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ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Straight Lines and Linear Programming (Set 4)

Take the ECAT Mathematics — Straight Lines and Linear Programming (Set 4): 30 multiple-choice questions (40 minutes) with instant scoring and a full…

What this test covers

Sample questions with answers

The first 5 questions of this test, with the correct answer and explanation.

  1. Q1. The slope of the line through (2, 3) and (6, 11) is

    • A. 4
    • B. −2
    • C. 1/2
    • D. 2 ✓

    Answer: D. 2

    Explanation: m = (11 − 3)/(6 − 2) = 8/4 = 2.

  2. Q2. The distance between (−1, 2) and (4, −10) is

    • A. √119
    • B. 13 ✓
    • C. 17
    • D. 7

    Answer: B. 13

    Explanation: d = √[(4 + 1)² + (−10 − 2)²] = √(25 + 144) = √169 = 13.

  3. Q3. The midpoint of the segment joining (3, −5) and (−1, 9) is

    • A. (2, 4)
    • B. (1, 4)
    • C. (2, 2)
    • D. (1, 2) ✓

    Answer: D. (1, 2)

    Explanation: Midpoint = ((3 − 1)/2, (−5 + 9)/2) = (1, 2).

  4. Q4. The x-intercept of the line 3x − 4y = 12 is

    • A. −4
    • B. −3
    • C. 3
    • D. 4 ✓

    Answer: D. 4

    Explanation: Put y = 0: 3x = 12, so x = 4.

  5. Q5. The line through (1, 3) with slope −2 has equation

    • A. 2x + y = 5 ✓
    • B. x + 2y = 7
    • C. 2x + y = 1
    • D. 2x − y = −1

    Answer: A. 2x + y = 5

    Explanation: y − 3 = −2(x − 1) gives y = −2x + 5, i.e. 2x + y = 5.

The other 25 questions

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  1. Q6. The distance from the point (2, 1) to the line 3x + 4y − 20 = 0 is

    • A. 14/5
    • B. 10
    • C. 2
    • D. 6

  2. Q7. The acute angle between the lines 2x − y = 3 and x + 3y = 1 is

    • A. tan⁻¹ 5
    • B. tan⁻¹ 7
    • C. tan⁻¹(1/7)
    • D. 45°

  3. Q8. The area of the triangle with vertices (1, 2), (4, 6) and (7, 2) is

    • A. 12
    • B. 10
    • C. 24
    • D. 6

  4. Q9. The centroid of the triangle with vertices (1, 2), (4, −1) and (7, 5) is

    • A. (4, 2)
    • B. (4, 3)
    • C. (12, 6)
    • D. (3, 2)

  5. Q10. The point dividing the segment from (1, 2) to (7, 8) internally in the ratio 1 : 2 is

    • A. (5, 6)
    • B. (2, 3)
    • C. (4, 5)
    • D. (3, 4)

  6. Q11. The line through (1, 2) parallel to 3x − y = 4 has equation

    • A. 3x − y − 1 = 0
    • B. 3x − y + 1 = 0
    • C. x + 3y − 7 = 0
    • D. 3x − y − 5 = 0

  7. Q12. The line through (2, 3) perpendicular to 2x + 3y = 6 has equation

    • A. 3x − 2y = 0
    • B. 3x + 2y = 12
    • C. 2x + 3y = 13
    • D. 2x − 3y + 5 = 0

  8. Q13. The lines 2x + ky = 3 and 3x − 4y = 7 are perpendicular when k equals

    • A. 3/2
    • B. 2/3
    • C. 8/3
    • D. −3/2

  9. Q14. The lines x + y = 3, 2x − y = 0 and 3x + ky = 11 are concurrent when k equals

    • A. 5
    • B. 4
    • C. 3
    • D. 2

  10. Q15. The points A(1, 2), B(4, k) and C(7, 10) are collinear when k equals

    • A. 5
    • B. 6
    • C. 8
    • D. 7

  11. Q16. The lines 2x + 3y = 12 and x − y = 1 intersect at (p, q). Then p + q equals

    • A. 6
    • B. 4
    • C. 5
    • D. 7

  12. Q17. The area of the region defined by 2x + y ≤ 6, x ≥ 0, y ≥ 0 is

    • A. 6
    • B. 12
    • C. 18
    • D. 9

  13. Q18. Maximise Z = 4x + 3y subject to 2x + y ≤ 10, x + y ≤ 7, x ≥ 0, y ≥ 0. The maximum value of Z is

    • A. 21
    • B. 24
    • C. 20
    • D. 28

  14. Q19. Minimise Z = 2x + 3y subject to x + y ≥ 5, x ≥ 1, y ≥ 1. The minimum value of Z is

    • A. 10
    • B. 11
    • C. 15
    • D. 14

  15. Q20. The acute angle between the lines represented by 2x² + 5xy + 2y² = 0 is

    • A. tan⁻¹(3/2)
    • B. 45°
    • C. tan⁻¹(4/3)
    • D. tan⁻¹(3/4)

  16. Q21. The lines x² − 4xy + 3y² = 0 together with the line x = 6 form a triangle of area

    • A. 12
    • B. 6
    • C. 18
    • D. 9

  17. Q22. The reflection of the point (4, 3) in the line 2x + y = 3 is

    • A. (−12/5, 1/5)
    • B. (12/5, 1/5)
    • C. (−12/5, −1/5)
    • D. (4/5, 7/5)

  18. Q23. Which pair of points lies on opposite sides of the line 2x − 3y + 6 = 0?

    • A. (0, 0) and (−4, 1)
    • B. (1, 0) and (5, 1)
    • C. (1, 1) and (2, 2)
    • D. (0, 0) and (1, 1)

  19. Q24. The distance between the parallel lines 3x + 4y − 5 = 0 and 6x + 8y + 20 = 0 is

    • A. 5
    • B. 3
    • C. 1
    • D. 3/2

  20. Q25. Among lines through (1, 2) that cut the positive axes, the one forming the triangle of least area with them is

    • A. x + y = 3
    • B. x + 2y = 5
    • C. 3x + y = 5
    • D. 2x + y = 4

  21. Q26. For the triangle with vertices (0, 0), (6, 0) and (0, 8), the distance between the centroid and the circumcentre is

    • A. 7/3
    • B. 4/3
    • C. 5/3
    • D. 1

  22. Q27. The lines y = x, y = 2x and x + y = 6 form a triangle of area

    • A. 3
    • B. 4
    • C. 6
    • D. 9

  23. Q28. Maximise Z = 5x + 4y subject to 6x + 4y ≤ 24, x + 2y ≤ 6, y ≤ 2, x ≥ 0, y ≥ 0. The maximum value of Z is

    • A. 24
    • B. 18
    • C. 21
    • D. 20

  24. Q29. The acute angle between the lines represented by 2x² − xy − y² = 0 is

    • A. tan⁻¹(1/3)
    • B. 60°
    • C. tan⁻¹ 3
    • D. 45°

  25. Q30. The distance from the point (1, 2) to the line through the origin and (3, 4) is

    • A. 2
    • B. 2/5
    • C. 4/5
    • D. 1/5