Undergraduate
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)
-
A.
1, 2
-
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 = [[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, 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
-
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)
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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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Q. If the amplitude of a wave is doubled, how does it affect the energy of the wave? (2020)
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A.
Remains the same
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B.
Doubles
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C.
Increases by four times
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D.
Halves
Solution
Energy of a wave is proportional to the square of the amplitude. If amplitude is doubled, energy increases by (2^2) = 4 times.
Correct Answer: C — Increases by four times
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Q. If the circumference of a circle is 31.4 cm, what is its radius? (Use π = 3.14) (2015)
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A.
5 cm
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B.
7 cm
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C.
10 cm
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D.
8 cm
Solution
Circumference = 2πr, so r = circumference / (2π) = 31.4 / (2 * 3.14) = 5 cm.
Correct Answer: A — 5 cm
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Q. If the circumference of a circle is 31.4 cm, what is the radius? (2020) 2020
-
A.
5 cm
-
B.
10 cm
-
C.
15 cm
-
D.
20 cm
Solution
Circumference = 2πr. Therefore, r = Circumference / (2π) = 31.4 cm / (2 * 3.14) = 5 cm.
Correct Answer: A — 5 cm
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Q. If the concentration of a reactant is tripled in a second-order reaction, how does the rate change? (2023)
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A.
Increases by 3 times
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B.
Increases by 6 times
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C.
Increases by 9 times
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D.
Increases by 12 times
Solution
For a second-order reaction, if concentration is tripled, the rate increases by 3² = 9 times.
Correct Answer: C — Increases by 9 times
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Q. If the concentration of Cu²⁺ in a cell is increased, what happens to the cell potential?
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A.
Increases
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B.
Decreases
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C.
Remains the same
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D.
Becomes zero
Solution
Increasing the concentration of Cu²⁺ increases the cell potential according to the Nernst equation.
Correct Answer: A — Increases
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Q. If the coordinates of point A are (1, 2) and point B are (4, 6), what is the length of AB? (2023)
Solution
Length AB = √[(4-1)² + (6-2)²] = √[9 + 16] = √25 = 5.
Correct Answer: A — 5
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Q. If the coordinates of point C are (5, 5) and point D are (5, 10), what is the length of CD? (2021)
Solution
Length CD = |10 - 5| = 5.
Correct Answer: A — 5
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Q. If the coordinates of point C are (5, 5) and the slope of the line is 1, what is the equation of the line? (2021) 2021
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A.
y = x
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B.
y = 5
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C.
y = 2x - 5
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D.
y = x + 5
Solution
Using point-slope form: y - 5 = 1(x - 5) => y = x.
Correct Answer: A — y = x
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Q. If the cost function is C(x) = 3x^2 + 12x + 5, find the minimum cost. (2020)
Solution
The minimum cost occurs at x = -b/(2a) = -12/(2*3) = -2. C(-2) = 3(-2)^2 + 12(-2) + 5 = 8.
Correct Answer: B — 8
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Q. If the cost function is C(x) = 3x^2 + 12x + 5, find the minimum cost. (2020) 2020
Solution
The minimum cost occurs at x = -b/(2a) = -12/(2*3) = -2. C(-2) = 3(-2)^2 + 12(-2) + 5 = 8.
Correct Answer: B — 8
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Q. If the cost function is C(x) = 5x^2 + 20x + 100, find the minimum cost. (2020)
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A.
100
-
B.
120
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C.
140
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D.
160
Solution
The minimum cost occurs at x = -b/(2a) = -20/(2*5) = -2. C(-2) = 5(-2)^2 + 20(-2) + 100 = 120.
Correct Answer: B — 120
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Q. If the current in an AC circuit is I(t) = 5√2 sin(100t + π/4), what is the RMS current? (2021)
-
A.
5 A
-
B.
2.5 A
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C.
7.07 A
-
D.
3.54 A
Solution
The RMS current is I_rms = I_peak / √2 = 5√2 / √2 = 5 A.
Correct Answer: B — 2.5 A
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