NCERT Exemplar Problems Class 11 Mathematics Chapter 12 Introduction to Three Dimensional Geometry

 

Short Answer Type Questions

Q1. Locate the following points:

(i) (1,-1, 3),
(ii) (-1,2,4)
(iii) (-2, -4, -7)
(iv) (-4,2, -5)
Sol: Given, coordinates
(i) (1,-1, 3),
(ii) (-1,2,4)
(iii) (-2, -4, -7)
(iv) (-4,2, -5)

Q2. Name the octant in which each of the following points lies.
(i) (1,2,3)
(ii) (4,-2, 3)
(iii) (4,-2,-5)
(iv)(4,2,-5)
(v) (-4,2,5)
(vi) (-3,-1,6)
(vii) (2,-4,-7)
(viii) (-4, 2,-5)

Sol: We know that the sign of the coordinates of a point determine the octant in which the point lies. The following table shows the signs of the coordinates in eight octants.

Q3. Let A, B, C be the feet of perpendiculars from a point P on the x, y,z-axes respectively. Find the coordinates of A, B and C in each of the following where the point P is:
(i) (3,4,2)
(ii) (-5,3,7)
(iii) (4,-3,-5)
Sol: We know that, on x-axis, y, z = 0, on y-axis, x, z = 0 and on z-axis, x,y = 0. Thus, the feet of perpendiculars from given point P on the axis are as follows.

(i) A(3,0,0),5(0,4,0),C(0,0,2)
(ii) A(-5, 0, 0), B(0, 3, 0), C(0, 0, 7)
(iii) A(4, 0, 0), 5(0, -3, 0), C(0,0, -5)

Q4. Let A, B, C be the feet of perpendiculars from a point P on the xy, yz and zx- planes respectively. Find the coordinates of A, B, C in each of the following where the point P is
(i) (3,4,5)
(ii) (-5,3,7)
(iii) (4,-3,-5).
Sol: We know that, on xy-plane z = 0, on yz-plane, x = 0 and on zx-plane, y = 0. Thus, the coordinates of feet of perpendicular on the xy, yz and zx-planes from the given point are as follows:
(i) A(3,4,0), 5(0,4, 5), C(3,0,5)
(ii) A(-5, 3,0), 5(0, 3, 7), C(-5, 0, 7)
(iii) A(4, -3, 0), 5(0, -3, -5), C(4,0, -5)

Q5. How far apart are the points (2,0, 0) and (-3, 0, 0)?
Sol: Given points are A (2, 0, 0) and 5(-3,0, 0).
AB = |2 – (-3)| = 5

Q6. Find the distance from the origin to (6, 6, 7).
Sol: Distance form origin to the point (6, 6, 7)

Q8. Show that the point ,4(1, -1, 3), 6(2, -4, 5) and (5, -13, 11) are collinear.
Sol: Given points are ,4(1, -1, 3), 6(2, -4, 5) and C(5, -13, 11).

Q9. Three consecutive vertices of a parallelogram ABCD are .4(6, -2,4), 6(2,4, -8), C(-2, 2, 4). Find the coordinates of the fourth vertex.
Sol: Let the coordinates of the fourth vertex D be (x, y, z).

Q10 .Show that the triangle ABC with vertices .4(0,4,1), 6(2,3, -1) and C(4, 5,0) is right angled.
Sol: The vertices of ∆ABC are A(0,4, 1), 5(2, 3, -1) and C(4, 5, 0).

Q11. Find the third vertex of triangle whose centroid is origin and two vertices are (2,4,6) and (0, -2, -5).
Sol: Let the third or unknown vertex of ∆ABC be A(x, y, z).
Other vertices of triangle are 5(2,4, 6) and C(0, -2, -5).
The centroid is G(0, 0, 0).

Q12. Find the centroid of a triangle, the mid-point of whose sides are D (1,2, – 3), E(3,0, l)and F(-l, 1,-4).
Sol:Given that, mid-points of sides of AABC are D(l, 2, -3), E(3, 0, 1) and F(-l, 1,-4).

Q14. Three vertices of a Parallelogram ABCD are A(\, 2, 3), B(-A, -2, -1) and C(2, 3, 2). Find the fourth vertex
Sol: Let the fourth vertex of the parallelogram D(x, y, z).
Mid-point of BD

Q15. Find the coordinate of the points which trisect the line segment joining the points .A(2, 1, -3) and B(5, -8, 3).

Q16. If the origin is the centroid of a triangle ABC having vertices A(a, 1, 3), B(-2, b, -5) and C(4, 7, c), find the values of a, b, c.
Sol: Vertices of AABC are A(a, 1, 3), B(-2, b, -5) and C(4, 7, c).
Also, the centroid is G(0, 0, 0).

Q17. Let A(2, 2, -3), 5(5, 6, 9) and C(2, 7, 9) be the vertices of a triangle. The internal bisector of the angle A meets BC at the point Find the coordinates of D.

Long Answer Type Questions

Q18. Show that the three points A(2, 3, 4), 5(-l, 2, -3) and C(-4, 1, -10) are collinear and find the ratio in which Cdivides
Sol: Given points are A(2, 3, 4), B(-1, 2, -3) and C(-4,1,-10)

Q19. The mid-point of the sides of a triangle are (1, 5, -1), (0,4, -2) and (2, 3,4). Find its vertices. Also, find the centroid of the triangle.
Sol: Given that mid-points of the sides of AABC are D( 1, 5, -1), E(0, 4, -2) and F(2, 3,4).

Q20. Prove that the points (0, -1, -7), (2, 1, -9) and (6, 5, -13) are collinear. Find the ratio in which the first point divides the join of the other two.
Sol: Given points are 4(0, -1, -7), 8(2, 1, -9) and C(6, 5, -13).

Q21. What are the coordinates of the vertices of a cube whose edge is 2 units, one of whose vertices coincides with the origin and the three edges passing through the origin, coincides with the positive direction of the axes through the origin?
Sol: The coordinate of the cube whose edge is 2 units, are:
(2, 0, 0), (2,2, 0), (0, 2, 0), (0, 2,2), (0, 0,2), (2,0, 2), (0, 0, 0) and (2,2, 2)

Objective Type Questions

Q22. The distance of point P(3,4, 5) from the yz-plane is
(a) 3 units
(b) 4 units
(c) 5 units
(d) 550
Sol: (a) Given point is P{3,4, 5).
Distance of P from yz-plane = |x coordinate of P| = 3

Q23. What is the length of foot of perpendicular drawn from the point P(3,4, 5) on y-axis?

Q24. Distance of the point (3,4, 5) from the origin (0, 0, 0) is

Q25. If the distance between the points (a,0,1) and (0,1,2) is √27, then the value of a is
(a) 5
(b) ± 5
(c) -5
(d) none of these

Q26. x-axis is the intersection of two planes
(a) xy and xz
(b) yz and zx
(c) xy and yz
(d) none of these
Sol: (a) We know that, on the xy and xz-planes, the line of intersection is x-axis.

Q27. Equation of Y-axis is considered as
(a) x = 0, y = 0
(b) y = 0, z = 0
(c) z = 0, x = 0
(d) none of these
Sol:(c) On the j-axis, x = 0 and z = 0.

Q28. The point (-2, -3, -4) lies in the
(a) First octant
(b) Seventh octant
(c) Second octant
(d) Eighth octant
Sol: (b) The point (-2, -3, -4) lies in seventh octant.

Q29. A plane is parallel to yz-plane so it is perpendicular to
(a) x-axis
(b) y-axis
(c) z-axis
(d) none of these
Sol: (a) A plane parallel to yz-plane is perpendicular to x-axis.

Q30. The locus of a point for which y = 0, z = 0 is
(a) equation of x-axis
(b) equation of y-axis
(c) equation at z-axis
(d) none of these
Sol: (a) We know that, equation of the x-axis is: y = 0, z = 0 So, the locus of the point is equation of x-axis.

Q31. The locus of a point for which x = 0 is
(a) xy-plane
(b) yz-plane
(c) zx-plane
(d) none of these
Sol: (b) On the yz-plane, x = 0, hence the locus of the point is yz-plane.

Q32. If a parallelepiped is formed by planes drawn through the points (5,8,10) and (3, 6, 8) parallel to the coordinate planes, then the length of diagonal of the parallelepiped is

Q33. L is the foot of the perpendicular drawn from a point P(3, 4, 5) on the xy-plane. The coordinates of point L are
(a) (3,0,0)
(b) (0,4,5)
(c) (3, 0, 5)
(d) none of these
Sol: (d) We know that on the xy-plane, z = 0.
Hence, the coordinates of the points L are (3,4, 0).

Q34. L is the foot of the perpendicular drawn from a point (3, 4, 5) on x-axis. The coordinates of L are
(a) (3,0,0)
(b) (0,4,0)
(c) (0, 0, 5)
(d) none of these
Sol: (a) On the x-axis, y = 0 and z = 0.
Hence, the required coordinates are (3, 0,0).

Fill in the Blanks Type Questions
Q35. The three axes OX, OY, OZ determine______ .
Sol: The three axes OX, OY and OZ determine three coordinate planes.

Q36. The three planes determine a rectangular parallelepiped which has____ of rectangular faces.

Q37. The coordinates of a point are the perpendicular distance from the _____ on the respective axes.
Sol: Given points

Q38. The three coordinate planes divide the space into _________parts.
Sol: Eight

Q39. If a point P lies in yz-plane, then the coordinates of a point on yz-plane is of the form_______.
Sol: We know that, on yz-plane, x = 0.So, the coordinates of the required point are (0, y, z).

Q40. The equation of yz-plane is ______ .
Sol: On yz-plane for any point x-coordinate is zero.
So, yz-plane is locus of point such that x = 0, which is its equation.

Q41. If the point P lies on z-axis, then coordinates of P are of the form_____.
Sol: On the z-axis, x = 0 and y = 0.
So, the required coordinates are of the form (0, 0, z).

Q42. The equation of z-axis, are ______.
Sol: Any point on the z-axis is taken as (0, 0, z).
So, for any point on z-axis, we have x = 0 and y = 0, which together represents its equation.
Q43. A line is parallel to xy-plane if all the points on the line have equal_________.
Sol: A line is parallel to xy-plane if each point P(x, y, z) on it is at same distance from xy-plane.
Distance of point P from xy plane is ‘z’
So, line is parallel to xy-plane if all the points on the line have equal z-coordinate.

Q44. A line is parallel to x-axis if all the points on the line have equal ______.
Sol: A line is parallel to x-axis if each point on it maintains constant distance from y-axis and z-axis.
So, each point has equal y and z-coordinates. .

Q45. x = a represents a plane parallel to .
Sol: Locus of point P(x, y, z) is x = a.
Therefore, each point P has constant x-coordinate.
Now, x is distance of point P from yz-plane.
So, here plane x = a is at constant distance ‘a’ from yz-plane and parallel to _yz-plane.

Q46. The plane parallel to yz-plane is perpendicular to_____ .
Sol: The plane parallel to yz-plane is perpendicular to x-axis.

Q47. 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 are______ .
Sol: Given dimensions are: a = 10, 6=13 andc = 8.
Required length of the string = yja² + b² + c² = ^100 + 169 + 64 = -7333

Q48. If the distance between the points (a, 2,1) and (1,-1,1) is 5, then a_______ .
Sol: Given points are (a, 2,1) and (1,-1,1).

Q49. If the mid-points of the sides of a triangle AB; BC; CA are D(l, 2, – 3), E( 3, 0, 1) and F(-l, 1, -4), then the centroid of the triangle ABC is________ .
Sol: Given that, mid-points of sides of AABC are D( 1, 2, -3), E(3, 0, 1) and F(-l, 1,-4).

Matching Column Type Questions

Q50. Match each item given under the column C₁ to its correct answer given under column C₂.

Column C,		Column C2
(a)	In xy-plane	(i)	1st octant
(b)	Point (2, 3,4) lies in the	(ii)	vz-plane
(c)	Locus of the points having x coordinate 0 is	(iii)	z-coordinate is zero
(d)	A line is parallel to x-axis if and only	(iv)	z-axis .
(e)	If x = 0, y = 0 taken together will represent the	(v)	plane parallel to xy-plane
(f)	z = c represent the plane	(vi)	if all the points on the line have equal y and z-coordinates.
(g)	Planes x = a, y = b represent the line	(vii)	from the point on the respective axis.
00	Coordinates of a point are the distances from the origin to the feet of perpendiculars	(viii)	parallel to z-axis
(i)	A ball is the solid region in the space	(ix)	disc
G)	Region in the plane enclosed by a circle is known as a	00	sphere

 

Sol: (a) In xy-plane, z-coordinate is zero.
(b) The point (2, 3,4) lies in 1st octant.
(c) Locus of the points having x-coordinate zero is yz-plane.
(d) A line is parallel to x-axis if and only if all the points on the line have equal y and z-coordinates.
(e)x = 0, y = 0 represent z-axis
(f) z = c represents the plane parallel to xy-plane.

(g) The plane x = a is parallel to yz-plane.
Plane y = b is parallel to xz-plane.
So, planes x = a and y = b is line of intersection of these planes.
Now, line of intersection of yz-plane and xz-plane is z-axis.
So, line of intersection of planes x = a andy = b is line parallel to z-axis.
(h) Coordinates of a point are the distances from the origin to the feet of perpendicular from the point on the respective axis.
(i) A ball is the solid region in the space enclosed by a sphere.
(j) The region in the plane enclosed by a circle is known as a disc.
Hence, the correct matches are:
(a) – (iii), (b) – (i), (c) – (ii), (d) – (vi), (e) – (iv),
(f) – (v), (g) – (viii), (h) – (vii), (i) – (x), (j) – (ix),