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NCERT Exemplar · Class 11 Mathematics Introduction to Three Dimensional Geometry

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EXERCISE 12.3 31–40 (part 4 of 5)

  1. Choose the correct answer from the given four options inidcated against each of the Exercises from $\displaystyle 22$ (M.C.Q.).

    Exercise 31

    The locus of a point for which x=0\displaystyle x=0 is
    (A)
    xy\displaystyle x y-plane
    (B)
    yz\displaystyle y z-plane
    (C)
    zx\displaystyle z x-plane
    (D)
    none of these

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    NCERT’s answer
    B
    (B) \(\displaystyle yz\)-planePoints with \[x = 0 \] are \(\displaystyle (0, y, z)\), which is the \(\displaystyle yz\)-plane. Answer: (B)

  2. Exercise 32

    If a parallelopiped is formed by planes drawn through the points (5,8,10)\displaystyle (5,8,10) and (3\displaystyle 3, 6\displaystyle 6, 8\displaystyle 8) parallel to the coordinate planes, then the length of diagonal of the parallelopiped is
    (A)
    23\displaystyle 2 \sqrt{3}
    (B)
    32\displaystyle 3 \sqrt{2}
    (C)
    2\displaystyle \sqrt{2}
    (D)
    3\displaystyle \sqrt{3}

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    NCERT’s answer
    A
    (A) \(\displaystyle 2\sqrt{3}\)The two given points are opposite vertices; the edges are the coordinate differences: \[5-3 = 2,\quad 8-6 = 2,\quad 10-8 = 2 \] \[d = \sqrt{2^2 + 2^2 + 2^2} = \sqrt{12} = 2\sqrt{3} \] Answer: (A) \(\displaystyle 2\sqrt{3}\)

  3. Exercise 33

    L is the foot of the perpendicular drawn from a point P (3\displaystyle 3, 4\displaystyle 4, 5\displaystyle 5) on the xy\displaystyle x y-plane. The coordinates of point L are
    (A)
    (3\displaystyle 3, 0\displaystyle 0, 0\displaystyle 0)
    (B)
    (0\displaystyle 0, 4\displaystyle 4, 5\displaystyle 5)
    (C)
    (3\displaystyle 3, 0\displaystyle 0, 5\displaystyle 5)
    (D)
    none of these

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    NCERT’s answer
    D
    (D) none of theseThe foot of the perpendicular on the \(\displaystyle xy\)-plane keeps \(\displaystyle x, y\) and sets \(\displaystyle z = 0\). \[L = (3,\,4,\,0) \] This is none of (A), (B), (C).

  4. Exercise 34

    L is the foot of the perpendicular drawn from a point (3\displaystyle 3, 4\displaystyle 4, 5\displaystyle 5) on x\displaystyle x-axis. The coordinates of L are
    (A)
    (3\displaystyle 3, 0\displaystyle 0, 0\displaystyle 0)
    (B)
    (0,4,0)\displaystyle (0,4,0)
    (C)
    (0\displaystyle 0, 0\displaystyle 0, 5\displaystyle 5)
    (D)
    none of these

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    NCERT’s answer
    A
    (A) \(\displaystyle (3, 0, 0)\)The foot on the \(\displaystyle x\)-axis keeps \(\displaystyle x\) and sets \(\displaystyle y = z = 0\). \[L = (3,\,0,\,0) \]
  5. Fill in the blanks in Exercises from $\displaystyle 35$ to 49.

    Exercise 35

    The three axes OX, OY, OZ determine ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    Three cordinates planes
    three mutually perpendicular planes (the coordinate planes)Taking the axes in pairs: \[OX, OY \to xy\text{-plane}: \; z = 0 \] \[OY, OZ \to yz\text{-plane}: \; x = 0 \] \[OZ, OX \to zx\text{-plane}: \; y = 0 \] Answer: three mutually perpendicular planes.

  6. Exercise 36

    The three planes determine a rectangular parallelopiped which has ____\displaystyle \_\_\_\_ of rectangular faces.

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    NCERT’s answer
    Three pairs
    three pairsThe coordinate planes and the planes through \(\displaystyle P(a, b, c)\) parallel to them bound the box. Its faces are \[x = 0,\; x = a \] \[y = 0,\; y = b \] \[z = 0,\; z = c \] Each line is a pair of opposite faces. Answer: three pairs.

  7. Exercise 37

    The coordinates of a point are the perpendicular distance from the ____\displaystyle \_\_\_\_ on the respectives axes.

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    NCERT’s answer
    given point
    given pointDraw perpendiculars \(\displaystyle PL, PM, PN\) from the given point \(\displaystyle P\) to the \(\displaystyle yz\)-, \(\displaystyle zx\)- and \(\displaystyle xy\)-planes. Their lengths, read along \(\displaystyle OX, OY, OZ\), are \[PL = a,\quad PM = b,\quad PN = c \] \[P = (a,\, b,\, c) \] Answer: given point.

  8. Exercise 38

    The three coordinate planes divide the space into ____\displaystyle \_\_\_\_ parts.

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    NCERT’s answer
    Eight
    eightEach of \(\displaystyle x, y, z\) is positive or negative, giving \[2 \times 2 \times 2 = 8 \] parts (octants). Answer: eight.

  9. Exercise 39

    If a point P lies in yz\displaystyle y z-plane, then the coordinates of a point on yz\displaystyle y z-plane is of the form ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    \(\displaystyle (0, y, z)\)
    \(\displaystyle (0, y, z)\)A point in the \(\displaystyle yz\)-plane is at distance zero from that plane, so its \(\displaystyle x\)-coordinate is \[x = 0 \] Answer: \(\displaystyle (0, y, z)\).

  10. Exercise 40

    The equation of yz\displaystyle y z-plane is ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    \(\displaystyle x=0\)
    \(\displaystyle x = 0\)A point lies in the \(\displaystyle yz\)-plane exactly when its \(\displaystyle x\)-coordinate is zero: \[x = 0 \] Answer: \(\displaystyle x = 0\).