GMAT Official Guide Quantitative Review 2019_ Book

5.5 Ci Answer Explanations

Geometry
Can we determine whether line k, which is
parallel to the y-axis, is tangent to the circle C?

( 1 ) Given that line k passes through the point

(-1,0), we can represent the scenario in the

following diagram, which is not drawn to

scale.

k

(-1.0)

y

(2)

x

The points (1,0) and (-1,0) are two units

apart on the x axis. We therefore know that

circle C, with center (1,0) and radius 2,

passes through the point (- 1 , 0 ) and that,

given that k passes through the point (- 1 ,0),

the circle intersects line k at this point.

Furthermore, the radial line from the center

(1,0) of the circle to point (-1,0) on the

circle rests on the x-axis and is therefore

perpendicular to the y-axis. Line k,

being parallel to the y-axis, must also be

perpendicular to this radial line. Therefore,

because line k both intersects the circle at a

point at which a radial line intersects the

circle and is perpendicular to this radial line,

we see that that line k must be tangent to

circle C; SUFFICIENT.

The sufficiency of statement 2 follows from

the sufficiency of statement 1. For, if k is

perpendicular to the y-axis and passes

through point (-1,-1), then k must also pass

through point (-1,0); k is simply the line

x = -1. The reasoning for statement 1 now

applies; SUFFICIENT.

DS14170

198. Company X's pro优s this year increased by 25% over
     last year's pro什ts. Was the dollar amount of Company
     X's pro什ts this year greater than the dollar amount of
     Company Y's?
     ',

..

'，l2

＇,

Algebra

Let Px and P;, respectively, be the profits of

Company X last year and this year, and let Py and

P{, respectively, be the profits of Company Y last

year and this year. Then P; = 1.2 5 Px·Is P; > P{?

(1)

(2)

Last year, the ratio of Company Y's profits to

Company X's profits was 5:2.

Company Y experienced a 40% drop in

profits from last year to this year.

Given that _y_ p^5

= -, it is not possible to

p X 2

determine whether P; > P; because nothing

is known about the value of P; other than

P; is positive; NOT sufficient.

Given that P; = 0.6Py, it is not possible

to determine whether P; > P; because

nothing is known that relates the profits of

Company X for either year to the profits of

Company Y for either year; NOT sufficient.

Taking (1) and (2) together, it is given that

笃= 1. 25 Px and from ( 1 ) it follows that杻三，

p X 2

orPx =—^2 凡，andthus P�= (^1 .2 5 2

) —Py. From

5

(2)it follows that P; = 0.6Py, m }� � 环，

and thus P{ = (1.2 5 尺）（归）. Since

the last equation expresses P�as a specific

number times P; 勹itfollows that it can be

determined whether or not P�> p;_ Note that

(^1.^2 5)(言）（点）＝言）（切(f)=¾, and so the

answer to the question "Is P�> P;" is no.

Th e correct answer ts C·

both statements together are sufficient.

Th e correct answer 1s D;

Each statement alone is sufficient.

,LL己乙a.5·