Q. What is the equation of a circle with center at (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
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 (3, -2) and radius 5? (2022)
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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
The standard form of a circle's equation is (x-h)² + (y-k)² = r². Here, h=3, k=-2, r=5, so (x-3)² + (y+2)² = 25.
Correct Answer: A — (x-3)² + (y+2)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
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C.
(x - h)^2 - (y - k)^2 = r^2
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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 a plane passing through the point (1, 2, 3) with normal vector (1, 1, 1)? (2022)
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A.
x + y + z = 6
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B.
x + y + z = 3
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C.
x + y + z = 1
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D.
x + y + z = 0
Solution
Equation of the plane: 1(x-1) + 1(y-2) + 1(z-3) = 0 => x + y + z = 6.
Correct Answer: A — x + y + z = 6
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Q. What is the equation of a plane passing through the point (1, 2, 3) with normal vector (2, -1, 3)? (2021)
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A.
2x - y + 3z = 12
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B.
2x + y - 3z = 0
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C.
2x - y + 3z = 0
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D.
2x + y + 3z = 12
Solution
Equation of the plane: 2(x-1) - 1(y-2) + 3(z-3) = 0 simplifies to 2x - y + 3z = 12.
Correct Answer: C — 2x - y + 3z = 0
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Q. What is the equation of an ellipse with foci at (±c, 0) and vertices at (±a, 0)?
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A.
x^2/a^2 + y^2/b^2 = 1
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B.
y^2/a^2 + x^2/b^2 = 1
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C.
x^2/b^2 + y^2/a^2 = 1
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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 motion for a damped harmonic oscillator?
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A.
m d²x/dt² + b dx/dt + kx = 0
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B.
m d²x/dt² + kx = 0
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C.
m d²x/dt² + b dx/dt = 0
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D.
m d²x/dt² + b dx/dt + kx = F(t)
Solution
The equation of motion for a damped harmonic oscillator is m d²x/dt² + b dx/dt + kx = 0.
Correct Answer: A — m d²x/dt² + b dx/dt + kx = 0
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Q. What is the equation of motion for a simple harmonic oscillator with amplitude A and angular frequency ω?
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A.
x(t) = A cos(ωt)
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B.
x(t) = A sin(ωt)
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C.
x(t) = A e^(ωt)
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D.
x(t) = A ωt
Solution
The equation of motion for SHM is x(t) = A cos(ωt) or x(t) = A sin(ωt).
Correct Answer: A — x(t) = A cos(ωt)
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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?
-
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 circle with center at (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
Standard form of a circle: (x-h)² + (y-k)² = r². Here, h=2, k=-3, r=5.
Correct Answer: A — (x-2)² + (y+3)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 (2, 5)? (2022)
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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 + 3
Solution
Since parallel lines have the same slope, the equation is y - 5 = 3(x - 2) which simplifies to y = 3x + 1.
Correct Answer: B — 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 + 4 that passes through the point (1, 2)? (2020)
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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
Parallel lines have the same slope. Using point-slope form: y - 2 = 3(x - 1) 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 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 = 3x - 4 that passes through the point (2, 1)? (2020)
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A.
y = 3x - 5
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B.
y = 3x + 1
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C.
y = 3x - 1
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D.
y = 3x + 4
Solution
Since parallel lines have the same slope, the equation is y - 1 = 3(x - 2) which simplifies to y = 3x - 5.
Correct Answer: C — y = 3x - 1
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Q. What is the equation of the line parallel to y = 3x - 5 and passing through (2, 1)?
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A.
y = 3x - 8
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B.
y = 3x + 5
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C.
y = 3x - 1
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D.
y = 3x + 1
Solution
Parallel lines have the same slope. The slope is 3. Using point-slope form: y - 1 = 3(x - 2) gives y = 3x - 8.
Correct Answer: A — y = 3x - 8
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Q. What is the equation of the line parallel to y = 3x - 5 that passes through the point (2, 1)?
-
A.
y = 3x - 8
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B.
y = 3x + 5
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C.
y = 3x - 1
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D.
y = 3x + 1
Solution
Since the lines are parallel, they have the same slope. Using point-slope form: y - 1 = 3(x - 2) gives y = 3x - 8.
Correct Answer: A — y = 3x - 8
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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)?
-
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 - 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 (0, 0) and (2, 4)? (2019)
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A.
y = 2x
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B.
y = x
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C.
y = 4x
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D.
y = 3x
Solution
Slope = (4-0)/(2-0) = 2, so the equation is y = 2x.
Correct Answer: A — y = 2x
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Q. What is the equation of the line passing through (0, 0) with a slope of 3? (2021)
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A.
y = 3x
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B.
y = x/3
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C.
y = 3/x
-
D.
y = 1/3x
Solution
Equation of line: y = mx + c; here m = 3, c = 0 => y = 3x
Correct Answer: A — y = 3x
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