Engineering Entrance
Q. Determine the minimum value of f(x) = x^2 - 4x + 5. (2021)
Solution
The vertex form gives the minimum at x = 2. f(2) = 2^2 - 4(2) + 5 = 1.
Correct Answer: A — 1
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Q. Determine the point of inflection for f(x) = x^4 - 4x^3 + 6. (2023)
-
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. Determine the roots of the equation x² + 2x - 8 = 0. (2023)
-
A.
-4 and 2
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B.
4 and -2
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C.
2 and -4
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D.
0 and 8
Solution
Factoring gives (x + 4)(x - 2) = 0, hence the roots are -4 and 2.
Correct Answer: A — -4 and 2
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Q. Evaluate the integral ∫ (3x^2 + 2x) dx. (2020)
-
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)
-
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 the limit: lim (x -> 0) (tan(x)/x) (2023)
-
A.
0
-
B.
1
-
C.
∞
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D.
Undefined
Solution
Using the limit property lim (x -> 0) (tan(x)/x) = 1, we find that the limit is 1.
Correct Answer: B — 1
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Q. Evaluate ∫ (2x + 3) dx. (2022)
-
A.
x^2 + 3x + C
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B.
x^2 + 3 + C
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C.
x^2 + 3x + 1
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D.
2x^2 + 3 + C
Solution
The integral is (2/2)x^2 + 3x + C = x^2 + 3x + C.
Correct Answer: A — x^2 + 3x + 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
-
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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Q. Find the area of the triangle formed by the points (0, 0), (4, 0), and (0, 3). (2022) 2022
Solution
Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.
Correct Answer: A — 6
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Q. Find the area of the triangle formed by the points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9). (2022)
Solution
The points are collinear, hence the area = 0.
Correct Answer: A — 0
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Q. Find the coefficient of x^2 in the expansion of (2x + 3)^6.
-
A.
540
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B.
720
-
C.
810
-
D.
960
Solution
The coefficient of x^2 is given by 6C2 * (2)^2 * (3)^4 = 15 * 4 * 81 = 4860.
Correct Answer: A — 540
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Q. Find the coefficient of x^2 in the expansion of (x + 4)^5. (2023)
-
A.
80
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B.
100
-
C.
120
-
D.
160
Solution
The coefficient of x^2 is C(5,2)(4)^3 = 10 * 64 = 640.
Correct Answer: A — 80
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Q. Find the coefficient of x^3 in the expansion of (2x - 3)^4. (2022)
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A.
-54
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B.
-108
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C.
108
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D.
54
Solution
The coefficient of x^3 is C(4,3) * (2)^3 * (-3)^1 = 4 * 8 * (-3) = -96.
Correct Answer: B — -108
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Q. Find the coefficient of x^4 in the expansion of (2x - 3)^6.
-
A.
540
-
B.
720
-
C.
810
-
D.
900
Solution
The coefficient of x^4 is C(6,4) * (2)^4 * (-3)^2 = 15 * 16 * 9 = 2160.
Correct Answer: A — 540
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Q. Find the coefficient of x^4 in the expansion of (3x + 2)^5. (2022)
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A.
240
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B.
360
-
C.
480
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D.
600
Solution
The coefficient of x^4 is C(5,4)(3)^4(2)^1 = 5 * 81 * 2 = 810.
Correct Answer: B — 360
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Q. Find the coefficient of x^5 in the expansion of (2x - 3)^6. (2022)
-
A.
-540
-
B.
540
-
C.
-720
-
D.
720
Solution
The coefficient of x^5 is C(6,5) * (2)^5 * (-3)^1 = 6 * 32 * (-3) = -576.
Correct Answer: A — -540
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Q. Find the coefficient of x^5 in the expansion of (x + 1)^7.
Solution
The coefficient of x^5 is C(7,5) = 21.
Correct Answer: C — 35
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Q. Find the constant term in the expansion of (x - 2/x)^6. (2022)
Solution
The constant term occurs when the power of x is zero. Setting 6 - 2k = 0 gives k = 3. The term is C(6,3)(-2)^3 = -64.
Correct Answer: A — -64
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Q. Find the coordinates of the midpoint of the line segment joining A(2, 3, 4) and B(4, 5, 6). (2023)
-
A.
(3, 4, 5)
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B.
(2, 3, 4)
-
C.
(4, 5, 6)
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D.
(5, 6, 7)
Solution
Midpoint M = ((2+4)/2, (3+5)/2, (4+6)/2) = (3, 4, 5).
Correct Answer: A — (3, 4, 5)
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Q. Find the derivative of f(x) = sin(x) + cos(x).
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A.
cos(x) - sin(x)
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B.
-sin(x) - cos(x)
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C.
sin(x) + cos(x)
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D.
-cos(x) + sin(x)
Solution
The derivative f'(x) = d/dx(sin(x) + cos(x)) = cos(x) - sin(x).
Correct Answer: A — cos(x) - sin(x)
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Q. Find the derivative of f(x) = tan(x). (2022) 2022
-
A.
sec^2(x)
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B.
csc^2(x)
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C.
sec(x)
-
D.
tan^2(x)
Solution
The derivative f'(x) = d/dx(tan(x)) = sec^2(x).
Correct Answer: A — sec^2(x)
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Q. Find the derivative of f(x) = x^5 + 3x^3 - 2x.
-
A.
5x^4 + 9x^2 - 2
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B.
5x^4 + 6x^2 - 2
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C.
3x^2 + 5x^4 - 2
-
D.
5x^4 + 3x^2 - 2
Solution
The derivative f'(x) = d/dx(x^5 + 3x^3 - 2x) = 5x^4 + 9x^2 - 2.
Correct Answer: A — 5x^4 + 9x^2 - 2
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Q. Find the derivative of f(x) = x^5 - 3x^3 + 2. (2022)
-
A.
5x^4 - 9x^2
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B.
5x^4 + 9x^2
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C.
3x^2 - 9x
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D.
5x^4 - 3x^2
Solution
The derivative f'(x) = d/dx(x^5 - 3x^3 + 2) = 5x^4 - 9x^2.
Correct Answer: A — 5x^4 - 9x^2
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Q. Find the determinant of E = [[3, 2], [1, 4]]. (2022)
Solution
Det(E) = (3*4) - (2*1) = 12 - 2 = 10.
Correct Answer: A — 10
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Q. Find the determinant of F = [[4, 5], [6, 7]]. (2020)
Solution
Det(F) = (4*7) - (5*6) = 28 - 30 = -2.
Correct Answer: A — -2
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Q. Find the determinant of H = [[3, 1], [2, 5]]. (2021)
Solution
Determinant of H = (3*5) - (1*2) = 15 - 2 = 13.
Correct Answer: A — 7
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Q. Find the determinant of J = [[5, 2], [1, 3]]. (2020)
Solution
The determinant of J is calculated as (5*3) - (2*1) = 15 - 2 = 13.
Correct Answer: A — 10
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