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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Question 31.
The planes \(\vec{r}\)(2\(\hat{i}\) + 3\(\hat{j}\) – 6\(\hat{k}\)) = 7 and
\(\vec{r}\)(\(\frac{-2}{7}\)\(\vec{i}\) – \(\frac{3}{j}\)\(\vec{j}\) + \(\frac{6}{7}\)\(\vec{k}\)) = 0 are
(a) parallel
(b) at right angles
(c) equidistant front origin
(d) None of these

Answer

Answer: (a) parallel

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Question 32.
The equation of the plane through point (1, 2, -3) which is parallel to the plane 3x- 5y + 2z = 11 is given by
(a) 3x – 5y + 2z – 13 = 0
(b) 5x – 3y + 2z + 13 = 0
(c) 3x – 2y + 5z + 13 = 0
(d) 3x – 5y + 2z + 13 = 0

Answer

Answer: (d) 3x – 5y + 2z + 13 = 0

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Question 33.
Distance of the point (a, β, γ) from y-axis is
(a) β
(b) |β|
(c) |β + γ|
(d) \(\sqrt{α^2+γ^2}\)

Answer

Answer: (d) \(\sqrt{α^2+γ^2}\)

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Question 34.
If the directions cosines of a line are A, k, k, then
(a) k > 0
(b) 0 < k < 1
(c) k = 1
(d) k = \(\frac{1}{√3}\) or –\(\frac{1}{√3}\)

Answer

Answer: (d) k = \(\frac{1}{√3}\) or –\(\frac{1}{√3}\)

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Question 35.
The distance of the plane \(\vec{r}\)(\(\frac{-2}{7}\)\(\hat{i}\) – \(\frac{3}{7}\)\(\hat{j}\) + \(\frac{6}{7}\)\(\hat{k}\)) = 0 from the orgin is
(a) 1
(b) 7
(c) \(\frac{1}{7}\)
(d) None of these

Answer

Answer: (a) 1

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Question 36.
The sine of the angle between the straight line \(\frac{x-2}{3}\) = \(\frac{y-3}{4}\) = \(\frac{z-4}{5}\) and the plane 2x – 2y + z = 5 is
(a) \(\frac{10}{6√5}\)
(b) \(\frac{4}{5√2}\)
(c) \(\frac{2√3}{5}\)
(d) \(\sqrt{\frac{√2}{10}}\)

Answer

Answer: (c) \(\frac{2√3}{5}\)

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Question 37.
The reflection of the point (a, β, γ) in the xy-plane is
(a) (α, β, 0)
(b) (0, 0, γ)
(c) (- α, – β, γ)
(d) (α, β, γ)

Answer

Answer: (d) (α, β, γ)

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Question 38.
The area of the quadrilateral ABCD, where A(0, 4, 1), B(2, 3, -1), C(4, 5, 0) and D(2, 6, 2) is equal to
(a) 9 sq. units
(b) 18 sq. units
(c) 27 sq. units
(d) 81 sq. units

Answer

Answer: (a) 9 sq. units

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Question 39.
The plane 2x – 3y + 6z – 11 = 0 makes an angle sin^-1 (α) with .e-axis. The value of a is equal to
(a) \(\frac{√3}{2}\)
(b) \(\frac{√2}{3}\)
(c) \(\frac{2}{7}\)
(d) \(\frac{3}{7}\)

Answer

Answer: (c) \(\frac{2}{7}\)

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Question 40.
The cosines of the angle between any two diagonals of a cube is
(a) \(\frac{1}{3}\)
(b) \(\frac{1}{2}\)
(c) \(\frac{2}{3}\)
(d) \(\frac{1}{√3}\)

Answer

Answer: (a) \(\frac{1}{3}\)

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