Mathematics MCQ on Three Dimensional Geometry for JEE and Engineering Exam 2022
MCQ on Three Dimensional Geometry: To be an expert in JEE Mathematics, it is absolutely necessary to practice and be familiar will all the concepts as well as the questions of different types. This is essential to gain mastery over the subject. We have also often heard the common saying, “Practice Makes a Man Perfect”, hence students have to practice, practice and practice till they master the subject.
In this post we are providing you MCQ on Three Dimensional Geometry, which will be beneficial for you in upcoming JEE and Engineering Exams.
MCQ on Three Dimensional Geometry
Q1. The direction ratios of the normal to the plane 7x + 4y – 2z + 5 = 0 are _______.
a) 7, 4,-2
b) 7, 4, 5
c) 7, 4, 2
d) 4, -2, 5
a) 7, 4,-2
Q2. If a line has direction ratios 2, – 1, – 2, determine its direction cosines _______.
a) ⅓, ⅔, -⅓
b) ⅔, -⅓, -⅔
c) -⅔, ⅓, ⅔
d) None of the above
b) ⅔, -⅓, -⅔
Q3. If the lines x-2/1 =y-2/1 =z-4/k and x-1/k = y-4/2 = z-5/1 are coplanar, then k can have _______.
a) Exactly two values
b) Exactly three values
c) Exactly one value
d) Any value
a) Exactly two values
Q4. The direction cosines of the y-axis are ______.
a) (9, 0, 0)
b) (1, 0, 0)
c) (0, 1, 0)
d) (0, 0, 1)
c) (0, 1, 0)
Q5. What are the direction cosines of the equation of the plane 2x + 3y – z = 5?
a) 1/√14, 3/√14, -2/√14
b) 2/√14, 3/√14, -1/√14
c) 2/√14, 1/√14, -1/√14
d) 2/√14, -2/√14, -3/√14
b) 2/√14, -2/√14, -3/√14
The equation of the plane, 2x + 3y – z = 5…. (1)
Direction ratio of the normal (2, 3, -1)
By using the formula,
√[(2)^{2} + (3)^{2} + (-1)^{2}] = √14
Now,
Divide both the sides of equation (1) by √14, we get
2x/(√14) + 3y/(√14) – z/(√14) = 5/√14
So this is of the form lx + my + nz = d
Where, l, m, n are the direction cosines and d is the distance
Therefore, the direction cosines are 2/√14, 3/√14, -1/√14
Q6. The vector equation for the line passing through the points (–1, 0, 2) and (3, 4, 6) is
a) i + 2k + λ(4i + 4j + 4k)
b) i – 2k + λ(4i + 4j + 4k)
c) -i + 2k+ λ(4i + 4j + 4k)
d) -i + 2k+ λ(4i – 4j – 4k)
c) -i + 2k+ λ(4i + 4j + 4k)
The vector equation of the line is given by:
r = a + λ (b – a), λ ∈ R
Let a = -i + 2k
And b = 3i + 4j + 6k
b – a = 4i + 4j + 4k
Let the vector equation be r, then;
r = -i + 2k + λ (4i + 4j + 4k)
Q7. Find the equation of the plane passing through the points P(1, 1, 1), Q(3, -1, 2), R(-3, 5, -4).
a) x + 2y = 0
b) x – y – 2 = 0
c) -x + 2y – 2 = 0
d) x + y – 2 = 0
d) x + y – 2 = 0
Q8. Direction ratio of line joining (2, 3, 4) and (−1, −2, 1), are _____.
a) (−3, −5, −3)
b) (−3, 1, −3)
c) (−1, −5, −3)
d) (−3, −5, 5)
a) (−3, −5, −3)
The direction ratio of the line joining A (2, 3, 4) and B (−1, −2, 1), are:
(−1−2), (−2−3), (1−4)
= (−3, −5, −3)
Q9. If l, m, n are the direction cosines of a line, then _____.
a) l^{2}+ m^{2}+ 2n^{2} = 1
b) l^{2}+ 2m^{2}+ n^{2} = 1
c) 2l^{2}+ m^{2}+ n^{2} = 1
d) l^{2}+ m^{2}+ n^{2} = 1
d) l^{2}+ m^{2}+ n^{2} = 1
Q10. The equation x² – x – 2 = 0 in three-dimensional space is represented by _______.
a) A pair of parallel planes
b) A pair of straight lines
c) A pair of the perpendicular plane
d) None of these
a) A pair of parallel planes
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