3D Geometry · JEE Main previous year questions

2020 15 questions

1. Q182020 · 6 Sept 2020 · Shift 1

   The shortest distance between the lines $$\frac{x-1}{0}=\frac{y+1}{-1}=\frac{z}{1}$$ and $x+y+z+1=0$, $2x-y+z+3=0$ is :

   1. A.1
   2. B.$\frac{1}{2}$
   3. C.$\frac{1}{\sqrt{2}}$
   4. D.$\frac{1}{\sqrt{3}}$
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   Answer: D

2. Q252020 · 6 Sept 2020 · Shift 1· Numerical answer

   The angle of elevation of the top of a hill from a point on the horizontal plane passing through the foot of the hill is found to be $45^{\circ }$. After walking a distance of 80 meters towards the top, up a slope inclined at an angle of $30^{\circ }$ to the horizontal plane, the angle of elevation of the top of the hill becomes $75^{\circ }$. Then the height of the hill (in meters) is ______ .

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   Answer: 80

3. Q132020 · 6 Sept 2020 · Shift 2

   Let L denote the line in the xy-plane with x and y intercepts as 3 and 1 respectively. Then the image of the point $(-1,-4)$ in this line is :

   1. A.$\left(\frac{11}{5},\frac{28}{5}\right)$
   2. B.$\left(\frac{29}{5},\frac{11}{5}\right)$
   3. C.$\left(\frac{29}{5},\frac{8}{5}\right)$
   4. D.$\left(\frac{8}{5},\frac{29}{5}\right)$
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   Answer: A

4. Q202020 · 6 Sept 2020 · Shift 2

   A plane P meets the coordinate axes at A, B and C respectively. The centroid of $\Delta \mathrm{ABC}$ is given to be $(1,1,2)$. Then the equation of the line through this centroid and perpendicular to the plane P is :

   1. A.$\frac{x-1}{1}=\frac{y-1}{1}=\frac{z-2}{2}$
   2. B.$\frac{x-1}{2}=\frac{y-1}{1}=\frac{z-2}{1}$
   3. C.$\frac{x-1}{2}=\frac{y-1}{2}=\frac{z-2}{1}$
   4. D.$\frac{x-1}{1}=\frac{y-1}{2}=\frac{z-2}{2}$
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   Answer: C

5. Q192020 · 3 Sept 2020 · Shift 1

   The foot of the perpendicular drawn from the point (4, 2, 3) to the line joining the points (1, –2, 3) and (1, 1, 0) lies on the plane :

   1. A.x – 2y + z = 1
   2. B.x + 2y – z = 1
   3. C.x – y – 2z = 1
   4. D.2x + y – z = 1
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   Answer: D

6. Q202020 · 3 Sept 2020 · Shift 2

   The plane which bisects the line joining, the points (4, –2, 3) and (2, 4, –1) at right angles also passes through the point :

   1. A.(4, 0, 1)
   2. B.(0, –1, 1)
   3. C.(0, 1, –1)
   4. D.(4, 0, –1)
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   Answer: D

7. Q242020 · 3 Sept 2020 · Shift 2· Numerical answer

   Let a plane P contain two lines $\vec{r}=\hat{i}+\lambda (\hat{i}+\hat{j}),\lambda \in \mathbb{R}$ and $\vec{r}=-\hat{j}+\mu (\hat{j}-\hat{k}),\mu \in \mathbb{R}$ If $\mathrm{Q}(\alpha ,\beta ,\gamma )$ is the foot of the perpendicular drawn from the point $\mathrm{M}(1,0,1)$ to P, then $3(\alpha +\beta +\gamma )$ equals _______.

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   Answer: 5

8. Q192020 · 2 Sept 2020 · Shift 1

   The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line, 2x = 3y, z = 1 also passes through the point :

   1. A.(0, 6, –2)
   2. B.(–2, 0, 1)
   3. C.(0, –6, 2)
   4. D.(2, 0 –1)
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   Answer: B

9. Q192020 · 2 Sept 2020 · Shift 2

   A plane passing through the point $(3,1,1)$ contains two lines whose direction ratios are $1,-2,2$ and $2,3,-1$ respectively. If this plane also passes through the point $(\alpha ,-3,5)$, then $\alpha$ is equal to:

   1. A.-10
   2. B.10
   3. C.5
   4. D.-5
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   Answer: C

10. Q112020 · 9 Jan 2020 · Shift 1

    If for some $\alpha$ and $\beta$ in $\mathbb{R}$, the intersection of the following three planes $x+4y-2z=1$ $x+7y-5z=\beta$ $x+5y+\alpha z=5$ is a line in ${\mathbb{R}}^3$, then $\alpha +\beta$ is equal to :

    1. A.-10
    2. B.0
    3. C.10
    4. D.2
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    Answer: C

11. Q242020 · 9 Jan 2020 · Shift 2· Numerical answer

    If the distance between the plane, $23x-10y-2z+48=0$ and the plane containing the lines $\frac{x+1}{2}=\frac{y-3}{4}=\frac{z+1}{3}$ and $\frac{x+3}{2}=\frac{y+2}{6}=\frac{z-1}{\lambda }(\lambda \in \mathbb{R})$ is equal to $\frac{k}{\sqrt{633}}$, then k is equal to _______.

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    Answer: 3

12. Q152020 · 8 Jan 2020 · Shift 1

    The shortest distance between the lines $$\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}\text{and}\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{4}\text{is :}$$

    1. A.3
    2. B.$\frac{7}{2}\sqrt{30}$
    3. C.$3\sqrt{30}$
    4. D.$2\sqrt{30}$
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    Answer: C

13. Q192020 · 8 Jan 2020 · Shift 2

    The mirror image of the point $(1,2,3)$ in a plane is $\left(-\frac{7}{3},-\frac{4}{3},-\frac{1}{3}\right)$. Which of the following points lies on this plane ?

    1. A.$(1,-1,1)$
    2. B.$(-1,-1,-1)$
    3. C.$(-1,-1,1)$
    4. D.$(1,1,1)$
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    Answer: A

14. Q192020 · 7 Jan 2020 · Shift 1

    Let P be a plane passing through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R be any point (2, 1, 6). Then the image of R in the plane P is :

    1. A.(4, 3, 2)
    2. B.(6, 5, - 2)
    3. C.(3, 4, -2)
    4. D.(6, 5, 2)
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    Answer: B

15. Q222020 · 7 Jan 2020 · Shift 2· Numerical answer

    If the foot of the perpendicular drawn from the point $(1,0,3)$ on a line passing through $(\alpha ,7,1)$ is $\left(\frac{5}{3},\frac{7}{3},\frac{17}{3}\right)$, then $\alpha$ is equal to _______.

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    Answer: 4