3D Geometry · JEE Main previous year questions

2019 19 questions

1. Q302019 · 12 Apr 2019 · Shift 1

   If the line $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-1}$ intersects the plane $2x+3y-z+13=0$ at a point P and the plane $3x+y+4z=16$ at a point Q, then PQ is equal to :

   1. A.$2\sqrt{7}$
   2. B.14
   3. C.$2\sqrt{14}$
   4. D.$\sqrt{14}$
   Show answer

   Answer: C

2. Q202019 · 12 Apr 2019 · Shift 2

   A plane which bisects the angle between the two given planes 2x – y + 2z – 4 = 0 and x + 2y + 2z – 2 = 0, passes through the point :

   1. A.(1, –4, 1)
   2. B.(1, 4, –1)
   3. C.(2, 4, 1)
   4. D.(2, –4, 1)
   Show answer

   Answer: D

3. Q242019 · 12 Apr 2019 · Shift 2

   The length of the perpendicular drawn from the point $(2,1,4)$ to the plane containing the lines $\vec{r}=(\hat{i}+\hat{j})+\lambda (\hat{i}+2\hat{j}-\hat{k})$ and $\vec{r}=(\hat{i}+\hat{j})+\mu (-\hat{i}+\hat{j}-2\hat{k})$ is :

   1. A.$\frac{1}{3}$
   2. B.$\frac{1}{\sqrt{3}}$
   3. C.3
   4. D.$\sqrt{3}$
   Show answer

   Answer: D

4. Q302019 · 12 Apr 2019 · Shift 2

   The angle of elevation of the top of a vertical tower standing on a horizontal plane is observed to be $45^{\circ }$ from a point A on the plane. Let B be the point 30 m vertically above the point A. If the angle of elevation of the top of the tower from B be $30^{\circ }$, then the distance (in m) of the foot of the tower from the point A is :

   1. A.$15\left(1+\sqrt{3}\right)$
   2. B.$15\left(3-\sqrt{3}\right)$
   3. C.$15\left(3+\sqrt{3}\right)$
   4. D.$15\left(5-\sqrt{3}\right)$
   Show answer

   Answer: C

5. Q262019 · 10 Apr 2019 · Shift 1

   If $\mathrm{Q}(0,-1,-3)$ is the image of the point P in the plane $3x-y+4z=2$ and R is the point (3, -1, -2), then the area (in sq. units) of $\Delta \mathrm{PQR}$ is :

   1. A.$\frac{\sqrt{65}}{2}$
   2. B.$2\sqrt{13}$
   3. C.$\frac{\sqrt{91}}{2}$
   4. D.$\frac{\sqrt{91}}{4}$
   Show answer

   Answer: C

6. Q242019 · 10 Apr 2019 · Shift 2

   If the plane $2x-y+2z+3=0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4x-2y+4z+\lambda =0$ and $2x-y+2z+\mu =0$, respectively, then the maximum value of $\lambda +\mu$ is equal to :

   1. A.13
   2. B.9
   3. C.5
   4. D.15
   Show answer

   Answer: A

7. Q302019 · 10 Apr 2019 · Shift 2

   A perpendicular is drawn from a point on the line $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z}{1}$ to the plane $x+y+z=3$ such that the foot of the perpendicular Q also lies on the plane $x-y+z=3$. Then the co-ordinates of Q are :

   1. A.(4, 0, - 1)
   2. B.(2, 0, 1)
   3. C.(1, 0, 2)
   4. D.(- 1, 0, 4)
   Show answer

   Answer: B

8. Q252019 · 9 Apr 2019 · Shift 1

   A plane passing through the points (0, −1, 0) and (0, 0, 1) and making an angle $\frac{\pi }{4}$ with the plane y − z + 5 = 0, also passes through the point

   1. A.$(\sqrt{2},1,4)$
   2. B.$(-\sqrt{2},1,-4)$
   3. C.$(-\sqrt{2},-1,-4)$
   4. D.$(\sqrt{2},-1,4)$
   Show answer

   Answer: A

9. Q262019 · 9 Apr 2019 · Shift 1

   If the line, $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}$ meets the plane, x + 2y + 3z = 15 at a point P, then the distance of P from the origin is :

   1. A.$\frac{\sqrt{5}}{2}$
   2. B.$2\sqrt{5}$
   3. C.9/2
   4. D.7/2
   Show answer

   Answer: C

10. Q282019 · 9 Apr 2019 · Shift 2

    Let P be the plane, which contains the line of intersection of the planes, $x+y+z-6=0$ and $2x+3y+z+5=0$ and it is perpendicular to the xy-plane. Then the distance of the point (0, 0, 256) from P is equal to :

    1. A.$205\sqrt{5}$
    2. B.$63\sqrt{5}$
    3. C.$11/\sqrt{5}$
    4. D.$17/\sqrt{5}$
    Show answer

    Answer: C

11. Q302019 · 8 Apr 2019 · Shift 2

    Two vertical poles of heights, 20 m and 80 m stand a part on a horizontal plane. The height (in meters) of the point of intersection of the lines joining the top of each pole to the foot of the other, from this horizontal plane is :

    1. A.12
    2. B.16
    3. C.15
    4. D.18
    Show answer

    Answer: B

12. Q272019 · 12 Jan 2019 · Shift 1

    The perpendicular distance from the origin to the plane containing the two lines, $$\frac{x+2}{3}=\frac{y-2}{5}=\frac{z+5}{7}\text{and}\frac{x-1}{1}=\frac{y-4}{4}=\frac{z+4}{7},\text{ is :}$$

    1. A.$6\sqrt{11}$
    2. B.$\frac{11}{\sqrt{6}}$
    3. C.11
    4. D.$11\sqrt{6}$
    Show answer

    Answer: B

13. Q302019 · 12 Jan 2019 · Shift 1

    A tetrahedron has vertices $P(1,2,1)$, $Q(2,1,3)$, $R(-1,1,2)$ and $O(0,0,0)$. The angle between the faces OPQ and PQR is :

    1. A.${\cos }^{-1}\left(\frac{17}{31}\right)$
    2. B.${\cos }^{-1}\left(\frac{9}{35}\right)$
    3. C.${\cos }^{-1}\left(\frac{19}{35}\right)$
    4. D.${\cos }^{-1}\left(\frac{7}{31}\right)$
    Show answer

    Answer: C

14. Q252019 · 12 Jan 2019 · Shift 2

    Let S be the set of all real values of $\lambda$ such that a plane passing through the points $(-{\lambda }^2,1,1)$, $(1,-{\lambda }^2,1)$ and $(1,1,-{\lambda }^2)$ also passes through the point $(-1,-1,1)$. Then S is equal to :

    1. A.$\{1,-1\}$
    2. B.$\{3,-3\}$
    3. C.$\{\sqrt{3}\}$
    4. D.$\{\sqrt{3},-\sqrt{3}\}$
    Show answer

    Answer: D

15. Q282019 · 12 Jan 2019 · Shift 2

    If an angle between the line, $\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}$ and the plane, $x-2y-kz=3$ is ${\cos }^{-1}\left(\frac{2\sqrt{2}}{3}\right)$, then a value of k is :

    1. A.$\sqrt{\frac{3}{5}}$
    2. B.$-\frac{5}{2}$
    3. C.$-\frac{3}{2}$
    4. D.$\sqrt{\frac{5}{3}}$
    Show answer

    Answer: D

16. Q242019 · 11 Jan 2019 · Shift 2

    Two lines $\frac{x-3}{1}=\frac{y+1}{3}=\frac{z-6}{-1}$ and $\frac{x+5}{7}=\frac{y-2}{-6}=\frac{z-3}{4}$ intersect at the point R. The reflection of R in the xy-plane has coordinates :

    1. A.$(2,4,7)$
    2. B.$(2,-4,-7)$
    3. C.$(2,-4,7)$
    4. D.$(-2,4,7)$
    Show answer

    Answer: B

17. Q272019 · 11 Jan 2019 · Shift 2

    If the point $(2,\alpha ,\beta )$ lies on the plane which passes through the points $(3,4,2)$ and $(7,0,6)$ and is perpendicular to the plane $2x-5y=15$, then $2\alpha -3\beta$ is equal to

    1. A.12
    2. B.7
    3. C.17
    4. D.5
    Show answer

    Answer: B

18. Q282019 · 9 Jan 2019 · Shift 1

    The equation of the line passing through $(-4,3,1)$, parallel to the plane $x+2y-z-5=0$ and intersecting the line $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z-2}{-1}$ is :

    1. A.$\frac{x+4}{3}=\frac{y-3}{-1}=\frac{z-1}{1}$
    2. B.$\frac{x+4}{1}=\frac{y-3}{1}=\frac{z-1}{3}$
    3. C.$\frac{x+4}{-1}=\frac{y-3}{1}=\frac{z-1}{1}$
    4. D.$\frac{x-4}{2}=\frac{y+3}{1}=\frac{z+1}{4}$
    Show answer

    Answer: A

19. Q292019 · 9 Jan 2019 · Shift 1

    The plane through the intersection of the planes x + y + z = 1 and 2x + 3y – z + 4 = 0 and parallel to y-axis also passes through the point :

    1. A.(–3, 0, -1)
    2. B.(3, 2, 1)
    3. C.(3, 3, -1)
    4. D.(–3, 1, 1)
    Show answer

    Answer: B