3D Geometry Practice Questions - DPP for JEE

 Page 1


PART-I (Single Correct MCQs)
1. The line,  intersects the curve xy = c
2
, z = 0 if c is
equal to
(a) ± 1
(b)
(c)
(d) None of these
2. From the point (1, –2, 3) lines are drawn to meet the sphere x
2
 + y
2
 + z
2
= 4 and they are divided internally in the ratio2 : 3. The locus of the
point of division is
(a)
(b)
(c)
(d)
Page 2


PART-I (Single Correct MCQs)
1. The line,  intersects the curve xy = c
2
, z = 0 if c is
equal to
(a) ± 1
(b)
(c)
(d) None of these
2. From the point (1, –2, 3) lines are drawn to meet the sphere x
2
 + y
2
 + z
2
= 4 and they are divided internally in the ratio2 : 3. The locus of the
point of division is
(a)
(b)
(c)
(d)
3. If two lines L
1
 and L
2 
in space, are defined by
,  and
, 
then L
1
 is perpendicular to L
2
, for all non-negative reals ? and µ, such that :
(a)
(b)
(c)
(d)
4. The locus of a point, such that the sum of the squares of its distances
from the planes x + y + z = 0, x – z =0 and x – 2y + z = 0 is 9, is
(a) (b)  
(c) (d)  
5. A variable plane passes through a fixed point (1, 2, 3). The locus of the
foot of the perpendicular from the origin to this plane is given by
(a) x
2
 + y
2
 + z
2
 – 14 = 0
(b) x
2
 + y
2
 + z
2
 + x + 2y + 3z = 0
(c) x
2
 + y
2
 + z
2
 – x – 2y – 3z = 0
(d) None of these
6. The direction cosines l, m, n, of one of the two lines connected by the
relations
 are
(a)
(b)
(c)
(d)
Page 3


PART-I (Single Correct MCQs)
1. The line,  intersects the curve xy = c
2
, z = 0 if c is
equal to
(a) ± 1
(b)
(c)
(d) None of these
2. From the point (1, –2, 3) lines are drawn to meet the sphere x
2
 + y
2
 + z
2
= 4 and they are divided internally in the ratio2 : 3. The locus of the
point of division is
(a)
(b)
(c)
(d)
3. If two lines L
1
 and L
2 
in space, are defined by
,  and
, 
then L
1
 is perpendicular to L
2
, for all non-negative reals ? and µ, such that :
(a)
(b)
(c)
(d)
4. The locus of a point, such that the sum of the squares of its distances
from the planes x + y + z = 0, x – z =0 and x – 2y + z = 0 is 9, is
(a) (b)  
(c) (d)  
5. A variable plane passes through a fixed point (1, 2, 3). The locus of the
foot of the perpendicular from the origin to this plane is given by
(a) x
2
 + y
2
 + z
2
 – 14 = 0
(b) x
2
 + y
2
 + z
2
 + x + 2y + 3z = 0
(c) x
2
 + y
2
 + z
2
 – x – 2y – 3z = 0
(d) None of these
6. The direction cosines l, m, n, of one of the two lines connected by the
relations
 are
(a)
(b)
(c)
(d)
7. A line makes the same angle a with each of the x and y axes. If the
angle ?, which it makes with the z-axis, is such that  sin
2
? = 2 sin
2
a ,
then what is the value of a ?
(a) p/4
(b) p/6
(c) p/3
(d) p/2
8. If Q is the image of the point P(2, 3, 4) under the reflection in the plane x – 2y +
5z = 6, then the equation of the line PQ is
(a)
(b)
(c)
(d)
9. The foot of the perpendicular from (2, 4, –1) to the line 
(a) (– 4, 1, – 3)
(b) (4, –1, –3)
(c) (–4, –1, 3)
(d) (– 4, – 1, –3)
10. The equation of two lines through the origin, which intersect the line 
 at angles of  each, are
(a)
(b)
(c)
Page 4


PART-I (Single Correct MCQs)
1. The line,  intersects the curve xy = c
2
, z = 0 if c is
equal to
(a) ± 1
(b)
(c)
(d) None of these
2. From the point (1, –2, 3) lines are drawn to meet the sphere x
2
 + y
2
 + z
2
= 4 and they are divided internally in the ratio2 : 3. The locus of the
point of division is
(a)
(b)
(c)
(d)
3. If two lines L
1
 and L
2 
in space, are defined by
,  and
, 
then L
1
 is perpendicular to L
2
, for all non-negative reals ? and µ, such that :
(a)
(b)
(c)
(d)
4. The locus of a point, such that the sum of the squares of its distances
from the planes x + y + z = 0, x – z =0 and x – 2y + z = 0 is 9, is
(a) (b)  
(c) (d)  
5. A variable plane passes through a fixed point (1, 2, 3). The locus of the
foot of the perpendicular from the origin to this plane is given by
(a) x
2
 + y
2
 + z
2
 – 14 = 0
(b) x
2
 + y
2
 + z
2
 + x + 2y + 3z = 0
(c) x
2
 + y
2
 + z
2
 – x – 2y – 3z = 0
(d) None of these
6. The direction cosines l, m, n, of one of the two lines connected by the
relations
 are
(a)
(b)
(c)
(d)
7. A line makes the same angle a with each of the x and y axes. If the
angle ?, which it makes with the z-axis, is such that  sin
2
? = 2 sin
2
a ,
then what is the value of a ?
(a) p/4
(b) p/6
(c) p/3
(d) p/2
8. If Q is the image of the point P(2, 3, 4) under the reflection in the plane x – 2y +
5z = 6, then the equation of the line PQ is
(a)
(b)
(c)
(d)
9. The foot of the perpendicular from (2, 4, –1) to the line 
(a) (– 4, 1, – 3)
(b) (4, –1, –3)
(c) (–4, –1, 3)
(d) (– 4, – 1, –3)
10. The equation of two lines through the origin, which intersect the line 
 at angles of  each, are
(a)
(b)
(c)
(d) None of the above
11. The planes 3x – y + z + 1= 0, 5x + y + 3z = 0 intersect inthe line PQ.
The equation of the plane through the point(2, 1, 4) and the
perpendicular to PQ is
(a)
(b) x + y + 2z = –5
(c)
(d) x + y – 2z = –5
12. The line  and the plane  meet in
(a) no point
(b) only one point
(c) infinitely many points
(d) None of these
13. If from a point P (a, b, c) perpendiculars PA and PB are drawn to yz and
zx planes, then the equation of the plane OAB is
(a) bcx + cay +abz = 0
(b) bcx + cay – abz = 0
(c) bcx – cay + abz = 0
(d) –bcx + cay + abz = 0
14. Under what condition do the planes
bx – ay = n, cy – bz = l , az – cx = m intersect in a line?
(a) a + b + c = 0
(b) a = b =
(c) al + bm + cn = 0
(d) l + m + n = 0
15. A variable plane which remains at a constant distance 3p from the
origin cut the coordinate axes at A, B and C. The locus of the centroid
of triangle ABC is
(a) x
–1
 + y
–1
 + z
–1
 = p
–1
(b) x
–2
 + y
–2
 + z
–2
 = p
–2
(c) x + y + z = p
(d) x
2
 + y
2
 + z
2
 = p
2
16. The radius of the sphere
Page 5


PART-I (Single Correct MCQs)
1. The line,  intersects the curve xy = c
2
, z = 0 if c is
equal to
(a) ± 1
(b)
(c)
(d) None of these
2. From the point (1, –2, 3) lines are drawn to meet the sphere x
2
 + y
2
 + z
2
= 4 and they are divided internally in the ratio2 : 3. The locus of the
point of division is
(a)
(b)
(c)
(d)
3. If two lines L
1
 and L
2 
in space, are defined by
,  and
, 
then L
1
 is perpendicular to L
2
, for all non-negative reals ? and µ, such that :
(a)
(b)
(c)
(d)
4. The locus of a point, such that the sum of the squares of its distances
from the planes x + y + z = 0, x – z =0 and x – 2y + z = 0 is 9, is
(a) (b)  
(c) (d)  
5. A variable plane passes through a fixed point (1, 2, 3). The locus of the
foot of the perpendicular from the origin to this plane is given by
(a) x
2
 + y
2
 + z
2
 – 14 = 0
(b) x
2
 + y
2
 + z
2
 + x + 2y + 3z = 0
(c) x
2
 + y
2
 + z
2
 – x – 2y – 3z = 0
(d) None of these
6. The direction cosines l, m, n, of one of the two lines connected by the
relations
 are
(a)
(b)
(c)
(d)
7. A line makes the same angle a with each of the x and y axes. If the
angle ?, which it makes with the z-axis, is such that  sin
2
? = 2 sin
2
a ,
then what is the value of a ?
(a) p/4
(b) p/6
(c) p/3
(d) p/2
8. If Q is the image of the point P(2, 3, 4) under the reflection in the plane x – 2y +
5z = 6, then the equation of the line PQ is
(a)
(b)
(c)
(d)
9. The foot of the perpendicular from (2, 4, –1) to the line 
(a) (– 4, 1, – 3)
(b) (4, –1, –3)
(c) (–4, –1, 3)
(d) (– 4, – 1, –3)
10. The equation of two lines through the origin, which intersect the line 
 at angles of  each, are
(a)
(b)
(c)
(d) None of the above
11. The planes 3x – y + z + 1= 0, 5x + y + 3z = 0 intersect inthe line PQ.
The equation of the plane through the point(2, 1, 4) and the
perpendicular to PQ is
(a)
(b) x + y + 2z = –5
(c)
(d) x + y – 2z = –5
12. The line  and the plane  meet in
(a) no point
(b) only one point
(c) infinitely many points
(d) None of these
13. If from a point P (a, b, c) perpendiculars PA and PB are drawn to yz and
zx planes, then the equation of the plane OAB is
(a) bcx + cay +abz = 0
(b) bcx + cay – abz = 0
(c) bcx – cay + abz = 0
(d) –bcx + cay + abz = 0
14. Under what condition do the planes
bx – ay = n, cy – bz = l , az – cx = m intersect in a line?
(a) a + b + c = 0
(b) a = b =
(c) al + bm + cn = 0
(d) l + m + n = 0
15. A variable plane which remains at a constant distance 3p from the
origin cut the coordinate axes at A, B and C. The locus of the centroid
of triangle ABC is
(a) x
–1
 + y
–1
 + z
–1
 = p
–1
(b) x
–2
 + y
–2
 + z
–2
 = p
–2
(c) x + y + z = p
(d) x
2
 + y
2
 + z
2
 = p
2
16. The radius of the sphere
, is 
(a)
(b)
(c)
(d)
17. Let the line  lie in the plane x + 3y – az + ß = 0.
Then (a, ß) equals
(a) (–6, 7)
(b) (5, –15)
(c) (–5, 5)
(d) (6, –17)
18. Equation of line in the plane p = 2x – y + z – 4 = 0 which is
perpendicular to the line l whose equation is
 and which passes through the point of intersection
of l and p is –
(a)
(b)
(c)
(d)
19. The equation of a plane passing through the line of intersection of the
planes x + 2y + 3z = 2 and x – y + z = 3 and at a distance from the
point (3, 1 ,–1) is
(a) 5x – 11y + z = 17
(b)