ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Straight Lines and Linear Programming (Set 2)

Take the ECAT Mathematics — Straight Lines and Linear Programming (Set 2): 30 multiple-choice questions (40 minutes) with instant scoring and a full…

What this test covers

• 30 ECAT Mathematics

Sample questions with answers

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

1. Q1. How far apart are the points P(1, −2) and Q(4, 2)?

   • A. √7
   • B. 7
   • C. 5 ✓
   • D. 25

   Answer: C. 5

   Explanation: d = √[(4 − 1)² + (2 + 2)²] = √(9 + 16) = 5.

2. Q2. The slope of the line through (−1, 4) and (3, −4) is:

   • A. −2 ✓
   • B. 0
   • C. −1/2
   • D. 2

   Answer: A. −2

   Explanation: m = (−4 − 4)/(3 + 1) = −8/4 = −2.

3. Q3. The line with slope 2 and y-intercept −3 is:

   • A. 2x + y + 3 = 0
   • B. 2x − y + 3 = 0
   • C. x − 2y − 3 = 0
   • D. 2x − y − 3 = 0 ✓

   Answer: D. 2x − y − 3 = 0

   Explanation: y = 2x − 3, i.e. 2x − y − 3 = 0.

4. Q4. The x- and y-intercepts of the line 4x − 3y = 12 are respectively:

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

   Answer: A. 3 and −4

   Explanation: Put y = 0: x = 3. Put x = 0: y = −4.

5. Q5. The line through (1, −1) perpendicular to 2x + y = 5 is:

   • A. x − 2y − 3 = 0 ✓
   • B. x + 2y + 1 = 0
   • C. 2x + y − 1 = 0
   • D. x − 2y + 3 = 0

   Answer: A. x − 2y − 3 = 0

   Explanation: The given slope is −2, so the required slope is 1/2: y + 1 = (x − 1)/2, i.e. x − 2y − 3 = 0.

The other 25 questions

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6. Q6. For what value of k are the lines kx + 2y = 1 and 3x − 6y = 4 parallel?

   • A. 4
   • B. 1
   • C. −4
   • D. −1

7. Q7. The distance of the origin from the line 5x + 12y = 26 is:

   • A. 2
   • B. 26/17
   • C. 13/2
   • D. 26

8. Q8. The distance between the parallel lines 6x − 8y + 3 = 0 and 3x − 4y − 1 = 0 is:

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

9. Q9. The acute angle between the lines y = 2x + 1 and y = −3x + 4 is:

   • A. 45°
   • B. 90°
   • C. 30°
   • D. 60°

10. Q10. The lines 2x + y = 7 and x − y = 2 intersect at:

    • A. (2, 3)
    • B. (1, 3)
    • C. (3, −1)
    • D. (3, 1)

11. Q11. The point dividing the join of A(2, 3) and B(6, 7) externally in the ratio 3 : 1 is:

    • A. (8, 9)
    • B. (5, 6)
    • C. (0, 1)
    • D. (10, 11)

12. Q12. The area of the triangle with vertices (1, 1), (4, 1) and (4, 5) is:

    • A. 7.5
    • B. 10
    • C. 12
    • D. 6

13. Q13. Find k if A(k, 2), B(1, 1) and C(3, 5) all lie on one straight line.

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

14. Q14. The line whose perpendicular distance from the origin is 4 and whose normal makes 60° with the x-axis is:

    • A. x + √3y = 8
    • B. x + √3y = 4
    • C. x − √3y = 8
    • D. √3x + y = 8

15. Q15. The centroid of the triangle with vertices (2, 3), (4, −1) and (6, 4) is:

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

16. Q16. The line through the origin with inclination 120° is:

    • A. y = x/√3
    • B. y = −x/√3
    • C. y = −√3x
    • D. y = √3x

17. Q17. The equation x² − 4y² = 0 represents the pair of lines:

    • A. x = 2y and x = −4y
    • B. x = ±4y
    • C. y = ±2x
    • D. x = ±2y

18. Q18. The joint equation of the lines y = x and y = 2x is:

    • A. 2x² − 3xy + y² = 0
    • B. 2x² − xy − y² = 0
    • C. 2x² + 3xy + y² = 0
    • D. x² − 3xy + 2y² = 0

19. Q19. Relative to the line 3x + 4y − 5 = 0, the point (2, −1):

    • A. lies on the line
    • B. lies on the opposite side from the origin
    • C. is at distance 3 from the line
    • D. lies on the same side as the origin

20. Q20. Maximise z = 2x + 3y subject to x + y ≤ 6, x ≤ 4, y ≤ 3, x ≥ 0, y ≥ 0. The maximum value of z is:

    • A. 18
    • B. 15
    • C. 14
    • D. 9

21. Q21. Minimise z = 3x + 2y subject to x + y ≥ 4, x ≥ 1, y ≥ 1. The minimum value of z is:

    • A. 11
    • B. 10
    • C. 8
    • D. 9

22. Q22. In a linear programming problem, a point of the feasible region where two boundary lines intersect is called:

    • A. a decision variable
    • B. a corner point (vertex)
    • C. an objective point
    • D. a constraint

23. Q23. Which point lies in the region x − 2y ≥ 4?

    • A. (0, 0)
    • B. (6, 0)
    • C. (0, −1)
    • D. (2, 1)

24. Q24. The y-intercept of the line 3x − 4y + 8 = 0 is:

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

25. Q25. The line through (3, −2) parallel to the y-axis is:

    • A. x = −2
    • B. y = −2
    • C. x = 3
    • D. x + y = 1

26. Q26. The line through (−1, 2) and (2, 8) is:

    • A. y = 2x
    • B. y = x + 6
    • C. y = 2x + 4
    • D. y = 3x + 5

27. Q27. The perpendicular bisector of the segment joining (0, 0) and (4, 2) is:

    • A. 2x + y = 5
    • B. 2x − y = 3
    • C. x + 2y = 4
    • D. x − 2y = 0

28. Q28. The foot of the perpendicular from the origin to the line x + y = 4 is:

    • A. (1, 3)
    • B. (0, 4)
    • C. (4, 0)
    • D. (2, 2)

29. Q29. The area of the triangle formed by the line 3x + 4y = 12 and the coordinate axes is:

    • A. 7
    • B. 24
    • C. 12
    • D. 6

30. Q30. For what value of k are the lines x + y = 3, x − y = 1 and 2x + ky = 5 concurrent?

    • A. 1
    • B. −1
    • C. 3
    • D. 2