ISEE Upper Level Quantitative : Variables

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

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Example Questions

Example Question #11 : How To Add Variables

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(a) and (b) are equal

(a) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(b) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

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 #12 : How To Add Variables

Define . The graph of  is a line with slope .

.

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) and (b) are equal

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

, 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 #13 : How To Add Variables

 and  are both positive.

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

It is impossible to determine which is greater from the information given

Correct answer:

(b) is the greater quantity

Explanation:

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

Example Question #1 : Variables And Exponents

Simplify:

Possible Answers:

The expression cannot be simplified further

Correct answer:

Explanation:

Group and combine like terms :

Example Question #2 : Variables And Exponents

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

Correct answer:

(b) is greater.

Explanation:

Since  and  have different signs,

, and, subsequently,

Therefore, 

This makes (b) the greater quantity.

Example Question #3 : Variables And Exponents

Assume that  and  are not both zero. Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Correct answer:

It is impossible to tell from the information given.

Explanation:

Simplify the expression in (a):

Therefore, whether (a) or (b) is greater depends on the values of  and , neither of which are known. 

Example Question #4 : Variables And Exponents

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

It is impossible to tell from the information given

(a) is greater

(a) and (b) are equal

(b) is greater

Correct answer:

It is impossible to tell from the information given

Explanation:

We give at least one positive value of  for which (a) is greater and at least one positive value of  for which (b) is greater.

Case 1: 

(a) 

(b) 

Case 2: 

(a) 

(b) 

Therefore, either (a) or (b) can be greater.

Example Question #5 : Variables And Exponents

Assume all variables to be nonzero. 

Simplify: 

Possible Answers:

None of the answer choices are correct.

Correct answer:

None of the answer choices are correct.

Explanation:

Any nonzero expression raised to the power of 0 is equal to 1. Therefore, 

.

None of the given expressions are correct.

Example Question #6 : Variables And Exponents

Simplify:

Possible Answers:

Correct answer:

Explanation:

Example Question #7 : Variables And Exponents

Which is greater?

(a) 

(b) 

Possible Answers:

(b) is greater

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

Correct answer:

(b) is greater

Explanation:

If , then  and 

 

, so by transitivity, , and (b) is greater

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