Three Dimensional Geometry Practice Questions
JEE · Mathematics · 2009 free MCQs with instant results and detailed explanations.
2009
Total
445
Easy
853
Medium
711
Hard
Start Practicing Three Dimensional Geometry
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Topics in Three Dimensional Geometry
Equation of Line
336
Skew Lines
369
Equation of Plane
330
Angle Between Line and Plane
331
Distance Formulas
277
Direction Cosines and Ratios
366
Sample Questions from Three Dimensional Geometry
Here are 10 sample questions. Start a quiz to get randomized questions with scoring.
Q1
Easy
What are the direction cosines of a vector that makes equal angles with all three axes?
A.
1/√3
B.
1
C.
0
D.
1/√2
Show Answer & Explanation
Correct Answer: A
For equal angles with the axes, direction cosines are equal and given by cos(θ) = 1/√3.
Q2
Easy
If the direction ratios of a line are (2, -3, 4), which of the following is a correct set of direction cosines?
A.
2/√29, -3/√29, 4/√29
B.
2, -3, 4
C.
1, 0, 0
D.
0, -1, 1
Show Answer & Explanation
Correct Answer: A
Direction cosines are obtained by normalizing the direction ratios using the formula x/√(x²+y²+z²).
Q3
Easy
The direction cosines of a line are l, m, n. What is the relationship between them?
A.
l² + m² + n² = 1
B.
l + m + n = 1
C.
l * m * n = 1
D.
l² + m² - n² = 0
Show Answer & Explanation
Correct Answer: A
The sum of the squares of the direction cosines equals 1 for any line in three-dimensional space.
Q4
Medium
If the coordinates of a point A are (2, 3, 5) and another point B is (4, 1, 7), find the direction ratios of the line joining A and B.
A.
(2, -2, 2)
B.
(1, -1, 1)
C.
(0, 0, 0)
D.
(3, -2, 2)
Show Answer & Explanation
Correct Answer: A
Direction ratios are found by subtracting the coordinates: (4-2, 1-3, 7-5) = (2, -2, 2).
Q5
Medium
A line passes through the point (1, 2, 3) and has direction ratios (4, 5, 6). What is the symmetric form of the line's equation?
A.
(x-1)/4 = (y-2)/5 = (z-3)/6
B.
(x+1)/4 = (y-2)/5 = (z-3)/6
C.
(x-1)/4 = (y+2)/5 = (z-3)/6
D.
(x+1)/4 = (y+2)/5 = (z+3)/6
Show Answer & Explanation
Correct Answer: A
The symmetric form is derived from the point and direction ratios by the formula (x-x0)/a = (y-y0)/b = (z-z0)/c.
Q6
Medium
If the direction cosines of a line are proportional to 2, -1, 3, what are the possible values of the direction ratios?
A.
2, -1, 3
B.
1, 0, 1
C.
2, 1, 3
D.
4, -2, 6
Show Answer & Explanation
Correct Answer: A
The direction ratios are proportional to the direction cosines, hence they can be taken as 2, -1, and 3.
Q7
Medium
If a line has direction ratios of 3, -2, and 4, what is the equation of the line in symmetric form?
A.
x/3 = y/-2 = z/4
B.
x/4 = y/-2 = z/3
C.
3x = -2y = 4z
D.
3x = -4y = 2z
Show Answer & Explanation
Correct Answer: A
The symmetric form of a line equation is given by the ratios of the coordinates to the direction ratios.
Q8
Hard
If the direction cosines of a line are proportional to (2, 3, -1), find the equation of the line in symmetric form.
A.
(x - x₀)/2 = (y - y₀)/3 = (z - z₀)/-1
B.
(x/x₀) = (y/y₀) = (z/z₀)
C.
(x - 1)/2 = (y - 2)/3 = (z + 1)/-1
D.
(x + 2)/2 = (y - 3)/3 = (z - 1)/1
Show Answer & Explanation
Correct Answer: A
The symmetric form relates the direction ratios with the coordinates of a point on the line.
Q9
Hard
A line is directed along the vector (4, -2, 1) and passes through the point (1, 1, 1). What is the parametric equation of this line?
A.
x = 1 + 4t, y = 1 - 2t, z = 1 + t
B.
x = 1 + 2t, y = 1 - 4t, z = 1 + t
C.
x = 1 + t, y = 1 - 2t, z = 1 - t
D.
x = 1 - 4t, y = 1 + 2t, z = 1 - t
Show Answer & Explanation
Correct Answer: A
The parametric equations of a line can be expressed as x = x₀ + at, y = y₀ + bt, z = z₀ + ct.
Q10
Hard
What is the condition for three vectors to be coplanar in three-dimensional space?
A.
The scalar triple product must be zero.
B.
The vector sum must equal zero.
C.
All three vectors must be parallel.
D.
The magnitudes must be equal.
Show Answer & Explanation
Correct Answer: A
Three vectors are coplanar if the scalar triple product of the vectors is zero.
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Three Dimensional Geometry — JEE Mathematics Practice Questions Online
This page contains 2009 practice MCQs for the chapter Three Dimensional Geometry in JEE Mathematics. The questions are organized by difficulty — 445 easy, 853 medium, 711 hard — 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. This chapter covers 6 topics, giving you comprehensive coverage of the entire chapter.