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 67 questions on this lesson, each with an explanation.
Find the equation of the plane through the origin with normal vector (1, −2, 2).
Correct answer: (A) x − 2y + 2z = 0
Explanation: Step 1: Plane through origin: a x + b y + c z = 0 (no constant term). Step 2: Normal (1, −2, 2) ⇒ x − 2y + 2z = 0. ∴ x − 2y + 2z = 0
Which of the following points lies on the plane x + 2y − z = 4 ?
Correct answer: (D) (1, 1, −1)
Explanation: Step 1: Substitute each point into x + 2y − z and check if it equals 4. Step 2: (1, 1, −1): 1 + 2(1) − (−1) = 1 + 2 + 1 = 4. ✓ Step 3: The others give 4? (1,1,1)→2, (2,0,1)→1, (0,0,4)→−4 — none equal 4. ∴ (1, 1, −1)
Find the distance from the point (4, −1, 2) to the plane 2x + y − 2z + 3 = 0.
Correct answer: (B) 2
Explanation: Step 1: Distance = |2(4) + 1(−1) − 2(2) + 3| / √(2²+1²+(−2)²). Step 2: = |8 − 1 − 4 + 3| / √9 = |6| / 3. ∴ distance = 2
Find the value of k for which the planes 2x + ky − z = 1 and x + 2y + 4z = 3 are perpendicular.
Correct answer: (D) 1
Explanation: Step 1: Two planes are perpendicular when their normals are perpendicular: n₁·n₂ = 0. Step 2: n₁ = (2, k, −1), n₂ = (1, 2, 4): n₁·n₂ = 2 + 2k − 4 = 2k − 2. Step 3: 2k − 2 = 0 ⇒ k = 1. ∴ k = 1
Find the equation of the plane passing through (1, 2, 3) and perpendicular to the line with direction (2, −1, 2).
Correct answer: (A) 2x − y + 2z − 6 = 0
Explanation: Step 1: A plane perpendicular to a line has that direction as its normal: n = (2, −1, 2). Step 2: 2(x − 1) − 1(y − 2) + 2(z − 3) = 0. Step 3: 2x − 2 − y + 2 + 2z − 6 = 0 ⇒ 2x − y + 2z − 6 = 0. ∴ 2x − y + 2z − 6 = 0
Find the distance between the parallel planes 2x − y + 2z − 6 = 0 and 2x − y + 2z − 12 = 0.
Correct answer: (A) 2
Explanation: Step 1: For parallel planes, distance = |d₁ − d₂| / √(a²+b²+c²). Step 2: = |(−6) − (−12)| / √(4+1+4) = 6 / 3. ∴ distance = 2
If the two planes 3x − y + kz = 2 and 6x + my − 4z = 7 are parallel, then k + m = ..........
Correct answer: (B) −4
Explanation: Step 1: Parallel planes have parallel normals: 3/6 = −1/m = k/(−4). Step 2: 3/6 = 1/2, so −1/m = 1/2 gives m = −2. Step 3: k/(−4) = 1/2 gives k = −2. Step 4: k + m = −4.
Find a direction vector of the line of intersection of the planes x + y + z = 1 and x − y + z = 3.
Correct answer: (A) (1, 0, −1)
Explanation: Step 1: The line of intersection is perpendicular to both normals, so its direction = n₁ × n₂. Step 2: n₁ = (1, 1, 1), n₂ = (1, −1, 1): n₁ × n₂ = (1·1−−, ...) = (2, 0, −2). Step 3: Simplify (2, 0, −2) = 2(1, 0, −1). ∴ direction = (1, 0, −1)
Determine the relationship between the line (x, y, z) = (0, 0, 1) + t(1, 2, 2) and the plane 2x − 2y + z = 5.
Correct answer: (C) the line is parallel to the plane (not in it)
Explanation: Step 1: d·n = (1)(2) + (2)(−2) + (2)(1) = 2 − 4 + 2 = 0, so the line is parallel to the plane. Step 2: Test the point (0, 0, 1): 2(0) − 2(0) + 1 = 1 ≠ 5, so it is not in the plane. ∴ the line is parallel to the plane (not lying in it)
Find the value of a for which the line (x, y, z) = (2, a, 1) + t(1, 2, −1) lies completely in the plane x + y + 3z = 8.
Correct answer: (D) 3
Explanation: Step 1: Direction (1, 2, −1)·normal (1, 1, 3) = 1 + 2 − 3 = 0, so the line is parallel to the plane for any a. Step 2: The line lies in the plane only if its point (2, a, 1) satisfies it: 2 + a + 3(1) = 8. Step 3: 5 + a = 8 ⇒ a = 3. ∴ a = 3
The first 10 questions in every subject are free, each with an explanation.