2019 3rd IUT Admission Test(SOL)
Math Examination (A TYPE) Solutions
Answer: ③
Key: The given inequality
implies . Thus its
negative integer solutions are and the number of them is .
Answer: ①
Key:
.
Answer: ⑤
Key: . So
1. [4 points]
Find the number of all negative integer solutions of
the inequality
.
①
②
③
④
⑤
2. [4 points]
Evaluate
.
①
②
③
④
⑤
3. [4 points]
Suppose that and . Find the
value of .
①
②
③
④
⑤
Answer: ⑤
Key: Since
, we have
. So,
.
Therefore,
.
Answer: ④
Key: The remainder is obtained by plugging to the polynomial
.
Answer: ④
Key:
.
Answer: ③
Key:
log
log
log
.
4. [4 points]
Find the smallest integer which is greater than
①
②
③
④
⑤
5. [4 points]
Find the remainder when a polynomial
is divided by .
① ② ③
④
⑤
6. [5 points]
Find
when
.
①
②
③
④
⑤
7. [5 points]
Evaluate
log
.
①
②
③
④
⑤
Answer: ③
Key: Since
, we have
Alternatively, since
, we have
.
Answer: ④
Key:
.
Answer: ①
Key: Multiplying
to the equation, we obtain
.
, that is, or . So the sum of the solutions is .
8. [5 points]
Let be a solution of
. Compute
.
①
②
③
④
⑤
9. [5 points]
Evaluate
where
.
①
②
③
④
⑤
10. [5 points]
Find the sum of all values of which satisfy the
equation
×
.
①
②
③
④
⑤
Answer: ④
Key: Put in
and we have
.
Answer: ②
Key: The system does not have a solution if the two lines represented by linear equations are
parallel and not equal. That is when the slope
of the first line is equal to the slope
of the second line. This happens when .
Alternatively, the system does not have a solution or infinitely many solutions when the
determinant of the matrix
of coefficients are zero. That is when , in
other words, when .
11. [5 points]
Suppose that a function satisfies
for every real number .
Evaluate .
①
②
③
④
⑤
12. [5 points]
Find such that the system of simultaneous equations
does not have a solution in and .
①
②
③
④
⑤
Answer: ④
Key:
Since
cos
sin
cos
sincos
cos
sincos
sin
sin sin
and sin on
, it is enough to find the maximum of
on
Since ′
and ′
on , the maximum value is
equal to
.
Answer: ②
Key: Since
,
we have
⋯
.
So
.
13. [5 points]
Find the maximum value of
cos
sin
on
.
①
②
③
④
⑤
14. [5 points]
The sequence
,
,
,… is defined by
and
( ≥ ). Evaluate
.
①
②
③
④
⑤
Answer: ⑤
Key: We have
and thus
.
Answer: ④
Key: The minimum occurs at the point on the line such that
the line passing through the origin and the point is perpendicular to .
The value of
is the distance between the origin an the line , which is
. Hence,
.
Answer: ⑤
Key: Note that the value of a -th term where
≤
is .
Since
×
and
×
, the th term is .
15. [5 points]
Suppose that
where
.
Find the sum
.
①
②
③
④
⑤
16. [6 points]
Find the minimum value of
when two real
numbers satisfy ≥ .
①
②
③
④
⑤
17. [6 points]
Consider a non-decreasing sequence of positive
integers
in which the integer appears times. Find the
th
term.
①
②
③
④
⑤
Answer: ③
Key: From the description of , the graph of consists of two branches which are
parts of convex-up parabolas which have vertices with coordinate equal to and .
So, the set has two elements and has three elements.
Answer: ①
Key:
sin
cos
18. [6 points]
For some quadratic functions , the function
≥
is continuous on the set of
real numbers. Suppose that the set has
elements if and only if . Also, ≥
for every real numbner . Find the value(s) of
such that the set has elements.
①
②
③
④ and
⑤ There is no such .
19. [6 points]
Find the area between two curves of sin and
where
≤ ≤
.
①
②
③
④
⑤
Answer: ②
Key: We have
lim
→
cos
cos
lim
→
cos
cos
×
lim
→
lim
→
cos
cos
×
lim
→
sin
×
×
20. [6 points]
Evaluate
lim
→
cos
cos
.
①
②
③
④
⑤
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