Conditions for perpendicularity of two lines MCQs With Answer

Introduction: Understanding the conditions for perpendicularity of two lines is essential for B. Pharm students who apply coordinate geometry and vector concepts in drug modeling, pharmacokinetics graphs, and instrumentation data analysis. This concise guide covers key ideas like slope product m1*m2 = -1, dot product of direction ratios equal to zero, conditions in general form ax+by+c=0, and extensions to 3D lines and planes. Clear definitions, representative examples, and focused terminology will help you master perpendicularity for exam MCQs and practical problem-solving. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the condition for two non-vertical lines with slopes m1 and m2 to be perpendicular?

• m1*m2 = -1
• m1 + m2 = 0
• m1 – m2 = 1
• m1 / m2 = -1

Correct Answer: m1*m2 = -1

Q2. For lines given by equations a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, what condition ensures they are perpendicular?

• a1a2 + b1b2 = 0
• a1b2 – a2b1 = 0
• a1/a2 = b1/b2
• a1 + a2 = b1 + b2

Correct Answer: a1a2 + b1b2 = 0

Q3. Two lines in vector form r = a + λd1 and r = b + μd2 are perpendicular if which condition holds (assuming they intersect)?

• d1 · d2 = 0
• d1 × d2 = 0
• d1 = d2
• d1 · (a – b) = 0

Correct Answer: d1 · d2 = 0

Q4. If one line has slope 2, what is the slope of a line perpendicular to it?

• -1/2
• 1/2
• -2
• 0.5

Correct Answer: -1/2

Q5. Which statement is true about a vertical line x = k and a horizontal line y = c?

• They are perpendicular to each other
• They are parallel to each other
• They never intersect
• They have equal slopes

Correct Answer: They are perpendicular to each other

Q6. For two lines with direction ratios (l1, m1, n1) and (l2, m2, n2) in 3D, what condition means the lines’ directions are perpendicular?

• l1l2 + m1m2 + n1n2 = 0
• l1/m1 = l2/m2
• l1 + l2 = 0
• l1l2 – m1m2 – n1n2 = 0

Correct Answer: l1l2 + m1m2 + n1n2 = 0

Q7. The angle θ between two lines with slopes m1 and m2 is given by tan θ = |(m2 – m1)/(1 + m1*m2)|. For θ = 90°, what denominator condition arises?

• 1 + m1*m2 = 0
• m2 – m1 = 0
• 1 – m1*m2 = 0
• m1 + m2 = 0

Correct Answer: 1 + m1*m2 = 0

Q8. If line L1 has equation 3x – 4y + 7 = 0, what is the slope of a line perpendicular to L1?

• 4/3
• -3/4
• -4/3
• 3/4

Correct Answer: 4/3

Q9. Two intersecting lines are perpendicular if the product of their direction cosines along x-axis equals what (for 2D representation)?

• The negative reciprocal of y-direction product
• Zero when combined appropriately
• One
• Negative one

Correct Answer: Zero when combined appropriately

Q10. Which of the following pairs of slopes represent perpendicular lines?

• m1 = 3, m2 = -1/3
• m1 = 2, m2 = 2
• m1 = -1, m2 = -1
• m1 = 0, m2 = 0

Correct Answer: m1 = 3, m2 = -1/3

Q11. For two lines in symmetric form (x – x1)/a = (y – y1)/b and (x – x2)/c = (y – y2)/d, which condition ensures perpendicular direction vectors?

• ac + bd = 0
• ad = bc
• ab + cd = 1
• a + c = b + d

Correct Answer: ac + bd = 0

Q12. If a line has slope m, the slope of a line perpendicular to it can be written as which expression?

• -1/m
• 1/m
• -m
• m + 1

Correct Answer: -1/m

Q13. Lines L1: y = (1/2)x + 3 and L2: y = -2x – 1 are:

• Perpendicular
• Parallel
• Coincident
• Neither perpendicular nor parallel

Correct Answer: Perpendicular

Q14. Which algebraic condition indicates two straight lines are perpendicular when given by angles θ1 and θ2?

• tan θ1 * tan θ2 = -1
• tan θ1 + tan θ2 = 0
• θ1 + θ2 = 0
• tan(θ1 + θ2) = 1

Correct Answer: tan θ1 * tan θ2 = -1

Q15. The direction vector of line ax + by + c = 0 is which of the following?

• (b, -a)
• (a, b)
• (-b, a)
• (a, -b)

Correct Answer: (b, -a)

Q16. Given lines L1: 2x + y – 5 = 0 and L2: x – 2y + 3 = 0, are they perpendicular?

• Yes, because 2*1 + 1*(-2) = 0
• No, because 2 + (-2) ≠ 0
• Yes, because slopes are equal
• No, because they are parallel

Correct Answer: Yes, because 2*1 + 1*(-2) = 0

Q17. For two lines to be perpendicular in plane geometry, which geometric property must hold at their intersection point?

• The angle formed is 90 degrees
• They do not intersect
• Their midpoints coincide
• They have equal length segments

Correct Answer: The angle formed is 90 degrees

Q18. If a line has direction ratios (3, -4, 0) and another has (4, 3, 0), what is their dot product and perpendicularity?

• 3*4 + (-4)*3 + 0*0 = 0, so perpendicular
• 12 – 12 = 24, not perpendicular
• Dot product is 1, so not perpendicular
• Dot product negative, so parallel

Correct Answer: 3*4 + (-4)*3 + 0*0 = 0, so perpendicular

Q19. Which of the following is a correct test for perpendicularity when slopes m1 and m2 are both non-zero?

• m1 = -1/m2
• m1 = m2
• m1 + m2 = 1
• m1 – m2 = -1

Correct Answer: m1 = -1/m2

Q20. If two skew lines in 3D have direction vectors d1 and d2 with d1·d2 = 0 but do not intersect, are the lines perpendicular to each other?

• Not perpendicular as lines unless they intersect; directions are perpendicular
• Yes, they are perpendicular lines in space
• They must be parallel
• They are coincident

Correct Answer: Not perpendicular as lines unless they intersect; directions are perpendicular

Q21. The perpendicular from a point to a line is the shortest distance. Which mathematical condition defines its foot?

• Vector from foot to point is orthogonal to line direction
• Foot divides line in equal halves
• Foot has same slope as the line
• Foot lies at infinity

Correct Answer: Vector from foot to point is orthogonal to line direction

Q22. Two lines are given in parametric form as L1: (x,y) = (1,2) + λ(2,3) and L2: (x,y) = (4,0) + μ(3,-2). Are their directions perpendicular?

• Yes, because 2*3 + 3*(-2) = 0
• No, because 2*3 – 3*2 ≠ 0
• Yes, because they intersect at a point
• No, because slopes are equal

Correct Answer: Yes, because 2*3 + 3*(-2) = 0

Q23. If the line y = mx + c is perpendicular to x = 5, what is m?

• 0
• Undefined
• 1
• -1

Correct Answer: 0

Q24. For two lines in plane with direction angles α and β, what must α + β equal for the lines to be perpendicular?

• 90° (π/2 radians)
• 180°
• 0°
• 45°

Correct Answer: 90° (π/2 radians)

Q25. Which condition indicates a line L is perpendicular to a plane whose normal is n?

• Direction vector of L is parallel to n
• Direction vector of L is perpendicular to n
• L lies entirely inside the plane
• L is parallel to the plane

Correct Answer: Direction vector of L is parallel to n

Q26. The general test for perpendicularity of two lines given in slope form y = m1x + c1 and y = m2x + c2 fails when which case occurs?

• One line is vertical (slope undefined)
• Both slopes are finite and non-zero
• Slopes satisfy m1*m2 = -1
• The lines intersect at origin

Correct Answer: One line is vertical (slope undefined)

Q27. If L1: y = (−3/4)x + 1, which of the following is an equation of a line perpendicular to L1?

• y = (4/3)x + 2
• y = (-3/4)x – 2
• y = (3/4)x + 1
• y = (-4/3)x + 5

Correct Answer: y = (4/3)x + 2

Q28. When two lines are perpendicular, what is true about their gradients on a pharmacokinetic concentration-time graph if axes are standard?

• The product of instantaneous slopes at intersection is -1
• Both slopes are equal
• Both slopes are zero
• The sum of slopes equals 1

Correct Answer: The product of instantaneous slopes at intersection is -1

Q29. If line L has equation 4x + 3y + 6 = 0, which equation represents a line perpendicular to L?

• 3x – 4y + 2 = 0
• 4x – 3y + 5 = 0
• -4x – 3y + 1 = 0
• 3x + 4y + 7 = 0

Correct Answer: 3x – 4y + 2 = 0

Q30. In coordinate geometry, two lines are perpendicular if the product of their direction vectors’ slopes equals:

• -1
• 0
• 1
• Undefined

Correct Answer: -1

Q31. Which of the following pairs of lines are perpendicular: L1: x + y = 0 and L2: x – y = 0?

• Yes, because normals satisfy a1a2 + b1b2 = 0
• No, they are parallel
• Yes, because both pass through origin but not perpendicular
• No, because slopes are equal

Correct Answer: Yes, because normals satisfy a1a2 + b1b2 = 0

Q32. For two lines to be perpendicular in plane, which trigonometric condition between their angles θ1 and θ2 is correct?

• θ1 – θ2 = ±90°
• θ1 + θ2 = 0°
• cos(θ1) = cos(θ2)
• sin(θ1) = sin(θ2)

Correct Answer: θ1 – θ2 = ±90°

Q33. Which operation on direction vectors yields a scalar whose zero value indicates perpendicularity?

• Dot product
• Cross product
• Vector addition
• Scalar division

Correct Answer: Dot product

Q34. If a line L2 is perpendicular to L1: y = -5x + 2 and passes through (1,1), its equation is:

• y = (1/5)x + 4/5
• y = -5x + 6
• y = (1/5)x – 4/5
• y = 5x + 1

Correct Answer: y = (1/5)x + 4/5

Q35. For two lines in 3D given by parametric forms to be perpendicular and intersecting, which two conditions must both hold?

• Direction vectors dot to zero and intersection point exists
• Direction vectors are parallel and intersection point exists
• Lines are skew and direction dot not zero
• They must be in different planes

Correct Answer: Direction vectors dot to zero and intersection point exists

Q36. Which of the following is a correct perpendicular slope to m = 0?

• Undefined (vertical line)
• 0
• 1
• -1

Correct Answer: Undefined (vertical line)

Q37. The product of slopes of two perpendicular lines in standard xy-plane is equal to:

• -1 for non-vertical lines
• 0 always
• 1 always
• Undefined for all cases

Correct Answer: -1 for non-vertical lines

Q38. If the direction ratios of two lines are (1, 2, -2) and (2, -1, 1), what is their dot product and are they perpendicular?

• 1*2 + 2*(-1) + (-2)*1 = 0, so perpendicular
• Dot product 6, so not perpendicular
• Dot product -1, so perpendicular
• Dot product 1, so not perpendicular

Correct Answer: 1*2 + 2*(-1) + (-2)*1 = 0, so perpendicular

Q39. For lines L1: y = 0 and L2: x = k, which statement is true?

• They are perpendicular for any real k
• They are parallel
• They never meet
• L2 is slope 0

Correct Answer: They are perpendicular for any real k

Q40. Which of the following is a necessary algebraic criterion for two non-coincident lines to be perpendicular in plane?

• Their slopes multiply to -1 or one is vertical and the other horizontal
• Their intercepts are equal
• Their constants sum to zero
• Their coefficients are proportional

Correct Answer: Their slopes multiply to -1 or one is vertical and the other horizontal

Q41. Given L1: 5x – y + 2 = 0 and L2: x + 5y – 3 = 0, are these lines perpendicular?

• Yes, because 5*1 + (-1)*5 = 0
• No, because coefficients don’t match
• Yes, because slopes are equal
• No, because they are skew

Correct Answer: Yes, because 5*1 + (-1)*5 = 0

Q42. The perpendicular to a given line through a point not on the line is unique because of which property?

• Shortest distance criterion gives a unique orthogonal projection
• Infinitely many lines pass through the point
• Lines cannot intersect at a single point
• Only parallel lines exist

Correct Answer: Shortest distance criterion gives a unique orthogonal projection

Q43. If two lines have slopes m1 = tan 30° and m2 = tan 120°, are they perpendicular?

• Yes, because 30° and 120° differ by 90°
• No, because 30° + 120° = 150°
• No, because slopes are equal
• Yes, because tan values multiply to 1

Correct Answer: Yes, because 30° and 120° differ by 90°

Q44. For a line given by y = mx, a line perpendicular through origin has equation:

• y = (-1/m) x
• y = m x
• x = m y
• y = (1/m) x

Correct Answer: y = (-1/m) x

Q45. Which pair of general form coefficients corresponds to perpendicular lines?

• Line1: 2x + 3y + 1 = 0, Line2: 3x – 2y + 2 = 0
• Line1: x + y = 0, Line2: 2x + 2y = 1
• Line1: x – y = 0, Line2: 2x – 2y = 4
• Line1: 3x + 4y = 0, Line2: 6x + 8y = 5

Correct Answer: Line1: 2x + 3y + 1 = 0, Line2: 3x – 2y + 2 = 0

Q46. If a line has slope -1/7, what is slope of a perpendicular line?

• 7
• -7
• 1/7
• -1/7

Correct Answer: 7

Q47. Two lines with equations y = 2x + 1 and 2y = -x + 4 are:

• Perpendicular because slopes 2 and -1/2 multiply to -1
• Parallel because slopes equal
• Coincident
• Neither perpendicular nor parallel

Correct Answer: Perpendicular because slopes 2 and -1/2 multiply to -1

Q48. A line perpendicular to 6x + 8y + 10 = 0 will have coefficients a and b such that which relation holds?

• 6a + 8b = 0
• a/6 = b/8
• 6/a + 8/b = 0
• a + b = 14

Correct Answer: 6a + 8b = 0

Q49. In applied B.Pharm tasks, when plotting two trend lines that should be orthogonal, which algebraic check confirms perpendicularity?

• Multiply their slopes and verify product is -1
• Check if their y-intercepts are equal
• Ensure both have positive slopes
• Confirm both cross the x-axis

Correct Answer: Multiply their slopes and verify product is -1

Q50. What combination of conditions ensures two spatial lines are truly perpendicular as lines (not only directions)?

• Direction vectors are orthogonal and lines intersect at a common point
• Direction vectors are orthogonal but lines are skew
• Lines have equal direction vectors
• Lines are parallel and at right distances

Correct Answer: Direction vectors are orthogonal and lines intersect at a common point

Download