Q. What is the value of the limit: lim (x -> ∞) (1/x)?
-
A.
0
-
B.
1
-
C.
Infinity
-
D.
Undefined
Solution
As x approaches infinity, 1/x approaches 0.
Correct Answer: A — 0
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Q. What is the value of the series 1 + 1/2 + 1/3 + ... + 1/n as n approaches infinity?
-
A.
0
-
B.
1
-
C.
∞
-
D.
undefined
Solution
The series diverges to infinity as n approaches infinity.
Correct Answer: C — ∞
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Q. What is the value of x if 2x + 3 = 11?
Solution
Subtract 3 from both sides: 2x = 8. Then divide by 2: x = 4.
Correct Answer: C — 4
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Q. What is the value of x if 3x - 5 = 16?
Solution
Solving for x: 3x = 21 => x = 7.
Correct Answer: A — 7
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Q. What is the value of x in the equation 2x^2 - 8x + 6 = 0?
Solution
Using the quadratic formula, x = [8 ± sqrt(64 - 48)] / 4 = [8 ± 4] / 4, giving x = 3 or x = 1.
Correct Answer: C — 3
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Q. What is the value of x in the equation 3x - 5 = 7?
Solution
Solving for x: 3x = 12, thus x = 4.
Correct Answer: A — 4
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Q. What is the value of x in the equation 5(x - 2) = 3x + 4?
Solution
Expanding gives 5x - 10 = 3x + 4. Rearranging gives 2x = 14, thus x = 7.
Correct Answer: A — -1
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Q. What is the value of x in the equation 5x - 3 = 2x + 12?
Solution
Rearranging gives 5x - 2x = 12 + 3 => 3x = 15 => x = 5.
Correct Answer: B — 4
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Q. What is the value of x in the equation 5x - 3 = 2x + 6?
Solution
Rearranging gives 5x - 2x = 6 + 3 => 3x = 9 => x = 3.
Correct Answer: B — 2
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Q. What is the value of z if z^2 = -1?
Solution
The solutions to z^2 = -1 are z = i and z = -i.
Correct Answer: A — i
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Q. What is the value of \( \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{vmatrix} \)?
Solution
The determinant is 0 because the first column is repeated.
Correct Answer: A — 0
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Q. What is the value of \( \tan(\tan^{-1}(3)) \)?
-
A.
0
-
B.
1
-
C.
3
-
D.
undefined
Solution
By definition, \( \tan(\tan^{-1}(x)) = x \). Therefore, \( \tan(\tan^{-1}(3)) = 3 \).
Correct Answer: C — 3
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Q. What is the value of |z| if z = -3 + 4i?
Solution
|z| = √((-3)^2 + (4)^2) = √(9 + 16) = √25 = 5.
Correct Answer: A — 5
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Q. What is the value of |z| if z = 4 - 3i?
Solution
|z| = √(4^2 + (-3)^2) = √(16 + 9) = √25 = 5.
Correct Answer: A — 5
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Q. What is the value of |z| if z = 4e^(iπ/3)?
Solution
The modulus |z| = 4, as it is the coefficient in the polar form.
Correct Answer: A — 4
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Q. What is the value of |z|^2 if z = 4 - 3i?
Solution
|z|^2 = 4^2 + (-3)^2 = 16 + 9 = 25.
Correct Answer: A — 25
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Q. What is the variance of the data set {4, 8, 6, 5, 3}?
Solution
Mean = 5.2; Variance = [(4-5.2)² + (8-5.2)² + (6-5.2)² + (5-5.2)² + (3-5.2)²] / 5 = 2.56.
Correct Answer: A — 2.5
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Q. What is the variance of the data set: 1, 2, 3, 4, 5?
Solution
Mean = 3. Variance = [(1-3)² + (2-3)² + (3-3)² + (4-3)² + (5-3)²] / 5 = 2.
Correct Answer: B — 1.5
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Q. What is the variance of the data set: 2, 4, 4, 4, 5, 5, 7, 9?
-
A.
2.5
-
B.
3.5
-
C.
4.5
-
D.
5.5
Solution
Mean = 5.0; Variance = [(2-5)^2 + (4-5)^2 + (4-5)^2 + (4-5)^2 + (5-5)^2 + (5-5)^2 + (7-5)^2 + (9-5)^2] / 8 = 2.5.
Correct Answer: A — 2.5
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Q. What is the variance of the following data set: {5, 7, 8, 9, 10}?
Solution
Mean = 7.8, Variance = [(5-7.8)² + (7-7.8)² + (8-7.8)² + (9-7.8)² + (10-7.8)²] / 5 = 3.
Correct Answer: B — 3
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Q. What is the vertex of the parabola defined by the equation y = -2(x - 1)^2 + 4?
-
A.
(1, 4)
-
B.
(1, -4)
-
C.
(4, 1)
-
D.
(-1, 4)
Solution
The vertex form of a parabola is y = a(x - h)^2 + k. Here, h = 1 and k = 4, so the vertex is (1, 4).
Correct Answer: A — (1, 4)
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Q. What is the vertex of the parabola given by the equation y = -2(x - 1)^2 + 4?
-
A.
(1, 4)
-
B.
(1, -4)
-
C.
(4, 1)
-
D.
(-1, 4)
Solution
The vertex form of a parabola is y = a(x - h)^2 + k. Here, h = 1 and k = 4, so the vertex is (1, 4).
Correct Answer: A — (1, 4)
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Q. What is the vertex of the parabola represented by the equation y = -2(x - 1)^2 + 4?
-
A.
(1, 4)
-
B.
(1, -4)
-
C.
(-1, 4)
-
D.
(-1, -4)
Solution
The vertex form of a parabola is y = a(x - h)^2 + k. Here, h = 1 and k = 4, so the vertex is (1, 4).
Correct Answer: A — (1, 4)
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Q. What is the vertex of the parabola represented by the equation y = 2x^2 - 8x + 5?
-
A.
(2, -3)
-
B.
(2, -7)
-
C.
(4, -3)
-
D.
(4, -7)
Solution
The vertex can be found using x = -b/(2a) = 4. Substituting x = 4 into the equation gives y = -3.
Correct Answer: A — (2, -3)
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Q. What is the weighted mean of the scores 70, 80, 90 with weights 1, 2, 3 respectively?
Solution
Weighted mean = (70*1 + 80*2 + 90*3) / (1 + 2 + 3) = 85.
Correct Answer: B — 85
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Q. What is the weighted mean of the scores 70, 80, and 90 with weights 1, 2, and 3 respectively?
Solution
Weighted mean = (70*1 + 80*2 + 90*3) / (1 + 2 + 3) = 85.
Correct Answer: B — 85
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Q. What is the weighted mean of the scores 80, 90, and 70 with weights 1, 2, and 1 respectively?
Solution
Weighted mean = (80*1 + 90*2 + 70*1) / (1 + 2 + 1) = 85.
Correct Answer: B — 85
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Q. What is the x-intercept of the line 3x + 4y - 12 = 0?
Solution
To find the x-intercept, set y = 0. Thus, 3x - 12 = 0 gives x = 4.
Correct Answer: B — 3
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Q. What is the y-intercept of the line 5x + 2y - 10 = 0?
Solution
Setting x = 0 in the equation gives 2y - 10 = 0, thus y = 5.
Correct Answer: C — 2
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Q. What is 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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