ECAT — Engineering College Admission Test (UET)
Take the ECAT Mathematics — Vectors (Set 5): 30 multiple-choice questions (40 minutes) with instant scoring and a full answer review. Covers 30 ECAT…
What this test covers
The first 5 questions of this test, with the correct answer and explanation.
Q1. Two vectors of magnitudes 7 and 8 are inclined at 60°. The magnitude of their sum is:
Answer: C. 13
Explanation: |a + b|² = 49 + 64 + 2·7·8·cos 60° = 169, so |a + b| = 13.
Q2. Forces F₁ = i + 2j, F₂ = 3i − j and F₃ = pi + qj keep a particle in equilibrium. Then p + q equals:
Answer: B. −5
Explanation: F₁ + F₂ + F₃ = 0 gives p = −4 and q = −1.
Q3. The points A(1, 0, 0), B(0, 1, 0), C(0, 0, 1) and D(1, 1, k) are coplanar when k equals:
Answer: A. −1
Explanation: The plane ABC is x + y + z = 1; D lies on it when 1 + 1 + k = 1, i.e. k = −1. (Equivalently [AB, AC, AD] = 1 + k = 0.)
Q4. If a·b = 8 and |a × b| = 6, then the product |a||b| equals:
Answer: C. 10
Explanation: |a||b|cos θ = 8 and |a||b|sin θ = 6, so (|a||b|)² = 64 + 36 = 100.
Q5. If [a b c] = 3, then the scalar triple product [a + b, b + c, c + a] equals:
Answer: B. 6
Explanation: Expanding the determinant of the coefficient matrix [[1,1,0],[0,1,1],[1,0,1]], whose value is 2, gives 2·3 = 6.
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Q6. The signed scalar component of a = 3i − j + 2k along the direction of b = i + 2j − 2k is:
Q7. The triangle with A(1, 2, 3), B(2, 4, 5) and C(3, 2, 4) has area equal to:
Q8. A force of magnitude 14 N acts in the direction of 2i + 3j + 6k and moves its point of application through the displacement 2i − j + k (in metres). The work done is:
Q9. A force F = 2i − j acts at the point A(1, 3, 0). Its moment about the point B(0, 1, 0) is:
Q10. A unit vector along the bisector of the angle between a = 3i + 4j and b = i is:
Q11. In what ratio does the xy-plane divide the segment joining A(1, 2, 3) and B(3, −2, −6)?
Q12. For the triangle with vertices A(1, 1, 1), B(2, 3, 3) and C(3, 2, −1), the right angle is at:
Q13. If a = 2i + j + k and b = i − j + 2k, then |a × b| equals:
Q14. The parallelepiped whose edges at a vertex are a = i + 2j + 3k, b = j + 4k and c = 5i + 6j has volume:
Q15. For distinct vectors a and b, the points with position vectors a, b and 2a − b are:
Q16. For non-zero vectors, if a·b = a·c and a × b = a × c with a ≠ 0, then:
Q17. If |a| = 3, |b| = 5 and |a + b| = 7, then |a − b| equals:
Q18. The component of a = 2i + j + 2k perpendicular to b = i + j has magnitude:
Q19. If G is the centroid of triangle ABC and P is any point, then PA + PB + PC (as vectors) equals:
Q20. If the diagonals of a parallelogram are perpendicular, the parallelogram must be a:
Q21. A vector of magnitude 6 perpendicular to both i + j and j + k is:
Q22. For any vectors a, b, c, the sum a × (b × c) + b × (c × a) + c × (a × b) equals:
Q23. A vector has direction cosines 1/3 and 2/3 with the x-axis and y-axis, and makes an acute angle with the z-axis. Its direction cosine with the z-axis is:
Q24. For a = i − 2j + 2k and b = 2i + 3j + 6k, cos θ (θ the angle between a and b) equals:
Q25. The projection of the vector i + j on the line through the origin in direction i − j is:
Q26. If a = i + 2j, b = j + 3k and c = 2i + k, then a·(b × c) equals:
Q27. A force 3j + 4k acts at the point (2, 0, 0). The magnitude of its moment about the origin is:
Q28. The distance of the point (3, 0, 0) from the line through the origin in the direction i + 2j + 2k is:
Q29. If a and b are unit vectors and |a + b| = √3 |a − b|, the angle between a and b is:
Q30. The value of the scalar triple product a·(a × b) + b·(a × b) is: