ECAT — Engineering College Admission Test (UET)

ECAT Mathematics — Straight Lines and Linear Programming (Set 7)

Take the ECAT Mathematics — Straight Lines and Linear Programming (Set 7): 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.

Q1. The line through the intersection of x + y = 4 and 2x − y = 2 and perpendicular to 3x + 2y = 1 is:
- A. 2x − 3y + 2 = 0 ✓
- B. 3x − 2y − 2 = 0
- C. 2x + 3y − 10 = 0
- D. 3x + 2y − 10 = 0
Answer: A. 2x − 3y + 2 = 0
Explanation: The lines meet at (2, 2). A line perpendicular to 3x + 2y = 1 has the form 2x − 3y + k = 0. Through (2, 2): 4 − 6 + k = 0, so k = 2.
Q2. The area of the triangle bounded by the lines y = x, y = 2x and x + y = 3 is:
- A. 1
- B. 3/4 ✓
- C. 3/2
- D. 9/4
Answer: B. 3/4
Explanation: The vertices are (0, 0), (1, 2) and (3/2, 3/2). Shoelace: ½|0·2 − 1·0 + 1·(3/2) − (3/2)·2 + 0| = ½|3/2 − 3| = 3/4.
Q3. The circumcentre of the triangle with vertices (1, 1), (5, 1) and (3, 5) is:
- A. (3, 7/3)
- B. (3, 3)
- C. (3, 5/2) ✓
- D. (3, 2)
Answer: C. (3, 5/2)
Explanation: The perpendicular bisector of the side from (1, 1) to (5, 1) is x = 3. Equating squared distances from (3, y) to (1, 1) and (3, 5): 4 + (y − 1)² = (y − 5)², so 8y = 20 and y = 5/2. (The centroid is (3, 7/3).)
Q4. The distance between the parallel lines 5x − 12y + 7 = 0 and 10x − 24y − 13 = 0 is:
- A. 20/26
- B. 1
- C. 27/26 ✓
- D. 20/13
Answer: C. 27/26
Explanation: Rewrite the second line as 5x − 12y − 13/2 = 0. The distance is |7 − (−13/2)|/√(25 + 144) = (27/2)/13 = 27/26.
Q5. The locus of points whose distances from the coordinate axes have a sum of 4 (|x| + |y| = 4) encloses an area of:
- A. 8
- B. 64
- C. 32 ✓
- D. 16
Answer: C. 32
Explanation: The locus is a square with vertices (±4, 0) and (0, ±4). Its diagonals are 8 each, so area = ½·8·8 = 32.

The other 25 questions

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Q6. For real numbers a, b, c in arithmetic progression, every line ax + by + c = 0 passes through the fixed point:
- A. (−1, 2)
- B. (1, 2)
- C. (1, −2)
- D. (2, −1)
Q7. Lines through (2, 3) make an angle of 45° with the line 2x − y = 1. Their slopes are:
- A. 1 and −1
- B. 3 and −1/3
- C. −3 and 3
- D. −3 and 1/3
Q8. The lines given by x² − y² = 0 together with the line x = 4 form a triangle whose area is:
- A. 32
- B. 8
- C. 16
- D. 4
Q9. Maximise Z = 5x + 4y subject to 2x + y ≤ 10, x + 2y ≤ 8, x ≥ 0, y ≥ 0. The maximum value of Z is:
- A. 25
- B. 30
- C. 28
- D. 16
Q10. Minimise Z = 2x + 5y subject to x + y ≥ 4, x + 3y ≥ 6, x ≥ 0, y ≥ 0. The minimum value of Z is:
- A. 12
- B. 11
- C. 10
- D. 20
Q11. In a linear programming problem with feasible region having corner points (5, 0) and (4, 2) on the line 2x + y = 10, the objective Z = px + qy (p, q > 0) attains the same maximum at both points. Then p : q equals:
- A. 3 : 2
- B. 1 : 1
- C. 2 : 1
- D. 1 : 2
Q12. Which of the following points satisfies both 3x + 4y < 12 and x − y > 1?
- A. (2, 0)
- B. (3, 3)
- C. (1, 2)
- D. (0, 4)
Q13. The points (1, 1), (2, 8) and (k, k³) are collinear. For k ≠ 1, 2, the value of k is:
- A. −1
- B. 3
- C. −3
- D. 0
Q14. The vertices A(1, 1), B(5, 2), C(4, 6), D(0, 4), taken in order, form a quadrilateral of area:
- A. 31/2
- B. 16
- C. 29/2
- D. 15
Q15. The axes are rotated through 45° (x = (X − Y)/√2, y = (X + Y)/√2). The curve xy = 1 becomes:
- A. X² − Y² = 2
- B. X² + Y² = 2
- C. XY = 2
- D. X² − Y² = 1
Q16. The points on the y-axis at distance 3 from the line 3x + 4y − 5 = 0 have y-coordinates whose sum is:
- A. 5
- B. −5/2
- C. 15/4
- D. 5/2
Q17. A right triangle has vertices (0, 0), (6, 0) and (0, 8). The coordinates of its incentre are:
- A. (2, 2)
- B. (3, 4)
- C. (2, 3)
- D. (3, 3)
Q18. The normal form x cos α + y sin α = p of the line √3x + y = 4 has:
- A. α = 30°, p = 4
- B. α = 60°, p = 2
- C. α = 60°, p = 4
- D. α = 30°, p = 2
Q19. A line through the point (4, 3) cuts the positive axes. The least possible area of the triangle it forms with the axes is:
- A. 25
- B. 24
- C. 12
- D. 32
Q20. The pair of lines 6x² − 5xy + y² = 0 has an acute angle θ between them with tan θ equal to:
- A. 1/7
- B. 1
- C. 5/7
- D. 1/6
Q21. The equation 3x² + 10xy + 8y² + 14x + 22y + λ = 0 represents a pair of straight lines for λ equal to:
- A. 8
- B. 15
- C. 12
- D. 20
Q22. The region defined by x ≥ 1, y ≥ 1 and x + y ≤ 5 has an area of:
- A. 9/2
- B. 4
- C. 3
- D. 9
Q23. Two lines pass through (2, 3) at distance 2 from the origin. One is x = 2. The other is:
- A. y = 3
- B. 5x + 12y − 46 = 0
- C. 12x − 5y − 9 = 0
- D. 5x − 12y + 26 = 0
Q24. The vertices of a triangle are (a, b + c), (b, c + a) and (c, a + b). Its area is:
- A. a + b + c
- B. 0
- C. ½(a + b + c)
- D. abc
Q25. The x-axis divides the segment joining (−2, 3) and (4, −6) in the ratio:
- A. 3 : 1
- B. 1 : 2
- C. 2 : 1
- D. 1 : 3
Q26. Which statement about a linear programming problem with a bounded, non-empty feasible region is correct?
- A. The objective is largest where the feasible region is widest
- B. The maximum always occurs at the corner point nearest the origin
- C. Both the maximum and the minimum of the objective occur at corner points
- D. The maximum occurs at an interior point when the coefficients are positive
Q27. The foot of the perpendicular from the point (1, 2) to the line x + 2y = 10 is:
- A. (4, 3)
- B. (0, 5)
- C. (2, 4)
- D. (3, 3.5)
Q28. A line through (3, 4) has intercepts on the axes whose sum is 14. The sum of all possible x-intercepts of such lines is:
- A. 14
- B. 6
- C. 7
- D. 13
Q29. The line through (0, 1) with inclination 45° is at what distance from the point (4, 1)?
- A. 4
- B. 2
- C. 2√2
- D. √2
Q30. The midpoints of the sides of a triangle are (1, 2), (3, 4) and (5, 0). The centroid of the triangle is:
- A. (3, 3)
- B. (9, 6)
- C. (3, 2)
- D. (2, 3)