CBSE Class 12 Maths - MCQ and Online Tests - Unit 11 - Three Dimensional Geometry

CBSE Class 12 Maths – MCQ and Online Tests – Unit 11 – Three Dimensional Geometry

Every year CBSE conducts board exams for 12th standard. These exams are very competitive to all the students. So our website provides online tests for all the 12th subjects. These tests are also very effective and useful for those who preparing for competitive exams like NEET, JEE, CA etc. It can boost their preparation level and confidence level by attempting these chapter wise online tests.

These online tests are based on latest CBSE Class 12 syllabus. While attempting these our students can identify the weak lessons and continuously practice those lessons for attaining high marks. It also helps to revise the NCERT textbooks thoroughly.

 

CBSE Class 12 Maths – MCQ and Online Tests – Unit 11 – Three Dimensional Geometry

Question 1.
The direction cosines of the y-axis are
(a) (6, 0, 0)
(b) (1, 0, 0)
(c) (0, 1, 0)
(d) (0, 0, 1)

Answer

Answer: (c) (0, 1, 0)

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Question 2.
The direction ratios of the line joining the points (x, y, z) and (x₂, y₂, z₁) are
(a) x₁ + x₂, y₁ + y₂, z₁ + z₂
(b) \(\sqrt{(x_1 – x_2)^2 + (y_1 – y_2)^2 + (z_1 + z_2)^2}\)
(c) \(\frac{x_1+x_2}{2}\), \(\frac{y_1+y_2}{2}\), \(\frac{z_1+z_2}{2}\)
(d) x₂ – x₁, y₂ – y₁, z₂ – z₁

Answer

Answer: (d) x₂ – x₁, y₂ – y₁, z₂ – z₁

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Question 3.
The coordinates of the midpoints of the line segment joining the points (2, 3, 4) and (8, -3, 8) are
(a) (10, 0, 12)
(b) (5, 6, 0)
(c) (6, 5, 0)
(d) (5, 0, 6)

Answer

Answer: (d) (5, 0, 6)

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Question 4.
If the planes a₁x + b, y + c, z + d₁ = 0 and a₂x + b, y + c₂z + d₂ = 0 are perpendicular to each other then
(a) \(\frac{a_1}{a_2}\) = \(\frac{b_1}{b_2}\) = \(\frac{c_1}{c_2}\)
(b) \(\frac{a_1}{a_2}\) + \(\frac{b_1}{b_2}\), \(\frac{c_1}{c_2}\)
(c) a₁a₂ + b₁b₂ + c₁c₂ = 0
(d) a\(_{1}^{2}\)a\(_{2}^{2}\) + b\(_{1}^{2}\)b\(_{2}^{2}\) + c\(_{1}^{2}\)c\(_{2}^{2}\) = 0

Answer

Answer: (c) a₁a₂ + b₁b₂ + c₁c₂ = 0

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Question 5.
The distance of the plane 2x – 3y + 6z + 7 = 0 from the point (2, -3, -1) is
(a) 4
(b) 3
(c) 2
(d) \(\frac{1}{5}\)

Answer

Answer: (c) 2

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Question 6.
The direction cosines of the normal to the plane 2x – 3y – 6z – 3 = 0 are
(a) \(\frac{2}{7}\), \(\frac{-3}{7}\), \(\frac{-6}{7}\)
(b) \(\frac{2}{7}\), \(\frac{3}{7}\), \(\frac{6}{7}\)
(c) \(\frac{-2}{7}\), \(\frac{-3}{7}\), \(\frac{-6}{7}\)
(d) None of these

Answer

Answer: (a) \(\frac{2}{7}\), \(\frac{-3}{7}\), \(\frac{-6}{7}\)

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Question 7.
If 2x + 5y – 6z + 3 = 0 be the equation of the plane, then the equation of any plane parallel to the given plane is
(a) 3x + 5y – 6z + 3 = 0
(b) 2x – 5y – 6z + 3 = 0
(c) 2x + 5y – 6z + k = 0
(d) None of these

Answer

Answer: (c) 2x + 5y – 6z + k = 0

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Question 8.
(2, – 3, – 1) 2x – 3y + 6z + 7 = 0
(a) 4
(b) 3
(c) 2
(d) \(\frac{1}{5}\)

Answer

Answer: (c) 2

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Question 9.
The length of the ⊥^er from the point (0, – 1, 3) to the plane 2x + y – 2z + 1 = 0 is
(a) 0
(b) 2√3
(c) \(\frac{2}{3}\)
(d) 2

Answer

Answer: (d) 2

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Question 10.
The shortest distance between the lines \(\vec{r}\) = \(\vec{a}\) + k\(\vec{b}\) and r = \(\vec{a}\) + l\(\vec{c}\) is (\(\vec{b}\) and \(\vec{c}\) are non-collinear)
(a) 0
(b) |\(\vec{b}\).\(\vec{c}\)|
(c) \(\frac{|\vec{b}×\vec{c}|}{|\vec {a}|}\)
(d) \(\frac{|\vec{b}.\vec{c}|}{|\vec {a}|}\)

Answer

Answer: (a) 0

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Question 11.
The equation xy = 0 in three dimensional space is represented by
(a) a plane
(b) two plane are right angles
(c) a pair of parallel planes
(d) a pair of st. line

Answer

Answer: (b) two plane are right angles

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Question 12.
The direction cosines of any normal to the xy plane are
(a) 1, 0 ,0
(b) 0, 1, 0
(c) 1, 1, 0
(d) 1, 1, 0

Answer

Answer: (d) 1, 1, 0

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Question 13.
How many lines through the origin in make equal angles with the coordinate axis?
(a) 1
(b) 4
(c) 8
(d) 2

Answer

Answer: (c) 8

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Question 14.
The direction cosines of the line joining (1, -1, 1) and (-1, 1, 1) are
(a) 2, -2, 0
(b) 1, -1, 0
(c) \(\frac{1}{√2}\), – \(\frac{1}{√2}\)
(d) None of these

Answer

Answer: (c) \(\frac{1}{√2}\), – \(\frac{1}{√2}\)

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Question 15.
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 perpendicular plane
(d) None of these

Answer

Answer: (a) A pair of parallel planes

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Question 16.
The distance of the point (-3, 4, 5) from the origin
(a) 50
(b) 5√2
(c) 6
(d) None of these

Answer

Answer: (b) 5√2

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Question 17.
If a line makes angles Q₁, Q₂₁ and Q₃ respectively with the coordinate axis then the value of cos² Q₁ + cos² Q₂ + cos² Q₃
(a) 2
(b) 1
(c) 4
(d) \(\frac{3}{2}\)

Answer

Answer: (b) 1

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Question 18.
The direction ratios of a line are 1,3,5 then its direction cosines are
(a) \(\frac{1}{\sqrt{35}}\), \(\frac{3}{\sqrt{35}}\), \(\frac{5}{\sqrt{35}}\)
(b) \(\frac{1}{9}\), \(\frac{1}{3}\), \(\frac{5}{9}\)
(c) \(\frac{5}{\sqrt{35}}\), \(\frac{3}{\sqrt{35}}\), \(\frac{1}{\sqrt{35}}\)
(d) None of these

Answer

Answer: (a) \(\frac{1}{\sqrt{35}}\), \(\frac{3}{\sqrt{35}}\), \(\frac{5}{\sqrt{35}}\)

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Question 19.
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

Answer

Answer: (a) 7, 4,-2

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Question 20.
The direction ratios of the line of intersection of the planes 3x + 2y – z = 5 and x – y + 2z = 3 are
(a) 3, 2, -1
(b) -3, 7, 5
(c) 1, -1, 2
(d) – 11, 4, -5

Answer

Answer: (b) -3, 7, 5

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Question 21.
The lines of intersection of the planes \(\vec{r}\)(3\(\hat{i}\) – \(\hat{j}\) + \(\hat{k}\)) = 1 and \(\vec{r}\)(\(\hat{i}\) +4\(\hat{j}\) – 2\(\hat{k}\)) = 2 is parallel to the vector
(a) 2\(\hat{i}\) + 7\(\hat{j}\) + 13\(\hat{k}\)
(b) -2\(\hat{i}\) + 7\(\hat{j}\) + 13\(\hat{k}\)
(c) 2\(\hat{i}\) – 7\(\hat{j}\) + 13\(\hat{i}\)
(b) -2\(\hat{i}\) – 7\(\hat{j}\) – 13\(\hat{k}\)

Answer

Answer: (b) -2\(\hat{i}\) + 7\(\hat{j}\) + 13\(\hat{k}\)

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Question 22.
The equation of the plane through the origin and parallel to the plane 3x – 4y + 5z + 6 = 0
(a) 3x – 4y – 5z – 6 = 0
(b) 3x – 4y + 5z + 6 = 0
(c) 3x – 4y + 5z = 0
(d) 3x + 4y – 5z + 6 = 0

Answer

Answer: (c) 3x – 4y + 5z = 0

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Question 23.
The locus of xy + yz = 0 is
(a) A pair of st. lines
(b) A pair of parallel lines
(c) A pair of parallel planes
(d) A pair of perpendicular planes

Answer

Answer: (d) A pair of perpendicular planes

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Question 24.
The plane x + y = 0
(a) is parallel to z-axis
(b) is perpendicular to z-axis
(c) passes through z-axis
(d) None of these

Answer

Answer: (c) passes through z-axis

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Question 25.
If α, β, γ are the angle which a half ray makes with the positive directions of the axis then sin²α + sin²β + sin²γ =
(a) 1
(b) 2
(c) 0
(d) -1

Answer

Answer: (b) 2

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Question 26.
If a line makes angles α, β, γ with the axis then cos 2α + cos 2β + cos 2γ =
(a) -2
(b) -1
(c) 1
(d) 2

Answer

Answer: (b) -1

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Question 27.
The line x = 1, y = 2 is
(a) parallel to x-axis
(b) parallel to y-axis
(c) parallel to z-axis
(d) None of these

Answer

Answer: (c) parallel to z-axis

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Question 28.
The points A (1, 1, 0), B(0, 1, 1), C(1, 0, 1) and D(\(\frac{2}{3}\), \(\frac{2}{3}\), \(\frac{2}{3}\))
(a) Coplanar
(b) Non-coplanar
(c) Vertices of a parallelogram
(d) None of these

Answer

Answer: (a) Coplanar

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Question 29.
The angle between the planes 2x – y + z = 6 and x + y + 2z = 7 is
(a) \(\frac{π}{4}\)
(b) \(\frac{π}{6}\)
(c) \(\frac{π}{3}\)
(d) \(\frac{π}{2}\)

Answer

Answer: (c) \(\frac{π}{3}\)

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Question 30.
The distance of the points (2, 1, -1) from the plane x- 2y + 4z – 9 is
(a) \(\frac{\sqrt{31}}{21}\)
(b) \(\frac{13}{21}\)
(c) \(\frac{13}{\sqrt{21}}\)
(d) \(\sqrt{\frac{π}{2}}\)

Answer

Answer: (c) \(\frac{13}{\sqrt{21}}\)

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