Engineering Entrance
Q. If f(x) = x^3 - 3x^2 + 4, find the critical points. (2022)
A.
1, 2
B.
0, 3
C.
2, 4
D.
1, 3
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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)
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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)
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Solution
Det(H) = (2*5) - (3*4) = 10 - 12 = -2.
Correct Answer: D — 7
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Q. If I = [[1, 0, 2], [0, 1, 3], [1, 1, 0]], find det(I). (2023)
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Solution
Using the determinant formula for 3x3 matrices, det(I) = 1(1*0 - 3*1) - 0(0 - 3*1) + 2(0 - 1*1) = 0 - 0 - 2 = -2.
Correct Answer: A — -1
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Q. If I = [[1, 2], [2, 4]], what is det(I)? (2021)
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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)
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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)
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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)
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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)
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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)
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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°
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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
C.
0.4
D.
0.2
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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
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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°
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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°
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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°
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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°
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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°
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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)
A.
Remains the same
B.
Doubles
C.
Increases by four times
D.
Halves
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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 charge on a capacitor is doubled, what happens to the energy stored? (2019)
A.
It remains the same
B.
It doubles
C.
It quadruples
D.
It halves
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Solution
The energy stored in a capacitor is proportional to the square of the charge (U = Q^2/(2C)). Doubling the charge increases the energy by a factor of 4.
Correct Answer: C — It quadruples
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Q. If the circumference of a circle is 31.4 cm, what is its radius? (2022)
A.
5 cm
B.
10 cm
C.
15 cm
D.
20 cm
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Solution
Circumference = 2πr; 31.4 = 2 * π * r; r = 5 cm.
Correct Answer: A — 5 cm
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Q. If the circumference of a circle is 31.4 cm, what is its radius? (Use π = 3.14) (2015)
A.
5 cm
B.
7 cm
C.
10 cm
D.
8 cm
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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
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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)
A.
Increases by 3 times
B.
Increases by 6 times
C.
Increases by 9 times
D.
Increases by 12 times
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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?
A.
Increases
B.
Decreases
C.
Remains the same
D.
Becomes zero
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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)
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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)
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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
A.
y = x
B.
y = 5
C.
y = 2x - 5
D.
y = x + 5
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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)
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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
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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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