ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Straight Lines and Linear Programming (Set 1)

Take the ECAT Mathematics — Straight Lines and Linear Programming (Set 1): 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 (−2, 1) and (4, 9) is:

   • A. 14
   • B. 100
   • C. 8
   • D. 10 ✓

   Answer: D. 10

   Explanation: d = √[(4 + 2)² + (9 − 1)²] = √(36 + 64) = 10.

2. Q2. The midpoint of the segment joining (2, −3) and (6, 5) is:

   • A. (4, 1) ✓
   • B. (2, 4)
   • C. (8, 2)
   • D. (4, −1)

   Answer: A. (4, 1)

   Explanation: Midpoint = ((2 + 6)/2, (−3 + 5)/2) = (4, 1).

3. Q3. The slope of the line through (1, 2) and (3, 8) is:

   • A. 3 ✓
   • B. 1/3
   • C. −3
   • D. 5

   Answer: A. 3

   Explanation: m = (8 − 2)/(3 − 1) = 6/2 = 3.

4. Q4. The slope of the line 2x + 3y − 6 = 0 is:

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

   Answer: A. −2/3

   Explanation: Write as y = −(2/3)x + 2, so the slope is −2/3 (in general −a/b).

5. Q5. The equation of the line through (1, 2) with slope 3 is:

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

   Answer: C. y = 3x − 1

   Explanation: Point–slope form: y − 2 = 3(x − 1) ⇒ y = 3x − 1.

The other 25 questions

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6. Q6. The line with x-intercept 3 and y-intercept 4 has equation:

   • A. x + y = 7
   • B. 4x − 3y = 12
   • C. 3x + 4y = 12
   • D. 4x + 3y = 12

7. Q7. For what value of k are the lines 2x + ky = 5 and x − 2y = 3 perpendicular?

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

8. Q8. The line through (2, 3) parallel to 3x − y + 1 = 0 is:

   • A. x + 3y − 11 = 0
   • B. 3x − y + 3 = 0
   • C. 3x + y − 9 = 0
   • D. 3x − y − 3 = 0

9. Q9. The perpendicular distance of the point (3, 4) from the line 3x + 4y − 5 = 0 is:

   • A. 4
   • B. 5
   • C. 4/5
   • D. 20

10. Q10. The distance between the parallel lines 3x + 4y = 5 and 3x + 4y = 15 is:

    • A. 10
    • B. 4
    • C. 2
    • D. 1

11. Q11. The acute angle between the lines y = x and y = √3 x is:

    • A. 15°
    • B. 60°
    • C. 45°
    • D. 30°

12. Q12. The point of intersection of x + y = 5 and x − y = 1 is:

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

13. Q13. The point dividing the join of A(1, 2) and B(7, 8) internally in the ratio 1 : 2 is:

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

14. Q14. The area of the triangle with vertices (0, 0), (8, 0) and (0, 5) is:

    • A. 10 square units
    • B. 13 square units
    • C. 20 square units
    • D. 40 square units

15. Q15. For what value of k are the points (1, 2), (3, 6) and (5, k) collinear?

    • A. 10
    • B. 8
    • C. 12
    • D. 9

16. Q16. When 3x + 4y = 10 is written in normal form x cos α + y sin α = p, the value of p is:

    • A. 10/7
    • B. 5
    • C. 2
    • D. 10

17. Q17. The centroid of the triangle with vertices (0, 0), (6, 0) and (0, 9) is:

    • A. (3, 3)
    • B. (3, 4.5)
    • C. (6, 9)
    • D. (2, 3)

18. Q18. The angle of inclination of the line y = −x + 2 is:

    • A. −45°
    • B. 120°
    • C. 135°
    • D. 45°

19. Q19. The homogeneous equation x² − 5xy + 6y² = 0 represents the pair of lines:

    • A. x = −2y and x = −3y
    • B. x = y and x = 6y
    • C. y = 2x and y = 3x
    • D. x = 2y and x = 3y

20. Q20. The lines represented by ax² + 2hxy + by² = 0 are perpendicular when:

    • A. h = 0
    • B. a + b = 0
    • C. h² = ab
    • D. a = b

21. Q21. Which point lies on the line 2x − 3y + 6 = 0?

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

22. Q22. The equation of the line through (2, 1) and (4, 5) is:

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

23. Q23. In a linear programming problem, the optimal value of the objective function (if it exists) occurs:

    • A. At the centre of the feasible region
    • B. At any interior point
    • C. At a corner point of the feasible region
    • D. Only at the origin

24. Q24. Maximise z = 3x + 2y subject to x + y ≤ 4, x ≥ 0, y ≥ 0. The maximum value of z is:

    • A. 8
    • B. 10
    • C. 12
    • D. 20

25. Q25. Minimise z = 2x + 5y subject to x + y ≥ 3, x ≥ 0, y ≥ 0. The minimum value of z is:

    • A. 6
    • B. 7
    • C. 0
    • D. 15

26. Q26. Which point satisfies both 2x + y ≤ 6 and x − y ≥ 0?

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

27. Q27. The constraints x ≥ 0 and y ≥ 0 restrict the feasible region to:

    • A. The first quadrant, including the axes
    • B. The first quadrant, excluding the axes
    • C. The second quadrant
    • D. The region below the x-axis

28. Q28. The graph of the inequality 3x + 2y < 6 is:

    • A. Only the line 3x + 2y = 6
    • B. The closed half-plane containing the origin, with a solid line
    • C. The open half-plane not containing the origin
    • D. The open half-plane containing the origin, with the line drawn dashed

29. Q29. Maximise z = x + 3y subject to x + 2y ≤ 8, x ≤ 4, x ≥ 0, y ≥ 0. The maximum of z is:

    • A. 12
    • B. 4
    • C. 10
    • D. 24

30. Q30. The orthocentre of the triangle with vertices (0, 0), (5, 0) and (0, 7) is:

    • A. (5/3, 7/3)
    • B. (0, 0)
    • C. (5, 7)
    • D. (2.5, 3.5)