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NCERT Exemplar · Class 11 Mathematics Introduction to Three Dimensional Geometry

50 questions · 50 still being checked

EXERCISE 12.3 41–50 (part 5 of 5)

  1. Fill in the blanks in Exercises from $\displaystyle 35$ to 49.

    Exercise 41

    If the point P lies on z\displaystyle z-axis, then coordinates of P are of the form ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    \(\displaystyle (0,0, z)\)
    \(\displaystyle (0,\,0,\,c)\), where \(\displaystyle c\) is any real number. The \(\displaystyle z\)-axis is the intersection of the \(\displaystyle yz\)-plane and the \(\displaystyle xz\)-plane, so both \(\displaystyle x\) and \(\displaystyle y\) vanish.\[x = 0 \quad \text{(on the } yz\text{-plane)} \] \[y = 0 \quad \text{(on the } xz\text{-plane)} \] \[P = (0,\,0,\,c) \]

  2. Exercise 42

    The equation of z\displaystyle z-axis, are ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    \(\displaystyle x=0, y=0\)
    \(\displaystyle x = 0,\ y = 0\). The \(\displaystyle z\)-axis is the line common to the \(\displaystyle yz\)-plane and the \(\displaystyle xz\)-plane.\[x = 0 \quad \text{(} yz\text{-plane)} \] \[y = 0 \quad \text{(} xz\text{-plane)} \] \[z \text{ is free, so } (0,\,0,\,z) \text{ lies on the line} \]

  3. Exercise 43

    A line is parallel to xy\displaystyle x y-plane if all the points on the line have equal ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    \(\displaystyle z\)- cordinates
    \(\displaystyle z\)-coordinate. Every plane parallel to the \(\displaystyle xy\)-plane has the form \(\displaystyle z = k\), and a line parallel to the \(\displaystyle xy\)-plane stays in one such plane.\[(x,\,y,\,z) = (x,\,y,\,k) \] \[\text{direction ratios } (l,\,m,\,0) \]

  4. Exercise 44

    A line is parallel to x\displaystyle x-axis if all the points on the line have equal ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    (\(\displaystyle y, z\) cordinates)
    \(\displaystyle y\)-coordinates and \(\displaystyle z\)-coordinates (both). A line parallel to the \(\displaystyle x\)-axis has direction ratios \(\displaystyle (l,\,0,\,0)\), so only \(\displaystyle x\) changes along it.\[(x,\,y,\,z) = (t,\,b,\,c),\quad t \in \mathbb{R} \] \[y = b, \quad z = c \]

  5. Exercise 45

    x=a\displaystyle x=a represent a plane parallel to ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    \(\displaystyle y z\)-plane
    \(\displaystyle yz\)-plane. The plane \(\displaystyle x = a\) consists of the points \(\displaystyle (a,\,y,\,z)\), each at the same distance from the \(\displaystyle yz\)-plane.\[\text{distance from the } yz\text{-plane} = |x| = |a| \] \[\text{the } yz\text{-plane is } x = 0 \]

  6. Exercise 46

    The plane parallel to yz\displaystyle y z - plane is perpendicular to ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    \(\displaystyle x\)-axis
    \(\displaystyle x\)-axis. A plane parallel to the \(\displaystyle yz\)-plane is \(\displaystyle x = a\), with normal direction \(\displaystyle (1,\,0,\,0)\), which is the direction of the \(\displaystyle x\)-axis.\[x = a \] \[\text{normal } (1,\,0,\,0) \parallel x\text{-axis} \]

  7. Exercise 47

    The length of the longest piece of a string that can be stretched straight in a rectangular room whose dimensions are 10\displaystyle 10, 13\displaystyle 13 and 8\displaystyle 8 units are ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    \(\displaystyle \sqrt{333}\)
    \(\displaystyle \sqrt{333} = 3\sqrt{37}\) units. The longest straight piece is the space diagonal of the room.\[d = \sqrt{10^2 + 13^2 + 8^2} \] \[d = \sqrt{100 + 169 + 64} \] \[d = \sqrt{333} = 3\sqrt{37} \]

  8. Exercise 48

    If the distance between the points (a,2,1)\displaystyle (a, 2,1) and (1,−1,1)\displaystyle (1,-1,1) is 5\displaystyle 5, then a\displaystyle a ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    \(\displaystyle a=5\) or -$\displaystyle 3$
    \(\displaystyle a = 5\) or \(\displaystyle a = -3\). Apply the distance formula and set the distance equal to 5.\[\sqrt{(a-1)^2 + (2+1)^2 + (1-1)^2} = 5 \] \[(a-1)^2 + 9 = 25 \] \[(a-1)^2 = 16 \] \[a - 1 = \pm 4 \] \[a = 5 \quad \text{or} \quad a = -3 \]

  9. Exercise 49

    If the mid-points of the sides of a triangle AB; BC; CA are D (1\displaystyle 1, 2\displaystyle 2, - 3\displaystyle 3), E (3\displaystyle 3, 0\displaystyle 0, 1\displaystyle 1) and F(−1,1,−4)\displaystyle \mathrm{F}(-1,1,-4), then the centriod of the triangle ABC is ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    ($\displaystyle 1$, $\displaystyle 1$, -$\displaystyle 2$)
    \(\displaystyle (1,\,1,\,-2)\)With \(\displaystyle D,E,F\) the mid-points of \(\displaystyle AB,BC,CA\): \[A+B=2D,\quad B+C=2E,\quad C+A=2F \] \[A+B+C = D+E+F \] So the centroid of \(\displaystyle \triangle ABC\) is the centroid of \(\displaystyle \triangle DEF\): \[G=\left(\frac{1+3-1}{3},\ \frac{2+0+1}{3},\ \frac{-3+1-4}{3}\right) \] \[G=(1,\,1,\,-2) \]

  10. Exercise 50

    Match each item given under the column C1\displaystyle \mathrm{C}_1 to its correct answer given under column C2\displaystyle \mathrm{C}_2.
    Column C1\displaystyle \mathrm{C}_1Column C2\displaystyle \mathrm{C}_2
    (a) In xy\displaystyle x y-plane(i) Ist octant
    (b) Point (2\displaystyle 2, 3,4\displaystyle 3,4) lies in the(ii) yz\displaystyle y z-plane
    (c) Locus of the points having x\displaystyle x coordinate 0\displaystyle 0 is(iii) z\displaystyle z-coordinate is zero
    (d) A line is parallel to x\displaystyle x-axis if and only(iv) z\displaystyle z-axis
    (e) If x=0,y=0\displaystyle x=0, y=0 taken together will represent the(v) plane parallel to xy\displaystyle x y-plane
    (f) z=c\displaystyle z=c represent the plane(vi) if all the points on the line have equal y\displaystyle y and z\displaystyle z-coordinates.
    (g) Planes x=a,y=b\displaystyle x=a, y=b represent the line(vii) from the point on the respective
    (h) Coordinates of a point are the distances from the origin to the feet of perpendiculars(viii) parallel to z\displaystyle z - axis.
    (i) A ball is the solid region in the space enclosed by a(ix) disc
    (j) Region in the plane enclosed by a circle is known as a(x) sphere

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    NCERT’s answer
    (a)
    \(\displaystyle \leftrightarrow\) (iii)
    (b)
    \(\displaystyle \leftrightarrow\) (i)
    (c)
    \(\displaystyle \leftrightarrow\) (ii)
    (d)
    \(\displaystyle \leftrightarrow\) (vi)
    (e)
    \(\displaystyle \leftrightarrow\) (iv)
    (f)
    \(\displaystyle \leftrightarrow\) (v)
    (g)
    \(\displaystyle \leftrightarrow\) (viii)
    (h)
    \(\displaystyle \leftrightarrow\) (vii)
    (i)
    \(\displaystyle \leftrightarrow\) (x)
    (j) \(\displaystyle \leftrightarrow\) (ix)
    (a)-(iii), (b)-(i), (c)-(ii), (d)-(vi), (e)-(iv), (f)-(v), (g)-(viii), (h)-(vii), (i)-(x), (j)-(ix)
    (a)
    On the \(\displaystyle xy\)-plane \(\displaystyle z=0\): (iii).
    (b)
    \(\displaystyle x,y,z>0\) for \(\displaystyle (2,3,4)\): first octant, (i).
    (c)
    \(\displaystyle x=0\) is the \(\displaystyle yz\)-plane: (ii).
    (d)
    Only \(\displaystyle x\) varies along such a line; \(\displaystyle y,z\) stay fixed: (vi).
    (e)
    \(\displaystyle x=0,\ y=0\) together give the \(\displaystyle z\)-axis: (iv).
    (f)
    \(\displaystyle z=c\) is a plane parallel to the \(\displaystyle xy\)-plane: (v).
    (g)
    \(\displaystyle x=a,\ y=b\) meet in a line parallel to the \(\displaystyle z\)-axis: (viii).
    (h)
    Feet of the perpendiculars lie on the respective axes: (vii).
    (i)
    The solid region enclosed by a sphere is a ball: (x).
    (j) The plane region enclosed by a circle is a disc: (ix).