ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Straight Lines and Linear Programming (Set 3)

Take the ECAT Mathematics — Straight Lines and Linear Programming (Set 3): 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. The distance between the points (−1, 3) and (4, −9) is:

   • A. 12
   • B. √119
   • C. 13 ✓
   • D. 15

   Answer: C. 13

   Explanation: d = √[(4 + 1)² + (−9 − 3)²] = √(25 + 144) = 13.

2. Q2. The slope and y-intercept of the line 3x − 4y + 12 = 0 are respectively:

   • A. −3/4 and 3
   • B. 3/4 and −3
   • C. 4/3 and 12
   • D. 3/4 and 3 ✓

   Answer: D. 3/4 and 3

   Explanation: Write y = (3/4)x + 3, so the slope is 3/4 and the y-intercept is 3.

3. Q3. The line through (2, −3) and (−4, −3) is:

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

   Answer: B. y = −3

   Explanation: Both points have y-coordinate −3, so the line is horizontal: y = −3.

4. Q4. The line passing through (−2, 0) and (0, 5) has equation:

   • A. 5x − 2y + 10 = 0 ✓
   • B. 5x + 2y + 10 = 0
   • C. 2x − 5y + 10 = 0
   • D. 5x − 2y − 10 = 0

   Answer: A. 5x − 2y + 10 = 0

   Explanation: x-intercept −2 and y-intercept 5: x/(−2) + y/5 = 1, i.e. 5x − 2y + 10 = 0 (check (0, 5): −10 + 10 = 0).

5. Q5. The line through (1, 1) parallel to 2x + 3y = 6 is:

   • A. 2x + 3y = 1
   • B. 2x + 3y = 5 ✓
   • C. 3x + 2y = 5
   • D. 3x − 2y = 1

   Answer: B. 2x + 3y = 5

   Explanation: A parallel line has the same x- and y-coefficients: 2x + 3y = c. At (1, 1): c = 5.

The other 25 questions

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6. Q6. The perpendicular distance of the point (3, −2) from the line 12x − 5y − 7 = 0 is:

   • A. 13/3
   • B. 3
   • C. 1/3
   • D. 39

7. Q7. The distance between the parallel lines 5x + 12y − 7 = 0 and 5x + 12y + 19 = 0 is:

   • A. 2
   • B. 13
   • C. 1
   • D. 26

8. Q8. The acute angle between the lines x − y = 2 and √3x + y = 1 is:

   • A. 75°
   • B. 45°
   • C. 60°
   • D. 15°

9. Q9. The lines 2x − y = 4 and x + 2y = 7 intersect at:

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

10. Q10. The point P divides the join of A(1, −2) and B(4, 7) internally in the ratio 1 : 2. P is:

    • A. (3, 4)
    • B. (2, 1)
    • C. (2, 3)
    • D. (5/2, 5/2)

11. Q11. The x-axis divides the join of (2, 3) and (4, −5) internally in the ratio:

    • A. 3 : 5
    • B. 1 : 1
    • C. 5 : 3
    • D. 3 : 8

12. Q12. The area of the triangle with vertices (2, 1), (5, 1) and (3, 7) is:

    • A. 18
    • B. 21/2
    • C. 9
    • D. 6

13. Q13. The points A(1, −1), B(2, 1) and C(k, 5) are collinear. Then k equals:

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

14. Q14. Two vertices of a triangle are (1, 2) and (3, 4) and its centroid is (3, 3). The third vertex is:

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

15. Q15. The circumcentre of the right-angled triangle with vertices (0, 0), (6, 0) and (0, 8) is:

    • A. (2, 8/3)
    • B. (0, 0)
    • C. (3, 4)
    • D. (6, 8)

16. Q16. The line whose perpendicular from the origin has length 5 and makes an angle of 30° with the positive x-axis has equation:

    • A. x + √3y = 10
    • B. √3x + y = 5
    • C. √3x + y = 10
    • D. x + y = 10

17. Q17. The length of the perpendicular from the origin to the line 3x − 4y + 10 = 0 is:

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

18. Q18. The equation x² − xy − 6y² = 0 represents the lines:

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

19. Q19. For what value of k does kx² + 5xy − 4y² = 0 represent two perpendicular lines?

    • A. 4
    • B. 5
    • C. −5
    • D. −4

20. Q20. The acute angle between the lines given by x² − 4xy + y² = 0 is:

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

21. Q21. The area of the region defined by x + y ≤ 4, x ≥ 0 and y ≥ 0 is:

    • A. 16
    • B. 8
    • C. 4
    • D. 12

22. Q22. Maximise z = 5x + 4y subject to x + y ≤ 5, x ≤ 3, x ≥ 0, y ≥ 0. The maximum value is:

    • A. 25
    • B. 23
    • C. 15
    • D. 20

23. Q23. Minimise z = 4x + 3y subject to x + y ≥ 5, x ≥ 1, y ≥ 0. The minimum value is:

    • A. 15
    • B. 20
    • C. 12
    • D. 16

24. Q24. The feasible region of x + y ≤ 4, x ≤ 3, x ≥ 0, y ≥ 0 has how many corner points?

    • A. 6
    • B. 4
    • C. 3
    • D. 5

25. Q25. Maximise z = x + y subject to 2x + y ≤ 8, x + 2y ≤ 8, x ≥ 0, y ≥ 0. The maximum value is:

    • A. 5
    • B. 16/3
    • C. 8
    • D. 4

26. Q26. In maximising z = 2x + 2y subject to x + y ≤ 6, x ≥ 0, y ≥ 0, the maximum value 12 is attained:

    • A. at every point of the segment x + y = 6 in the first quadrant
    • B. only at (3, 3)
    • C. only at (6, 0)
    • D. only at (0, 6)

27. Q27. A line through (3, 4) has equal non-zero intercepts on the two axes. Its equation is:

    • A. x + y = 1
    • B. x − y = −1
    • C. x + y = 7
    • D. 4x + 3y = 24

28. Q28. The line through (2, −1) inclined at 135° to the positive x-axis has equation:

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

29. Q29. The point on the x-axis that is equidistant from (1, 2) and (5, 4) is:

    • A. (4, 0)
    • B. (9/2, 0)
    • C. (5, 0)
    • D. (3, 0)

30. Q30. The reflection of the point (2, 3) in the line x + y = 1 is:

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