1. The slope of a line passing through points (x₁, y₁) and (x₂, y₂) is given by:
A. (y₂ - y₁) / (x₂ - x₁)
B. (x₂ - x₁) / (y₂ - y₁)
C. (y₂ + y₁) / (x₂ + x₁)
D. (y₁ - x₁) / (y₂ - x₂)
Show Answer
Answer: A. (y₂ - y₁) / (x₂ - x₁)
Explanation: Slope m = tan θ = (y₂ - y₁) / (x₂ - x₁) provided x₁ ≠ x₂.
2. What is the slope of a line parallel to the x-axis?
A. 0
B. 1
C. Undefined
D. -1
Show Answer
Answer: A. 0
Explanation: A line parallel to the x-axis makes an angle θ = 0° with the x-axis. Slope m = tan 0° = 0.
3. The slope of a line perpendicular to the x-axis (parallel to y-axis) is:
A. Undefined
B. 0
C. 1
D. -1
Show Answer
Answer: A. Undefined
Explanation: Angle with positive x-axis is 90°. tan 90° is undefined.
4. Two non-vertical lines with slopes m₁ and m₂ are perpendicular if and only if:
A. m₁ · m₂ = -1
B. m₁ = m₂
C. m₁ + m₂ = 0
D. m₁ · m₂ = 1
Show Answer
Answer: A. m₁ · m₂ = -1
Explanation: The condition for perpendicularity of two lines is m₁ · m₂ = -1.
5. Two lines with slopes m₁ and m₂ are parallel if and only if:
A. m₁ = m₂
B. m₁ · m₂ = -1
C. m₁ + m₂ = 0
D. m₁ · m₂ = 1
Show Answer
Answer: A. m₁ = m₂
Explanation: Parallel lines make equal inclination with the x-axis, so their slopes are equal.
6. The equation of a line in slope-intercept form is:
A. y = mx + c
B. x/a + y/b = 1
C. y - y₁ = m(x - x₁)
D. Ax + By + C = 0
Show Answer
Answer: A. y = mx + c
Explanation: Here 'm' represents the slope and 'c' represents the y-intercept of the line.
7. The equation of a line in intercept form with x-intercept 'a' and y-intercept 'b' is:
A. x/a + y/b = 1
B. ax + by = 1
C. x/b + y/a = 1
D. a/x + b/y = 1
Show Answer
Answer: A. x/a + y/b = 1
Explanation: Standard intercept form of a straight line equation is x/a + y/b = 1.
8. The equation of a line passing through (x₁, y₁) with slope m is:
A. y - y₁ = m(x - x₁)
B. x - x₁ = m(y - y₁)
C. y + y₁ = m(x + x₁)
D. y = m(x - x₁)
Show Answer
Answer: A. y - y₁ = m(x - x₁)
Explanation: Point-slope form of a straight line is y - y₁ = m(x - x₁).
9. The slope of the line 3x - 4y + 10 = 0 is:
A. 3/4
B. -3/4
C. 4/3
D. -4/3
Show Answer
Answer: A. 3/4
Explanation: Rewriting in y = mx + c form: 4y = 3x + 10 ⇒ y = (3/4)x + 5/2. Thus slope m = 3/4.
10. The distance of a point (x₁, y₁) from the line Ax + By + C = 0 is:
A. |Ax₁ + By₁ + C| / √(A² + B²)
B. (Ax₁ + By₁ + C) / (A + B)
C. |Ax₁ + By₁| / √(A² + B²)
D. |Ax₁ - By₁ + C| / (A² + B²)
Show Answer
Answer: A. |Ax₁ + By₁ + C| / √(A² + B²)
Explanation: Standard formula for perpendicular distance from point (x₁, y₁) to line Ax + By + C = 0.
11. Distance between two parallel lines Ax + By + C₁ = 0 and Ax + By + C₂ = 0 is:
A. |C₁ - C₂| / √(A² + B²)
B. |C₁ + C₂| / √(A² + B²)
C. |C₁ - C₂| / (A² + B²)
D. (C₁ - C₂) / √(A + B)
Show Answer
Answer: A. |C₁ - C₂| / √(A² + B²)
Explanation: Formula for perpendicular distance between parallel lines is d = |C₁ - C₂| / √(A² + B²).
12. Find the slope of the line passing through (3, -2) and (-1, 4):
A. -3/2
B. 3/2
C. -2/3
D. 2/3
Show Answer
Answer: A. -3/2
Explanation: Slope m = (4 - (-2)) / (-1 - 3) = 6 / -4 = -3/2.
13. Acute angle θ between two lines with slopes m₁ and m₂ is given by:
A. tan θ = |(m₂ - m₁) / (1 + m₁m₂)|
B. tan θ = (m₁ + m₂) / (1 - m₁m₂)
C. tan θ = (m₂ - m₁) / (1 - m₁m₂)
D. cos θ = |(m₂ - m₁) / (1 + m₁m₂)|
Show Answer
Answer: A. tan θ = |(m₂ - m₁) / (1 + m₁m₂)|
Explanation: Standard trigonometric formula for the acute angle between two intersecting lines.
14. What is the equation of the x-axis?
A. y = 0
B. x = 0
C. x + y = 0
D. y = 1
Show Answer
Answer: A. y = 0
Explanation: Every point on the x-axis has a y-coordinate equal to 0, so its equation is y = 0.
15. What is the equation of the y-axis?
A. x = 0
B. y = 0
C. x + y = 1
D. x = 1
Show Answer
Answer: A. x = 0
Explanation: Every point on the y-axis has an x-coordinate equal to 0, so its equation is x = 0.
16. The equation of a line parallel to 2x + 3y + 5 = 0 is of the form:
A. 2x + 3y + k = 0
B. 3x - 2y + k = 0
C. 2x - 3y + k = 0
D. 3x + 2y + k = 0
Show Answer
Answer: A. 2x + 3y + k = 0
Explanation: Parallel lines have equal slopes, so the coefficients of x and y remain proportional.
17. The equation of a line perpendicular to 2x + 3y + 5 = 0 is of the form:
A. 3x - 2y + k = 0
B. 2x + 3y + k = 0
C. 2x - 3y + k = 0
D. 3x + 2y + k = 0
Show Answer
Answer: A. 3x - 2y + k = 0
Explanation: For perpendicular lines, swap x and y coefficients and change the sign between them (m₁m₂ = -1).
18. Distance between parallel lines 3x - 4y + 7 = 0 and 3x - 4y + 2 = 0 is:
Show Answer
Answer: A. 1
Explanation: d = |7 - 2| / √(3² + (-4)²) = 5 / √25 = 5/5 = 1.
19. If points (a, 0), (0, b) and (x, y) are collinear, then:
A. x/a + y/b = 1
B. x/a - y/b = 1
C. ax + by = 1
D. a/x + b/y = 1
Show Answer
Answer: A. x/a + y/b = 1
Explanation: Points lie on the line with x-intercept 'a' and y-intercept 'b', which is x/a + y/b = 1.
20. Distance of the origin (0,0) from the line 4x + 3y - 12 = 0 is:
A. 12/5
B. 5/12
C. 12
D. 7/5
Show Answer
Answer: A. 12/5
Explanation: d = |4(0) + 3(0) - 12| / √(4² + 3²) = |-12| / √25 = 12/5.
21. The x-intercept and y-intercept of line 2x - 3y = 6 are respectively:
A. 3 and -2
B. 3 and 2
C. -3 and 2
D. 2 and -3
Show Answer
Answer: A. 3 and -2
Explanation: Divide by 6: x/3 + y/(-2) = 1. Therefore, x-intercept = 3 and y-intercept = -2.
22. Find inclination of line whose slope is 1:
A. 45°
B. 30°
C. 60°
D. 90°
Show Answer
Answer: A. 45°
Explanation: tan θ = 1 ⇒ θ = 45°.
23. Inclination of line with slope -1/√3 is:
A. 150°
B. 120°
C. 135°
D. 30°
Show Answer
Answer: A. 150°
Explanation: tan θ = -1/√3 ⇒ tan θ = tan(180° - 30°) = tan 150° ⇒ θ = 150°.
24. Equation of line passing through origin and having slope m is:
A. y = mx
B. y = mx + c
C. x = my
D. x + y = m
Show Answer
Answer: A. y = mx
Explanation: Since y-intercept c = 0 for origin passing lines, y = mx + 0 ⇒ y = mx.
25. Line passing through (1, 2) and perpendicular to x - 2y + 1 = 0 is:
A. 2x + y - 4 = 0
B. 2x - y = 0
C. x + 2y - 5 = 0
D. x - 2y + 3 = 0
Show Answer
Answer: A. 2x + y - 4 = 0
Explanation: Slope of given line = 1/2. Perpendicular slope = -2. Equation: y - 2 = -2(x - 1) ⇒ 2x + y - 4 = 0.
26. Area of triangle formed by lines x = 0, y = 0, and x/a + y/b = 1 is:
A. 1/2 |ab|
B. |ab|
C. 1/4 |ab|
D. 2|ab|
Show Answer
Answer: A. 1/2 |ab|
Explanation: Right triangle with base |a| and height |b|. Area = 1/2 × base × height = 1/2 |ab|.
27. Line x - y + 2 = 0 makes an angle with the positive direction of x-axis equal to:
A. 45°
B. 135°
C. 60°
D. 30°
Show Answer
Answer: A. 45°
Explanation: y = x + 2 ⇒ slope m = 1 ⇒ tan θ = 1 ⇒ θ = 45°.
28. Two lines a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 are coincident if:
A. a₁/a₂ = b₁/b₂ = c₁/c₂
B. a₁/a₂ = b₁/b₂ ≠ c₁/c₂
C. a₁ a₂ + b₁ b₂ = 0
D. a₁ b₂ - a₂ b₁ = 1
Show Answer
Answer: A. a₁/a₂ = b₁/b₂ = c₁/c₂
Explanation: Coincident lines represent identical linear equations, so corresponding coefficients are proportional.
29. Midpoint of line segment joining (2, 4) and (6, 8) lies on which line?
A. x + y = 10
B. x - y = 2
C. 2x - y = 1
D. x + 2y = 8
Show Answer
Answer: A. x + y = 10
Explanation: Midpoint = ((2+6)/2, (4+8)/2) = (4, 6). Substituting into x + y = 10 gives 4 + 6 = 10.
30. Intersecting point of lines x + y = 5 and x - y = 1 is:
A. (3, 2)
B. (2, 3)
C. (4, 1)
D. (1, 4)
Show Answer
Answer: A. (3, 2)
Explanation: Adding equations: 2x = 6 ⇒ x = 3. Subtracting equations: 2y = 4 ⇒ y = 2. Point is (3, 2).