Pure Mathematics — First: Algebra · Mathematics. These are 10 questions from the Magmo3 bank on this lesson, from easy to hard. Solve each one, then open "Show answer and explanation". Magmo3 has 37 questions on this lesson, each with an explanation.
Find the magnitude of the vector v = 2î − 3ĵ + 6k̂.
Correct answer: (B) 7
Explanation: Step 1: ‖v‖ = √(x² + y² + z²). Step 2: = √(2² + (−3)² + 6²) = √(4 + 9 + 36) = √49. Step 3: √49 = 7. ∴ ‖v‖ = 7
If ‖a‖ = 4, find ‖3a‖.
Correct answer: (C) 12
Explanation: Step 1: For a scalar λ, ‖λa‖ = |λ| · ‖a‖. Step 2: ‖3a‖ = 3 · ‖a‖ = 3 · 4. Step 3: = 12. ∴ ‖3a‖ = 12
Find ‖v‖ for v = î − 2ĵ + 2k̂.
Correct answer: (D) 3
Explanation: Step 1: ‖v‖ = √(1² + (−2)² + 2²) = √(1 + 4 + 4) = √9. Step 2: √9 = 3. ∴ ‖v‖ = 3
Which of the following can NOT be the measures of the direction angles (θx, θy, θz) of a vector in space?
Correct answer: (C) (30°, 45°, 60°)
Explanation: Step 1: Direction angles must satisfy cos²θx + cos²θy + cos²θz = 1. Step 2: (30°, 45°, 60°): 3/4 + 1/2 + 1/4 = 3/2 ≠ 1. Step 3: The others give 1/2 + 1/2 + 0, 1/4 + 1/2 + 1/4 and 0 + 3/4 + 1/4, each equal to 1. ∴ (30°, 45°, 60°) is impossible.
A vector A whose norm is 12 units makes angles of measures 60°, 45° and 120° with the positive directions of the x, y and z axes respectively. Then A = ..........
Correct answer: (B) 6î + 6√2 ĵ − 6k̂
Explanation: Step 1: A = ‖A‖(cos θx î + cos θy ĵ + cos θz k̂). Step 2: cos 60° = 1/2, cos 45° = √2/2, cos 120° = −1/2. Step 3: A = 12(1/2 î + (√2/2) ĵ − 1/2 k̂) = 6î + 6√2 ĵ − 6k̂. ∴ 6î + 6√2 ĵ − 6k̂ (cos 120° is negative; using sines gives the 6√3 option).
Find the direction cosines of the vector AB, where A(3, −1, 2) and B(1, 3, −2).
Correct answer: (C) (−1/3, 2/3, −2/3)
Explanation: Step 1: AB = B − A = (−2, 4, −4). Step 2: ‖AB‖ = √(4 + 16 + 16) = 6. Step 3: Direction cosines = (−2/6, 4/6, −4/6) = (−1/3, 2/3, −2/3). ∴ (−1/3, 2/3, −2/3)
If (x² − 5, 2y + 1, 3) = (4, y − 2, z² − 1), find the value of x² + y + z².
Correct answer: (A) 10
Explanation: Step 1: Equal vectors have equal components: x² − 5 = 4, so x² = 9. Step 2: 2y + 1 = y − 2, so y = −3. Step 3: 3 = z² − 1, so z² = 4. Step 4: x² + y + z² = 9 − 3 + 4 = 10. ∴ 10
The vector A = (m² − 9, m + 1, 4) is parallel to the yz-plane and ‖A‖ = 2√5. Find m.
Correct answer: (D) −3
Explanation: Step 1: A is parallel to the yz-plane, so its x-component is 0: m² − 9 = 0 and m = ±3. Step 2: m = 3 gives A = (0, 4, 4) and ‖A‖ = 4√2, which is rejected. Step 3: m = −3 gives A = (0, −2, 4) and ‖A‖ = √20 = 2√5. ∴ m = −3
A vector A has norm 14 units, cos θx = 2/7 and cos θy = 3/7, and it makes an obtuse angle with the positive direction of the z-axis. Then A = ..........
Correct answer: (D) 4î + 6ĵ − 12k̂
Explanation: Step 1: cos²θz = 1 − 4/49 − 9/49 = 36/49, so cos θz = ±6/7. Step 2: θz is obtuse, so cos θz = −6/7. Step 3: A = 14(2/7 î + 3/7 ĵ − 6/7 k̂) = 4î + 6ĵ − 12k̂. ∴ 4î + 6ĵ − 12k̂ (+12k̂ ignores the obtuse angle; −4k̂ forgets to square; 2î + 3ĵ − 6k̂ has norm 7, not 14).
If A(1, −2, 3) and B(3, 0, 2), find the vector of norm 15 units in the direction opposite to AB.
Correct answer: (D) (−10, −10, 5)
Explanation: Step 1: AB = B − A = (2, 2, −1) and ‖AB‖ = √(4 + 4 + 1) = 3. Step 2: The unit vector opposite to AB is −(1/3)(2, 2, −1). Step 3: The required vector = 15 × (−1/3)(2, 2, −1) = −5(2, 2, −1) = (−10, −10, 5). ∴ (−10, −10, 5)
The first 10 questions in every subject are free, each with an explanation.