Q. Find the maximum value of f(x) = -x^2 + 4x + 1.
Solution
The maximum occurs at x = 2. f(2) = -2^2 + 4(2) + 1 = 5.
Correct Answer: A — 5
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Q. Find the maximum value of f(x) = -x^2 + 4x.
Solution
The vertex form gives maximum at x = 2. f(2) = -2^2 + 4*2 = 4.
Correct Answer: A — 4
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Q. Find the maximum value of the function f(x) = -2x^2 + 8x - 3.
Solution
The function is a downward-opening parabola. The maximum occurs at x = -b/(2a) = 8/(2*2) = 2. f(2) = -2(2^2) + 8(2) - 3 = 8.
Correct Answer: B — 8
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Q. Find the maximum value of the function f(x) = -2x^2 + 8x - 5.
Solution
The function is a downward-opening parabola. The maximum occurs at x = -b/(2a) = 8/(2*2) = 2. f(2) = -2(2^2) + 8(2) - 5 = 9.
Correct Answer: C — 9
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Q. Find the maximum value of the function f(x) = -x^2 + 4x + 1.
Solution
The vertex occurs at x = 2. f(2) = -2^2 + 4(2) + 1 = 9, which is the maximum value.
Correct Answer: A — 5
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Q. Find the maximum value of the function f(x) = -x^2 + 6x - 8.
Solution
The vertex occurs at x = 3. f(3) = -3^2 + 6(3) - 8 = 6.
Correct Answer: C — 8
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Q. Find the median of the following numbers: 1, 3, 3, 6, 7, 8, 9.
Solution
Arranging the numbers: 1, 3, 3, 6, 7, 8, 9. Median = 6 (middle value).
Correct Answer: B — 6
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Q. Find the median of the following set of numbers: 1, 3, 3, 6, 7, 8, 9.
Solution
Arranging the numbers: 1, 3, 3, 6, 7, 8, 9. Median = 6 (middle value).
Correct Answer: B — 6
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Q. Find the median of the following set of numbers: 3, 1, 4, 2.
Solution
Arranging the numbers: 1, 2, 3, 4. Median = (2 + 3) / 2 = 2.5.
Correct Answer: B — 2.5
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Q. Find the midpoint of the line segment joining the points (1, 2) and (3, 4).
-
A.
(2, 3)
-
B.
(1, 2)
-
C.
(3, 4)
-
D.
(4, 5)
Solution
Midpoint = ((1+3)/2, (2+4)/2) = (2, 3).
Correct Answer: A — (2, 3)
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Q. Find the midpoint of the line segment joining the points (2, 3) and (4, 7).
-
A.
(3, 5)
-
B.
(2, 5)
-
C.
(4, 3)
-
D.
(5, 6)
Solution
Midpoint M = ((x1 + x2)/2, (y1 + y2)/2) = ((2 + 4)/2, (3 + 7)/2) = (3, 5).
Correct Answer: A — (3, 5)
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Q. Find the minimum value of the function f(x) = 3x^2 - 12x + 7.
Solution
The vertex occurs at x = -b/(2a) = 12/6 = 2. f(2) = 3(2^2) - 12(2) + 7 = 1.
Correct Answer: B — 1
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Q. Find the minimum value of the function f(x) = x^2 - 4x + 5.
Solution
The vertex of the parabola occurs at x = 2. f(2) = 2^2 - 4(2) + 5 = 1, which is the minimum value.
Correct Answer: A — 1
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Q. Find the minimum value of the function f(x) = x^4 - 8x^2 + 16.
Solution
f'(x) = 4x^3 - 16x. Setting f'(x) = 0 gives x = 0, ±2. f(2) = 0, which is the minimum value.
Correct Answer: A — 0
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Q. Find the particular solution of dy/dx = 2x with the initial condition y(0) = 1.
-
A.
y = x^2 + 1
-
B.
y = x^2 - 1
-
C.
y = 2x + 1
-
D.
y = 2x - 1
Solution
Integrating gives y = x^2 + C. Using the initial condition y(0) = 1, we find C = 1.
Correct Answer: A — y = x^2 + 1
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Q. Find the particular solution of dy/dx = x + y, given y(0) = 1.
-
A.
y = e^x + 1
-
B.
y = e^x - 1
-
C.
y = x + 1
-
D.
y = x + e^x
Solution
The general solution is y = e^x + C. Using the initial condition y(0) = 1, we find C = 1.
Correct Answer: A — y = e^x + 1
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Q. Find the point of inflection for the function f(x) = x^3 - 6x^2 + 9x.
-
A.
(1, 4)
-
B.
(2, 3)
-
C.
(3, 0)
-
D.
(0, 0)
Solution
f''(x) = 6x - 12. Setting f''(x) = 0 gives x = 2. The point of inflection is (2, f(2)) = (2, 3).
Correct Answer: C — (3, 0)
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Q. Find the point of inflection for the function f(x) = x^4 - 4x^3 + 6.
-
A.
(1, 3)
-
B.
(2, 2)
-
C.
(3, 1)
-
D.
(0, 6)
Solution
f''(x) = 12x^2 - 24x. Setting f''(x) = 0 gives x(x - 2) = 0, so x = 0 or x = 2. The point of inflection is at (2, f(2)) = (2, 2).
Correct Answer: A — (1, 3)
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Q. Find the point of intersection of the lines y = 2x + 1 and y = -x + 4.
-
A.
(1, 3)
-
B.
(2, 5)
-
C.
(3, 7)
-
D.
(4, 9)
Solution
Setting 2x + 1 = -x + 4 gives 3x = 3, thus x = 1. Substituting x back gives y = 3, so the point is (1, 3).
Correct Answer: A — (1, 3)
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Q. Find the point of intersection of the lines y = x + 1 and y = -x + 5.
-
A.
(2, 3)
-
B.
(3, 2)
-
C.
(1, 2)
-
D.
(0, 1)
Solution
Set x + 1 = -x + 5. Solving gives x = 2, y = 3. Thus, the point is (2, 3).
Correct Answer: A — (2, 3)
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Q. Find the projection of vector A = (2, 3) onto vector B = (1, 1).
Solution
Projection of A onto B = (A · B) / |B|^2 * B; A · B = 2*1 + 3*1 = 5; |B|^2 = 1^2 + 1^2 = 2; Projection = (5/2)(1, 1) = (2.5, 2.5).
Correct Answer: A — 1
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Q. Find the projection of vector A = (3, 4) onto vector B = (1, 2).
Solution
Projection of A onto B = (A · B) / |B|^2 * B. A · B = 3*1 + 4*2 = 11, |B|^2 = 1^2 + 2^2 = 5. Thus, projection = (11/5) * (1, 2) = (11/5, 22/5).
Correct Answer: B — 2
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Q. Find the range of the data set: 10, 15, 20, 25, 30.
Solution
Range = Maximum - Minimum = 30 - 10 = 20.
Correct Answer: A — 15
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Q. Find the range of the data set: 12, 15, 20, 22, 30.
Solution
Range = Maximum - Minimum = 30 - 12 = 18.
Correct Answer: C — 18
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Q. Find the range of the data set: 12, 15, 22, 30, 5.
Solution
Range = max - min = 30 - 5 = 25.
Correct Answer: A — 25
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Q. Find the range of the data set: 8, 12, 15, 20, 25.
Solution
Range = Maximum - Minimum = 25 - 8 = 17.
Correct Answer: A — 12
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Q. Find the real part of the complex number z = 2 + 3i.
Solution
The real part of z = 2 + 3i is 2.
Correct Answer: A — 2
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Q. Find the real part of the complex number z = 2e^(iπ/3).
Solution
The real part is 2 * cos(π/3) = 2 * 1/2 = 1.
Correct Answer: B — 2
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Q. Find the real part of the complex number z = 3 + 4i.
Solution
The real part of z is 3.
Correct Answer: A — 3
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Q. Find the real part of the complex number z = 4 + 3i.
Solution
The real part of z = 4 + 3i is 4.
Correct Answer: A — 4
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