Q. In triangle DEF, if DE = 12 cm, EF = 16 cm, and DF = 20 cm, what is the semi-perimeter? (2021)
-
A.
24 cm
-
B.
28 cm
-
C.
30 cm
-
D.
20 cm
Solution
Semi-perimeter s = (12 + 16 + 20) / 2 = 48 / 2 = 24 cm.
Correct Answer:
B
— 28 cm
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Q. In triangle DEF, if DE = 5 cm, EF = 12 cm, and DF = 13 cm, is triangle DEF a right triangle? (2019)
-
A.
Yes
-
B.
No
-
C.
Cannot be determined
-
D.
Only if angle D is 90 degrees
Solution
Using the Pythagorean theorem, 5^2 + 12^2 = 25 + 144 = 169 = 13^2. Thus, triangle DEF is a right triangle.
Correct Answer:
A
— Yes
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Q. In triangle GHI, if angle G = 30 degrees and side GH = 10 cm, what is the length of side HI if angle H = 60 degrees? (2023)
-
A.
5 cm
-
B.
10 cm
-
C.
15 cm
-
D.
20 cm
Solution
Using the sine rule, HI/sin(60) = GH/sin(30). Therefore, HI = (10 * sin(60)) / sin(30) = 10 * (√3/2) / (1/2) = 10√3 cm.
Correct Answer:
C
— 15 cm
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Q. In triangle GHI, if angle G = 90 degrees and GH = 9 cm, HI = 12 cm, what is the length of side GI? (2022)
-
A.
15 cm
-
B.
10 cm
-
C.
12 cm
-
D.
9 cm
Solution
Using the Pythagorean theorem, GI = √(HI² - GH²) = √(12² - 9²) = √(144 - 81) = √63 = 15 cm.
Correct Answer:
A
— 15 cm
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Q. In triangle GHI, if GH = 10 cm, HI = 24 cm, and GI = 26 cm, what is the area of the triangle? (2019)
-
A.
120 cm²
-
B.
130 cm²
-
C.
140 cm²
-
D.
150 cm²
Solution
Using Heron's formula, semi-perimeter s = (10 + 24 + 26) / 2 = 30. Area = √(s(s-a)(s-b)(s-c)) = √(30*20*6*4) = 120 cm².
Correct Answer:
A
— 120 cm²
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Q. In triangle GHI, if GH = 10 cm, HI = 24 cm, and GI = 26 cm, what is the length of the longest side? (2022)
-
A.
10 cm
-
B.
24 cm
-
C.
26 cm
-
D.
Cannot be determined
Solution
The longest side of triangle GHI is GI, which measures 26 cm.
Correct Answer:
C
— 26 cm
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Q. In triangle GHI, if GH = 10 cm, HI = 24 cm, and GI = 26 cm, what is the perimeter? (2019)
-
A.
50 cm
-
B.
60 cm
-
C.
70 cm
-
D.
80 cm
Solution
Perimeter = GH + HI + GI = 10 + 24 + 26 = 60 cm.
Correct Answer:
A
— 50 cm
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Q. In triangle GHI, if the lengths of sides GH, HI, and GI are in the ratio 3:4:5, what type of triangle is it? (2019)
-
A.
Equilateral
-
B.
Isosceles
-
C.
Scalene
-
D.
Right-angled
Solution
Since the sides are in the ratio 3:4:5, it follows the Pythagorean theorem, indicating it is a right-angled triangle.
Correct Answer:
D
— Right-angled
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Q. In triangle JKL, if angle J = 90 degrees and JK = 15 cm, KL = 20 cm, what is the length of side JL? (2023)
-
A.
10 cm
-
B.
12 cm
-
C.
15 cm
-
D.
25 cm
Solution
Using Pythagoras theorem: JL = √(KL² - JK²) = √(20² - 15²) = √(400 - 225) = √175 = 12 cm.
Correct Answer:
B
— 12 cm
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Q. In triangle JKL, if angle J = 90 degrees and JK = 15 cm, KL = 20 cm, what is the length of JL? (2022)
-
A.
10 cm
-
B.
12 cm
-
C.
15 cm
-
D.
25 cm
Solution
Using the Pythagorean theorem, JL = √(KL² - JK²) = √(20² - 15²) = √(400 - 225) = √175 = 12.25 cm.
Correct Answer:
B
— 12 cm
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Q. In triangle JKL, if JK = 15 cm, KL = 20 cm, and JL = 25 cm, what is the semi-perimeter? (2022)
-
A.
30 cm
-
B.
25 cm
-
C.
20 cm
-
D.
15 cm
Solution
Semi-perimeter = (JK + KL + JL) / 2 = (15 + 20 + 25) / 2 = 30 cm.
Correct Answer:
A
— 30 cm
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Q. In triangle JKL, if JK = 15 cm, KL = 20 cm, and JL = 25 cm, what is the semi-perimeter of the triangle? (2022)
-
A.
30 cm
-
B.
25 cm
-
C.
20 cm
-
D.
35 cm
Solution
Semi-perimeter = (JK + KL + JL) / 2 = (15 + 20 + 25) / 2 = 30 cm.
Correct Answer:
A
— 30 cm
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Q. In triangle JKL, if JK = 15 cm, KL = 20 cm, and JL = 25 cm, what is the type of triangle? (2023)
-
A.
Equilateral
-
B.
Isosceles
-
C.
Scalene
-
D.
Right-angled
Solution
Since 15² + 20² = 25² (225 + 400 = 625), triangle JKL is a right-angled triangle.
Correct Answer:
D
— Right-angled
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Q. In triangle JKL, if JK = 5 cm, KL = 12 cm, and JL = 13 cm, what is the type of triangle? (2023)
-
A.
Equilateral
-
B.
Isosceles
-
C.
Scalene
-
D.
Right-angled
Solution
Since 5² + 12² = 13² (25 + 144 = 169), triangle JKL is a right-angled triangle.
Correct Answer:
D
— Right-angled
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Q. In triangle MNO, if MN = 8 cm, NO = 15 cm, and MO = 17 cm, what is the type of triangle? (2023)
-
A.
Equilateral
-
B.
Isosceles
-
C.
Scalene
-
D.
Right-angled
Solution
Using the Pythagorean theorem, 8² + 15² = 64 + 225 = 289 = 17². Hence, it is a right-angled triangle.
Correct Answer:
D
— Right-angled
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Q. In triangle PQR, if PQ = 10 cm, PR = 24 cm, and QR = 26 cm, what is the area of the triangle? (2019)
-
A.
120 cm²
-
B.
240 cm²
-
C.
60 cm²
-
D.
80 cm²
Solution
Using Heron's formula, s = (10 + 24 + 26) / 2 = 30. Area = √(30(30-10)(30-24)(30-26)) = √(30*20*6*4) = √(4800) = 120 cm².
Correct Answer:
A
— 120 cm²
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Q. In triangle PQR, if PQ = 8 cm, QR = 6 cm, and PR = 10 cm, what is the perimeter of the triangle? (2022)
-
A.
24 cm
-
B.
26 cm
-
C.
22 cm
-
D.
20 cm
Solution
Perimeter = PQ + QR + PR = 8 + 6 + 10 = 24 cm.
Correct Answer:
A
— 24 cm
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Q. In triangle PQR, if PQ = 8 cm, QR = 6 cm, and PR = 10 cm, what is the semi-perimeter? (2022)
-
A.
12 cm
-
B.
14 cm
-
C.
16 cm
-
D.
18 cm
Solution
Semi-perimeter = (PQ + QR + PR) / 2 = (8 + 6 + 10) / 2 = 12 cm.
Correct Answer:
B
— 14 cm
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Q. In triangle PQR, if PQ = 8 cm, QR = 6 cm, and PR = 10 cm, what is the semi-perimeter of the triangle? (2022)
-
A.
12 cm
-
B.
14 cm
-
C.
16 cm
-
D.
18 cm
Solution
Semi-perimeter = (PQ + QR + PR) / 2 = (8 + 6 + 10) / 2 = 12 cm.
Correct Answer:
B
— 14 cm
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Q. In triangle XYZ, if angle X = 45 degrees and angle Y = 45 degrees, what is the length of side XY if side XZ = 10 cm? (2020)
-
A.
5√2 cm
-
B.
10 cm
-
C.
10√2 cm
-
D.
20 cm
Solution
In an isosceles right triangle, the sides opposite the 45-degree angles are equal. Therefore, XY = XZ / √2 = 10 / √2 = 5√2 cm.
Correct Answer:
A
— 5√2 cm
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Q. In triangle XYZ, if angle X = 45 degrees and angle Y = 45 degrees, what is the ratio of the sides opposite to these angles? (2021)
-
A.
1:1
-
B.
1:√2
-
C.
√2:1
-
D.
2:1
Solution
In an isosceles triangle with angles 45-45-90, the sides opposite the equal angles are equal, hence the ratio is 1:1.
Correct Answer:
A
— 1:1
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Q. In triangle XYZ, if angle X = 45 degrees and angle Y = 45 degrees, what is the ratio of the lengths of sides opposite to these angles? (2021)
-
A.
1:1
-
B.
1:√2
-
C.
√2:1
-
D.
2:1
Solution
In an isosceles triangle with angles 45 degrees, the sides opposite these angles are equal, hence the ratio is 1:1.
Correct Answer:
A
— 1:1
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Q. In triangle XYZ, if XY = 12 cm, YZ = 16 cm, and XZ = 20 cm, what is the semi-perimeter? (2020)
-
A.
24 cm
-
B.
28 cm
-
C.
30 cm
-
D.
32 cm
Solution
Semi-perimeter s = (XY + YZ + XZ)/2 = (12 + 16 + 20)/2 = 24 cm.
Correct Answer:
A
— 24 cm
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Q. In triangle XYZ, if XY = 8 cm, YZ = 15 cm, and XZ = 17 cm, what is the length of the longest side? (2020)
-
A.
8 cm
-
B.
15 cm
-
C.
17 cm
-
D.
Not determinable
Solution
The longest side in triangle XYZ is XZ, which measures 17 cm.
Correct Answer:
C
— 17 cm
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Q. Is the function f(x) = 1/(x-1) continuous at x = 1?
-
A.
Yes
-
B.
No
-
C.
Only left continuous
-
D.
Only right continuous
Solution
The function f(x) = 1/(x-1) is not defined at x = 1, hence it is discontinuous at that point.
Correct Answer:
B
— No
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Q. Is the function f(x) = sqrt(x) continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only from the right
-
D.
Only from the left
Solution
The function f(x) = sqrt(x) is continuous at x = 0 as it is defined and the limit exists.
Correct Answer:
A
— Yes
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Q. Is the function f(x) = |x| continuous at x = 0?
-
A.
Yes
-
B.
No
-
C.
Only left continuous
-
D.
Only right continuous
Solution
The function f(x) = |x| is continuous at x = 0 because the left limit, right limit, and f(0) all equal 0.
Correct Answer:
A
— Yes
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Q. Solve for x: 3x - 7 = 2x + 5. (2021)
Solution
Subtract 2x from both sides: x - 7 = 5. Then add 7: x = 12.
Correct Answer:
A
— 12
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Q. Solve the differential equation dy/dx = 2y.
-
A.
y = Ce^(2x)
-
B.
y = 2Ce^x
-
C.
y = Ce^(x/2)
-
D.
y = 2x + C
Solution
This is a separable equation. Separating variables and integrating gives ln|y| = 2x + C, hence y = Ce^(2x).
Correct Answer:
A
— y = Ce^(2x)
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Q. Solve the differential equation dy/dx = 5 - 2y.
-
A.
y = 5/2 + Ce^(-2x)
-
B.
y = 5/2 - Ce^(-2x)
-
C.
y = 2.5 + Ce^(2x)
-
D.
y = 2.5 - Ce^(2x)
Solution
Rearranging gives dy/(5 - 2y) = dx. Integrating both sides leads to y = 5/2 + Ce^(-2x).
Correct Answer:
A
— y = 5/2 + Ce^(-2x)
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