Free practice › Class 11 › Mathematics › Chapter 11
Introduction to Three Dimensional Geometry — JEE MCQs with answers
Topics in this chapter
- 11.1 Introduction
- 11.2 Coordinate Axes and Coordinate Planes in Three Dimensional Space
- 11.3 Coordinates of a Point in Space
- 11.4 Distance between Two Points
Free practice questions
What definite conclusion, about how many numbers are needed, does §11.1 draw right after the room example?
- A. We need two or three numbers, depending on the room
- B. We need at least four numbers
- C. We need not only two but three numbers
- D. We need exactly two numbers, as in the plane
Show answer
Answer: C — Option C is correct, quoting the text verbatim: “Therefore, we need not only two but three numbers.” Option A hedges with “two or three,” which the text never does — it commits to three. Option B says “at least four,” overstating the requirement. Option D says “exactly two,” the opposite of what the sentence concludes.
we need not only two but three numbers
Among the following four pairs of points, which pair is separated by the SMALLEST distance?
W: P(-5, 5, 0), Q(0, 4, 1); X: P(1, -3, 1), Q(-3, -5, 3); Y: P(1, 0, 2), Q(0, -3, -3); Z: P(-1, 5, -3), Q(2, 0, -5).
- A. Pair Z
- B. Pair W
- C. Pair Y
- D. Pair X
Show answer
Answer: D — Option D is correct. Computing $PQ^2=(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2$ for each pair: W: $PQ^2=27$, $PQ=3\sqrt{3}$; X: $PQ^2=24$, $PQ=2\sqrt{6}$; Y: $PQ^2=35$, $PQ=\sqrt{35}$; Z: $PQ^2=38$, $PQ=\sqrt{38}$. Pair X has the smallest value of $PQ^2$ (and hence of PQ, since the square root is increasing), so it is the pair with the smallest distance. Each of the other three options names a pair whose distance is not the smallest of the four, by the same computed values.
Class 11 · Mathematics · Ch 11 · §11.4 · p.212 · ¶1 ("This gives us the distance between two points (x1, y1, z1) and (x2, y2, z2).")
What are the coordinates of the origin O?
- A. (0, 0, 0) only in the octant XOYZ
- B. (0, 0, 0)
- C. cannot be determined without more information
- D. same as the coordinates of any point on the z-axis
Show answer
Answer: B — Option B is correct because the Note states the coordinates of the origin O are (0,0,0). Option A is incorrect: the origin O is the common vertex of all eight octants; its coordinates (0,0,0) do not depend on any particular octant. Option C is incorrect: the Note fixes the origin's coordinates as (0,0,0) unconditionally. Option D is incorrect: a general point on the z-axis has coordinates (0,0,z) with z possibly nonzero; only the origin itself is (0,0,0).
The coordinates of the origin O are (0,0,0).
Into how many octants do the three coordinate planes divide space, according to §11.2?
- A. six
- B. eight
- C. sixteen
- D. four
Show answer
Answer: B — Option B is correct: §11.2 states "The three coordinate planes divide the space into eight parts known as octants." — eight. "Six" and "four" undercount (four is the number of quadrants a single plane is divided into by two axes, not the three-dimensional count); "sixteen" overcounts, e.g. by doubling eight without basis in the text.
The three coordinate planes divide the space into eight parts known as octants.
If P is the origin O(0,0,0), find the distance OQ where Q is the point (2, 3, 6).
- A. $2\sqrt{10}$
- B. $7$
- C. $\sqrt{53}$
- D. $3\sqrt{5}$
Show answer
Answer: B — Option B is correct. Putting $x_1=y_1=z_1=0$ in the distance formula gives $OQ = \sqrt{x_2^2+y_2^2+z_2^2}$. Here $x_2=2$, $y_2=3$, $z_2=6$, so $OQ^2 = 4+9+36 = 49$, and $OQ = 7$. $3\sqrt{5}$ is wrong: it results if one dropped the x-term (used only the other two squared differences); that sum is 45, not 49, so it does not satisfy $OQ^2=49$. $2\sqrt{10}$ is wrong: it results if one dropped the y-term (used only the other two squared differences); that sum is 40, not 49, so it does not satisfy $OQ^2=49$. $\sqrt{53}$ is wrong: it results if one double-counted the x-term (added its squared difference twice); that sum is 53, not 49, so it does not satisfy $OQ^2=49$.
Class 11 · Mathematics · Ch 11 · §11.4 · p.212 · ¶1 ("which gives the distance between the origin O and any point Q (x2, y2, z2).")
In Fig 11.3, planes through P parallel to the coordinate planes meet the x-axis, y-axis and z-axis at A, B and C respectively. If P = (-7, -3, 2), what are OA, OB and OC respectively, taking distances along OX′, OY′ and OZ′ as negative as §11.2 does?
- A. OA = -3, OB = -7, OC = 2
- B. OA = -7, OB = 2, OC = -3
- C. OA = 2, OB = -3, OC = -7
- D. OA = -7, OB = -3, OC = 2
Show answer
Answer: D — Option D is correct because the text sets OA = x, OB = y and OC = z, so for P = (-7, -3, 2), OA = -7, OB = -3, OC = 2. Option A is incorrect: this swaps the x- and y-values between OA and OB; OA must equal x (-7), not -3. Option B is incorrect: this swaps the y- and z-values between OB and OC; OB must equal y (-3), not 2. Option C is incorrect: this swaps the x- and z-values between OA and OC; OA must equal x (-7), not 2.
Let OA = x, OB = y and OC = z.
In §11.2 the eight octants are named by priming the axis letters, and are then denoted by Roman numerals I to VIII in the order in which the section lists them. Which numeral denotes the octant XOY′Z′?
- A. I
- B. VI
- C. VIII
- D. V
Show answer
Answer: C — Option C is correct: §11.2 lists the eight octant names in the order XOYZ, X′OYZ, X′OY′Z, XOY′Z, XOYZ′, X′OYZ′, X′OY′Z′ and XOY′Z′, and denotes them I, II, III, ..., VIII respectively, so XOY′Z′ - the entry in position 8 of that list - is denoted VIII. Option A is wrong: I denotes the octant XOYZ (position 1), not XOY′Z′ - it is where you land if you count 8 places from the wrong end of the list. Option B is wrong: VI denotes the octant X′OYZ′ (position 6), not XOY′Z′ - it is 2 places earlier in the list. Option D is wrong: V denotes the octant XOYZ′ (position 5), not XOY′Z′ - it is 3 places earlier in the list.
These octants could be named as XOYZ, X′OYZ, X′OY′Z, XOY′Z, XOYZ′, X′OYZ′, X′OY′Z′ and XOY′Z′. and denoted by I, II, III, ..., VIII , respectively.
Which part of the ceiling fan does §11.1 use as its second room example?
- A. The edge of one blade of the ceiling fan
- B. The central tip of a table fan
- C. The central tip of a wall-mounted clock
- D. The central tip of the ceiling fan in a room
Show answer
Answer: D — Option D is correct: the text says “the position of the central tip of the ceiling fan in a room.” Option A substitutes “the edge of one blade,” a different, unstated point on the fan. Option B and Option C both replace “ceiling fan” with an unrelated appliance not mentioned in the section.
the position of the central tip of the ceiling fan in a room
$ABCD$ has vertices $A(-6, 2, -1)$, $B(-6, 7, -3)$, $C(-10, 10, -1)$ and $D(-9, 6, 1)$. Find the length of the diagonal $AC$.
- A. $\sqrt{26}$
- B. $4 \sqrt{5}$
- C. $2 \sqrt{29}$
- D. $\sqrt{29}$
Show answer
Answer: B — The two diagonals of $ABCD$ are $AC$ and $BD$; here $AC^2=80$ and $BD^2=26$, and the sides are $AB=\sqrt{29}$, $BC=\sqrt{29}$. Option B is correct: $AC^2=(-10+6)^2+(10-2)^2+(-1+1)^2=80$, so $AC=4 \sqrt{5}$. Option A is wrong: that is the other diagonal $BD$, whose square is 26, not 80. Option C is wrong: that is $AB+BC=\sqrt{29}+\sqrt{29}\approx 10.77$, the route round two sides, not the straight diagonal $AC\approx 8.944$. Option D is wrong: that is the side $AB$ (square 29), not the diagonal $AC$.
Now, it is required to prove that ABCD is not a rectangle. For this, we show that
P is the point (−5, 4, −8). What is the perpendicular distance of P from the ZX-plane?
- A. 5
- B. −5
- C. 4
- D. 8
Show answer
Answer: C — Option C is correct. §11.3 states that x, y and z are the perpendicular distances of P from the YZ, ZX and XY planes respectively, so the distance of P from the ZX-plane is given by its y-coordinate, 4. Option A is incorrect: 5 is |x|, the size of P's distance from the YZ-plane, not from the ZX-plane. Option B is incorrect: −5 is the x-coordinate of P, which goes with the YZ-plane, not the ZX-plane. Option D is incorrect: 8 is |z|, the size of P's distance from the XY-plane, not from the ZX-plane.
x, y and z are perpendicular distances from YZ, ZX and XY planes, respectively.
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