Conditions for parallelism of two lines MCQs With Answer

Understanding the Conditions for parallelism of two lines is essential for B. Pharm students studying applied mathematics in pharmaceutics, biostatistics and modelling. This topic covers algebraic and vector criteria—equal slopes in Cartesian form, proportional coefficients in general form ax + by + c = 0, and proportional direction ratios or zero cross-product in three dimensions. Mastery helps interpret calibration curves, chromatographic alignment and spatial models used in drug formulation and delivery. These MCQs focus on derivations, proofs, computations and practical examples, building conceptual depth and problem-solving skills. Now let’s test your knowledge with 50 MCQs on this topic.

Q1. What is the basic condition for two lines in the plane to be parallel?

• Their slopes are equal
• Their slopes multiply to -1
• Their intercepts are equal
• They pass through the origin

Correct Answer: Their slopes are equal

Q2. For lines given by a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, which condition indicates they are parallel?

• a1/a2 = c1/c2
• a1/a2 = b1/b2
• b1/b2 = c1/c2
• a1 + a2 = b1 + b2

Correct Answer: a1/a2 = b1/b2

Q3. In vector form r = a + λb and r = c + μd, when are the two lines parallel?

• If a = c
• If b and d are proportional
• If λ = μ for some values
• If a·c = 0

Correct Answer: If b and d are proportional

Q4. For three-dimensional lines, which vector condition ensures parallelism?

• b × d = 0
• b · d = 0
• b + d = 0

Correct Answer: b × d = 0

Q5. Two lines y = m1x + c1 and y = m2x + c2 are parallel when:

• m1 = -m2
• m1 = m2
• c1 = c2
• m1c2 = m2c1

Correct Answer: m1 = m2

Q6. Which statement best describes direction ratios for parallel straight lines?

• Direction ratios are perpendicular
• Direction ratios are proportional
• Direction ratios are equal to zero
• Direction ratios sum to one

Correct Answer: Direction ratios are proportional

Q7. How can you distinguish two parallel but distinct lines from coincident lines in general form?

• a1/a2 = b1/b2 = c1/c2 indicates distinct parallel lines
• a1/a2 = b1/b2 but c1/c2 different indicates distinct parallel lines
• If a1 = a2 and b1 = b2 they must be distinct
• Distinct lines always have different slopes

Correct Answer: a1/a2 = b1/b2 but c1/c2 different indicates distinct parallel lines

Q8. The slope of a line ax + by + c = 0 is:

• a/b
• -a/b
• b/a
• -b/a

Correct Answer: -a/b

Q9. Using the angle formula tanθ = |(m1 – m2)/(1 + m1 m2)|, when is the angle between two lines zero?

• When m1 m2 = -1
• When m1 = m2
• When 1 + m1 m2 = 0
• When m1 = -m2

Correct Answer: When m1 = m2

Q10. What is the distance between two parallel lines ax + by + c1 = 0 and ax + by + c2 = 0?

• |c1 – c2| / (a + b)
• |c1 – c2| / sqrt(a^2 + b^2)
• |c1 + c2| / sqrt(a^2 – b^2)
• |c1 – c2| * sqrt(a^2 + b^2)

Correct Answer: |c1 – c2| / sqrt(a^2 + b^2)

Q11. Are the lines with slopes 2 and -1/2 parallel, perpendicular, or neither?

• Parallel
• Perpendicular
• Coincident
• Neither

Correct Answer: Neither

Q12. Which condition describes skew lines in 3D compared to parallel lines?

• Direction vectors proportional and intersecting
• Direction vectors not proportional and no intersection
• Direction vectors proportional and same point
• Direction vectors orthogonal and intersecting

Correct Answer: Direction vectors not proportional and no intersection

Q13. Using determinants, two lines are parallel if which determinant equals zero?

Correct Answer: |a1 b1; a2 b2| = 0

Q14. For symmetric form (x – x1)/l = (y – y1)/m = (z – z1)/n, when are two lines parallel?

• If l1 = l2 and m1 = m2 and n1 = n2
• If l1:l2 = m1:m2 = n1:n2
• If x1 = x2 and y1 = y2 and z1 = z2
• If l1 + l2 = m1 + m2 = n1 + n2

Correct Answer: If l1:l2 = m1:m2 = n1:n2

Q15. In vector notation, r = a + t b and r = c + s d are parallel when:

• (a – c) × b = 0 and b × d = 0
• b · d = 0 and a = c
• b × d = 0 regardless of points a and c
• a · c = 0

Correct Answer: b × d = 0 regardless of points a and c

Q16. Determine the relation between the lines y = 3x + 2 and 6x – 2y + 4 = 0.

• Parallel distinct
• Perpendicular
• Coincident
• Skew

Correct Answer: Coincident

Q17. Which of the following is the condition for two lines to be perpendicular (contrast with parallelism)?

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

Correct Answer: m1 * m2 = -1

Q18. If two lines have identical direction cosines, they are:

• Perpendicular
• Parallel
• Intersecting at origin
• Skew

Correct Answer: Parallel

Q19. For ax + by + c = 0 and a’x + b’y + c’ = 0, the condition a b’ – a’ b = 0 implies:

• The lines are perpendicular
• The lines are parallel
• The lines intersect at origin
• The lines are skew

Correct Answer: The lines are parallel

Q20. Which statement is true for lines y = m x + c1 and y = m x + c2 in relation to pharmaceutical calibration curves?

• Same slope and same intercept indicate different methods
• Same slope different intercept implies parallel calibration curves
• Different slopes always indicate parallel curves
• Calibration curves cannot be parallel

Correct Answer: Same slope different intercept implies parallel calibration curves

Q21. Two 3D lines are parallel but do not coincide. Which is true?

• Their direction vectors are not proportional
• Their direction vectors are proportional but points differ
• They must intersect
• They are necessarily skew

Correct Answer: Their direction vectors are proportional but points differ

Q22. Given points P(1,2) and Q(3,6), and R(0,0) and S(2,4), are PQ and RS parallel?

• Yes, because slopes are equal
• No, they are perpendicular
• They intersect at (1,2)
• Not enough information

Correct Answer: Yes, because slopes are equal

Q23. In chromatography data alignment, two peak baseline trends represented by lines with same slope but different intercepts indicate:

• Identical retention times
• Parallel baselines requiring offset correction
• Perpendicular baselines
• No relationship

Correct Answer: Parallel baselines requiring offset correction

Q24. If direction ratios of one line are 2: -3: 4 and another line has ratios 4: -6: 8, what is their relation?

• Perpendicular
• Parallel
• Intersecting but not parallel
• Skew

Correct Answer: Parallel

Q25. Which algebraic test confirms two planar lines are parallel using coefficients?

• a1b2 – a2b1 = 0
• a1c2 – a2c1 = 0
• b1c2 – b2c1 = 1
• a1 + b2 = 0

Correct Answer: a1b2 – a2b1 = 0

Q26. For lines given as x/2 = y/3 and x/4 = y/6, are they parallel?

• No, because intercepts differ
• Yes, because direction ratios proportional
• No, because slopes are negative
• Yes, because they intersect at origin

Correct Answer: Yes, because direction ratios proportional

Q27. Which condition shows two 2D lines are coincident rather than just parallel?

• Same slope and different intercepts
• a1/a2 = b1/b2 = c1/c2
• a1/a2 = b1/b2 but c1/c2 different
• a1 + a2 = 0

Correct Answer: a1/a2 = b1/b2 = c1/c2

Q28. If two lines have slopes m and m, and intercepts c1 and c2, what operation gives the perpendicular distance between them?

• |c1 + c2| / sqrt(1 + m^2)
• |c1 – c2| / sqrt(1 + m^2)
• |m(c1 – c2)|
• |c1 – c2| * sqrt(1 + m^2)

Correct Answer: |c1 – c2| / sqrt(1 + m^2)

Q29. Two lines intersect if which of the following is true?

• Their slopes are equal
• Their slopes are not equal
• They have proportional direction ratios
• They have same intercepts but different slopes

Correct Answer: Their slopes are not equal

Q30. Vertical lines x = a and x = b are:

• Parallel if a ≠ b
• Perpendicular if a ≠ b
• Coincident for any a and b
• Skew

Correct Answer: Parallel if a ≠ b

Q31. Horizontal lines y = k1 and y = k2 are parallel when:

• k1 = k2
• k1 ≠ k2
• k1 * k2 = 1
• k1 + k2 = 0

Correct Answer: k1 ≠ k2

Q32. Two 3D lines with direction vectors (1,2,3) and (2,4,6) are:

• Skew
• Parallel
• Perpendicular
• Intersecting only

Correct Answer: Parallel

Q33. For lines ax + by + c = 0 and a’x + b’y + c’ = 0, parallelism implies which equality involving a and b?

• a/b = a’/b’
• a + b = a’ + b’
• a^2 + b^2 = a’^2 + b’^2
• a – b = a’ – b’

Correct Answer: a/b = a’/b’

Q34. In 3D, two lines with direction vectors b and d satisfy b × d = 0. What does this physically imply?

• The lines are orthogonal
• The lines are parallel or collinear
• The lines intersect at right angle
• The lines lie on different planes and are skew

Correct Answer: The lines are parallel or collinear

Q35. Determine relationship: y = 2x + 3 and 4x – 2y + 1 = 0.

• Coincident
• Parallel distinct
• Perpendicular
• Intersecting at a single point

Correct Answer: Parallel distinct

Q36. If two direction vectors are scalar multiples, what is guaranteed about the corresponding lines?

• They are perpendicular
• They are parallel or collinear
• They intersect at the origin only
• They cannot be in the same plane

Correct Answer: They are parallel or collinear

Q37. Which computational check is efficient to test parallelism of ax + by + c = 0 and dx + ey + f = 0?

• Compute a/e – b/d and check if zero
• Compute a*e – b*d and check if zero
• Compute c – f and check if zero
• Check if a + b = d + e

Correct Answer: Compute a*e – b*d and check if zero

Q38. In modelling drug diffusion along parallel channels, why is identifying parallel lines useful?

• Parallel channels guarantee identical flux
• Parallel trends indicate consistent directional behavior and allow simplified modelling
• Parallel lines mean channels intersect
• It is irrelevant to diffusion models

Correct Answer: Parallel trends indicate consistent directional behavior and allow simplified modelling

Q39. If line1 has equation 2x – 3y + 5 = 0 and line2 has equation 4x – 6y + k = 0, for which k are they coincident?

• k = 10
• k = 5
• k = -10
• Any k will do

Correct Answer: k = 10

Q40. Two lines are parallel and one passes through point P. How many points of intersection exist between the two lines?

• Exactly one
• Infinitely many if coincident
• None if distinct parallel
• Both b and c are possible depending on relation

Correct Answer: Both b and c are possible depending on relation

Q41. If slopes of two lines are undefined (vertical), what is their relationship?

• They are parallel if x-intercepts differ
• They are perpendicular
• They must coincide
• They cannot be compared

Correct Answer: They are parallel if x-intercepts differ

Q42. In the context of analytical geometry, proportionality of coefficients implies parallelism. Which proportional set confirms that?

• a1:a2 = b1:b2 = c1:c2 implies distinct parallel
• a1:a2 = b1:b2 with c1:c2 different implies distinct parallel
• a1:b1 = a2:b2 = c2:b1 implies parallel
• Only equality of c’s matters

Correct Answer: a1:a2 = b1:b2 with c1:c2 different implies distinct parallel

Q43. For two lines in 2D, setting their slopes equal leads to what algebraic condition between coefficients?

• a1/a2 + b1/b2 = 0
• a1 b2 = a2 b1
• c1 = c2
• a1 + b1 = a2 + b2

Correct Answer: a1 b2 = a2 b1

Q44. Given parametric lines L1: (x,y) = (1,0) + t(2,3) and L2: (x,y) = (4,1) + s(4,6), are they parallel?

• No, direction vectors differ
• Yes, because (4,6) is 2*(2,3)
• No, they intersect
• Yes, because points (1,0) and (4,1) are proportional

Correct Answer: Yes, because (4,6) is 2*(2,3)

Q45. When fitting linear models to two datasets, identical slopes but different intercepts imply:

• Models are identical
• Models are parallel and indicate consistent response per unit change
• Models are perpendicular
• One model is a translation and rotation of the other

Correct Answer: Models are parallel and indicate consistent response per unit change

Q46. For lines given by vector direction ratios (l1,m1,n1) and (l2,m2,n2), proportional ratios imply parallelism. Which test checks this?

• Check l1*l2 + m1*m2 + n1*n2 = 0
• Check l1/l2 = m1/m2 = n1/n2
• Check l1 + l2 = m1 + m2 = n1 + n2
• Check l1^2 = l2^2 etc.

Correct Answer: Check l1/l2 = m1/m2 = n1/n2

Q47. If two planar lines satisfy a1x + b1y + c1 = 0 and k(a1x + b1y) + c2 = 0 for some k ≠ 0, what does this mean?

• They are perpendicular
• They are parallel
• They intersect at origin only
• They are skew

Correct Answer: They are parallel

Q48. In the lab, two instrument calibration lines have equations y = 0.85x + 0.2 and y = 0.85x + 1.1. What corrective action is implied?

• No correction needed since slopes differ
• Apply an intercept offset to align baselines
• Replace one instrument because slopes must differ
• Rotate one calibration curve

Correct Answer: Apply an intercept offset to align baselines

Q49. Which of the following is NOT a valid condition for two lines to be parallel?

• Their slopes are equal
• Their direction vectors are scalar multiples
• Their cross product is zero in 3D
• Their slopes multiply to -1

Correct Answer: Their slopes multiply to -1

Q50. For robust identification, which combined checks confirm two lines are coincident rather than just parallel?

• Equal slopes and equal intercepts
• Direction ratios proportional and a point of one lies on the other
• a1/a2 = b1/b2 = c1/c2
• All of the above

Correct Answer: All of the above

Download