ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Straight Lines and Linear Programming (Set 5): 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 point on the y-axis that is equidistant from (2, 0) and (0, 4) is:
Answer: D. (0, 3/2)
Explanation: For (0, k): 4 + k² = (k − 4)², so 4 = −8k + 16 and k = 3/2.
Q2. The equation x² − y² + 2x + 1 = 0 represents two lines meeting at:
Answer: A. (−1, 0)
Explanation: (x + 1)² = y², so x + 1 = ±y, i.e. x − y + 1 = 0 and x + y + 1 = 0. They meet where y = 0, x = −1.
Q3. The slope of the bisector of the acute angle between the x-axis and the line 4y = 3x is:
Answer: B. 1/3
Explanation: If tan θ = 3/4 (cos θ = 4/5), tan(θ/2) = sin θ/(1 + cos θ) = (3/5)/(9/5) = 1/3.
Q4. The line through the intersection of x + y = 4 and x − y = 2 and parallel to 2x + 3y = 1 is:
Answer: B. 2x + 3y = 9
Explanation: The lines meet at (3, 1). A parallel line is 2x + 3y = c with c = 6 + 3 = 9.
Q5. In triangle OBC with O(0, 0), B(4, 0), C(1, 3), the altitude from O to side BC has length:
Answer: D. 2√2
Explanation: BC has slope −1, so its line is x + y = 4. Distance from the origin: 4/√2 = 2√2.
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Q6. The orthocentre of the triangle with vertices (0, 0), (4, 0) and (1, 3) is:
Q7. The circumcentre of the triangle with vertices (0, 0), (6, 0) and (2, 4) is:
Q8. A line through (4, 3) has non-zero intercepts, and its x-intercept is twice its y-intercept. Its equation is:
Q9. The area of the quadrilateral with vertices (0, 0), (4, 0), (5, 3) and (1, 5), taken in that order, is:
Q10. The slope of a line inclined at 105° to the positive x-axis is:
Q11. Eliminating t from x = 2 + 3t, y = 1 + 4t gives the line:
Q12. The origin is shifted to (2, −1) without rotating the axes. The new coordinates of the point (5, 3) are:
Q13. The origin is shifted to (1, −1). The line 2x − y = 3 then has equation:
Q14. Axes are rotated anticlockwise through 90° (X = x cos θ + y sin θ, Y = −x sin θ + y cos θ). The point (3, 2) then has coordinates:
Q15. The axes are rotated through 45° (anticlockwise). The equation xy = 8 becomes:
Q16. The equation x² + kxy + 9y² = 0 represents coincident lines when k equals:
Q17. For the lines given by 3x² + 10xy + 3y² = 0, the tangent of the acute angle between them is:
Q18. Maximise z = 3x + 5y subject to x ≤ 4, y ≤ 6, 3x + 2y ≤ 18, x, y ≥ 0. The maximum value of z is:
Q19. A workshop makes chairs (x) and tables (y) with profits of Rs 40 and Rs 50. Carpentry time is 2x + 3y ≤ 120 hours and x + y ≤ 50 (finishing). The maximum profit is:
Q20. Minimise z = 4x + 6y subject to 2x + y ≥ 10, x + 3y ≥ 15, x, y ≥ 0. The minimum value of z is:
Q21. For the problem: maximise z = x + y subject to x − y ≤ 1, x ≥ 0, y ≥ 0, which statement is correct?
Q22. Maximise z = 3x + 6y subject to x + 2y ≤ 10, x ≤ 8, x, y ≥ 0. The maximum value 30 is attained:
Q23. The region 1 ≤ x ≤ 4, 0 ≤ y ≤ x is bounded by four lines. Its area in square units is:
Q24. Maximise z = 5x + 3y subject to x + y ≤ 6, 2x + y ≤ 10, x, y ≥ 0. The maximum value of z is:
Q25. Maximise z = x + y subject to x + y ≤ 2, x + y ≥ 5, x, y ≥ 0. Which statement is correct?
Q26. The median from A(0, 0) of the triangle with vertices A(0, 0), B(4, 2) and C(2, 6) has equation:
Q27. A point P moves so that PA² − PB² = 16, where A = (0, 0) and B = (4, 0). The locus of P is the line:
Q28. The points on the line y = x that are at distance √2 from the line x + y = 2 are:
Q29. For every real k, the line kx − y + 3 − 2k = 0 passes through the fixed point:
Q30. The distance of the origin from the line x/3 + y/4 = 1 is: