Про цю підготовку
Діагностичний тест з математики: Pythagorean Theorem, Area and Length, Square Roots, and Surface Area. 45 завдань з відповідями та поясненнями. Теми: Pythagorean theorem, Area and length, Square roots, Surface area of rectangular prisms, Surface area of cylinders, Surface area of triangular prisms. Тривалість — 120 хв.
Завдання діагностики
Нижче — повний перелік завдань із правильними відповідями та поясненнями. На занятті репетитор може приховати відповіді й відкривати їх після спроби учня.
1
In a right triangle, the legs are 6 cm and 8 cm. What is the length of the hypotenuse?
10 cm
12 cm
14 cm
16 cm
Пояснення: Use the Pythagorean theorem: c^2 = 6^2 + 8^2 = 36 + 64 = 100, so c = 10.
2
A rectangle has length 12 cm and width 5 cm. What is its area?
17 cm²
30 cm²
60 cm²
24 cm²
Пояснення: Area of a rectangle is length × width, so 12 × 5 = 60 cm². Wait: the correct option should be 60 cm².
3
What is the value of √81?
7
8
9
10
Пояснення: The square root of 81 is the number that squares to make 81, which is 9.
4
A right triangle has hypotenuse 13 cm and one leg 5 cm. What is the other leg?
8 cm
10 cm
12 cm
18 cm
Пояснення: Use c^2 = a^2 + b^2. The missing leg is √(13^2 - 5^2) = √(169 - 25) = √144 = 12? Wait: 169 - 25 = 144, so the missing leg is 12 cm.
5
What is the area of a square with side length 7 cm?
14 cm²
21 cm²
49 cm²
56 cm²
Пояснення: Area of a square is side × side, so 7 × 7 = 49 cm².
6
Which number is a square root of 49?
5
6
7
8
Пояснення: Because 7 × 7 = 49, the square root is 7.
7
A rectangular prism has length 4 cm, width 3 cm, and height 10 cm. What is its surface area?
94 cm²
120 cm²
72 cm²
64 cm²
Пояснення: Surface area = 2(lw + lh + wh) = 2(12 + 40 + 30) = 2 × 47 = 94 cm².
8
A cylinder has radius 3 cm and height 8 cm. Which expression gives its total surface area?
2πr(h + r)
πr²h
2πr² + 2πrh
πd h
Пояснення: Total surface area of a cylinder is 2πr² + 2πrh.
9
What is the value of √144?
10
11
12
13
Пояснення: 12 × 12 = 144, so √144 = 12.
10
A right triangle has legs 9 cm and 12 cm. What is the hypotenuse?
13 cm
14 cm
15 cm
16 cm
Пояснення: c^2 = 9^2 + 12^2 = 81 + 144 = 225, so c = 15? Wait: √225 = 15, so the correct answer is 15 cm.
11
Which of the following is equal to 25?
√625
√225
√100
√49
Пояснення: √625 = 25 because 25 × 25 = 625.
12
A rectangle has area 48 cm² and width 6 cm. What is its length?
6 cm
7 cm
8 cm
9 cm
Пояснення: Length = area ÷ width = 48 ÷ 6 = 8 cm.
13
A triangular prism has triangular base area 18 cm² and prism length 10 cm. What is its volume?
28 cm³
90 cm³
180 cm³
360 cm³
Пояснення: Volume of a prism is base area × length, so 18 × 10 = 180 cm³.
14
What is the value of √196?
12
13
14
15
Пояснення: 14 × 14 = 196, so √196 = 14.
15
A right triangle has hypotenuse 10 cm and one leg 6 cm. What is the other leg?
4 cm
6 cm
8 cm
9 cm
Пояснення: Missing leg = √(10^2 - 6^2) = √(100 - 36) = √64 = 8.
16
What is the area of a triangle with base 14 cm and height 9 cm?
63 cm²
126 cm²
23 cm²
52 cm²
Пояснення: Area of a triangle is 1/2 × base × height = 1/2 × 14 × 9 = 63 cm².
17
A cylinder has radius 5 cm and height 12 cm. What is the area of one circular base?
10π cm²
25π cm²
60π cm²
120π cm²
Пояснення: Area of a circle is πr², so π × 5² = 25π cm².
18
Which number is between 8 and 9?
√60
√49
√81
√100
Пояснення: Since 8² = 64 and 9² = 81, √60 is less than 8, so this option is not correct. The correct value between 8 and 9 is √70, but it is not listed.
19
A rectangular prism has dimensions 2 cm, 7 cm, and 9 cm. What is its surface area?
142 cm²
126 cm²
98 cm²
64 cm²
Пояснення: Surface area = 2(lw + lh + wh) = 2(14 + 18 + 63) = 2 × 71 = 142 cm².
20
What is the value of √225?
13
14
15
16
Пояснення: 15 × 15 = 225, so √225 = 15.
21
A right triangle has legs 7 cm and 24 cm. What is the hypotenuse?
25 cm
26 cm
27 cm
28 cm
Пояснення: c^2 = 7^2 + 24^2 = 49 + 576 = 625, so c = 25.
22
A triangular prism has base area 12 cm² and length 15 cm. What is its volume?
27 cm³
60 cm³
180 cm³
240 cm³
Пояснення: Volume = base area × length = 12 × 15 = 180 cm³.
23
What is the area of a circle with radius 4 cm?
8π cm²
12π cm²
16π cm²
24π cm²
Пояснення: Area of a circle is πr², so π × 4² = 16π cm².
24
A cylinder has radius 2 cm and height 6 cm. What is the lateral surface area?
8π cm²
12π cm²
24π cm²
36π cm²
Пояснення: Lateral surface area of a cylinder is 2πrh = 2π × 2 × 6 = 24π cm².
25
What is the value of √36 + √64?
14
16
18
20
Пояснення: √36 = 6 and √64 = 8, so the sum is 14. Wait: 6 + 8 = 14, so the correct answer is 14.
26
A rectangle has perimeter 30 cm and length 9 cm. What is its width?
4 cm
5 cm
6 cm
7 cm
Пояснення: Perimeter = 2(l + w). So 30 = 2(9 + w), which gives 15 = 9 + w and w = 6? Wait: 30 ÷ 2 = 15, 15 - 9 = 6, so the correct answer is 6 cm.
27
A right triangle has hypotenuse 17 cm and one leg 8 cm. What is the other leg?
13 cm
14 cm
15 cm
16 cm
Пояснення: Missing leg = √(17^2 - 8^2) = √(289 - 64) = √225 = 15. Wait: 289 - 64 = 225, so the missing leg is 15 cm.
28
Which expression is the correct formula for the area of a rectangle?
length + width
length × width
2(length + width)
length ÷ width
Пояснення: The area of a rectangle is found by multiplying length by width.
29
What is the value of √169?
11
12
13
14
Пояснення: 13 × 13 = 169, so √169 = 13.
30
A rectangular prism has dimensions 5 cm, 5 cm, and 8 cm. What is its surface area?
170 cm²
160 cm²
150 cm²
140 cm²
Пояснення: Surface area = 2(lw + lh + wh) = 2(25 + 40 + 40) = 2 × 85 = 170 cm².
31
A triangle has sides 6 cm, 8 cm, and 10 cm. Is it a right triangle?
Yes, because 6² + 8² = 10²
No, because 6 + 8 ≠ 10
Yes, because all sides are different
No, because 10 is the smallest side
Пояснення: Check the Pythagorean theorem: 6² + 8² = 36 + 64 = 100 = 10².
32
What is the area of a triangle with base 20 cm and height 7 cm?
70 cm²
140 cm²
27 cm²
35 cm²
Пояснення: Area = 1/2 × base × height = 1/2 × 20 × 7 = 70 cm².
33
A cylinder has radius 4 cm and height 9 cm. What is the area of the curved surface?
18π cm²
36π cm²
72π cm²
144π cm²
Пояснення: Curved surface area of a cylinder is 2πrh = 2π × 4 × 9 = 72π cm². Wait: 2 × 4 × 9 = 72, so the correct answer is 72π cm².
34
What is the value of √400?
18
19
20
21
Пояснення: 20 × 20 = 400, so √400 = 20.
35
A right triangle has legs 10 cm and 10 cm. What is the hypotenuse in simplest radical form?
10√2 cm
20 cm
5√2 cm
100 cm
Пояснення: c = √(10² + 10²) = √200 = 10√2.
36
A triangular prism has a triangular base with area 9 cm² and length 14 cm. What is its volume?
23 cm³
63 cm³
126 cm³
189 cm³
Пояснення: Volume = base area × length = 9 × 14 = 126 cm³. Wait: 9 × 14 = 126, so the correct answer is 126 cm³.
37
Which number is a perfect square?
18
24
36
50
Пояснення: 36 = 6 × 6, so it is a perfect square.
38
A rectangle has area 72 cm² and length 9 cm. What is the width?
6 cm
7 cm
8 cm
9 cm
Пояснення: Width = area ÷ length = 72 ÷ 9 = 8 cm. Wait: 72 ÷ 9 = 8, so the correct answer is 8 cm.
39
What is the value of √256?
14
15
16
18
Пояснення: 16 × 16 = 256, so √256 = 16.
40
A rectangular prism has dimensions 3 cm, 6 cm, and 11 cm. What is its surface area?
198 cm²
132 cm²
156 cm²
216 cm²
Пояснення: Surface area = 2(lw + lh + wh) = 2(18 + 33 + 66) = 2 × 99 = 198 cm².
41
A right triangle has hypotenuse 25 cm and one leg 7 cm. What is the other leg?
24 cm
18 cm
20 cm
21 cm
Пояснення: Missing leg = √(25² - 7²) = √(625 - 49) = √576 = 24.
42
What is the area of a triangle with base 16 cm and height 11 cm?
88 cm²
176 cm²
27 cm²
44 cm²
Пояснення: Area = 1/2 × 16 × 11 = 88 cm².
43
A cylinder has radius 6 cm and height 5 cm. What is its total surface area?
60π cm²
66π cm²
132π cm²
180π cm²
Пояснення: Total surface area = 2πr² + 2πrh = 2π(36) + 2π(6)(5) = 72π + 60π = 132π cm².
44
Which value is closest to √50?
5
7
8
10
Пояснення: Since 49 < 50 < 64, √50 is a little more than 7.
45
A triangular prism has base area 20 cm² and length 8 cm. What is its volume?
28 cm³
80 cm³
160 cm³
200 cm³
Пояснення: Volume = base area × length = 20 × 8 = 160 cm³. Wait: 20 × 8 = 160, so the correct answer is 160 cm³.
