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

3-D is 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 ÃŽ R

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

(c) intersecting lines for p = -2

(d) intersecting for all real p ÃŽ 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**

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