Q. What is the domain of the function f(x) = 1/(x-3)?
-
A.
x ≠ 3
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B.
x > 3
-
C.
x < 3
-
D.
All real numbers
Solution
The function is undefined at x = 3, so the domain is x ≠ 3.
Correct Answer: A — x ≠ 3
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Q. What is the domain of the function f(x) = sqrt(x - 1)?
-
A.
x >= 1
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B.
x > 1
-
C.
x <= 1
-
D.
x < 1
Solution
The expression under the square root must be non-negative, so x - 1 >= 0, hence x >= 1.
Correct Answer: A — x >= 1
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Q. What is the dot product of vectors (1, 2) and (3, 4)?
Solution
Dot product = 1*3 + 2*4 = 3 + 8 = 11
Correct Answer: A — 11
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Q. What is the dot product of vectors A = (2, 3, 4) and B = (1, 0, -1)?
Solution
A . B = 2*1 + 3*0 + 4*(-1) = 2 + 0 - 4 = -2.
Correct Answer: B — 5
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Q. What is the eccentricity of a hyperbola defined by the equation x^2/a^2 - y^2/b^2 = 1?
-
A.
1
-
B.
√2
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C.
√(1 + b^2/a^2)
-
D.
√(1 - b^2/a^2)
Solution
The eccentricity e of a hyperbola is given by e = √(1 + b^2/a^2).
Correct Answer: C — √(1 + b^2/a^2)
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Q. What is the equation of a circle with center at (-1, 2) and radius 4?
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A.
(x + 1)² + (y - 2)² = 16
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B.
(x - 1)² + (y + 2)² = 16
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C.
(x + 1)² + (y + 2)² = 16
-
D.
(x - 1)² + (y - 2)² = 16
Solution
Using the standard form, the equation is (x + 1)² + (y - 2)² = 4² = 16.
Correct Answer: A — (x + 1)² + (y - 2)² = 16
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Q. What is the equation of a circle with center at (2, -3) and radius 5?
-
A.
(x - 2)² + (y + 3)² = 25
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B.
(x + 2)² + (y - 3)² = 25
-
C.
(x - 2)² + (y - 3)² = 25
-
D.
(x + 2)² + (y + 3)² = 25
Solution
The standard form of a circle's equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
Correct Answer: A — (x - 2)² + (y + 3)² = 25
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Q. What is the equation of a circle with center at (h, k) and radius r?
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A.
(x - h)^2 + (y - k)^2 = r^2
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B.
(x + h)^2 + (y + k)^2 = r^2
-
C.
(x - h)^2 - (y - k)^2 = r^2
-
D.
(x + h)^2 - (y + k)^2 = r^2
Solution
The equation of a circle with center at (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2.
Correct Answer: A — (x - h)^2 + (y - k)^2 = r^2
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Q. What is the equation of an ellipse with foci at (±c, 0) and vertices at (±a, 0)?
-
A.
x^2/a^2 + y^2/b^2 = 1
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B.
y^2/a^2 + x^2/b^2 = 1
-
C.
x^2/b^2 + y^2/a^2 = 1
-
D.
y^2/b^2 + x^2/a^2 = 1
Solution
The standard form of the equation of an ellipse with horizontal major axis is x^2/a^2 + y^2/b^2 = 1.
Correct Answer: A — x^2/a^2 + y^2/b^2 = 1
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Q. What is the equation of the circle with center (2, -3) and radius 4?
-
A.
(x-2)² + (y+3)² = 16
-
B.
(x+2)² + (y-3)² = 16
-
C.
(x-2)² + (y-3)² = 16
-
D.
(x+2)² + (y+3)² = 16
Solution
Equation of circle: (x-h)² + (y-k)² = r² => (x-2)² + (y+3)² = 4² = 16.
Correct Answer: A — (x-2)² + (y+3)² = 16
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Q. What is the equation of the circle with center (2, -3) and radius 5?
-
A.
(x-2)² + (y+3)² = 25
-
B.
(x+2)² + (y-3)² = 25
-
C.
(x-2)² + (y-3)² = 25
-
D.
(x+2)² + (y+3)² = 25
Solution
Equation of circle: (x-h)² + (y-k)² = r² => (x-2)² + (y+3)² = 25.
Correct Answer: A — (x-2)² + (y+3)² = 25
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Q. What is the equation of the circle with center (3, -2) and radius 5?
-
A.
(x-3)² + (y+2)² = 25
-
B.
(x+3)² + (y-2)² = 25
-
C.
(x-3)² + (y-2)² = 25
-
D.
(x+3)² + (y+2)² = 25
Solution
Equation of circle: (x-h)² + (y-k)² = r² => (x-3)² + (y+2)² = 5² = 25.
Correct Answer: A — (x-3)² + (y+2)² = 25
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Q. What is the equation of the directrix of the parabola x^2 = 8y?
-
A.
y = -2
-
B.
y = 2
-
C.
x = -4
-
D.
x = 4
Solution
The directrix of the parabola x^2 = 8y is y = -2.
Correct Answer: A — y = -2
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Q. What is the equation of the ellipse with center at the origin, semi-major axis 5, and semi-minor axis 3?
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A.
x^2/25 + y^2/9 = 1
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B.
x^2/9 + y^2/25 = 1
-
C.
x^2/15 + y^2/5 = 1
-
D.
x^2/5 + y^2/15 = 1
Solution
The equation of the ellipse is x^2/25 + y^2/9 = 1.
Correct Answer: A — x^2/25 + y^2/9 = 1
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Q. What is the equation of the line parallel to y = 2x + 1 that passes through the point (3, 4)?
-
A.
y = 2x + 2
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B.
y = 2x + 1
-
C.
y = 2x + 3
-
D.
y = 2x - 2
Solution
Parallel lines have the same slope, so y - 4 = 2(x - 3) => y = 2x - 2.
Correct Answer: A — y = 2x + 2
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Q. What is the equation of the line parallel to y = 2x + 3 that passes through the point (1, 1)?
-
A.
y = 2x - 1
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B.
y = 2x + 1
-
C.
y = 2x + 3
-
D.
y = 2x - 3
Solution
Parallel lines have the same slope: y - 1 = 2(x - 1) => y = 2x - 1.
Correct Answer: A — y = 2x - 1
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Q. What is the equation of the line parallel to y = 3x + 2 that passes through the point (1, 1)?
-
A.
y = 3x - 2
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B.
y = 3x + 1
-
C.
y = 3x + 2
-
D.
y = 3x - 1
Solution
Parallel lines have the same slope, so y - 1 = 3(x - 1) => y = 3x - 1.
Correct Answer: D — y = 3x - 1
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Q. What is the equation of the line parallel to y = 3x + 4 that passes through the point (0, -2)?
-
A.
y = 3x - 2
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B.
y = -3x - 2
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C.
y = 3x + 2
-
D.
y = -3x + 4
Solution
Parallel lines have the same slope. The slope is 3, so using point-slope form: y + 2 = 3(x - 0) => y = 3x - 2.
Correct Answer: A — y = 3x - 2
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Q. What is the equation of the line parallel to y = 3x - 2 and passing through the point (2, 5)?
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A.
y = 3x + 1
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B.
y = 3x - 1
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C.
y = 3x + 2
-
D.
y = 3x - 2
Solution
The slope of the given line is 3. Using point-slope form: y - 5 = 3(x - 2) gives y = 3x + 1.
Correct Answer: A — y = 3x + 1
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Q. What is the equation of the line parallel to y = 3x - 2 that passes through the point (2, 5)?
-
A.
y = 3x + 1
-
B.
y = 3x - 1
-
C.
y = 3x + 2
-
D.
y = 3x - 2
Solution
Since parallel lines have the same slope, the equation is y - 5 = 3(x - 2) which simplifies to y = 3x + 1.
Correct Answer: A — y = 3x + 1
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Q. What is the equation of the line parallel to y = 4x - 5 and passing through (2, 3)?
-
A.
y = 4x - 5
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B.
y = 4x - 1
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C.
y = 4x + 5
-
D.
y = 4x + 3
Solution
Parallel lines have the same slope. Using point-slope form: y - 3 = 4(x - 2) => y = 4x - 5.
Correct Answer: B — y = 4x - 1
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Q. What is the equation of the line parallel to y = 4x - 5 that passes through the point (2, 3)?
-
A.
y = 4x - 5
-
B.
y = 4x - 1
-
C.
y = 4x + 5
-
D.
y = 4x + 3
Solution
Parallel lines have the same slope. Using point-slope form: y - 3 = 4(x - 2) => y = 4x - 8 + 3 => y = 4x - 5.
Correct Answer: B — y = 4x - 1
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Q. What is the equation of the line parallel to y = 5x - 2 and passing through the point (2, 3)?
-
A.
y = 5x - 7
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B.
y = 5x + 7
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C.
y = 5x - 2
-
D.
y = 5x + 2
Solution
Parallel lines have the same slope. Using point-slope form: y - 3 = 5(x - 2) gives y = 5x - 7.
Correct Answer: A — y = 5x - 7
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Q. What is the equation of the line passing through the points (1, 2, 3) and (4, 5, 6)?
-
A.
x = 1 + 3t, y = 2 + 3t, z = 3 + 3t
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B.
x = 1 + t, y = 2 + t, z = 3 + t
-
C.
x = 1 + t, y = 2 + 2t, z = 3 + 3t
-
D.
x = 1 + 3t, y = 2 + 2t, z = 3 + t
Solution
Direction ratios = (3, 3, 3), hence the line equation is x = 1 + 3t, y = 2 + 3t, z = 3 + 3t.
Correct Answer: A — x = 1 + 3t, y = 2 + 3t, z = 3 + 3t
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Q. What is the equation of the line that is perpendicular to y = 3x + 1 and passes through the point (2, 3)?
-
A.
y = -1/3x + 4
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B.
y = 3x - 3
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C.
y = -3x + 9
-
D.
y = 1/3x + 2
Solution
The slope of the given line is 3, so the perpendicular slope is -1/3. Using point-slope form: y - 3 = -1/3(x - 2) gives y = -1/3x + 4.
Correct Answer: A — y = -1/3x + 4
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Q. What is the equation of the line that is perpendicular to y = 3x + 2 and passes through the point (2, 3)?
-
A.
y = -1/3x + 4
-
B.
y = 3x - 3
-
C.
y = -3x + 9
-
D.
y = 1/3x + 2
Solution
The slope of the perpendicular line is -1/3. Using point-slope form: y - 3 = -1/3(x - 2) gives y = -1/3x + 4.
Correct Answer: A — y = -1/3x + 4
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Q. What is the equation of the line that is perpendicular to y = 3x + 4 and passes through the origin?
-
A.
y = -1/3x
-
B.
y = 3x
-
C.
y = -3x
-
D.
y = 1/3x
Solution
The slope of the given line is 3. The slope of the perpendicular line is -1/3. Thus, the equation is y = -1/3x.
Correct Answer: A — y = -1/3x
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Q. What is the equation of the line that passes through the point (2, 3) and has a slope of -1?
-
A.
y = -x + 5
-
B.
y = -x + 3
-
C.
y = x + 1
-
D.
y = -x + 2
Solution
Using point-slope form: y - 3 = -1(x - 2) => y = -x + 5.
Correct Answer: A — y = -x + 5
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Q. What is the equation of the line with slope 2 passing through the point (1, 2)?
-
A.
y = 2x + 1
-
B.
y = 2x - 2
-
C.
y = 2x + 2
-
D.
y = 2x - 1
Solution
Using point-slope form: y - 2 = 2(x - 1) => y = 2x - 2 + 2 => y = 2x - 1.
Correct Answer: D — y = 2x - 1
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Q. What is the equation of the line with slope 3 passing through the point (1, 2)?
-
A.
y = 3x + 2
-
B.
y = 3x - 1
-
C.
y = 3x + 1
-
D.
y = 3x - 2
Solution
Using point-slope form: y - 2 = 3(x - 1) => y = 3x - 1.
Correct Answer: C — y = 3x + 1
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