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General Education
Deparo High School
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MMC 2019 Grade 10 Eliminations
____________ 1 What is the distance between − 11 6 and −
3
4 on the number line?
____________ 2 In a class, the ratio of the number of girls to boys is 5 ∶ 4. If only 15 girls had enrolled in the class, the ratio would have been 25 ∶ 8. How many students are in the class?
____________ 3 Twice a certain number is the average of
7
8 a nd 1
3
4.
What is the number?
____________ 4 If 0 .1 5 a = 0. 33 b, what is the x in the proportion a ∶ bx = 11 ∶ 10?
____________ 5 Chris drives from City A to City B on an expressway. After driving for 2
1
2
hours, Chris saw a sign that reads “City A - 22 5 km; City B - 19 8 km”. Continuing at the same constant speed, what is his total travel time from City A to City B? ____________ 6 A cylindrical bottle with radius 1 5 cm is half-filled with water. A circular stone is dropped inside the bottle (completely submerging into the water) and the water level rises by 2 .5 cm without spilling over. What is the radius of the stone? ____________ 7 Let A = { 0 , 1 , 2 ,... , 10 } and B = {x ∣ x is a prime number less than 20 }. What is A ∩ B? ____________ 8 Using sets A and B in the previous item, what is the cardinality of (A − B) ∪ (B − A)? ____________ 9 Solve for x in (x − 1 ) − 2 (x − 2 ) + 3 (x − 3 ) − 4 (x − 4 ) + ⋯ − 10 (x − 10 ) = 0.
____________ 10 A military camp has provisions for 6 0 days. If, after 1 5 days, 5 00 men strengthen them and the food lasts 40 days longer, how many men were there in the camp originally?
____________ 11 The average of two numbers is x + y. If one number is x, find the other number.
____________ 12 In 6 years, Kate will be thrice as old as she was 1 0 years ago. How old was Kate 1 0 years ago?
____________ 13 Two consecutive angles of a quadrilateral are 98 ○ and 112 ○. Find the obtuse angle between the bisectors of the other two angles.
____________ 14 If 16 workers can finish a job is 4 hours, how long should it take 1 0 workers to finish the same job?
____________ 15 What is the least integer that satisfies the inequality
2
3
( 7 − x) < 15?
m
ℓ
112 ○
65 ○
53 ○
θ
Problem 16
D
A B
C
F
E
G
Problem 17
D
A B
C
E
Problem 1 8
____________ 16 In the figure above, what is θ so that lines ℓ and m are parallel?
____________ 17 In the figure above, ABCD is a rectangle, AE ∶ EB = 3 ∶ 1 , CF = FD, and AG = GD. If the area of ABCD is 32 cm 2 , what is the area of △EFG?
____________ 18 In the figure above, ABCD is a rectangle with DE ⊥ AC. If AB = 4 cm and BC = 3 cm, how long is AE?
____________ 19 Last week, Mark computed the average height of his tomato seedlings, and it was 2. 2 cm. After one week, each seedling grew 1 cm more. What is the new average height of the seedlings?
____________ 20 A linear function f satisfies f ( 0 ) = − 1 and f ( 1 ) = −4. If f (x) = m(x − 2 ) + b, what is b?
____________ 21 In tossing three fair coins, what is the probability of getting at least two heads?
____________ 22 What is P(x) if ( 2 x + 1 )P(x) + ( 3 x − 1 )( 2 x + 1 ) = 2 x 2 + 5 x + 2?
____________ 23 Factor completely: 26 a 2 b 2 − 12 a 3 b − 12 ab 3
1
MMC 2019 Grade 10 Eliminations
____________ 24 If 3 x −
2
1 + 2 x
=
P(x) Q(x) , where P(x) and Q(x) are polynomials, what is P(x) + Q(x)?
____________ 25 In the sequence 7 , 6 , 3 , 9 , 2 ,.. ., each term after the second is equal to the units digit of the sum of the two preceding terms. What is the 2018 th term?
____________ 26 The vertices of a triangle have coordinates (− 2 , 3 ), ( 1 , − 2 ), and ( 6 , 2 ). Is the triangle equilateral, isosceles, or scalene?
____________ 27 The point (h, − 6 ) is equidistant to both points ( 2 , 2 ) and ( 7 , − 1 ). What is h?
____________ 28 The lines 3y = 2 x + 12 and 3 x + y + 7 = 0 intersect at (h, k). What is h + 4 k?
____________ 29 For what values of x is the inequality 2 x 2 < 7 x true?
____________ 30 If L 11 / 4 is 50 % larger than L 5 / 2 , what is 50 % of L?
____________ 31 Find all values of x such that ∣x − 2 ∣ = 2 − x.
____________ 32 Find the area of the closed region determined by the lines x + y = 7 , 2 y = 5 x, and 2 x = 5 y.
____________ 33 If 6 p = 2 , 6 q = 5, and 6 r = 1 5, write r in terms of p and q.
____________ 34 A box contains 6 red balls and 4 blue balls. Drawing three balls from the box randomly, what is the probability that exactly two of these balls are red?
____________ 35 The vertex of a parabola is at (− 3 , 2 ). If the parabola passes through the point (− 4 , 4 ), what are the coordinates of its y-intercept?
____________ 36 Let a 1 , a 2 , a 3 ,... be an arithmetic sequence. If a 4 = − 6 and a 20 = 12 , what is a 25?
____________ 37 If a polynomial is divided by 2 x + 5, the remainder is 12. What is the remainder when the same polynomial is divided by 4 x + 10?
____________ 38 The sum of the roots of 4 x 2 = 8 x − d is twice their difference. Find the value of d.
____________ 39 Factor completely: a 3 c + 8 ab 3 − 8 b 3 c − a 4
____________ 40 By joining the midpoints of a triangle, a smaller triangle is constructed. If the area of the larger triangle is 1 8 cm 2 , what is the area of the smaller triangle?
____________ 41 A circle of radius 10 cm circumscribes a triangle having an area of 96 cm 2. If one side of the triangle is 2 0 cm, how long is its shortest side?
____________ 42 Let S(a, b) denote the sum of the integers from a to b. Find S( 100 , 5 00 ).
____________ 43 An isosceles trapezoid is circumscribed about a circle. If the parallel sides are 1 8 and 6 cm, what is the radius of the circle?
____________ 44 The product of the 3rd, 9 th, and 15 th terms of a geometric sequence is 337 5. Find the 9 th term.
____________ 45 Chord AB is bisected at M by chord CD. If CM = 8 cm and MD = 1 8 cm, how long is AB?
____________ 46 If 8x 3 + 2 x 2 − 61 x + 15 = ( 2 x + a)(x + b)( 4 x + c), what is 2a 2 bc + 4 ab 2 c + abc 2?
____________ 47 The sides of a triangle are 6 , 10 , and 12 cm long. If θ is its smallest angle, what is sin θ?
____________ 48 Let P be a point inside rectangle ABCD. If PD = 12 , PC = 7 , and PA = 10 cm, find PB.
____________ 49 Simplify:
1
2!
+
2
3!
+
3
4!
+ ⋯ +
2018
2019!
____________ 50 You have 6 coins of distinct weights and a beam balance with two pans. What is the minimum number of weightings that could identify with certainty the second lightest coin?
2
MTAP competitions
Subject: General Education
School: Deparo High School
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