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Indian Navy 10+2 B.Tech Cadet Surface Areas and Volumes — practice questions

Explore 5 free Surface Areas and Volumes questions in Indian Navy 10+2 B.Tech Cadet Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Surface Areas and Volumes questions and exercises with answers

Showing questions 1–5 of 5.

A line passes through the points $(1, 2, 3)$ and $(4, 6, 9)$. The direction ratios of the line are:Read the answer and worked solution →Two lines are parallel if their direction ratios are:Read the answer and worked solution →Equation of straight line passing through two points A(1,2,3) and B(4,5,6):Read the answer and worked solution →A solid metallic cylinder of radius 6 cm and height 30 cm is melted and made into solid cubes of side 6 cm. The number of cubes isRead the answer and worked solution →A spherical ball of iron of radius 6 cm is dropped into a cylinder of radius 12 cm partially full of water. The rise in water level isRead the answer and worked solution →

Worked practice questions

A line passes through the points $(1, 2, 3)$ and $(4, 6, 9)$. The direction ratios of the line are:

  1. $(4, 6, 9)$
  2. $(3, 4, 6)$
  3. $(5, 8, 12)$
  4. $(1, 2, 3)$
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Correct answer: B. $(3, 4, 6)$

Direction ratios of the line joining two points are the differences in coordinates: $(x_2 - x_1, y_2 - y_1, z_2 - z_1)$. Here: $(4 - 1, 6 - 2, 9 - 3) = (3, 4, 6)$. Direction ratios are not unique. $(6, 8, 12)$ or $(1.5, 2, 3)$ would also be valid since multiplying by any non-zero scalar preserves the direction. The standard answer just uses the raw differences. Direction cosines (the normalised version) are $(3, 4, 6) / \sqrt{9 + 16 + 36} = (3, 4, 6) / \sqrt{61}$.

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Two lines are parallel if their direction ratios are:

  1. Perpendicular
  2. Proportional (a₁/a₂ = b₁/b₂ = c₁/c₂)
  3. Random
  4. Equal
Show answer and solution

Correct answer: B. Proportional (a₁/a₂ = b₁/b₂ = c₁/c₂)

Two lines parallel iff their direction ratios are proportional. Two lines perpendicular iff dot product of direction vectors = 0: a₁a₂ + b₁b₂ + c₁c₂ = 0.

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Equation of straight line passing through two points A(1,2,3) and B(4,5,6):

  1. (x-1)/3 = (y-2)/3 = (z-3)/3 (direction = B-A = (3,3,3))
  2. x = y = z
  3. (x-1)/4 = (y-2)/5 = (z-3)/6
  4. x + y + z = 6
Show answer and solution

Correct answer: A. (x-1)/3 = (y-2)/3 = (z-3)/3 (direction = B-A = (3,3,3))

Direction = B - A = (3, 3, 3) or any scalar multiple (1, 1, 1). Through A(1,2,3): (x-1)/1 = (y-2)/1 = (z-3)/1, equivalently (x-1)/3 = (y-2)/3 = (z-3)/3.

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