Q. Determine the angle between the lines y = 2x + 1 and y = -1/2x + 3. (2021)
A.
90 degrees
B.
45 degrees
C.
60 degrees
D.
30 degrees
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Solution
The slopes are m1 = 2 and m2 = -1/2. The angle θ = tan⁻¹(|(m1 - m2) / (1 + m1*m2)|) = tan⁻¹(5/3), which is approximately 90 degrees.
Correct Answer:
A
— 90 degrees
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Q. Determine the coefficient of x^5 in the expansion of (3x - 4)^7.
A.
252
B.
336
C.
672
D.
840
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Solution
The coefficient of x^5 in (3x - 4)^7 is C(7, 5) * (3)^5 * (-4)^2 = 21 * 243 * 16 = 68016.
Correct Answer:
A
— 252
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Q. Determine the continuity of the function f(x) = { x^2, x < 1; 2, x = 1; x + 1, x > 1 } at x = 1.
A.
Continuous
B.
Not continuous
C.
Depends on the limit
D.
Only left continuous
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Solution
The left limit as x approaches 1 is 1, the right limit is 2, and f(1) = 2. Since the left and right limits do not match, f(x) is not continuous at x = 1.
Correct Answer:
B
— Not continuous
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Q. Determine the continuity of the function f(x) = { x^2, x < 1; 2x - 1, x ≥ 1 } at x = 1.
A.
Continuous
B.
Discontinuous
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
At x = 1, f(1) = 2(1) - 1 = 1 and lim x→1- f(x) = 1, lim x→1+ f(x) = 1. Thus, f(x) is continuous at x = 1.
Correct Answer:
A
— Continuous
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Q. Determine the continuity of the function f(x) = |x| at x = 0. (2020)
A.
Continuous
B.
Not continuous
C.
Depends on the limit
D.
Only left continuous
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Solution
The function f(x) = |x| is continuous at x = 0 since both the left-hand limit and right-hand limit equal f(0) = 0.
Correct Answer:
A
— Continuous
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Q. Determine the critical points of the function f(x) = x^2 - 4x + 4. (2022)
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Solution
f'(x) = 2x - 4; Setting f'(x) = 0 gives x = 2 as the critical point.
Correct Answer:
C
— 2
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Q. Determine the derivative of f(x) = x^3 - 4x + 7. (2023)
A.
3x^2 - 4
B.
3x^2 + 4
C.
x^2 - 4
D.
3x^2 - 7
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Solution
Using the power rule, f'(x) = 3x^2 - 4.
Correct Answer:
A
— 3x^2 - 4
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Q. Determine the derivative of f(x) = x^5 - 3x^3 + 2x. (2023)
A.
5x^4 - 9x^2 + 2
B.
5x^4 - 9x + 2
C.
5x^4 - 3x^2 + 2
D.
5x^4 - 3x^3
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Solution
Using the power rule, f'(x) = 5x^4 - 9x^2 + 2.
Correct Answer:
A
— 5x^4 - 9x^2 + 2
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Q. Determine the distance between the points (-1, -1) and (2, 2).
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Solution
Using the distance formula: d = √[(2 - (-1))² + (2 - (-1))²] = √[(2 + 1)² + (2 + 1)²] = √[9 + 9] = √18 = 3√2 ≈ 4.24.
Correct Answer:
C
— 5
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Q. Determine the distance between the points (0, 0) and (0, 8).
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Solution
Using the distance formula: d = √[(0 - 0)² + (8 - 0)²] = √[0 + 64] = √64 = 8.
Correct Answer:
A
— 8
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Q. Determine the distance between the points (1, 2) and (4, 6). (2022)
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Solution
Using the distance formula: d = √[(4 - 1)² + (6 - 2)²] = √[9 + 16] = √25 = 5.
Correct Answer:
A
— 5
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Q. Determine the distance between the points (2, 3) and (2, -1).
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Solution
Using the distance formula: d = √[(2 - 2)² + (-1 - 3)²] = √[0 + 16] = √16 = 4.
Correct Answer:
A
— 4
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Q. Determine the distance from the point (1, 2) to the line 2x + 3y - 6 = 0. (2023)
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Solution
Using the formula for distance from a point to a line, the distance is |2(1) + 3(2) - 6| / sqrt(2^2 + 3^2) = 1.
Correct Answer:
B
— 2
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Q. Determine the local maxima and minima of f(x) = x^2 - 4x + 3.
A.
Maxima at x=2
B.
Minima at x=2
C.
Maxima at x=1
D.
Minima at x=1
Show solution
Solution
f'(x) = 2x - 4. Setting f'(x) = 0 gives x = 2. f''(x) = 2 > 0 indicates a local minimum at x = 2.
Correct Answer:
B
— Minima at x=2
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Q. Determine the local maxima and minima of f(x) = x^4 - 8x^2 + 16. (2023)
A.
Maxima at x = 0
B.
Minima at x = 2
C.
Maxima at x = 2
D.
Minima at x = 0
Show solution
Solution
f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x = 0, ±2. f''(x) = 12x^2 - 16. Minima at x = 0.
Correct Answer:
D
— Minima at x = 0
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Q. Determine the local maxima of f(x) = -x^2 + 4x. (2022)
A.
(2, 4)
B.
(0, 0)
C.
(4, 0)
D.
(1, 1)
Show solution
Solution
f'(x) = -2x + 4. Setting f'(x) = 0 gives x = 2. f(2) = -2^2 + 4(2) = 4.
Correct Answer:
A
— (2, 4)
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Q. Determine the local maxima or minima of f(x) = -x^2 + 4x. (2019)
A.
Maxima at x=2
B.
Minima at x=2
C.
Maxima at x=4
D.
Minima at x=4
Show solution
Solution
f'(x) = -2x + 4. Setting f'(x) = 0 gives x = 2. Since f''(x) = -2 < 0, it is a maxima.
Correct Answer:
A
— Maxima at x=2
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Q. Determine the maximum value of f(x) = -2x^2 + 4x + 1. (2023)
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Solution
The vertex is at x = -4/(2*(-2)) = 1. The maximum value is f(1) = -2(1)^2 + 4(1) + 1 = 3.
Correct Answer:
C
— 3
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Q. Determine the maximum value of f(x) = -x^2 + 4x. (2020)
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Solution
f'(x) = -2x + 4. Setting f'(x) = 0 gives x = 2. f(2) = -2^2 + 4(2) = 8.
Correct Answer:
A
— 4
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Q. Determine the maximum value of the function f(x) = -x^2 + 6x - 8. (2022)
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Solution
The vertex is at x = -6/(2*(-1)) = 3. The maximum value is f(3) = -3^2 + 6*3 - 8 = 1.
Correct Answer:
B
— 4
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Q. Determine the median of the following numbers: 9, 7, 5, 3, 1.
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Solution
Arrange the numbers: 1, 3, 5, 7, 9. The median is the middle value, which is 5.
Correct Answer:
A
— 5
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Q. Determine the median of the following set: 1, 2, 3, 4, 5, 6, 7, 8. (2020)
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Solution
Arrange the numbers: 1, 2, 3, 4, 5, 6, 7, 8. The median is the average of the 4th and 5th numbers: (4 + 5) / 2 = 4.5.
Correct Answer:
B
— 4.5
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Q. Determine the minimum value of f(x) = x^2 - 6x + 10. (2019)
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Solution
The minimum occurs at x = -b/(2a) = 6/(2*1) = 3. f(3) = 3^2 - 6(3) + 10 = 3.
Correct Answer:
B
— 3
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Q. Determine the minimum value of the function f(x) = x^2 - 4x + 6. (2020)
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Solution
The function is a upward-opening parabola. The minimum occurs at x = -b/(2a) = 4/(2*1) = 2. f(2) = 2^2 - 4(2) + 6 = 2.
Correct Answer:
A
— 2
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Q. Determine the mode of the following data: {1, 2, 2, 3, 4, 4, 4, 5, 5}.
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Solution
The mode is 4, as it appears 3 times, more than any other number.
Correct Answer:
C
— 4
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Q. Determine the slope of the tangent line to f(x) = x^2 at x = 3. (2023)
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Solution
f'(x) = 2x; thus, f'(3) = 2(3) = 6.
Correct Answer:
B
— 6
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Q. Determine the solution of the differential equation dy/dx = y^2 - 1.
A.
y = tan(x + C)
B.
y = 1/(C - x)
C.
y = 1/(C + x)
D.
y = e^(x + C)
Show solution
Solution
This is separable. Separating and integrating gives y = 1/(C - x).
Correct Answer:
B
— y = 1/(C - x)
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Q. Determine the x-intercept of the line given by the equation 5x + 2y - 10 = 0. (2023)
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Solution
Setting y = 0 in the equation gives 5x = 10, thus x = 2. The x-intercept is 2.
Correct Answer:
C
— 5
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Q. Determine the y-intercept of the line given by the equation 5x + 2y - 10 = 0. (2021)
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Solution
Setting x = 0 in the equation gives 2y = 10, thus y = 5. The y-intercept is 5.
Correct Answer:
B
— 2
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Q. Differentiate f(x) = 4x^2 * e^x. (2022)
A.
4e^x + 4x^2e^x
B.
4x^2e^x + 4xe^x
C.
4e^x + 2x^2e^x
D.
8xe^x
Show solution
Solution
Using the product rule, f'(x) = 4e^x + 4x^2e^x.
Correct Answer:
A
— 4e^x + 4x^2e^x
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