Lines and Planes in 3D Practice Questions
SPM (Malaysia) · SPM Mathematics · 151 free MCQs with instant results and detailed explanations.
151
Total
46
Easy
76
Medium
29
Hard
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Sample Questions from Lines and Planes in 3D
Here are 10 sample questions. Start a quiz to get randomized questions with scoring.
Q1
Easy
What is the distance between the point (3, 4, 5) and the origin (0, 0, 0) in 3D space?
A.
7
B.
5.74
C.
6.16
D.
8
Show Answer & Explanation
Correct Answer: C
The distance is calculated using the formula โ(xยฒ + yยฒ + zยฒ). For the point (3, 4, 5), the distance is โ(3ยฒ + 4ยฒ + 5ยฒ) = โ(9 + 16 + 25) = โ50 โ 7.07.
Q2
Easy
Which of the following represents a plane in 3D space?
A.
x + y + z = 1
B.
xยฒ + yยฒ + zยฒ = 1
C.
z = 3
D.
x + y = z + 2
Show Answer & Explanation
Correct Answer: A
The equation x + y + z = 1 is a linear equation in three variables and represents a plane in 3D space, whereas xยฒ + yยฒ + zยฒ = 1 is a sphere equation.
Q3
Easy
If the line defined by the parametric equations x = 1 + 2t, y = 3 - t, z = 4 + 3t intersects the plane defined by the equation 2x + y - z = 1, what is the value of t at the intersection?
A.
1
B.
0
C.
2
D.
-1
Show Answer & Explanation
Correct Answer: A
To find the intersection, substitute the parametric equations into the plane equation. When t = 1, substituting gives valid coordinates that satisfy the plane equation.
Q4
Medium
What is the equation of a plane that passes through the point (1, 2, 3) and is perpendicular to the vector (2, -1, 4)?
A.
2x - y + 4z = 10
B.
2x - y + 4z = 5
C.
2x - y + 4z = 2
D.
2x + y - 4z = 3
Show Answer & Explanation
Correct Answer: A
The equation of a plane can be expressed as Ax + By + Cz = D. To find D, substitute the point (1, 2, 3) into the equation formed by the normal vector (2, -1, 4). This gives 2(1) - 1(2) + 4(3) = 10, hence the equation is 2x - y + 4z = 10.
Q5
Medium
Find the distance from the point P(3, 4, 5) to the plane defined by the equation x + 2y + 3z = 12.
A.
1
B.
2
C.
3
D.
4
Show Answer & Explanation
Correct Answer: B
The distance d from a point (x0, y0, z0) to the plane Ax + By + Cz + D = 0 is given by the formula d = |Ax0 + By0 + Cz0 + D| / sqrt(A^2 + B^2 + C^2). Here, A=1, B=2, C=3, D=-12. Substituting gives d = |1*3 + 2*4 + 3*5 - 12| / sqrt(1^2 + 2^2 + 3^2) = 2.
Q6
Medium
Which of the following lines represented by parametric equations intersects with the plane x + y + z = 6?
A.
x = 1 + t, y = 2 + t, z = 3 + t
B.
x = 2t, y = t, z = 6 - t
C.
x = 0, y = 0, z = 6 - t
D.
x = t, y = 2t, z = 3t
Show Answer & Explanation
Correct Answer: B
To check intersection, substitute the parametric equations into the plane equation. For option B, substituting gives t + 2t + (6 - t) = 6, confirming that they intersect.
Q7
Medium
Determine the angle between the two planes given by the equations 2x - 3y + z = 1 and 4x + y - z = 2.
A.
30 degrees
B.
60 degrees
C.
90 degrees
D.
45 degrees
Show Answer & Explanation
Correct Answer: B
The angle ฮธ between two planes can be found using the normal vectors. The normals are (2, -3, 1) and (4, 1, -1). Using the dot product formula, cos(ฮธ) = (n1ยทn2) / (|n1| |n2|) results in cos(ฮธ) = 0.5, giving ฮธ = 60 degrees.
Q8
Hard
Given the equations of two planes: P1: 2x - 3y + z = 7 and P2: x + y - 4z = -5. Determine whether these planes are parallel, intersecting, or identical.
A.
The planes are parallel.
B.
The planes are intersecting.
C.
The planes are identical.
D.
The planes are perpendicular.
Show Answer & Explanation
Correct Answer: B
The planes are intersecting because their normal vectors (2, -3, 1) and (1, 1, -4) are not scalar multiples of each other, indicating they are not parallel or identical.
Q9
Hard
Given the points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9), determine if the points are collinear.
A.
Yes, they are collinear.
B.
No, they are not collinear.
C.
They are collinear only in the XY plane.
D.
They are collinear only in the XZ plane.
Show Answer & Explanation
Correct Answer: A
The points A, B, and C lie on the same straight line in 3D space, thus making them collinear. This is confirmed by checking the direction ratios from A to B and B to C, which are proportional.
Q10
Hard
A plane is defined by the equation 2x + 3y - z = 12. What is the distance from the point P(1, -1, 4) to this plane?
A.
5 units
B.
6 units
C.
7 units
D.
8 units
Show Answer & Explanation
Correct Answer: B
The distance from a point to a plane can be found using the formula |Ax + By + Cz + D| / sqrt(A^2 + B^2 + C^2). For the point P(1, -1, 4), substitute the values into the formula to calculate the distance.
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Lines and Planes in 3D โ SPM (Malaysia) SPM Mathematics Practice Questions Online
This page contains 151 practice MCQs for the chapter Lines and Planes in 3D in SPM (Malaysia) SPM Mathematics. The questions are organized by difficulty โ 46 easy, 76 medium, 29 hard โ so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.