All ISEE Upper Level Quantitative Resources
Example Questions
Example Question #51 : Operations
Which is the greater quantity?
(a)
(b)
(a) is the greater quantity
(a) and (b) are equal
It cannot be determined which of (a) and (b) is greater
(b) is the greater quantity
(a) is the greater quantity
Suppose is nonnegative.
Then .
Consequently,
,
which must be positive,
and
,
which is the opposite of and consequently must be negative. Therefore, (a) is greater.
Suppose is negative.
Then .
Consequently,
,
and
.
, so
,
and (a) is greater.
(a) is the greater quantity either way.
Example Question #52 : Variables
Define . The graph of
is a line with slope
.
.
Which is the greater quantity?
(a)
(b)
It is impossible to determine which is greater from the information given
(a) is the greater quantity
(b) is the greater quantity
(a) and (b) are equal
(a) is the greater quantity
, so
.
, so, by definition,
, or
.
The graph of is a line through the point with coordinates
and with slope
. The equation of the line can be determined by setting
in the slope-intercept form:
.
The equation of the line is , which makes this the definition of
. By setting
,
.
Therefore,
Example Question #51 : Operations
;
and
are both positive.
Which is the greater quantity?
(a)
(b)
It is impossible to determine which is greater from the information given
(b) is the greater quantity
(a) is the greater quantity
(a) and (b) are equal
(b) is the greater quantity
If ,
then
.
The absolute value of a negative number is its (positive) opposite, so
Also, if and
are both positive, then
is positive; the absolute value of a positive number is the number itself, so
. Since
, it follows that
. Therefore,
Since is given to be positive,
.
and
All ISEE Upper Level Quantitative Resources
