## Important Questions in 3-D for JEE Main 2019

3-D is a very important topic for JEE Mains, every year 2 questions are asked from three-dimensional geometry. Generally, these questions are very easy to solve. Students who had prepared for their board exams can also solve questions from 3-D.

Students are advised to solve previous year JEE Main Questions

Here are some very important questions in 3-D for JEE Main and JEE Advanced as well

## Three-Dimensional Geometry for JEE Main

Q 1. Angle between the lines 3x + 2y + z – 5 = 0 = x + y -2z – 3 and x-y + 4z = 0 = x + y-4z is equal to

(a)\[{{\cos }^{-1}}\left( \frac{29}{5\sqrt{51}} \right)\]

(b)\[{{\cos }^{-1}}\left( \frac{51}{5\sqrt{29}} \right)\]

(c)\[{{\cos }^{-1}}\left( \frac{1}{3} \right)\]

(d)\[\frac{\pi }{2}\]

Q 2. Line passes through (3, 4, 5) & (4, 6, 3) has a projection on line passing through (-1,2, 4) and (1, 0, 5) will be

(a) – 4/3

(b) 2/3

(c) 1/3

(d) 1/2

Q 3. A line is making a, b, g angles from X, Y, Z-axes. If value of a, b, g are respectively q, 60°, 30°, then sin q is equal to

(a) 1

(b) -2

(c) 0

(d) 1/2

Q 4. The plane YZ divides the joining of A(- 2, 3,5) and B(a, 2, 5) in the ratio of 2 : 3, then a is equal to

(a) 3

(b) -1

(c) 2

(d) None of these

Q 5. The angle between the lines whose direction cosines are given by l + m + n = 0 and l2 + m2 + n2 is

(a) pi/4

(b) pi/3

(c) pi/6

(d) pi/2

Q 6. If lines \[\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}and\frac{x-1}{3k}=\frac{y-5}{1}=\frac{z-6}{-5}\] perpendicular to each other, then k is equal to

(a) 5/7

(b) 7/5

(c) -7/10

(d) -10/7

Q 7. The angle between the lines 2x = 3y = – z and 6x = – y = – 4z is

(a) pi/6

(b) pi/3

(c) pi/4

(d) pi/2

Q 8. If the straight lines \[\frac{x-1}{k}=\frac{y-2}{2}=\frac{z-3}{3}and\frac{x-2}{3}=\frac{y-3}{k}=\frac{z-1}{2}\] intersect at a point, then k is ?

(a) -5

(b) 5

(c) 2

(d) -2

Q 9. The vector equations of the two lines L1 and L2 are given

\[{{L}_{1}}:\vec{r}=(2\hat{i}+9\hat{j}+13\hat{k})+\lambda (\hat{i}+2\hat{j}+3\hat{k})\] \[{{L}_{2}}:\vec{r}=(-3\hat{i}+7\hat{j}+p\hat{k})+\mu (-\hat{i}+2\hat{j}-3\hat{k})\],

then the lines L1 and L2 are

(a) skew lines for all p Î R

(b) intersecting for all p Î R and the point of intersection is (-1, 3, 4)

(c) intersecting lines for p = -2

(d) intersecting for all real p Î R

Q 10. The equation of plane which passes through (2, – 3,1) and is normal to the line joining the points (3, 4, -1) and (2, -1, 5), then

(a) x + 5y – 6z +19 = 0

(b) x – 5y + 6z – 19 = 0

(c) x + 5y + 6z + 19 = 0

(d) x – 5y – 6z – 19 = 0

Q 11. Equation of plane passing through (1, -3,-2) & perpendicular to planes x + 2y + 3z = 5 & 3x + 3y + 2z = 8 ?

(a) 5x – 7y + 3z + 20 = 0

(b) 2x-4y-3z + 8 = 0

(c) 2x + 4y + 3z + 8 = 0

(d) None

Q 12. The reflection of the point (2, -1, 3) in the plane 3x – 2y – z = 9 is

(a) \[\left( \frac{26}{7},\frac{15}{7},\frac{17}{7} \right)\]

(b) \[\left( \frac{26}{7},-\frac{15}{7},\frac{17}{7} \right)\]

(c) \[\left( \frac{15}{7},\frac{26}{7},-\frac{17}{7} \right)\]

(d) \[\left( \frac{26}{7},\frac{17}{7},-\frac{15}{7} \right)\]

Q 13. The distance of the point (-1, – 5, -10) from the point of intersection of the line, \[\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}\] and the plane x – y + z = 5, is

(a) 10

(b) 11

(c) 12

(d) 13

Q 14. The distance of the point (1, -2, 3) from the plane x – y +z =5 measured parallel to the line \[\frac{x}{2}=\frac{y}{3}=\frac{z}{-6}\]is

(a) 1

(b) 2

(c) 3

(d) None of these

Q 15. The equation of line x + y+ z – 1 = 0, 4x + y-2z + 2 = 0 written in the symmetrical form is

(a) \[\frac{x+1}{1}=\frac{y-2}{-2}=\frac{z}{1}\]

(b) \[\frac{x}{1}=\frac{y}{-2}=\frac{z}{1}\]

(c) \[\frac{x}{1}=\frac{y}{-2}=\frac{z+1}{2}\]

(d) \[\frac{x-\frac{1}{2}}{1}=\frac{y-1}{-2}=\frac{z-\frac{1}{2}}{1}\]

Q 16. The acute angle that the vector \[2\hat{i}-2\hat{j}+\hat{k}\]makes with the plane contained by the two vectors \[2\hat{i}+3\hat{j}-\hat{k}\]and \[\hat{i}-\hat{j}+2\hat{k}\]is given by

(a) \[{{\cos }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\]

(b) \[{{\sin }^{-1}}\left( \frac{1}{\sqrt{2}} \right)\]

(c) \[{{\tan }^{-1}}\left( \sqrt{2} \right)\]

(d) \[{{\cot }^{-1}}\left( \sqrt{2} \right)\]

Q 17. A plane meets the coordinate axes in A, B, C and (a, b, g) is the centroid of the triangle ABC. Then, the equation of plane is

(a) \[\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=3\]

(b) \[\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=1\]

(c) \[\frac{3x}{\alpha }+\frac{3y}{\beta }+\frac{3z}{\gamma }=1\]

(d) \[\frac{x}{\alpha }+\frac{y}{\beta }+\frac{z}{\gamma }=2\]

Q 18. Equation of plane which passes through the point of intersection of lines \[\frac{x-1}{3}=\frac{y-2}{1}=\frac{z-3}{2}and\frac{x-3}{1}=\frac{y-1}{2}=\frac{z-2}{3}\], and at greatest distance from the point (0, 0, 0) is

(a) 4x + 3y + 5z = 25

(b) 4x + 3y + 5z = 50

(c) 3x + 4y + 5z = 49

(d) x + 7y – 5z = 2

Q 19. Angle between a plane 2x – y + z = 6 and a plane perpendicular to the planes x + y + 2z = 7 and x – y = 3 is

(a) \[\frac{\pi }{4}\]

(b) \[\frac{\pi }{3}\]

(c) \[\frac{\pi }{6}\]

(d) \[\frac{\pi }{2}\]

Q 20. If the lines \[\frac{x}{1}=\frac{y}{2}=\frac{z}{3},\frac{x-1}{3}=\frac{y-2}{-1}=\frac{z-3}{4}and\frac{x+k}{3}=\frac{y-1}{2}=\frac{z-2}{h}\] are concurrent, then

(a) h = -2 and k = -6

(b) h= 1/2 and 6k = 2

(c) h = 6 and k = 2

(d) h = 2and k = 1/2

Q 21. The direction ratios of a normal to the plane through (1, 0, 0), (0, 1, 0) which makes an angle of pi/4 with the plane x + y = 3 are

(a) \[(1,\sqrt{2},1)\]

(b) \[(1,1,\sqrt{2})\]

(c) \[(1,1,\sqrt{3})\]

(d) \[(\sqrt{2},1,1)\]

Q 22. The equation of the line passing through (1, 1, 1) and parallel to plane 2x + 3y + z + 5 = 0

(a) \[\frac{x-1}{1}=\frac{y-1}{2}=\frac{z-1}{1}\]

(b) \[\frac{x-1}{-1}=\frac{y-1}{1}=\frac{z-1}{1}\]

(c) \[\frac{x-1}{3}=\frac{y-1}{2}=\frac{z-1}{1}\]

(d) \[\frac{x-1}{2}=\frac{y-1}{3}=\frac{z-1}{1}\]

Q 23. Length of perpendicular from (1, 2, 3) to line \[\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}\]

(a) 4

(b) 5

(c) 6

(d) 7

Q 24. A line with direction cosines proportional to 2, 1, 2 meets each of the lines x = y + a = z and x + a =2y =2z. The coordinates of each of the points of intersection are given by

(a) (2a, 3a, a), (2a, a, a)

(b) (3a, 2a, 3a), (a, a, a)

(c) (3a, 2a, 3a), (a, a, 2a)

(d) (3a, 3a, 3a), (a, a, a)

Q 25. The two lines x =ay + b, z =cy + d and x = a’y + b’, z =c’y + d’ will be perpendicular if and only if

(a) aa’ + cc’ + 1 = 0

(b) aa’ + bb’ +cc’=1

(c) aa’ + bb’ + cc’ = 0

(d) (a + b’)(b+ b’)(c +c’)= 0

Q 26. The value of l such that the system x – 2y + z = -4, 2x – y + 2z = 2, x + y + lz = 4 has no solution, is

(a) 0

(b) 1

(c) -1

(d) 3

Q 27. A plane which passes through the point (3, 2, 0) and contains the line \[\frac{x-3}{1}=\frac{y-6}{5}=\frac{z-4}{4}\]is

(a) x – y + z = 1

(b) x + y + z = 5

(c) x + 2y – z = 1

(d) None of these

Q 28. The distance between the line \[\vec{r}=2i-2\hat{j}+3\hat{k}+\lambda (\hat{i}-\hat{j}+4\hat{k})\]and the plane \[\vec{r}.(\hat{i}+5\hat{j}+\hat{k})=5\]is

(a) \[\frac{10}{9}\]

(b) \[\frac{10}{3\sqrt{3}}\]

(c) \[\frac{10}{3}\]

(d) \[\frac{11}{3}\]

Q 29. Equation of plane that contains lines \[\vec{r}=(\hat{i}+\hat{j})+\lambda (\hat{i}+2\hat{j}+\hat{k})\]and \[\vec{r}=(\hat{i}+\hat{j})+\mu (-\hat{i}+\hat{j}-2\hat{k})\]

(a) \[\vec{r}.(-\hat{i}+\hat{j}+\hat{k})=0\]

(b) \[\vec{r}.(\hat{i}+\hat{j}+\hat{k})=0\]

(c) \[\vec{r}.(2\hat{i}+\hat{j}-3\hat{k})=-4\]

(d) \[\vec{r}\times (-\hat{i}+\hat{j}+\hat{k})=0\]

Q 30. The lines \[\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k}and\frac{x-1}{k}=\frac{y-4}{2}=\frac{z-5}{1}\]are coplanar, if

(a) k = 3,-3

(b) k = 0, -1

(c) k = 1, -1

(d) k = 0, -3

Q 31. A tetrahedron has vertices O (0, 0, 0), A(1,2,1), B(2,1,3), C(-1,1,2). Then, the angle between the faces OAB and ABC will be

(a) \[{{\cos }^{-1}}\left( \frac{19}{31} \right)\]

(b) \[{{\cos }^{-1}}\left( \frac{19}{35} \right)\]

(c) \[{{\cos }^{-1}}\left( \frac{17}{31} \right)\]

(d) \[{{\cos }^{-1}}\left( \frac{17}{35} \right)\]

Q 32. The value of k such that \[\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-k}{2}\] lies on the plane 2x – 4y + 7 =7 is

(a) 7

(b) -7

(c) 4

(d) no real value

Q 33. The directions ratios of the normal to the plane passing through the points (1,-2, 3), (-1, 2, -1) and parallel to line \[\frac{x-2}{2}=\frac{y+1}{3}=\frac{z}{4}\]

(a) 2,3,4

(b) 4,0,7

(c) -2,0,-1

(d) 2,0,-1

Q 34. If the straight lines =1 + s, y = -3 – ls, z = 1 + ls and x = t/2, y = 1 + t, z =2 – t with parameters s and t respectively, are co-planar, then l equals

(a) 0

(b) -1

(c) -1/2

(d) -2

Q 35. Let A (0, 0,1), B (0,1, 0) and C (1,1,1) are the points in a plane. Then, the equation of plane perpendicular to the plane ABC and passing through A and B is

(a) –x + y + z + 1 = 0

(b) x – y – z – 1 = 0

(c) –x + y + z – 1 = 0

(d) 2x + y + z – 1 = 0

Q 36. A ray of light is sent through the point P (1,2, 3) and is reflected on the XY-plane. If the reflected ray passes through the point Q (3,2, 5), then the equation of the reflected ray is

(a) \[\frac{x-3}{1}=\frac{y-2}{0}=\frac{z-5}{1}\]

(b) \[\frac{x-3}{1}=\frac{y-2}{0}=\frac{z-5}{-4}\]

(c) \[\frac{x-3}{1}=\frac{y-2}{0}=\frac{z-5}{4}\]

(d) \[\frac{x-1}{1}=\frac{y-2}{0}=\frac{z-3}{4}\]

Q 37. The equation of the plane containing the line 2x – 5y + z = 3, x + y + 4z = 5 and parallel to the plane x + 3y + 6z = 1 is

(a) 2x+ 6y+ 12z = 13

(b) x + 3y+ 6z= -7

(c) x + 3y + 6z = 7

(d) 2x + 6y + 12z = -13

## Answers 3D Geometry for JEE Main-2019

1. a 2. a 3. a 4. a 5. b 6. d 7. d 8. a 9. c 10. a 11. a 12. b 13. d

14. a 15. a 16. d 17. a 18. b 19. d 20. d 21. b 22. b

23. d 24. b 25. a 26. b 27. a 28. b 29. a 30. d 31. b 32. a 33. d 34. d

35. d 36. c 37. c