Sample QuestionsIntroduction to Three Dimensional Geometry questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If the distance between the points $(a, 0, 1)$ and $(0, 1, 2)$ is $\sqrt{27}$ then the value of $a$ is:
- A
$5$
- ✓
$\underline{+}5$
- C
$-5$
- D
Answer: B.
View full solution →If $A = (1, 2, 3), B = (2, 3, 4)$ and $AB$ is produced upto $C$ such that $\text{2AB = BC}$ then $C =$
- A
$(5, 4, 6)$
- B
$(6, 2, 4)$
- ✓
$(4, 5, 6)$
- D
Answer: C.
View full solution →There is one and only one sphere through:
- ✓
$4$ points not in the same plane
- B
$4$ points not lie in the same straight line
- C
- D
$3$ points not lie in the same line
Answer: A.
View full solution →The vector equation of a sphere having centre at origin and radius $5$ is:
Answer: A.
View full solution →The perpendicular distance of the point $P(3, 3, 4)$ from the $x-$axis is
Answer: B.
View full solution →Directions: In the following questions, the Assertions $(A)$ and Reason(s) $(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The points $A(3, -1, 2), B(1, 2, -4), C(-1, 1, 2)$ and $D(1, -2, 8)$ are the vertices of a parallelogram.
Reason: Coordinates of mid$-$point of a line joining the points $A(x_1, y_1, z_1)$ and $B(x_1, y_2, z_2)$ is $\Big(\frac{\text{x}_{1}+\text{x}_{2}}{2},\frac{\text{y}_1+\text{y}_2}{2},\frac{\text{z}_{1}+\text{z}_{2}}{2}\Big).$
- ✓
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
Answer: A.
View full solution →Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The points $A(1, -1, 3), B(2, -4, 5)$ and $C(5, -13, 11)$ are collinear.
Reason: If $\text{AB + BC = AC,}$ then $\text{A, B, C}$ are collinear.
- ✓
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
Answer: A.
View full solution →Directions: In the following questions, the Assertions $(A)$ and Reason(s) $(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: Coordinates of centroid of a triangle formed by the vertices $A(3, 2, 0), B(5, 3, 2)$ and $C(0, 2, 4)$ is $\Big(\frac{8}{3},\frac{8}{3},\frac{8}{3}\Big).$
Reason: Coordinates of centroid of a triangle with vertices $A(x_1, y_1, z_1), B(x_2, y_2, z_2)$ and $C(x_3, y_3, z_3)$ is $\Big(\frac{\text{x}_{1}+\text{x}_{2}+\text{x}_{3}}{3},\frac{\text{y}_{1}+\text{y}_{2}+\text{y}_{3}}{3},\frac{\text{z}_{1}+\text{z}_{2}+\text{z}_{3}}{3}\Big).$
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- ✓
Assertion is wrong statement but Reason is correct statement.
Answer: D.
View full solution →Directions: In the following questions, the Assertions $(A)$ and Reason$(s) (R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The foot of perpendicular drawn from the point $A(1, 2, 8)$ on the $xy -$ plane is $(1, 2, 0).$
Reason: Equation of $xy -$ plane is $y = 0.$
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- ✓
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
Answer: C.
View full solution →Directions: In the following questions, the Assertions $(A)$ and Reason(s) $(R)$ have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: The distance between the points $ (1+\sqrt{11}, 0, 0) $ and $(1, -2, 3)$ is $2\sqrt{6}$ units.
Reason: Distance between any two points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$
$\mid\text{AB}\mid=\sqrt{(\text{x}_{2}+\text{x}_{1})^{2}+(\text{y}_{2}+\text{y}_{1})^{2}(\text{z}_{2}+\text{z}_{1})^{2}}.$
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- ✓
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
Answer: C.
View full solution →Find the distance between (2, -1, 3) and (-2, 1, 3) pairs of points.
Find the distance between (-1, 3, -4) and (1, -3, 4) pairs of points.
Find the distance between (-3, 7, 2) and (2, 4, -1) pairs of points.
Find the distance between (2, 3, 5) and (4, 3, 1) pairs of points.
Name the octants in which the following points lie:
(1, 2, 3), (4, -2, 3), (4, -2, -5), (4, 2, -5), (-4, 2, -5), (-4, 2, 5), (-3, -1, 6), (-2, -4, -7)
A point R with x-coordinate 4 lies on the line segment joining the points P(2, -3, 4) and Q (8, 0, 10). Find the coordinates of the point R.
[Hint Suppose R divides PQ in the ratio k : 1. The coordinates of the point R are given by $\left(\frac{8 k+2}{k+1}, \frac{-3}{k+1}, \frac{10 k+4}{k+1}\right)$].
View full solution →Find the coordinates of a point on $y-$axis which are at a distance of $5 \sqrt2$ from the point $P(3, -2, 5).$
View full solution →If the origin is the centriod of the triangle PQR with vertices P(2a, 2, 6), Q(-4, 3b, -10) and R(8, 14, 2c), then find the values of a, b and c.
Three vertices of a parallelogram ABCD are A(3, -1, 2), B(1, 2, -4) and C(-1, 1, 2). Find the coordinates of the fourth vertex.
Find the equation of the set of points which are equidistance from the points $(1, 2, 3)$ and $(3, 2, -1).$
View full solution →If $A$ and $B$ be the points $(3, 4, 5)$ and $(-1, 3, -7),$ respectively, find the equation of the set of points $P$ such that $PA^2 + PB^2 = k^2,$ where $k$ is a constant.
View full solution →Find the length of the medians of the triangle with vertices A (0, 0, 6), B (0, 4, 0) and C (6, 0, 0).
Find the equation of the set of the point $P$, the sum of whose distance from $A(4, 0, 0)$ and $B(-4, 0, 0)$ is equal to $10.$
View full solution →Find the equation of the set of points $P$ such that $PA^2 + PB^2 =2k^2,$ where $A$ and $B$ are the points $(3, 4, 5) $and $(-1, 3, -7),$ respectively.
View full solution →Are the points $A(3, 6, 9), B(10, 20, 30)$ and $C(25, -41, 5),$ the vertices of a right-angled triangle$?$
View full solution →Fill in the blanks.
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 centriod of the triangle ABC is ________.
Fill in the blanks.
A line is parallel to x-axis if all the points on the line have equal ________.
Fill in the blanks.
The coordinates of a point are the perpendicular distance from the ________ on the respectives axes.
Fill in the blanks:
Coordinate planes divide the space into ______ octants.
Fill in the blanks:
The coordinates of points in the XY-plane are of the form _______.