Na+ or F- is bigger. Since F- has an added electron, the added electrons will repel in the outer shell (like charges repel), thereby making the F- radius larger than the F radius. Since the Na+ ion has lost an electron, the Na nucleus is now attracting the remaining electrons with greater force, thereby decreasing the Na+ ion radius relative to Na. By this logic, the F- radius must be greater than the Na+ radius, since the F has increased its radius, whereas the Na has decreased its radius. Combining all this information, the final answer is Li+, Na+, F-

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