ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Straight Lines and Linear Programming (Set 6): 30 multiple-choice questions (40 minutes) with instant scoring and a full…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. The foot of the perpendicular from the point (4, 1) to the line x + y = 3 is:
Answer: A. (3, 0)
Explanation: The perpendicular direction is (1, 1). Foot = (4, 1) − [(4 + 1 − 3)/2](1, 1) = (4 − 1, 1 − 1) = (3, 0).
Q2. The reflection of the point (4, 1) in the line y = x + 1 is:
Answer: B. (0, 5)
Explanation: Reflection in y = x + 1 maps (a, b) to (b − 1, a + 1). So (4, 1) → (0, 5). Check: the midpoint (2, 3) lies on the line and the segment direction (−4, 4) is perpendicular to it.
Q3. The lines 2x + y = 5, x − y = 1 and kx + 2y = 7 are concurrent. Then k equals:
Answer: D. 5/2
Explanation: The first two meet at (2, 1) (adding gives 3x = 6). The third passes through it: 2k + 2 = 7 ⇒ k = 5/2.
Q4. The acute angle between the lines 3x − y = 2 and x − 2y = 5 is:
Answer: B. 45°
Explanation: The slopes are 3 and 1/2. tan θ = |(3 − 1/2)/(1 + 3/2)| = (5/2)/(5/2) = 1, so θ = 45°.
Q5. The perpendicular bisector of the segment joining (1, 2) and (3, −2) has the equation:
Answer: C. x − 2y − 2 = 0
Explanation: Midpoint (2, 0); the segment has slope −2, so the bisector has slope 1/2: y = (x − 2)/2, i.e. x − 2y − 2 = 0.
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Q6. The line through the intersection of x + y = 4 and 2x − y = 2 and through the point (0, 1) has the equation:
Q7. The perpendicular distance from the origin to the line through (3, 0) and (0, 4) is:
Q8. A line through (4, 3) has an x-intercept twice its y-intercept (both positive). Its equation is:
Q9. The inclination (angle with the positive x-axis) of the line 3x + √3y = 5 is:
Q10. The points (k, 2), (3, −1) and (5, −4) are collinear. Then k equals:
Q11. The pair of lines x² − 5xy + 6y² = 0 has an acute angle between them equal to:
Q12. The equation 2x² − 3xy + ky² = 0 represents two perpendicular lines. Then k equals:
Q13. The point on the line y = x that is equidistant from (1, 0) and (0, 3) is:
Q14. The x-axis divides the line segment joining (2, −3) and (5, 6) in the ratio:
Q15. The distance of the point (2, 3) from the line through (1, 1) and (4, 5) is:
Q16. A line through (1, 2) meets the positive axes and forms a triangle of area 4 with them. Its y-intercept is:
Q17. The lines kx + 3y = 2 and 2x + (k + 1)y = 3 are parallel for a positive value of k equal to:
Q18. A triangle has vertices (0, 0), (6, 0) and (0, 8). Its perimeter is:
Q19. The perpendicular distance from the origin to the line √3x + y = 4 is:
Q20. In a linear programme the objective z = px + qy (p, q > 0) takes its maximum value at both corner points (2, 3) and (4, 1) of the feasible region. Then:
Q21. Which of the following points satisfies all of 2x + y ≤ 8, x + 2y ≤ 7, x ≥ 0, y ≥ 0?
Q22. Minimise C = 4x + 3y subject to 2x + y ≥ 8, x + 2y ≥ 8, x ≥ 0, y ≥ 0. The minimum value is:
Q23. A workshop earns Rs. 40 per chair and Rs. 30 per table. A chair needs 2 hours of lathe time and 1 hour of polishing; a table needs 1 hour of lathe time and 2 hours of polishing. Lathe time is limited to 100 hours and polishing to 80 hours. The maximum profit in rupees is:
Q24. The centroid of the triangle with vertices (1, 2), (3, −4) and (5, 8) is:
Q25. The distance between the parallel lines y = 2x + 3 and y = 2x − 4 is:
Q26. The triangle formed by the lines y = x, y = 2x and x + y = 6 has area:
Q27. The circumcentre of the triangle with vertices (0, 0), (4, 0) and (1, 3) is:
Q28. The area enclosed by the graph of |x|/3 + |y|/2 = 1 is:
Q29. Minimise z = 3x + 5y subject to x + 2y ≥ 6, 2x + y ≥ 6, x ≥ 0, y ≥ 0. The minimum value of z is:
Q30. The feasible region of a linear programme has vertices (0, 0), (4, 0), (3, 2) and (0, 3). The maximum of z = x + 4y over the region is: