Cracking the SAT Subject Test in Math 2, 2nd Edition (E) 21 1 44 If, in the figure above, the magnitude of vector u is 9 and the magnitude of vector v is 7, what is the magnitude of vector (u + v) ? ([.]
(E) 21.1 44 If, in the figure above, the magnitude of vector u is 9 and the magnitude of vector v is 7, what is the magnitude of vector (u + v) ? (A) 5.79 (B) 7.00 (C) 11.40 (D) 12.26 (E) 15.05 LOGIC Every now and then, as you proceed innocently through the SAT Subject Test in Math 2, you will come upon a question asked in simple English that seems to have nothing at all to do with math This is a logic question Here’s a typical example 24 If every precious stone is harder than glass, which of the following statements must also be true? (A) Glass can be a precious stone (B) Every stone harder than glass is a precious stone (C) No stone is exactly as hard as glass (D) Some stones softer than glass are precious stones (E) Every stone softer than glass is not a precious stone Here’s How to Crack It This is madness There’s no math here at all However, there is a rule here for you to work with The rule states that given one statement, there’s only one other statement that is logically necessary, the contrapositive This is what the contrapositive states: The Contrapositive Given the statement A → B, you also know ~B → ~A In English, that means that the statement “If A, then B” also tells you that “If not B, then not A.” To find the contrapositive of any statement, switch the order of the first and second parts of the original statement, and negate their meaning But you can’t be sure of anything else For example, “If not A” doesn’t necessarily mean “then not B.” And “if B” doesn’t necessarily mean “then A.” This is how you’d find the contrapositive of the statement, “Every precious stone is harder than glass.” Start by making sure that you clearly see what the two parts of the original statement are stone is precious → stone harder than glass Then switch the order of the statement’s parts, and negate their meanings: stone not harder than glass → stone is not precious This is the contrapositive Once you’ve found it, just check the answer choices for a statement with an equivalent meaning In this case, (E) is equivalent to the contrapositive Trap answers will typically say things like “Every stone that is harder than glass is precious” or “Every stone that is not precious is softer than glass.” Almost all logic questions test your understanding of the contrapositive There are just a couple of other points that might come up in logic questions • If you see the statement “Some A are B,” then you also know that “Some B are A.” For example, “Some teachers are pretty cool people” also means that “Some pretty cool people are teachers.” • To disprove the claim, “X might be true,” or “X is possible,” you must show that X is never, ever true, in any case, anywhere • To disprove the claim, “X is true,” you only need to show that there’s one exception, somewhere, sometime In other words, a statement that something may be true is very hard to disprove; you’ve got to demonstrate conclusively that there’s no way it could be true On the other hand, a statement that something is definitely true is easy to disprove; all you have to is find one exception If you remember the three bullet points above and the contrapositive, you’ll be prepared for any logic question on the SAT Subject Test in Math 2 DRILL 9: LOGIC Exercise your powers of logic on these practice questions The answers can be found in Part IV 28 At Legion High School in a certain year, no sophomore received failing grades Which of the following statements must be true? (A) There were failures in classes other than the sophomore class (B) Sophomores had better study skills than other students that year (C) No student at Legion High School received failing grades that year (D) Any student who received failing grades was not a sophomore (E) There were more passing grades in the sophomore class than in other classes ... prepared for any logic question on the SAT Subject Test in Math 2 DRILL 9: LOGIC Exercise your powers of logic on these practice questions The answers can be found in Part IV 28 At Legion High School in a certain year, no sophomore... “If not B, then not A.” To find the contrapositive of any statement, switch the order of the first and second parts of the original statement, and negate their meaning But you can’t be sure of anything else For... glass.” Start by making sure that you clearly see what the two parts of the original statement are stone is precious → stone harder than glass Then switch the order of the statement’s parts, and negate their meanings: stone not harder than glass → stone is not precious