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MP Board Class 12 (MPBSE) Three Dimensional Geometry — practice questions — page 2

Explore 54 free Three Dimensional Geometry questions in MP Board Class 12 (MPBSE) Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Three Dimensional Geometry questions and exercises with answers

Showing questions 31–54 of 54.

True or false: direction cosinesRead the answer and worked solution →Direction cosines or direction ratios?Read the answer and worked solution →Order the steps: direction ratios to direction cosinesRead the answer and worked solution →Skew linesRead the answer and worked solution →Tap every pair of lines that could be skewRead the answer and worked solution →Match each equation to what it describesRead the answer and worked solution →Complete the definition of direction cosinesRead the answer and worked solution →Complete the definition of skew linesRead the answer and worked solution →True or false: angles between linesRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Spot the one that does not belongRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Reading the equationsRead the answer and worked solution →Flip and pair: symbol with meaningRead the answer and worked solution →Same plane or different planes?Read the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →Order the steps: angle between two lines from direction ratiosRead the answer and worked solution →Tap each statement that is true of a line's equationRead the answer and worked solution →True or false: reading the chapter's scopeRead the answer and worked solution →Match each condition to what it meansRead the answer and worked solution →The normalising relationRead the answer and worked solution →Spot the odd one outRead the answer and worked solution →Assertion and reason: equations of a lineRead the answer and worked solution →Which quantity is which?Read the answer and worked solution →

Worked practice questions

True or false: direction cosines

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Questions
  1. Statement is true
    Yes
    Statement
    For any line, l² + m² + n² = 1.
  2. Statement is true
    No
    Statement
    Direction ratios must also satisfy a² + b² + c² = 1.
  3. Statement is true
    Yes
    Statement
    Direction cosines are the cosines of the angles made with the axes.
  4. Statement is true
    No
    Statement
    A line has exactly one set of direction ratios.
  5. Statement is true
    Yes
    Statement
    Direction ratios are proportional to the direction cosines.
Direction cosines are normalised, direction ratios are not. That single difference decides most questions here. Source: syllabus Maths Part II, Ch 11 Three Dimensional Geometry.

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Direction cosines or direction ratios?

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Items
  1. Item
    Written l, m, n
    Category
    Direction cosines
  2. Item
    Satisfy l² + m² + n² = 1
    Category
    Direction cosines
  3. Item
    Are actual cosines of angles
    Category
    Direction cosines
  4. Item
    Written a, b, c
    Category
    Direction ratios
  5. Item
    Any proportional triple will do
    Category
    Direction ratios
  6. Item
    Need dividing by √(a²+b²+c²) to normalise
    Category
    Direction ratios
Categories
  1. Item
    Direction cosines
  2. Item
    Direction ratios
Cosines are fixed and normalised ratios are any proportional triple. Source: syllabus Maths Part II, Ch 11 Three Dimensional Geometry.

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Order the steps: direction ratios to direction cosines

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Steps
  1. Start from the direction ratios a, b, c
  2. Square each of them and add the three squares
  3. Take the square root of that sum
  4. Divide each of a, b and c by that root
  5. The results are the direction cosines l, m, n
Your syllabus gives l = a/√(a²+b²+c²) and similarly for m and n, which is exactly this normalising procedure. Source: syllabus Maths Part II, Ch 11 Three Dimensional Geometry.

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