All GMAT Math Resources
Example Questions
Example Question #61 : Dsq: Understanding Functions
What is the -intercept of the graph of
?
Statement 1: .
Statement 2: The graphs of and
intersect only at the point
.
STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.
BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.
BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.
STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.
EITHER STATEMENT ALONE provides sufficient information to answer the question.
BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.
The -intercept of the graph of
is the point
at which the graph intersects the
-axis. At that point,
, or, equivalently,
.
Therefore, we need to find the value for which
.
Between the two statements, we only know that and
. The value of
for which
cannot be determined.
Example Question #61 : Functions/Series
True or false: ,
is an arithmetic sequence.
Statement 1:
Statement 2:
STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.
EITHER STATEMENT ALONE provides sufficient information to answer the question.
BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.
STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.
BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.
STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.
An arithmetic sequence is one in which the difference of each term in the sequence and the one preceding is constant.
Assume Statement 1 alone.
,
meaning that at least two such differences are unequal, and proving the sequence is not arithmetic.
Statement 2 alone only proves that two such differences are equal, but says nothing about any of the other (infinitely many) such differences. Therefore, it leaves the question unresolved.
Example Question #63 : Dsq: Understanding Functions
Define and
.
Is it true that ?
Statement 1:
Statement 2:
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
For to be each other's inverse, it must be true that
and
We can look at the first condition.
For this to be true, it must hold that:
and
Since both statements violate these conditions, it is impossible for , even if you are only given one of them.
The answer is that either statement alone is sufficient to answer the question.
Example Question #3001 : Gmat Quantitative Reasoning
Evaluate the expression
1)
2)
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is not sufficient.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient.
Simplify the expression:
Therefore, we only need to know - If we know
, we calculate that
The answer is that Statement 2 alone is sufficient to answer the question, but Statement 1 is not.
Example Question #3002 : Gmat Quantitative Reasoning
Evaluate:
Statement 1:
Statement 2:
BOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Therefore, you only need to know the value of to evaluate this; knowing the value of
is neither necessary nor helpful.
Example Question #1 : Dsq: Simplifying Algebraic Expressions
Evaluate the expression for positive :
Statement 1:
Statement 2:
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Cancel the from both halves:
As can be seen by the simplification, it turns out that only the value of , which is given only in Statement 2, affects the value of the expression.
Example Question #3003 : Gmat Quantitative Reasoning
What is the value of ?
(1)
(2)
EACH statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but the other statement alone is not sufficient
BOTH statements TOGETHER are not sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
Statement 1 ALONE is sufficient to answer the question, but the other statement alone is not sufficient
EACH statement ALONE is sufficient to answer the question.
(1) Add to both sides to make
. Then divide through by 7 to get
. This statement is sufficient.
(2) Divide both sides by 13. The equation becomes . This statement is sufficient.
Example Question #3011 : Gmat Quantitative Reasoning
Is ?
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
BOTH statements TOGETHER are not sufficient to answer the question.
EACH statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.
Statement 2 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.
Statement 1 ALONE is sufficient to answer the question, but the other statement alone is not sufficient.
(1) Since 5 must be added to to make it equal to
, it follows that
. This statement is sufficient.
(2) Multiply both sides by 6 to obtain . Thus, whether
or
depends on the value selected for
and
. For instance,
implies
, (such that
) but
implies
,
(such that ). Therefore, this statement is insufficient.
Example Question #1 : Simplifying Algebraic Expressions
True or false?
Statement 1:
Statement 2:
BOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Simplify each expression.
The inequality, therefore, is equivalent to
,
the truth or falsity of which depends only on the value of
Example Question #1 : Dsq: Simplifying Algebraic Expressions
True or false?
Statement 1:
Statement 2:
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Simplify both expressions algebraically.
Using similar algebra, you can simplify the other expression:
The question, assuming the variables have nonzero values, is equivalent to asking whether is true. Since we need to know the values of both variables to answer this, both statements are necessary and sufficient.
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