3D Geometry · JEE Main previous year questions

2026 13 questions

1. Q142026 · 6 Apr 2026 · Shift 2

   The shortest distance between the lines $\frac{x-4}{1}=\frac{y-3}{2}=\frac{z-2}{-3}$ and $\frac{x+2}{2}=\frac{y-6}{4}=\frac{z-5}{-5}$ is :

   1. A.$\frac{5\sqrt{6}}{6}$
   2. B.$2\sqrt{5}$
   3. C.$3\sqrt{5}$
   4. D.$4\sqrt{5}$
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   Answer: C

2. Q242026 · 6 Apr 2026 · Shift 2· Numerical answer

   Let the image of the point $\mathrm{P}(0,-5,0)$ in the line $\frac{x-1}{2}=\frac{y}{1}=\frac{z+1}{-2}$ be the point $\mathrm{R}$ and the image of the point $\mathrm{Q}\left(0,\frac{-1}{2},0\right)$ in the line $\frac{x-1}{-1}=\frac{y+9}{4}=\frac{z+1}{1}$ be the point $\mathrm{S}$. Then the square of the area of the parallelogram $\mathrm{PQRS}$ is ____ .

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   Answer: 162

3. Q152026 · 5 Apr 2026 · Shift 2

   If the distance of the point $(a,2,5)$ from the image of the point $(1,2,7)$ in the line $\frac{x}{1}=\frac{y-1}{1}=\frac{z-2}{2}$ is 4, then the sum of all possible values of $a$ is equal to :

   1. A.$11$
   2. B.$9$
   3. C.$6$
   4. D.$4$
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   Answer: C

4. Q142026 · 4 Apr 2026 · Shift 1

   A line with direction ratios $1,-1,2$ intersects the lines $\frac{x}{2}=\frac{y}{3}=\frac{z+1}{3}$ and $\frac{x+1}{-1}=\frac{y-2}{1}=\frac{z}{4}$ at the points P and Q, respectively. If the length of the line segment PQ is $\alpha$, then $225{\alpha }^2$ is equal to:

   1. A.1024
   2. B.1014
   3. C.1104
   4. D.1204
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   Answer: B

5. Q132026 · 4 Apr 2026 · Shift 2

   The shortest distance between the lines $$\vec{r}=\left(\frac{1}{3}\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\frac{8}{3}\hat{\mathrm{k}}\right)+\lambda (2\hat{\mathrm{i}}-5\hat{\mathrm{j}}+6\hat{\mathrm{k}})$$ and $\vec{r}=\left(-\frac{2}{3}\hat{\mathrm{i}}-\frac{1}{3}\hat{\mathrm{k}}\right)+\mu (\hat{\mathrm{j}}-\hat{\mathrm{k}})$, $\lambda ,\mu \in \mathbb{R}$, is:

   1. A.$\sqrt{5}$
   2. B.3
   3. C.$2\sqrt{3}$
   4. D.$\sqrt{15}$
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   Answer: B

6. Q132026 · 2 Apr 2026 · Shift 1

   If the point of intersection of the lines $\frac{x+1}{3}=\frac{y+a}{5}=\frac{z+b+1}{7}$ and $\frac{x-2}{1}=\frac{y-b}{4}=\frac{z-2a}{7}$ lies on xy-plane, then the value of $a+b$ is:

   1. A.2
   2. B.5
   3. C.7
   4. D.9
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   Answer: C

7. Q162026 · 2 Apr 2026 · Shift 2

   Two adjacent sides of a parallelogram PQRS are given by $\overrightarrow{\mathrm{PQ}}=\hat{j}+\hat{k}$ and $\overrightarrow{\mathrm{PS}}=\hat{i}-\hat{j}$. If the side PS is rotated about the point P by an acute angle $\alpha$ in the plane of the parallelogram so that it becomes perpendicular to the side PQ, then ${\sin }^2\left(\frac{5\alpha }{2}\right)-{\sin }^2\left(\frac{\alpha }{2}\right)$ is equal to :

   1. A.$\frac{1}{2}$
   2. B.$\frac{\sqrt{3}}{2}$
   3. C.$\frac{\sqrt{3}}{4}$
   4. D.$\frac{2\sqrt{3}}{5}$
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   Answer: B

8. Q72026 · 28 Jan 2026 · Shift 2

   Let P be a point in the plane of the vectors $\overrightarrow{AB}=3\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{AC}=\hat{i}-\hat{j}+3\hat{k}$ such that P is equidistant from the lines AB and AC. If $|\overrightarrow{AP}|=\frac{\sqrt{5}}{2}$, then the area of the triangle ABP is :

   1. A.$\frac{3}{2}$
   2. B.$\frac{\sqrt{30}}{4}$
   3. C.$\frac{\sqrt{26}}{4}$
   4. D.2
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   Answer: B

9. Q232026 · 28 Jan 2026 · Shift 2· Numerical answer

   If the distance of the point $P(43,\alpha ,\beta ),\beta <0$, from the line $\vec{r}=4\hat{i}-\hat{k}+\mu (2\hat{i}+3\hat{k}),\mu \in \mathbb{R}$ along a line with direction ratios $3,-1,0$ is $13\sqrt{10}$, then ${\alpha }^2+{\beta }^2$ is equal to ________

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   Answer: 170

10. Q222026 · 24 Jan 2026 · Shift 1· Numerical answer

    Let a line L passing through the point $\mathrm{P}(1,1,1)$ be perpendicular to the lines $\frac{x-4}{4}=\frac{y-1}{1}=\frac{z-1}{1}$ and $\frac{x-17}{1}=\frac{y-71}{1}=\frac{z}{0}$. Let the line L intersect the $yz-$ plane at the point Q . Another line parallel to L and passing through the point $\mathrm{S}(1,0,-1)$ intersects the $yz$-plane at the point R . Then the square of the area of the parallelogram PQRS is equal to ____ .

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    Answer: 6

11. Q32026 · 23 Jan 2026 · Shift 1

    Let the direction cosines of two lines satisfy the equations : $4l+m-n=0$ and $2mn+10nl+3lm=0$. Then the cosine of the acute angle between these lines is :

    1. A.$\frac{10}{7\sqrt{38}}$
    2. B.$\frac{10}{\sqrt{38}}$
    3. C.$\frac{10}{3\sqrt{38}}$
    4. D.$\frac{20}{3\sqrt{38}}$
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    Answer: C

12. Q212026 · 23 Jan 2026 · Shift 2· Numerical answer

    If the image of the point $P(a,2,a)$ in the line $\frac{x}{2}=\frac{y+a}{1}=\frac{z}{1}$ is $Q$ and the image of $Q$ in the line $\frac{x-2b}{2}=\frac{y-a}{1}=\frac{z+2b}{-5}$ is $P$, then $a+b$ is equal to _____.

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    Answer: 3

13. Q172026 · 22 Jan 2026 · Shift 1

    Let $\mathrm{P}(\alpha ,\beta ,\gamma )$ be the point on the line $\frac{x-1}{2}=\frac{y+1}{-3}=z$ at a distance $4\sqrt{14}$ from the point $(1,-1,0)$ and nearer to the origin. Then the shortest distance, between the lines $\frac{x-\alpha }{1}=\frac{y-\beta }{2}=\frac{z-\gamma }{3}$ and $\frac{x+5}{2}=\frac{y-10}{1}=\frac{z-3}{1}$, is equal to

    1. A.$4\sqrt{\frac{7}{5}}$
    2. B.$7\sqrt{\frac{5}{4}}$
    3. C.$4\sqrt{\frac{5}{7}}$
    4. D.$2\sqrt{\frac{7}{4}}$
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    Answer: A