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.
The point (0, −3, 5) lies in:
Correct answer: (D) the yz-plane
Explanation: Step 1: Its x-coordinate is 0. Step 2: All points of the form (0, y, z) lie in the yz-plane, whose equation is x = 0. ∴ the yz-plane
Find the distance between the points A(1, 2, 3) and B(4, 6, 3).
Correct answer: (C) 5
Explanation: Step 1: AB = B − A = (4−1, 6−2, 3−3) = (3, 4, 0). Step 2: ‖AB‖ = √(3² + 4² + 0²) = √(9 + 16 + 0) = √25. Step 3: √25 = 5. ∴ The distance = 5
Find the distance from the origin O to the point P(2, 3, 6).
Correct answer: (A) 7
Explanation: Step 1: OP = √(2² + 3² + 6²) = √(4 + 9 + 36) = √49. Step 2: √49 = 7. ∴ The distance = 7
Find the point on the z-axis that is equidistant from A(3, 1, 2) and B(1, −1, −2).
Correct answer: (C) (0, 0, 1)
Explanation: Step 1: A point on the z-axis has the form P = (0, 0, z). Step 2: PA² = 9 + 1 + (z − 2)² and PB² = 1 + 1 + (z + 2)². Step 3: PA = PB gives 10 + z² − 4z + 4 = 2 + z² + 4z + 4, so 8z = 8 and z = 1. Step 4: Check: PA² = 9 + 1 + 1 = 11 and PB² = 1 + 1 + 9 = 11. ∴ P = (0, 0, 1)
C(1, −2, 3) is the midpoint of AB, where A(x, 4, −1) and B(5, y, z). Find the value of x + y + z.
Correct answer: (B) −4
Explanation: Step 1: (x + 5)/2 = 1, so x = −3. Step 2: (4 + y)/2 = −2, so y = −8. Step 3: (−1 + z)/2 = 3, so z = 7. Step 4: x + y + z = −3 − 8 + 7 = −4. ∴ −4
The distance between the point P(k, 3, 2k) and the y-axis is 5√5 length units. Find the positive value of k.
Correct answer: (B) 5
Explanation: Step 1: The foot of the perpendicular from P to the y-axis is (0, 3, 0). Step 2: Distance = √(k² + (2k)²) = √(5k²) = √5 k for k positive. Step 3: √5 k = 5√5, so k = 5. ∴ k = 5
Find the perimeter of the triangle whose vertices are O(0, 0, 0), A(2, 3, 6) and B(2, 3, 0).
Correct answer: (B) 13 + √13
Explanation: Step 1: OA = √(4 + 9 + 36) = 7. Step 2: OB = √(4 + 9 + 0) = √13. Step 3: AB = √(0 + 0 + 36) = 6. Step 4: Perimeter = 7 + √13 + 6 = 13 + √13. ∴ 13 + √13 length units
The points A(−1, 2, 0), B(1, 3, 2) and C(5, 5, 6):
Correct answer: (B) are collinear, and B lies between A and C
Explanation: Step 1: AB = √(4 + 1 + 4) = 3. Step 2: BC = √(16 + 4 + 16) = 6. Step 3: AC = √(36 + 9 + 36) = 9. Step 4: AB + BC = 3 + 6 = 9 = AC, so the three points lie on one straight line with B between A and C. ∴ collinear, B between A and C
The points A(4, 1, 1), B(1, 4, 1) and C(1, 1, k) are the vertices of a triangle. Find the value(s) of k that make the triangle ABC equilateral.
Correct answer: (C) k = 4 or k = −2
Explanation: Step 1: AB² = (1 − 4)² + (4 − 1)² + 0² = 18. Step 2: AC² = 9 + 0 + (k − 1)² and BC² = 0 + 9 + (k − 1)², so AC = BC for every k. Step 3: Equilateral needs AC² = AB²: 9 + (k − 1)² = 18, so (k − 1)² = 9 and k − 1 = ±3. ∴ k = 4 or k = −2
A lies on the x-axis, B on the y-axis and C on the z-axis. If (2, 3, 0) is the midpoint of AB and (0, 3, 4) is the midpoint of BC, find the length of AC.
Correct answer: (A) 4√5
Explanation: Step 1: Let A = (a, 0, 0), B = (0, b, 0), C = (0, 0, c). Step 2: Midpoint of AB = (a/2, b/2, 0) = (2, 3, 0), so a = 4 and b = 6. Step 3: Midpoint of BC = (0, b/2, c/2) = (0, 3, 4), so c = 8. Step 4: AC = √((0 − 4)² + 0² + (8 − 0)²) = √80 = 4√5. ∴ AC = 4√5 length units
The first 10 questions in every subject are free, each with an explanation.