Undergraduate
Q. A wave travels with a speed of 340 m/s and has a frequency of 170 Hz. What is its wavelength? (2022)
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A.
2.0 m
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B.
1.0 m
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C.
0.5 m
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D.
3.0 m
Solution
Wavelength λ = v/f = 340 m/s / 170 Hz = 2.0 m.
Correct Answer: A — 2.0 m
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Q. A wheel rotates with a constant angular acceleration α. If its initial angular velocity is ω₀, what is its angular velocity after time t? (2023)
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A.
ω₀ + αt
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B.
ω₀ - αt
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C.
αt²
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D.
ω₀t
Solution
Using the equation of motion for rotation, ω = ω₀ + αt.
Correct Answer: A — ω₀ + αt
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Q. An object is dropped from a height of 80 m. How long will it take to reach the ground? (Take g = 10 m/s²) (2023)
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A.
4 s
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B.
8 s
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C.
10 s
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D.
12 s
Solution
Using the formula: h = 0.5 * g * t^2. Rearranging gives t = sqrt(2h/g) = sqrt(2*80/10) = sqrt(16) = 4 s.
Correct Answer: B — 8 s
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Q. An object moves with a constant velocity of 10 m/s. What is the net force acting on it? (2023)
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A.
0 N
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B.
10 N
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C.
20 N
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D.
100 N
Solution
If the object moves with constant velocity, the net force acting on it is 0 N.
Correct Answer: A — 0 N
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Q. Calculate the coefficient of x^4 in the expansion of (x + 1/2)^6. (2021)
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A.
15/8
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B.
45/8
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C.
5/8
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D.
1/8
Solution
The coefficient of x^4 is C(6,4)(1/2)^2 = 15 * 1/4 = 15/4.
Correct Answer: B — 45/8
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Q. Calculate the coefficient of x^4 in the expansion of (x + 3)^6. (2021)
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A.
54
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B.
81
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C.
108
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D.
729
Solution
The coefficient of x^4 is C(6,4)(3)^2 = 15 * 9 = 135.
Correct Answer: C — 108
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Q. Calculate the derivative of f(x) = ln(x^2 + 1).
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A.
2x/(x^2 + 1)
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B.
1/(x^2 + 1)
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C.
2/(x^2 + 1)
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D.
x/(x^2 + 1)
Solution
Using the chain rule, f'(x) = d/dx(ln(x^2 + 1)) = (2x)/(x^2 + 1).
Correct Answer: A — 2x/(x^2 + 1)
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Q. Calculate the determinant of D = [[3, 2, 1], [1, 0, 2], [0, 1, 3]]. (2023)
Solution
Det(D) = 3(0*3 - 2*1) - 2(1*3 - 0*2) + 1(1*1 - 0*0) = 3(0 - 2) - 2(3) + 1(1) = -6 - 6 + 1 = -11.
Correct Answer: A — 1
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Q. Calculate the determinant of J = [[1, 2, 1], [0, 1, 2], [1, 0, 1]]. (2023)
Solution
Det(J) = 1(1*1 - 2*0) - 2(0*1 - 1*1) + 1(0*0 - 1*1) = 1(1) - 2(-1) + 1(-1) = 1 + 2 - 1 = 2.
Correct Answer: C — 2
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Q. Calculate the determinant of the matrix \( C = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \). (2020)
Solution
The determinant is \( 5*8 - 6*7 = 40 - 42 = -2 \).
Correct Answer: A — -2
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Q. Calculate the determinant of the matrix \( H = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix} \). (2020)
Solution
The determinant of an upper triangular matrix is the product of its diagonal elements: \( 1*1*1 = 1 \).
Correct Answer: A — 1
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Q. Calculate the distance from the point P(1, 2, 3) to the origin O(0, 0, 0). (2023)
Solution
Distance = √[(1-0)² + (2-0)² + (3-0)²] = √[1 + 4 + 9] = √14.
Correct Answer: B — √14
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Q. Calculate the perimeter of a square with side length 4 cm. (2015)
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A.
16 cm
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B.
12 cm
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C.
8 cm
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D.
20 cm
Solution
Perimeter = 4 × side = 4 × 4 cm = 16 cm.
Correct Answer: A — 16 cm
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Q. Calculate the pH of a 0.05 M NH4Cl solution (Kb for NH3 = 1.8 x 10^-5).
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A.
4.75
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B.
5.25
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C.
5.75
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D.
6.25
Solution
Using the formula for weak bases, pH = 14 - 0.5(pKb - logC) = 14 - 0.5(4.74 - log(0.05)) = 5.25.
Correct Answer: B — 5.25
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Q. Calculate the pH of a 0.1 M NaOH solution.
Solution
pOH = -log[OH-] = -log(0.1) = 1, thus pH = 14 - pOH = 14 - 1 = 13.
Correct Answer: C — 14
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Q. Calculate the pH of a 0.2 M solution of KOH.
Solution
pOH = -log(0.2) = 0.7, thus pH = 14 - 0.7 = 13.3.
Correct Answer: B — 13
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Q. Calculate the value of 12 × 8 - 24. (2021)
Solution
12 × 8 = 96, then 96 - 24 = 72.
Correct Answer: A — 72
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Q. Calculate the value of 5! (5 factorial). (2020)
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A.
120
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B.
100
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C.
60
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D.
24
Solution
5! = 5 × 4 × 3 × 2 × 1 = 120.
Correct Answer: A — 120
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Q. Determine the coefficient of x^4 in the expansion of (2x - 3)^6.
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A.
540
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B.
720
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C.
810
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D.
960
Solution
The coefficient of x^4 is given by 6C4 * (2)^4 * (-3)^2 = 15 * 16 * 9 = 2160.
Correct Answer: B — 720
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Q. Determine the coordinates of the centroid of the triangle with vertices A(0, 0, 0), B(4, 0, 0), C(0, 3, 0). (2023)
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A.
(1, 1, 0)
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B.
(2, 1, 0)
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C.
(4/3, 1, 0)
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D.
(0, 1, 0)
Solution
Centroid G = ((0+4+0)/3, (0+0+3)/3, (0+0+0)/3) = (4/3, 1, 0).
Correct Answer: B — (2, 1, 0)
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Q. Determine the critical points of f(x) = e^x - 2x. (2021)
Solution
f'(x) = e^x - 2. Setting f'(x) = 0 gives e^x = 2, so x = ln(2).
Correct Answer: B — 1
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Q. Determine the intervals where f(x) = x^4 - 4x^3 has local minima. (2020)
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A.
(0, 2)
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B.
(1, 3)
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C.
(2, 4)
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D.
(0, 1)
Solution
f'(x) = 4x^3 - 12x^2. Setting f'(x) = 0 gives x = 0, 3. Testing intervals shows local minima at (0, 2).
Correct Answer: A — (0, 2)
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Q. Determine the local maxima of f(x) = -x^3 + 3x^2 + 1. (2021)
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A.
(0, 1)
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B.
(1, 3)
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C.
(2, 5)
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D.
(3, 4)
Solution
f'(x) = -3x^2 + 6x. Setting f'(x) = 0 gives x = 0 or x = 2. f(2) = 5 is a local maximum.
Correct Answer: B — (1, 3)
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Q. Determine the point of inflection for f(x) = x^4 - 4x^3 + 6. (2023)
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A.
(1, 3)
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B.
(2, 2)
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C.
(0, 6)
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D.
(3, 0)
Solution
f''(x) = 12x^2 - 24x. Setting f''(x) = 0 gives x(12x - 24) = 0, so x = 0 or x = 2. Check f(1) = 3.
Correct Answer: A — (1, 3)
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Q. Determine the product of the roots of the equation x² + 6x + 9 = 0. (2021)
Solution
The product of the roots is c/a = 9/1 = 9.
Correct Answer: A — 9
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Q. Evaluate the integral ∫ (3x^2 + 2x) dx. (2020)
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A.
x^3 + x^2 + C
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B.
x^3 + x^2 + 2C
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C.
x^3 + x^2 + 1
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D.
x^3 + 2x + C
Solution
The integral is (3/3)x^3 + (2/2)x^2 + C = x^3 + x^2 + C.
Correct Answer: A — x^3 + x^2 + C
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Q. Evaluate the integral ∫(3x^2 + 2)dx. (2022)
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A.
x^3 + 2x + C
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B.
x^3 + 2x^2 + C
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C.
x^3 + 2x^3 + C
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D.
3x^3 + 2x + C
Solution
Integrating term by term, ∫3x^2dx = x^3 and ∫2dx = 2x. Thus, ∫(3x^2 + 2)dx = x^3 + 2x + C.
Correct Answer: A — x^3 + 2x + C
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Q. Evaluate ∫ (4x^3 - 2x) dx. (2019)
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A.
x^4 - x^2 + C
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B.
x^4 - x^2 + 2C
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C.
x^4 - x + C
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D.
4x^4 - 2x^2 + C
Solution
The integral is (4/4)x^4 - (2/2)x^2 + C = x^4 - x^2 + C.
Correct Answer: A — x^4 - x^2 + C
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Q. Evaluate ∫ (5 - 3x) dx. (2022)
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A.
5x - (3/2)x^2 + C
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B.
5x - (3/3)x^2 + C
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C.
5x - (3/4)x^2 + C
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D.
5x - (3/5)x^2 + C
Solution
The integral is 5x - (3/2)x^2 + C.
Correct Answer: A — 5x - (3/2)x^2 + C
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Q. Evaluate ∫(5x^4)dx. (2020)
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A.
(5/5)x^5 + C
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B.
(1/5)x^5 + C
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C.
(5/4)x^4 + C
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D.
(1/4)x^4 + C
Solution
The integral of 5x^4 is (5/5)x^5 + C = x^5 + C.
Correct Answer: A — (5/5)x^5 + C
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