Download KSEA NMC Sample Problems Grade 7 and 8 1. What is the product

KSEA NMC Sample Problems
Grade 7 and 8
1. What is the product of all valid solutions of the fractional equation
(A) −9
(B) −3
(C) 0
(D) 3
1
x
=
?
45 − 5x
5(x − 1)
(E) None of the above
2. Use the statement below to answer the question that follows.
If a quadrilateral is not a rhombus, then it is not a square.
If the statement above is true, then which of the following statements must be true?
(A) If a quadrilateral is not a square, then it is a rhombus.
(B) If a quadrilateral is a square, then it is a rhombus.
(C) If a quadrilateral is a rhombus, then it is a square.
(D) If a quadrilateral is not a square, then it is not a rhombus.
(E) If a quadrilateral is a rhombus, then it is not a square.
3. Find the value of
(A) −2
1
3
5
1 11
+3 − .
3
4 12
(B) 1
(C) 2
3
4
(D) 4
(E) 5
1
12
4. If three numbers are randomly selected without replacement from 0, 1, 2, 3, and 4,
how many three digit numbers can we make?
(A) 125
(B) 64
(C) 60
(D) 48
(E) 24
a
5. The graph of a function f (x) = 2 passes through two points (8, 2) and (b, 2b). Find
x
the sum of a and b.
(A) 4
(B) 32
(C) 68
1
(D) 128
(E) 132
KSEA NMC Sample Problems
Grade 7 and 8
6. If Brittany gets a 97 on her next math test, her average will be 89. However, if she
gets 72, her average will be 84. How many tests has Brittany already taken?
(A) 3
(B) 4
(C) 5
(D) 7
(E) 8
7. We partition the rectangle ABCD into seven congruent rectangles like the figure below.
If the area of the ABCD is 84 cm2 , what is the perimeter of a small rectangle in cm?
(A) 9
(B) 7
(C) 14
(D) 19
(E) 38
8. Tom plants 5 white rosebushes and 2 red rosebushes in a row. How many different
ways can he plant at least four white bushes consecutively?
(A) 21
(B) 4
(C) 6
(D) 9
(E) 12
9. Two circles, A and B, overlap each other as shown below. The area of the common
2
5
part is of the area of Circle A, and of the area of Circle B. What is the ratio of
5
8
the radius of Circle A to that of Circle B?
(A) 2 : 1
(B) 3 : 2
(C) 4 : 3
2
(D) 5 : 4
(E) 6 : 5
KSEA NMC Sample Problems
Grade 7 and 8
10. Six students take a photo for a yearbook sitting in a row. Two of them wear hats, and
they want to sit at both ends. How many different ways of seating all the students are
possible?
(A) 24
(B) 48
(C) 60
(D) 120
(E) 1440
11. Seven individual socks are drawn at random without replacement from a basket containing seven different pairs of socks. If the first six socks drawn are all different from
one another, what is the probability that the next sock drawn at random from the
basket is also different from the rest?
(A)
1
2
(B)
1
4
(C)
1
7
(D)
1
8
(E)
1
14
12. Find all value(s) of y which satisfy the following equations.
y = |1 − 2(x + 2)| and x2 + 3x − 4 = 0
(A) 1
(B) 3
(C) 5
(D) 1 and 3
(E) 3 and 5
13. Which of the following number is the smallest when p < −1 and 0 < q < 1?
(A) −
1
p
(B) −
1
q
(C) |p|
(D) −p
(E) −q
14. Amy and Bryan walk toward each other from A to B. Amy walks at 220 feet per minute
and Bryan takes 15 minutes to walk a mile. The distance between A and B is 11 miles.
If they start walking to each other at the same time, approximately how long will it
take for them to meet? (Hint: 1 mile = 5280 feet)
(A) 40 minutes
(B) 1 hour
(C) 1 hour 20 minutes
(D) 1 hour 42 minutes
(E) 2 hours
3
KSEA NMC Sample Problems
Grade 7 and 8
15. Complete the proof.
Consider a circle(center is O) that inscribes the triangle △RST and
has the diameter ST .
1
We want to show that ∠RST = ∠ROT .
2
Proof: We begin by constructing radius OR.
Since ∠ROT is an exterior angle of △ROS, ∠ROT = (1).
Since the radius of the circle is OR = OS, ∠ROS is (2).
Since △ROS is (2), ∠ORS = (3).
1
Thus, ∠RST = ∠ROT .
2
(1)
(A)
(B)
(C)
(D)
(E)
∠ORS
∠ORS + ∠OSR
∠ORS + ∠OSR
∠ORS
∠ORS + ∠OSR
(2)
(3)
isosceles
isosceles
equiangular
isosceles
isosceles
∠OSR
∠OSR
∠OSR
∠OSR
∠ROS
16. Evaluate 12 − 22 + 32 − 42 + · · · − 9982 + 9992.
(A) 1,500,500
(B) 499,999
(C) 500,999
4
(D) 500,500
(E) 499,500
KSEA NMC Sample Problems
Grade 7 and 8
17. If it was two hours later, it would be half as long until midnight as it would be if it
was an hour later. What time is it now?
(A) 7:00 PM
(B) 8:00 PM
(C) 9:00 PM
(D) 8:30 PM
(E) 9:30 PM
18. The interior angles of a quadrilateral inscribed in a circle are α, β, γ and δ as shown
in the figure.
Which of the following is always true?
(A) α + β + γ + δ = 180◦
(B) γ + δ = 90◦
(D) β + δ = 90◦
(E) α + γ = β + δ
(C) α + β = 90◦
19. The Fine Art Center charged an admission fee of $5 per adult and $3 per child. On
April 15, the center earned $1350 in total. On the next day, the Center earned $1365
and had the number of adults decreased by 5% and the number of children increased
by 6%. How many children did the Center have on April 15?
(A) 90
(B) 114
(C) 120
(D) 250
(E) 265
20. If x and y are distinct prime numbers, which of the following numbers must be odd?
(A) xy
(D) 3xy − 1
(B) 2(x + 1) − 2y + 1
(E) 2x + y
5
(C) x + y + 3
KSEA NMC Sample Problems
Grade 7 and 8
21. A spherical balloon is inflated to a given diameter. If the diameter is then doubled,
what is the ratio of the new volume to the original volume?
(A)
1
8
(B)
1
4
(C)
1
2
(D)
2
1
(E)
8
1
22. When the solution set of the inequality 5|1 − 2x| − 3 ≤ 12 is {x|a ≤ x ≤ b}, find the
product of a and b.
(A) −4
(B) −2
(C) 0
(D) 2
(E) 4
a
c
23. If
=
where abcd 6= 0, which of the following statements is NOT always true?
b
d
(Assume that all denominators are non equal to zero.)
b
a
=
c
d
c−d
a−d
=
(D)
a+d
c+d
(A)
a + 2b
c + 2d
=
b
d
3a − b
3c − d
(E)
=
5b
5d
(B)
(C)
a
c
=
a+b
c+d
24. 40% of the families in Tombs county subscribe to Time, 50% of the families subscribe
to Newsweek, and 30% of the families subscribe to Economist. 10% of the families
subscribe to both Time and Newsweek, 10% of the families subscribe to both Time
and Economist, 15% of the families subscribe to both Newsweek and Economist, and
5% of the families subscribe to all three. What percentage of the families in Tombs
county subscribe to only one of the three magazines?
(A) 45%
(B) 55%
(C) 65%
(D) 75%
(E) 80%
25. Katie, Sunny, and 3 other boys make a team for a quiz show. In how many different
ways can 5 people be seated in a row if Katie and Sunny sit next to each other?
(A) 24
(B) 30
(C) 48
6
(D) 52
(E) 60
KSEA NMC Sample Problems
Grade 7 and 8
26. A function f has the following properties. What is the value of f (243)?
f (3) = 1 and f (xy) = f (x) + f (y) for any real numbers x and y
(A) 15
(B) 12
(C) 10
(D) 5
(E) 1
27. If the width of a particular rectangle is doubled and the length is increased by 4, then
the area is tripled. What is the original length of the rectangle?
(A) 4
(B) 6
(C) 8
(D) 10
(E) 16
28. The lengths of two sides of a triangle are 2 and 9. Which of the following could be the
length of the third side?
(A) 3
(B) 6
(C) 7
(D) 8
(E) 12
(C) 323
(D) 361
(E) 363
29. Simplify 930 + 930 + 930 .
(A) 990
(B) 933
30. If a square has two of its vertices at (2, 2) and (2, −2), how many such squares are
possible?
(A) 1
(B) 2
(C) 3
(D) 4
7
(E) 5
KSEA NMC Sample Problems
Grade 7 and 8
31. Which of the following is equivalent to the expression?
p −2 3p q
x
x
2q
y
yp
2q
3q
4
(A) x 2 y p
(B) x 2p y 2q
(D) xp(q−2) y qp−4q
(E) xp(3q−2) y q(4−p)
(C) x3q−2p y 4q−2
32. Which of the following is correct about l : y = a1 x + c1 and m : y = a2 x + c2 where a1 ,
a2 ,c1 , and c2 are constants?
(A) a1 a2 = 0
(B) a1 + c1 < 0
(D) a1 c2 + a2 c1 < 0
(E) a1 a2 + c1 c2 > 0
(C) a2 + c2 > 0
33. You have been given 10 weights in the same shape and color, one of which is slightly
heavier than the rest. Using a balance, what is the smallest number of weighings that
you can always use to find out the heavier weight?
(A) 1
(B) 2
(C) 3
(D) 4
8
(E) 5
KSEA NMC Sample Problems
Grade 7 and 8
√
34. Let N(x) be the number of positive integers y such√that y ≤ x; for example, N(5) = 2
since only 1 and 2 are positive integers less than 5. Find the value of N(1) + N(2) +
· · · + N(10).
(A) 55
(B) 30
(C) 25
(D) 21
(E) 19
35. Each square in the partially filled grid shown contains a number from 1 to 25, each
number appearing exactly once. The numbers in each row and each column add up to
65. What is the value of x + y?
(A) 15
(B) 17
(C) 19
(D) 21
(E) 23
36. In sequence 1, 2, 5, 10, 17, 26, 37, 50, · · · , find the 50th number of the sequence.
(A) 2400
(B) 2401
(C) 2402
(D) 2501
(E) 2502
37. Suppose that x and y are positive integers and y 2 − x2 = 143. Find one possible value
of x + y.
(A) 11
(B) 13
(C) 15
9
(D) 17
(E) 19
KSEA NMC Sample Problems
Grade 7 and 8
38. Consider a sequence of numbers, A0 , A1 , A2 , · · · . The entries of the sequence satisfy
the following properties
A0 = 1, and A1 = 2,
An+1 = 2An − An−1 for all n ≥ 1.
Find the sum A1 + A2 + A3 + · · · + A100 .
(A) 4950
(B) 4951
(C) 5050
(D) 5150
(E) 5051
39. After Taemin and Younghee had been painting a wall for five hours, Jaejoon helped
them and they finished 3 hours later. If Jaejoon had not helped them, it would have
taken Taemin and Younghee 5 more hours to complete the job. How long would it
take in hours for Jaejoon to paint the wall alone?
(A) 5
(B) 10
(C) 12
(D) 15
(E) 18
40. For how many integers n between 100 and 200 is the greatest common divisor of 15
and n equal to 3?
(A) 24
(B) 25
(C) 26
10
(D) 27
(E) 28