Q. What is the equation of the circle with center (2, -3) and radius 4?
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A.
(x-2)² + (y+3)² = 16
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B.
(x+2)² + (y-3)² = 16
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C.
(x-2)² + (y-3)² = 16
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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?
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A.
(x-2)² + (y+3)² = 25
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B.
(x+2)² + (y-3)² = 25
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C.
(x-2)² + (y-3)² = 25
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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?
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A.
(x-3)² + (y+2)² = 25
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B.
(x+3)² + (y-2)² = 25
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C.
(x-3)² + (y-2)² = 25
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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?
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A.
y = -2
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B.
y = 2
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C.
x = -4
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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
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C.
x^2/15 + y^2/5 = 1
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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)?
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A.
y = 2x + 2
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B.
y = 2x + 1
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C.
y = 2x + 3
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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)?
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A.
y = 2x - 1
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B.
y = 2x + 1
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C.
y = 2x + 3
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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)?
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A.
y = 3x - 2
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B.
y = 3x + 1
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C.
y = 3x + 2
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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)?
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A.
y = 3x - 2
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B.
y = -3x - 2
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C.
y = 3x + 2
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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
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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)?
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A.
y = 3x + 1
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B.
y = 3x - 1
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C.
y = 3x + 2
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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)?
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A.
y = 4x - 5
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B.
y = 4x - 1
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C.
y = 4x + 5
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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)?
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A.
y = 4x - 5
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B.
y = 4x - 1
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C.
y = 4x + 5
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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)?
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A.
y = 5x - 7
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B.
y = 5x + 7
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C.
y = 5x - 2
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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)?
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A.
x = 1 + 3t, y = 2 + 3t, z = 3 + 3t
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B.
x = 1 + t, y = 2 + t, z = 3 + t
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C.
x = 1 + t, y = 2 + 2t, z = 3 + 3t
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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)?
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A.
y = -1/3x + 4
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B.
y = 3x - 3
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C.
y = -3x + 9
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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)?
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A.
y = -1/3x + 4
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B.
y = 3x - 3
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C.
y = -3x + 9
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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?
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A.
y = -1/3x
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B.
y = 3x
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C.
y = -3x
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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?
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A.
y = -x + 5
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B.
y = -x + 3
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C.
y = x + 1
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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)?
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A.
y = 2x + 1
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B.
y = 2x - 2
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C.
y = 2x + 2
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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)?
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A.
y = 3x + 2
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B.
y = 3x - 1
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C.
y = 3x + 1
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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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Q. What is the equation of the line with slope 3 that passes through the point (1, 2)?
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A.
y = 3x + 2
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B.
y = 3x - 1
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C.
y - 2 = 3(x - 1)
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D.
y = 2x + 1
Solution
Using point-slope form: y - y1 = m(x - x1) => y - 2 = 3(x - 1).
Correct Answer: C — y - 2 = 3(x - 1)
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Q. What is the equation of the line with slope 5 that passes through the point (1, 2)?
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A.
y = 5x - 3
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B.
y = 5x + 2
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C.
y = 5x + 1
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D.
y = 5x - 2
Solution
Using point-slope form: y - 2 = 5(x - 1) gives y = 5x - 3.
Correct Answer: C — y = 5x + 1
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Q. What is the equation of the parabola that opens upwards with vertex at the origin and passes through the point (2, 8)?
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A.
y = 2x^2
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B.
y = x^2
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C.
y = 4x^2
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D.
y = 8x^2
Solution
The vertex form of a parabola is y = ax^2. Since it passes through (2, 8), we have 8 = a(2^2) => 8 = 4a => a = 2. Thus, the equation is y = 4x^2.
Correct Answer: C — y = 4x^2
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Q. What is the equation of the parabola with focus at (0, 2) and directrix y = -2?
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A.
x^2 = 8y
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B.
x^2 = -8y
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C.
y^2 = 8x
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D.
y^2 = -8x
Solution
The distance from the focus to the directrix is 4, so the equation is y = (1/4)(x - 0)^2 + 0, which simplifies to x^2 = 8y.
Correct Answer: A — x^2 = 8y
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Q. What is the equation of the parabola with focus at (0, 3) and directrix y = -3?
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A.
x^2 = 12y
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B.
y^2 = 12x
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C.
y = 3x^2
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D.
x = 3y^2
Solution
The distance from the focus to the directrix is 6, so p = 3. The equation is y^2 = 4px = 12y.
Correct Answer: A — x^2 = 12y
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Q. What is the equation of the tangent line to the curve y = x^2 + 2x at the point where x = 1?
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A.
y = 3x - 2
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B.
y = 2x + 1
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C.
y = 2x + 2
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D.
y = x + 3
Solution
f'(x) = 2x + 2. At x = 1, f'(1) = 4. The point is (1, 3). The tangent line is y - 3 = 4(x - 1) => y = 4x - 1.
Correct Answer: A — y = 3x - 2
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Q. What is the equation of the tangent line to the curve y = x^2 + 2x at the point (1, 3)?
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A.
y = 2x + 1
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B.
y = 2x + 2
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C.
y = 3x
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D.
y = x + 2
Solution
f'(x) = 2x + 2. At x = 1, f'(1) = 4. The tangent line is y - 3 = 4(x - 1) => y = 4x - 1.
Correct Answer: A — y = 2x + 1
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Q. What is the equilibrium constant expression for the reaction N2(g) + 3H2(g) ⇌ 2NH3(g)?
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A.
Kc = [NH3]^2 / ([N2][H2]^3)
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B.
Kc = [N2][H2]^3 / [NH3]^2
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C.
Kc = [NH3]^2 / [N2][H2]
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D.
Kc = [N2][H2] / [NH3]^2
Solution
The equilibrium constant Kc is given by the ratio of the concentration of products to reactants, raised to the power of their coefficients.
Correct Answer: A — Kc = [NH3]^2 / ([N2][H2]^3)
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Q. What is the equilibrium constant expression for the reaction: 2A + B ⇌ C?
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A.
[C]/([A]^2[B])
-
B.
[A]^2[B]/[C]
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C.
[C]/[A][B]
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D.
[A][B]/[C]
Solution
The equilibrium constant K is given by the expression K = [C]/([A]^2[B]) for the reaction 2A + B ⇌ C.
Correct Answer: A — [C]/([A]^2[B])
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