Q. What is the equation of the line passing through (2, 3) with a slope of 2? (2021)
A.
y = 2x - 1
B.
y = 2x + 1
C.
y = 2x + 3
D.
y = 2x - 3
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Solution
Using point-slope form: y - 3 = 2(x - 2) => y = 2x - 4 + 3 => y = 2x - 1.
Correct Answer: B — y = 2x + 1
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Q. What is the equation of the line passing through the points (1, 2) and (3, 4)?
A.
y = x + 1
B.
y = 2x
C.
y = x + 2
D.
y = 2x - 2
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Solution
The slope m = (4 - 2) / (3 - 1) = 1. Using point-slope form, y - 2 = 1(x - 1) gives y = x + 1.
Correct Answer: A — y = x + 1
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Q. What is the equation of the line passing through the points (1, 2) and (3, 6)?
A.
y = 2x
B.
y = 3x - 1
C.
y = x + 1
D.
y = 4x - 2
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Solution
The slope m = (6 - 2) / (3 - 1) = 2. Using point-slope form, y - 2 = 2(x - 1) gives y = 2x.
Correct Answer: A — y = 2x
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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
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
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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 perpendicular to y = 3x + 1 that passes through (2, 3)?
A.
y = -1/3x + 4
B.
y = 3x - 3
C.
y = -3x + 9
D.
y = 1/3x + 2
Show solution
Solution
The slope of the perpendicular line is -1/3. Using point-slope form, we find y - 3 = -1/3(x - 2) which simplifies to 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 + 1 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
Show solution
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
Show solution
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
Show solution
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 origin and has a slope of -1?
A.
y = -x
B.
y = x
C.
y = -2x
D.
y = 2x
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Solution
The equation of a line through the origin with slope m is y = mx. Thus, y = -1x or y = -x.
Correct Answer: A — y = -x
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Q. What is the equation of the line that passes through the origin and has a slope of -4? (2023)
A.
y = -4x
B.
y = 4x
C.
y = -x/4
D.
y = 1/4x
Show solution
Solution
Using the slope-intercept form y = mx + b, with m = -4 and b = 0, the equation is y = -4x.
Correct Answer: A — y = -4x
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Q. What is the equation of the line that passes through the origin and has a slope of -5?
A.
y = -5x
B.
y = 5x
C.
y = -x/5
D.
y = 5/x
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Solution
The equation of a line through the origin with slope m is y = mx. Thus, y = -5x.
Correct Answer: A — y = -5x
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Q. What is the equation of the line that passes through the origin and has a slope of -3? (2022)
A.
y = -3x
B.
y = 3x
C.
y = -x/3
D.
y = 1/3x
Show solution
Solution
The equation of a line through the origin with slope m is y = mx. Thus, y = -3x.
Correct Answer: A — y = -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
Show solution
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
Show solution
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
Show solution
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)?
A.
y = 3x + 2
B.
y = 3x - 1
C.
y - 2 = 3(x - 1)
D.
y = 2x + 1
Show solution
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)?
A.
y = 5x - 3
B.
y = 5x + 2
C.
y = 5x + 1
D.
y = 5x - 2
Show solution
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)?
A.
y = 2x^2
B.
y = x^2
C.
y = 4x^2
D.
y = 8x^2
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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?
A.
x^2 = 8y
B.
x^2 = -8y
C.
y^2 = 8x
D.
y^2 = -8x
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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?
A.
x^2 = 12y
B.
y^2 = 12x
C.
y = 3x^2
D.
x = 3y^2
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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 plane passing through the point (1, 2, 3) with normal vector (1, -1, 1)? (2023)
A.
x - y + z = 0
B.
x + y + z = 6
C.
x - y + z = 1
D.
x + y - z = 0
Show solution
Solution
Equation of the plane: 1(x-1) - 1(y-2) + 1(z-3) = 0 => x - y + z = 1.
Correct Answer: C — x - y + z = 1
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Q. What is the equation of the plane passing through the points (1, 2, 3), (4, 5, 6), and (7, 8, 9)? (2021)
A.
0 = 0
B.
x + y + z = 12
C.
x + y + z = 10
D.
x + y + z = 9
Show solution
Solution
The points are collinear, hence the equation of the plane is 0 = 0.
Correct Answer: A — 0 = 0
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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)?
A.
y = 2x + 1
B.
y = 2x + 2
C.
y = 3x
D.
y = x + 2
Show solution
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 equation of the tangent line to the curve y = x^2 + 2x at the point where x = 1?
A.
y = 3x - 2
B.
y = 2x + 1
C.
y = 2x + 2
D.
y = x + 3
Show solution
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 equilibrium constant (K) for the dissociation of acetic acid (CH3COOH) in water? (2023)
A.
1.8 x 10^-5
B.
1.0
C.
10
D.
100
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Solution
The dissociation constant (K) for acetic acid is approximately 1.8 x 10^-5, indicating it is a weak acid.
Correct Answer: A — 1.8 x 10^-5
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Q. What is the equilibrium constant (Kc) expression for the reaction: 2A + B ⇌ C?
A.
Kc = [C] / ([A]^2[B])
B.
Kc = [A]^2[B] / [C]
C.
Kc = [C] / [A]^2
D.
Kc = [A]^2 / ([B][C])
Show solution
Solution
The equilibrium constant Kc is defined as the ratio of the concentration of products to the concentration of reactants, each raised to the power of their coefficients in the balanced equation. For the reaction 2A + B ⇌ C, Kc = [C] / ([A]^2[B]).
Correct Answer: A — Kc = [C] / ([A]^2[B])
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Q. What is the equilibrium constant (Kc) for the reaction 2A + B ⇌ C? (2022)
A.
Kc = [C] / ([A]^2[B])
B.
Kc = [A]^2[B] / [C]
C.
Kc = [B] / ([A]^2[C])
D.
Kc = [C] / ([B][A])
Show solution
Solution
The equilibrium constant Kc is defined as the concentration of the products raised to the power of their coefficients divided by the concentration of the reactants raised to the power of their coefficients. For the reaction 2A + B ⇌ C, Kc = [C] / ([A]^2[B]).
Correct Answer: A — Kc = [C] / ([A]^2[B])
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Q. What is the equilibrium constant (Kc) for the reaction: A + B ⇌ C + D? (2022)
A.
Kc = [C][D]/[A][B]
B.
Kc = [A][B]/[C][D]
C.
Kc = [C][D][A][B]
D.
Kc = [A][B][C][D]
Show solution
Solution
The equilibrium constant Kc is defined as the ratio of the concentrations of the products to the reactants, each raised to the power of their coefficients in the balanced equation.
Correct Answer: A — Kc = [C][D]/[A][B]
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Q. What is the equilibrium constant (Kc) for the reaction: N2(g) + 3H2(g) ⇌ 2NH3(g)? (2022) 2022
A.
1.0 × 10^-5
B.
1.0 × 10^-2
C.
1.0 × 10^2
D.
1.0 × 10^5
Show solution
Solution
The equilibrium constant Kc for the formation of ammonia from nitrogen and hydrogen is typically around 1.0 × 10^5 at standard conditions.
Correct Answer: D — 1.0 × 10^5
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Q. What is the equilibrium constant expression for the reaction N2(g) + 3H2(g) ⇌ 2NH3(g)?
A.
Kc = [NH3]^2 / ([N2][H2]^3)
B.
Kc = [N2][H2]^3 / [NH3]^2
C.
Kc = [NH3]^2 / [N2][H2]
D.
Kc = [N2][H2] / [NH3]^2
Show solution
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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