For
, let 
be
defined by 
What is the
value of 
?A)
B) 
C) 2 D) 
E) 5Solution
Special Symbols Up
Some
questions contain the symbols and the rules you have to follow.
Example
1
For
all positive integers a and b, let a ^ b be defined
as a ^ b = a b - ( a + b )
Compare 5 ^ 2 and
2 ^ 5, and choose:
A) if 5 ^ 2 > 2 ^ 5; B) if 5 ^ 2 < 2 ^ 5;
C)
if 5 ^ 2 = 2 ^ 5;
D) if the relationship cannot be determined
Solution
Step 1
Find 5 ^ 2 = 5 x 2 - ( 5 + 2 ) = 10 - 7 = 3
Step 2
Find 2 ^ 5 = 2 x 5 - ( 2 + 5 ) = 10 - 7 = 3
Step 3
Compare 3 = 3
Answer is "C"
Example 2
For
, let
be
defined by
What is the
value of ?
A)
B)
C) 2 D)
E) 5
Solution
Example 3
Two numbers form a "couple" if the sum of their reciprocals equals 2.
For example, 
form a "couple" because 
.
If x and y form a
"couple" and 
what is the value of y ?
Solution
Step 1
Follow the given rule:
( use reciprocal for 
) 
Step 2
Solve for 
Step
3 Solve for y ( take a reciprocal )
Example 4
For
all integers x, let «x»= x² when x is an even integer,
and
«x»= x² -
1 when x is an odd integer. What is
the value of «5» - «4» ?
Solution
Step
1 Find «5» using the pattern:
Since 5 is an odd
integer «5»= 5² - 1 = 24
Step
2 Find «4» using the pattern:
Since 4 is an even
integer «4»= 4² = 16
Step
3 Find «5» - «4»:
«5» - «4» = 24 - 16 =
8
Example 5
For all values of a, b, s and t, let the
expression 
equals at - bs.
For what
value of a will 
equal 0 ?
A) 0 B) 
C) 
D) 8 E) 10
Solution
Step 1 Follow the rule at - bs = 0. Use given data.
Step 2 Solve for a
2a = 20; a =
10 Answer is "E"
Example 6
A perfect
number is defined as one which is equals to the sum of all its
divisors including 1, but
not including the number itself. Which is the perfect
number
?
A) 26 B) 28 C) 12 D) 10 E) 9
Solution
Step 1
Find all divisors of 26, not including 26
1, 2, 13
Step 2
Find the sum of divisors and match with 26
1 + 2 + 13 = 16 ; 16 
26
Step 3
Find all divisors of 28
1, 2, 4, 7, 14
Step 4
Find their sum and match with 28
1 + 2 + 4 + 7 + 14 = 28
Answer is
"B"
Up