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NCERT Exemplar · Class 11 Mathematics Introduction to Three Dimensional Geometry

50 questions · 50 still being checked

EXERCISE 12.3 21–30 (part 3 of 5)

  1. Exercise 21

    What are the coordinates of the vertices of a cube whose edge is 2\displaystyle 2 units, one of whose vertices coincides with the origin and the three edges passing through the origin, coincides with the positive direction of the axes through the origin?

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    NCERT’s answer
    ($\displaystyle 2,0,0$), ($\displaystyle 2,2,0$), ($\displaystyle 0,2,0$), ($\displaystyle 0,2,2$) ($\displaystyle 0,0,2$) ($\displaystyle 2,0,2$), ($\displaystyle 0,0,0$), ($\displaystyle 2,2,2$)
    Edges of length $\displaystyle 2$ lie along the positive \(\displaystyle x\), \(\displaystyle y\), \(\displaystyle z\)-axes from \(\displaystyle O\), so every coordinate of a vertex is \(\displaystyle 0\) or \(\displaystyle 2\). \[O(0,0,0),\quad A(2,0,0),\quad B(0,2,0),\quad C(0,0,2) \] \[D(2,2,0),\quad E(2,0,2),\quad F(0,2,2),\quad G(2,2,2) \] NCERT_Solution_Class11_Maths_Exemplar_Ch12_Ex12-3_Q21 Answer: \(\displaystyle (0,0,0),(2,0,0),(0,2,0),(0,0,2),(2,2,0),(2,0,2),(0,2,2),(2,2,2)\)
  2. Choose the correct answer from the given four options inidcated against each of the Exercises from $\displaystyle 22$ (M.C.Q.).

    Exercise 22

    The distance of point P(3,4,5)\displaystyle \mathrm{P}(3,4,5) from the yz\displaystyle y z-plane is
    (A)
    3\displaystyle 3 units
    (B)
    4\displaystyle 4 units
    (C)
    5\displaystyle 5 units
    (D)
    550\displaystyle 550

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    NCERT’s answer
    A
    (A) $\displaystyle 3$ units. The distance of a point from the \(\displaystyle yz\)-plane is the absolute value of its \(\displaystyle x\)-coordinate. \[d=|x|=|3|=3 \]

  3. Exercise 23

    What is the length of foot of perpendicular drawn from the point P(3,4,5)\displaystyle \mathrm{P}(3,4,5) on y\displaystyle y-axis
    (A)
    41\displaystyle \sqrt{41}
    (B)
    34\displaystyle \sqrt{34}
    (C)
    5\displaystyle 5 (D) none of these

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    NCERT’s answer
    B
    (B) \(\displaystyle \sqrt{34}\). The foot of the perpendicular from \(\displaystyle P(3,4,5)\) on the \(\displaystyle y\)-axis is \(\displaystyle M(0,4,0)\). \[PM=\sqrt{(3-0)^2+(4-4)^2+(5-0)^2}=\sqrt{34} \]

  4. Exercise 24

    Distance of the point (3\displaystyle 3, 4\displaystyle 4, 5\displaystyle 5) from the origin (0\displaystyle 0, 0\displaystyle 0, 0\displaystyle 0) is
    (A)
    50\displaystyle \sqrt{50}
    (B)
    3\displaystyle 3 (C) 4\displaystyle 4 (D) 5\displaystyle 5

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    NCERT’s answer
    A
    (A) \(\displaystyle \sqrt{50}\). Distance formula from the origin: \[OP=\sqrt{3^2+4^2+5^2}=\sqrt{9+16+25}=\sqrt{50} \]

  5. Exercise 25

    If the distance between the points (a,0,1)\displaystyle (a, 0,1) and (0,1,2)\displaystyle (0,1,2) is 27\displaystyle \sqrt{27}, then the value of a\displaystyle a is
    (A)
    5\displaystyle 5 (B) ±5\displaystyle \pm 5
    (C)
    5\displaystyle 5
    (D) none of these

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    NCERT’s answer
    B
    (B) \(\displaystyle \pm 5\)Distance formula: \[(a-0)^2 + (0-1)^2 + (1-2)^2 = (\sqrt{27})^2 \] \[a^2 + 1 + 1 = 27 \] \[a^2 = 25 \] \[a = \pm 5 \] Answer: (B) \(\displaystyle \pm 5\)

  6. Exercise 26

    x\displaystyle x-axis is the intersection of two planes
    (A)
    xy\displaystyle x y and xz\displaystyle x z
    (B)
    yz\displaystyle y z and zx\displaystyle z x
    (C)
    xy\displaystyle x y and yz\displaystyle y z
    (D)
    none of these

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    NCERT’s answer
    A
    (A) \(\displaystyle xy\) and \(\displaystyle xz\)On the \(\displaystyle x\)-axis \(\displaystyle y = 0\) and \(\displaystyle z = 0\). \[xy\text{-plane}: z = 0 \] \[xz\text{-plane}: y = 0 \] Both hold together exactly on the \(\displaystyle x\)-axis. Answer: (A)

  7. Exercise 27

    Equation of y\displaystyle y-axis is considered as
    (A)
    x=0,y=0\displaystyle x=0, y=0
    (B)
    y=0,z=0\displaystyle y=0, z=0
    (C)
    z=0,x=0\displaystyle z=0, x=0
    (D)
    none of these

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    (C) \(\displaystyle z=0,\ x=0\)Every point of the \(\displaystyle y\)-axis has \(\displaystyle x\)- and \(\displaystyle z\)-coordinates zero: \[(0, y, 0) \Rightarrow x = 0,\ z = 0 \] Answer: (C)

  8. Exercise 28

    The point (-2\displaystyle 2, -3\displaystyle 3, -4\displaystyle 4) lies in the
    (A)
    First octant
    (B)
    Seventh octant
    (C)
    Second octant
    (D)
    Eighth octant

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    NCERT’s answer
    B
    (B) Seventh octantSigns of the coordinates: \[(-2, -3, -4) \to (-, -, -) \] The octant with signs \(\displaystyle (-,-,-)\) is the seventh. Answer: (B) Seventh octant

  9. Exercise 29

    A plane is parallel to yz\displaystyle y z-plane so it is perpendicular to :
    (A)
    x\displaystyle x-axis
    (B)
    y\displaystyle y-axis
    (C)
    z\displaystyle z-axis
    (D)
    none of these

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    NCERT’s answer
    A
    (A) \(\displaystyle x\)-axisThe \(\displaystyle yz\)-plane is \(\displaystyle x = 0\), so a parallel plane is \[x = k \] Every point on it has the same \(\displaystyle x\); the plane meets the \(\displaystyle x\)-axis at right angles. Answer: (A) \(\displaystyle x\)-axis

  10. Exercise 30

    The locus of a point for which y=0,z=0\displaystyle y=0, z=0 is
    (A)
    equation of x\displaystyle x-axis
    (B)
    equation of y\displaystyle y-axis
    (C)
    equation at z\displaystyle z-axis
    (D)
    none of these

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    NCERT’s answer
    A
    (A) equation of \(\displaystyle x\)-axisPoints with \[y = 0,\ z = 0 \] are \(\displaystyle (x, 0, 0)\), which is the \(\displaystyle x\)-axis. Answer: (A)