NCERT Exemplar Solutions Class 11 Maths Introduction to Three Dimensional Geometry — Fill in the Blanks and Matching Q35–50

NCERT Exemplar Problems, Class 11 Mathematics — Introduction to Three Dimensional Geometry, Fill in the Blanks and Matching questions 35 to 50. All 16 questions solved.

Fifteen fill-in-the-blanks (35 to 49) and one ten-part matching question (50) that together revise the vocabulary of the chapter: coordinate planes, octants, equations of axes and planes, and a couple of short calculations (47, 48, 49). Nothing here needs more than the distance formula and the rule that each axis is where two coordinates vanish. Question 49 uses the centroid, which is no longer part of the current rationalised NCERT chapter, but it is a one-line average.

None of the printed answers in the key is wrong on this page.

Key insight. The three coordinate planes are $x=0$ ($yz$-plane), $y=0$ ($zx$-plane) and $z=0$ ($xy$-plane). Each axis is the intersection of two of them, and a plane $x=a$ is parallel to the $yz$-plane because it fixes only $x$. Once that dictionary is clear, every blank here is a lookup.

Question 35

The three axes $OX$, $OY$, $OZ$ determine ____.

Solution. Taken two at a time, the three axes span the $xy$-plane, the $yz$-plane and the $zx$-plane.

Question 36

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

Solution. A rectangular box has six faces, which come in three pairs of opposite faces, one pair for each of the three planes.

Question 37

The coordinates of a point are the perpendicular distance from the ____ on the respective axes.

Solution. The book’s wording is clumsy. What it intends is that the $x$-, $y$- and $z$-coordinates of a point are the signed perpendicular distances from the given point to the $yz$-, $zx$- and $xy$-planes, measured along the respective axes. The printed answer is the given point.

Question 38

The three coordinate planes divide the space into ____ parts.

Solution. Each coordinate has two possible signs, so there are $2\times2\times2=8$ regions, the eight octants.

Question 39

If a point $P$ lies in the $yz$-plane, then the coordinates of a point on the $yz$-plane are of the form ____.

Solution. The $yz$-plane is $x=0$, with $y$ and $z$ free.

Question 40

The equation of the $yz$-plane is ____.

Solution. Every point of the $yz$-plane has zero distance from it in the $x$ direction.

Question 41

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

Solution. On the $z$-axis both $x$ and $y$ are zero.

Question 42

The equation of the $z$-axis are ____.

Solution. The $z$-axis is the intersection of the $yz$-plane ($x=0$) and the $zx$-plane ($y=0$).

Question 43

A line is parallel to the $xy$-plane if all the points on the line have equal ____.

Solution. The $xy$-plane is $z=0$. A line parallel to it never rises or falls, so every point on it has the same height.

Question 44

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

Solution. A line parallel to the $x$-axis moves only in the $x$ direction, so $y$ and $z$ stay constant.

Question 45

$x=a$ represents a plane parallel to ____.

Solution. The equation fixes $x$ and leaves $y$ and $z$ free, just like the $yz$-plane $x=0$; it is that plane shifted by $a$.

Question 46

The plane parallel to the $yz$-plane is perpendicular to ____.

Solution. The $yz$-plane has the $x$-axis as its normal, and parallel planes share a normal.

Question 47

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

Solution. The longest straight line in a box joins opposite corners, i.e. the space diagonal: $\sqrt{10^2+13^2+8^2}=\sqrt{100+169+64}=\sqrt{333}=3\sqrt{37}$ units.

Question 48

If the distance between the points $(a,2,1)$ and $(1,-1,1)$ is $5$, then $a=$ ____.

Solution. Squaring: $(a-1)^2+(2+1)^2+(1-1)^2=25$, so $(a-1)^2=16$ and $a-1=\pm4$. The two roots are $a=5$ and $a=-3$.

Question 49

If the mid-points of the sides of a triangle $AB$, $BC$, $CA$ are $D(1,2,-3)$, $E(3,0,1)$ and $F(-1,1,-4)$, then the centroid of the triangle $ABC$ is ____.

Solution. As in Question 12, the centroid of the triangle of midpoints equals the centroid of $ABC$: $\left(\frac{1+3-1}{3},\frac{2+0+1}{3},\frac{-3+1-4}{3}\right)=(1,1,-2)$.

Question 50

Match each item given under the column $C_1$ to its correct answer given under column $C_2$.

Solution. Take the rows in turn.

• (a) In the $xy$-plane the height is zero, so (iii) $z$-coordinate is zero.
• (b) $(2,3,4)$ has all three coordinates positive, so (i) first octant.
• (c) $x=0$ is the (ii) $yz$-plane.
• (d) A line is parallel to the $x$-axis exactly when $y$ and $z$ are the same at every point, (vi).
• (e) $x=0$ together with $y=0$ is the (iv) $z$-axis.
• (f) $z=c$ is a (v) plane parallel to the $xy$-plane.
• (g) Two planes $x=a$ and $y=b$ meet in a line along which only $z$ varies, so (viii) parallel to the $z$-axis.
• (h) The book’s phrase for this item is the same clumsy sentence as in Question 37; the printed pairing is (vii) “from the point on the respective”.
• (i) A solid ball is the region enclosed by a (x) sphere.
• (j) The region of a plane enclosed by a circle is a (ix) disc.

Common mistakes

• Question 36: answering “six” when the blank says “of rectangular faces” and the book wants “three pairs”. Both describe the same box, so match the wording of the sentence.
• Question 42: writing $z=0$ for the $z$-axis. $z=0$ is the $xy$-plane; the $z$-axis is where both $x$ and $y$ vanish.
• Question 44: answering “$x$-coordinates”. A line parallel to the $x$-axis moves along $x$, so it is $y$ and $z$ that must stay constant.
• Question 47: adding the three dimensions, $10+13+8=31$, instead of taking the diagonal $\sqrt{10^2+13^2+8^2}$.
• Question 48: stopping at $a=5$ after $a-1=4$. Squaring gives two roots, and $a=-3$ is just as valid.
• Question 50 (g): matching $x=a,\ y=b$ to a plane. Two non-parallel planes meet in a line, and since $z$ is unrestricted it runs parallel to the $z$-axis.

Practise next

• More from this unit: Short Answer Type Q1–17 and Long Answer and Objective Type Q18–34.
• NCERT Class 11 Maths Exercise 11.1 for the coordinate axes and planes.
• NCERT Class 11 Maths Exercise 11.2 for the distance formula.
• NCERT Class 11 Maths Miscellaneous Exercise.