Note: The values for x and y used in (2) above
can be found by solving (x - 3 )^2 = 3 - x, which
can be rewritten as夕- 6x + 9 = 3 - x, or
J?- - 5x^ + 6 = 0, or (x - 3)(x - 2) = 0.
Th e correct answer 1s A·，
statement 1 alone is sufficient.
DS06810
- What is the probability that Lee will make exactly 5
errors on a certain typing test?
(1) The prob ab小ty that Lee will make 5 or more
errors on the test is 0.27.
(2) The probability that Lee will make 5 or fewer
errors on the test is 0.85.
Arithmetic
(1) Given that 0.27 is小e probability that
Lee will make 5 or more errors on the
test, it is clearly not possible to determine
the probability that Lee will make exactly
5 errors on the test; NOT sufficient.
(2) Given that 0.85 is the probability that
Lee will make 5 or fewer errors on the
test, it is clearly not possible to determine
the probability that Lee will make exactly
5 errors on the test; NOT sufficient.
Taking (1) and (2) together, let Ebe the event
that Lee will make 5 or more errors on the test
and let Fbe the event that Lee will make 5 or
fewer errors on the test. Then P(E or F) = l, since
it will always be the case that, when ta如ngthe
test, Lee will make at least 5 errors or at most
5 errors. Also, (1) and (2) can be expressed as
P(E) = 0.27 and P(F) = 0.85, and the question
asks for the value of P(E and乃Using the
identity P(E or F) = P(E) + P(F) - P(E and凡 it
follows that 1 = 0.27 + 0.85 -P(E and乃，or P(E
and F) = 0.27 + 0.85 -1 = 0.12. Therefore, the
probability that Lee will make exactly 5 errors on
the test is 0.12.
Th e correct answer 1s C·，
both statements together are sufficient.
DS19208
- If p is a positive integer, is 2P + 1 a prime number?
(l) p is a prime number.
(2) p is an even number.
5.5 t..i Sutfic1t>1 Answer Explanations
Arithmetic
Given that p is a positive integer, can we
determine whether穸+1 is a prime number?
(1) Given that pis a prime number, we don't
have enough information to determine
whether 21 + l is a prime number. To see
this, it best to consider some cases. If p = 2,
then 2/'+ 1 = 2气1 =2x2+1=4+1= 5,
which is prime. And if p = 3, then 2/ + 1 =
23 + 1 = 2 x 2 x 2 + 1 = 8 + 1 = 9, which is
not prime (it is equal to 3 x 3); NOT
sufficient.
(2) Given that pis an even number, we can
again consider cases and see that it is
impossible to determine whether 21 + l is a
prime number. If p = 2 then 2/ + 1 = 2红1
= 4 + 1 = 5, which, again, is prime. And if p
= 6, then 21 + l = 2丘1=2x2x2x2x2
X 2 + 1 = (2 X 2 X 2) X (2 X 2 X 2) + 1 = 8 X
8 + 1= 64 + 1 = 65, which is not prime (it is
equal to 13 x 5); NOT sufficient.
Considering the two statements together, we
have that p is both prime and even. The only
even number that is not divisible by some other
positive integer besides 1 is 2. That is, the only
prime even integer is 2. 21 + l is therefore
equal to 2红1= 5, which is prime.
Th e correct answer 1s C·
both statements together are sufficient.
DS02741
- In the xy-plane, point (r,s) lies on a circle with center at
the origin. What is the value of r^2 + s^2?
(1) The circle has radius 2.
(2) The point (我，一立） lies on the circle.
Geometry
Let R be the radius of the circle. A right triangle
with legs oflengths Ir! and Isl can be formed so
that the line segment with endpoints (r,s) and
(O,O) is the hypotenuse. Since the length of the
hypotenuse is R, the Pythagorean theorem for
如s right triangle gives R^2 = ?-+ s2.1herefore, to
determine the value of户+s2, it is sufficient to
determine the value of R.
(1) It is given that R = 2; SUFFICIENT.
