Engineering Entrance
Q. If a student guesses on a multiple-choice test with 4 options, what is the probability of guessing correctly? (2022)
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A.
1/4
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B.
1/2
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C.
1/3
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D.
1/5
Solution
Probability of guessing correctly = 1/4.
Correct Answer: A — 1/4
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Q. If a student is selected at random from a group of 20 students, 12 of whom are girls, what is the probability that the student is a girl?
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A.
3/5
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B.
2/5
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C.
1/2
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D.
1/3
Solution
Probability of selecting a girl = 12/20 = 3/5.
Correct Answer: A — 3/5
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Q. If a wave has a frequency of 50 Hz and a wavelength of 2 m, what is its speed? (2018)
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A.
100 m/s
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B.
50 m/s
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C.
25 m/s
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D.
200 m/s
Solution
Wave speed = frequency × wavelength = 50 Hz × 2 m = 100 m/s.
Correct Answer: A — 100 m/s
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Q. If a wave has a frequency of 50 Hz, what is its period? (2022)
-
A.
0.02 s
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B.
0.5 s
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C.
2 s
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D.
20 s
Solution
Period (T) = 1/Frequency = 1/50 Hz = 0.02 s.
Correct Answer: A — 0.02 s
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Q. If C = [[0, 1], [1, 0]], what is det(C)? (2022)
Solution
Determinant of C = (0*0) - (1*1) = 0 - 1 = -1.
Correct Answer: C — -1
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Q. If cos(θ) = 0.5, what is the value of θ in degrees? (2017)
-
A.
30°
-
B.
45°
-
C.
60°
-
D.
90°
Solution
cos(60°) = 0.5, so θ = 60°
Correct Answer: C — 60°
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Q. If cos(θ) = 0.5, what is the value of θ? (2020)
-
A.
0°
-
B.
30°
-
C.
60°
-
D.
90°
Solution
cos(60°) = 0.5, so θ = 60°.
Correct Answer: C — 60°
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Q. If cos(θ) = 0.6, what is sin(θ) using Pythagorean identity? (2017)
-
A.
0.4
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B.
0.5
-
C.
0.6
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D.
0.8
Solution
sin²(θ) + cos²(θ) = 1; sin²(θ) = 1 - 0.6² = 0.64; sin(θ) = √0.64 = 0.8.
Correct Answer: A — 0.4
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Q. If cos(θ) = 1/2, what is θ in degrees?
-
A.
30°
-
B.
45°
-
C.
60°
-
D.
90°
Solution
cos(60°) = 1/2, so θ = 60°
Correct Answer: C — 60°
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Q. If E = [[2, 1, 3], [1, 0, 2], [4, 1, 1]], what is det(E)? (2020)
Solution
The determinant of E can be calculated using the rule of Sarrus or cofactor expansion, resulting in 0.
Correct Answer: A — -1
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Q. If F = [[1, 2, 3], [0, 1, 4], [5, 6, 0]], what is det(F)? (2021)
Solution
Det(F) = 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1.
Correct Answer: A — -14
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Q. If F = [[2, 0], [0, 3]], what is det(F)? (2020)
Solution
The determinant of F is calculated as (2*3) - (0*0) = 6.
Correct Answer: B — 6
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Q. If F = [[2, 1, 3], [1, 0, 2], [3, 4, 1]], find det(F). (2022)
Solution
Using the determinant formula, det(F) = 2*(0*1 - 2*4) - 1*(1*1 - 2*3) + 3*(1*4 - 0*3) = 2*(-8) - 1*(-5) + 3*4 = -16 + 5 + 12 = 1.
Correct Answer: A — -10
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Q. If f(x) = x^3 - 3x^2 + 4, find the critical points. (2022)
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A.
1, 2
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B.
0, 3
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C.
2, 4
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D.
1, 3
Solution
f'(x) = 3x^2 - 6x. Setting f'(x) = 0 gives x(3x - 6) = 0, so x = 0 or x = 2.
Correct Answer: A — 1, 2
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Q. If H = [[1, 2, 1], [0, 1, 0], [2, 1, 1]], find det(H). (2021)
Solution
The determinant of H is calculated as 1(1*1 - 0*1) - 2(0*1 - 0*2) + 1(0*1 - 1*2) = 1 - 0 - 2 = -1.
Correct Answer: B — 1
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Q. If H = [[2, 3], [4, 5]], find det(H). (2022)
Solution
Det(H) = (2*5) - (3*4) = 10 - 12 = -2.
Correct Answer: D — 7
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Q. If I = [[1, 2], [2, 4]], what is det(I)? (2021)
Solution
The determinant of I is 0 because the rows are linearly dependent.
Correct Answer: A — 0
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Q. If J = [[1, 1], [1, 1]], what is det(J)? (2019)
Solution
Det(J) = (1*1) - (1*1) = 1 - 1 = 0.
Correct Answer: A — 0
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Q. If J = [[1, 2, 1], [0, 1, 3], [2, 1, 0]], calculate det(J). (2023)
Solution
Using the determinant formula, det(J) = 1*(1*0 - 3*1) - 2*(0*0 - 3*2) + 1*(0*1 - 1*2) = 1*(-3) - 2*(-6) + 1*(-2) = -3 + 12 - 2 = 7.
Correct Answer: A — -4
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Q. If J = [[1, 2], [2, 4]], what is det(J)? (2022)
Solution
Det(J) = (1*4) - (2*2) = 4 - 4 = 0.
Correct Answer: A — 0
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Q. If one root of the equation x² - 7x + k = 0 is 3, find k. (2023)
Solution
Using the root, substitute x = 3: 3² - 7*3 + k = 0, which gives k = 10.
Correct Answer: A — 10
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Q. If one root of the equation x² - 7x + k = 0 is 3, what is the value of k? (2020)
Solution
Using the root, substitute x = 3: 3² - 7*3 + k = 0, which gives k = 9.
Correct Answer: D — 9
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Q. If sin(θ) = 0.5, what is θ in degrees? (2014)
-
A.
30°
-
B.
45°
-
C.
60°
-
D.
90°
Solution
sin(30°) = 0.5, so θ = 30°.
Correct Answer: A — 30°
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Q. If sin(θ) = 0.8, what is cos(θ) using Pythagorean identity? (2020)
-
A.
0.6
-
B.
0.8
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C.
0.4
-
D.
0.2
Solution
Using sin²(θ) + cos²(θ) = 1, cos²(θ) = 1 - 0.64 = 0.36, cos(θ) = 0.6
Correct Answer: A — 0.6
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Q. If sin(θ) = 0.8, what is cos(θ)? (2022)
-
A.
0.6
-
B.
0.8
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C.
0.4
-
D.
0.2
Solution
Using sin²(θ) + cos²(θ) = 1, cos(θ) = √(1 - 0.8²) = √(0.36) = 0.6.
Correct Answer: A — 0.6
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Q. If sin(θ) = 0.866, what is θ in degrees? (2020)
-
A.
30°
-
B.
45°
-
C.
60°
-
D.
90°
Solution
sin(60°) = √3/2 ≈ 0.866, so θ = 60°.
Correct Answer: C — 60°
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Q. If sin(θ) = 1, what is the value of θ? (2023)
-
A.
0°
-
B.
90°
-
C.
180°
-
D.
270°
Solution
sin(90°) = 1, so θ = 90°.
Correct Answer: B — 90°
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Q. If sin(θ) = 1/2, what is the possible value of θ? (2022)
-
A.
30°
-
B.
60°
-
C.
90°
-
D.
45°
Solution
sin(30°) = 1/2, so θ = 30°.
Correct Answer: A — 30°
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Q. If sin(θ) = 1/√2, what is θ in degrees?
-
A.
30°
-
B.
45°
-
C.
60°
-
D.
90°
Solution
sin(45°) = 1/√2, so θ = 45°
Correct Answer: B — 45°
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Q. If tan(θ) = 1, what is the value of θ? (2020)
-
A.
0°
-
B.
30°
-
C.
45°
-
D.
60°
Solution
tan(45°) = 1, so θ = 45°.
Correct Answer: C — 45°
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