Q. Find the weighted mean of the numbers 10, 20, and 30 with weights 1, 2, and 3 respectively.
Solution
Weighted mean = (10*1 + 20*2 + 30*3) / (1 + 2 + 3) = (10 + 40 + 90) / 6 = 140 / 6 = 23.33.
Correct Answer: B — 25
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Q. Find the x-coordinate of the point where the function f(x) = 2x^3 - 9x^2 + 12x has a local maximum.
Solution
f'(x) = 6x^2 - 18x + 12. Setting f'(x) = 0 gives x = 1 and x = 2. f''(1) < 0 indicates a local maximum at x = 1.
Correct Answer: B — 2
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Q. Find the x-coordinate of the point where the function f(x) = x^2 - 4x + 5 has a minimum.
Solution
The vertex occurs at x = -b/(2a) = 4/2 = 2, which is the x-coordinate of the minimum point.
Correct Answer: A — 2
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Q. Find the x-coordinate of the point where the function f(x) = x^2 - 4x + 5 has a local minimum.
Solution
The vertex occurs at x = -b/(2a) = 4/2 = 2. This is where the local minimum occurs.
Correct Answer: B — 2
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Q. Find the y-intercept of the line represented by the equation 5x - 2y = 10.
Solution
Set x = 0: -2y = 10 => y = -5. The y-intercept is (0, -5).
Correct Answer: B — 2
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Q. For f(x) = { x^2, x < 1; 2x - 1, x ≥ 1 }, is f differentiable at x = 1?
-
A.
Yes
-
B.
No
-
C.
Only left
-
D.
Only right
Solution
f'(1) from left = 2, from right = 2; hence f is differentiable at x = 1.
Correct Answer: B — No
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Q. For the data set 10, 20, 30, 40, 50, what is the mean deviation?
Solution
Mean = 30; Mean Deviation = (|10-30| + |20-30| + |30-30| + |40-30| + |50-30|) / 5 = 10.
Correct Answer: B — 15
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Q. For the data set {10, 12, 23, 23, 16, 23, 21}, what is the mode?
Solution
The mode is the number that appears most frequently, which is 23.
Correct Answer: C — 23
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Q. For the data set {12, 15, 20, 22, 25}, what is the mode?
-
A.
12
-
B.
15
-
C.
20
-
D.
No mode
Solution
There is no mode as all values appear only once.
Correct Answer: D — No mode
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Q. For the data set {2, 4, 6, 8, 10}, what is the mean deviation?
Solution
Mean = 6; Mean deviation = (|2-6| + |4-6| + |6-6| + |8-6| + |10-6|)/5 = (4 + 2 + 0 + 2 + 4)/5 = 12/5 = 2.4.
Correct Answer: B — 1.6
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Q. For the data set {4, 8, 6, 5, 3}, what is the mean?
-
A.
4.5
-
B.
5.5
-
C.
6.0
-
D.
5.0
Solution
Mean = (4 + 8 + 6 + 5 + 3) / 5 = 26 / 5 = 5.0.
Correct Answer: D — 5.0
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Q. For the data set: 1, 2, 3, 4, 5, what is the interquartile range?
Solution
Q1 = 2, Q3 = 4; Interquartile Range = Q3 - Q1 = 4 - 2 = 2.
Correct Answer: B — 2
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Q. For the data set: 5, 7, 8, 9, 10, what is the mean absolute deviation?
Solution
Mean = 7.5; MAD = (|5-7.5| + |7-7.5| + |8-7.5| + |9-7.5| + |10-7.5|) / 5 = 1.
Correct Answer: B — 2
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Q. For the data set: 5, 7, 8, 9, 10, what is the standard deviation?
Solution
Mean = 7.5; Variance = [(5-7.5)^2 + (7-7.5)^2 + (8-7.5)^2 + (9-7.5)^2 + (10-7.5)^2] / 5 = 2; Standard Deviation = sqrt(2) = 1.41
Correct Answer: B — 2
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Q. For the ellipse defined by the equation 9x^2 + 16y^2 = 144, what are the lengths of the semi-major and semi-minor axes?
-
A.
3, 4
-
B.
4, 3
-
C.
6, 8
-
D.
8, 6
Solution
The semi-major axis is 4 and the semi-minor axis is 3 after rewriting the equation in standard form.
Correct Answer: A — 3, 4
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Q. For the function f(x) = 2x^3 - 9x^2 + 12x, find the inflection point.
-
A.
(1, 1)
-
B.
(2, 2)
-
C.
(3, 3)
-
D.
(4, 4)
Solution
f''(x) = 12x - 18. Setting f''(x) = 0 gives x = 1.5. The inflection point is (1.5, f(1.5)).
Correct Answer: B — (2, 2)
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Q. For the function f(x) = 2x^3 - 9x^2 + 12x, find the intervals where the function is increasing.
-
A.
(-∞, 1)
-
B.
(1, 3)
-
C.
(3, ∞)
-
D.
(0, 3)
Solution
f'(x) = 6x^2 - 18x + 12. Setting f'(x) = 0 gives x = 1 and x = 3. Testing intervals shows f is increasing on (1, 3).
Correct Answer: B — (1, 3)
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Q. For the function f(x) = 2x^3 - 9x^2 + 12x, find the local maxima.
-
A.
(1, 5)
-
B.
(2, 0)
-
C.
(3, 0)
-
D.
(0, 0)
Solution
f'(x) = 6x^2 - 18x + 12. Setting f'(x) = 0 gives x = 1 and x = 2. f(1) = 5 is a local maximum.
Correct Answer: A — (1, 5)
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Q. For the function f(x) = 3x^2 - 12x + 7, find the coordinates of the vertex.
-
A.
(2, -5)
-
B.
(2, -1)
-
C.
(3, -2)
-
D.
(1, 1)
Solution
The vertex is at x = -b/(2a) = 12/(2*3) = 2. f(2) = 3(2^2) - 12(2) + 7 = -1. So, the vertex is (2, -1).
Correct Answer: B — (2, -1)
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Q. For the function f(x) = 3x^3 - 12x^2 + 9, find the x-coordinates of the inflection points.
Solution
f''(x) = 18x - 24. Setting f''(x) = 0 gives x = 4/3. This is the inflection point.
Correct Answer: B — 2
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Q. For the function f(x) = 3x^3 - 12x^2 + 9x, the number of local maxima and minima is:
Solution
Finding f'(x) = 9x^2 - 24x + 9 and solving gives two critical points. The second derivative test confirms one maximum and one minimum.
Correct Answer: C — 2
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Q. For the function f(x) = e^x - x^2, the point of inflection occurs at:
-
A.
x = 0
-
B.
x = 1
-
C.
x = 2
-
D.
x = -1
Solution
To find the point of inflection, we compute f''(x) = e^x - 2. Setting f''(x) = 0 gives e^x = 2, leading to x = ln(2). The closest integer is x = 1.
Correct Answer: B — x = 1
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Q. For the function f(x) = ln(x), find the point where it is not differentiable.
-
A.
x = 0
-
B.
x = 1
-
C.
x = -1
-
D.
x = 2
Solution
f(x) = ln(x) is not defined for x ≤ 0, hence not differentiable at x = 0.
Correct Answer: A — x = 0
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Q. For the function f(x) = sin(x) + cos(x), find the x-coordinate of the maximum point in the interval [0, 2π].
-
A.
π/4
-
B.
3π/4
-
C.
5π/4
-
D.
7π/4
Solution
f'(x) = cos(x) - sin(x). Setting f'(x) = 0 gives tan(x) = 1, so x = π/4 + nπ. In [0, 2π], the maximum occurs at x = 3π/4.
Correct Answer: B — 3π/4
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Q. For the function f(x) = x^2 + 2x + 1, what is f'(x)?
-
A.
2x + 1
-
B.
2x + 2
-
C.
2x
-
D.
x + 1
Solution
f'(x) = 2x + 2.
Correct Answer: B — 2x + 2
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Q. For the function f(x) = x^2 + 2x + 3, find the point where it is not differentiable.
-
A.
x = -1
-
B.
x = 0
-
C.
x = 1
-
D.
It is differentiable everywhere
Solution
The function is a polynomial and is differentiable everywhere.
Correct Answer: D — It is differentiable everywhere
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Q. For the function f(x) = x^2 + kx + 1 to be differentiable at x = -1, what must k be?
Solution
Setting the derivative f'(-1) = 0 gives k = 1 for differentiability.
Correct Answer: C — 1
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Q. For the function f(x) = x^2 - 2x + 1, find the slope of the tangent line at x = 1.
Solution
f'(x) = 2x - 2. Thus, f'(1) = 2(1) - 2 = 0.
Correct Answer: A — 0
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Q. For the function f(x) = x^2 - 4x + 4, find the point where it is not differentiable.
-
A.
x = 0
-
B.
x = 2
-
C.
x = 4
-
D.
It is differentiable everywhere
Solution
As a polynomial, f(x) is differentiable everywhere, including at x = 2.
Correct Answer: D — It is differentiable everywhere
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Q. For the function f(x) = x^2 - 4x + 5, find the minimum value.
Solution
The vertex occurs at x = 2. f(2) = 2^2 - 4*2 + 5 = 1, which is the minimum value.
Correct Answer: B — 2
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