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Yukti Vault

Vectors & 3D

3D Geometry: JEE Main & Advanced PYQs

461 previous year questions on 3D Geometry from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 1-50 of 461

  1. 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 …
  2. If $\left(2\alpha+1,\alpha^{2}-3\alpha,\frac{\alpha-1}{2}\right)$ is the image of $(\alpha,2\alpha,1)$ in the line $\frac{x-2}{3}=\frac{y-1}{2}=\frac{z}{1}$, then the …
  3. The equation of the plane passing through the point $(1,2,-3)$ and perpendicular to the planes $3x+y-2z=5$ and $2x-5y-z=7$, is :
  4. Let a triangle $PQR$ be such that $P$ and $Q$ lie on the line $\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2}$ and are at a distance of $6$ units from $R(1,2,3)$. If …
  5. The distance of the point $(1,1,9)$ from the point of intersection of the line $\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2}$ and the plane $x+y+z=17$ is :
  6. 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 :
  7. Let the point $A$ be the foot of perpendicular drawn from the point $P(a,b,0)$ on the line $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-\alpha}{3}$. If the midpoint of the line …
  8. Let the foot of perpendicular from the point $(\lambda,2,3)$ on the line $\frac{x-4}{1}=\frac{y-9}{2}=\frac{z-5}{1}$ be the point $(1,\mu,2)$. Then the distance between …
  9. The shortest distance between the lines $\vec{r}=\left(\frac{1}{3}\hat{i}+2\hat{j}+\frac{8}{3}\hat{k}\right)+\lambda\left(2\hat{i}-5\hat{j}+6\hat{k}\right)$ and …
  10. Let $P$ be the point of intersection of the line $\frac{x+3}{3}=\frac{y+2}{1}=\frac{1-z}{2}$ and the plane $x+y+z=2$. If the distance of the point $P$ from the plane …
  11. Let $L$ be the line of intersection of planes $\vec{r}\cdot\left(\hat{i}-\hat{j}+2\hat{k}\right)=2$ and $\vec{r}\cdot\left(2\hat{i}+\hat{j}-\hat{k}\right)=2$. If …
  12. Let $a,b\in\mathbf{R}$. If the mirror image of the point $P(a,6,9)$ with respect to the line $\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}$ is $(20,b,-a-9)$, then $|a+b|$ …
  13. If the equation of the plane passing through the line of intersection of the planes $2x-y+z=3$, $4x-3y+5z+9=0$ and parallel to the line …
  14. Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $l+m-n=0$ and $l^{2}+m^{2}-n^{2}=0$. Then the value of …
  15. The equation of the line through the point $(0,1,2)$ and perpendicular to the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}$ is :
  16. The shortest distance between the lines $\frac{x+2}{1}=\frac{y}{-2}=\frac{z-5}{2}$ and $\frac{x-4}{1}=\frac{y-1}{2}=\frac{z+3}{0}$ is
  17. The area of the quadrilateral $ABCD$ with vertices $A(2,1,1)$, $B(1,2,5)$, $C(-2,-3,5)$ and $D(1,-6,-7)$ is equal to
  18. If equation of the plane that contains the point $(-2,3,5)$ and is perpendicular to each of the planes $2x+4y+5z=8$ and $3x-2y+3z=5$ is $\alpha x+\beta y+\gamma z+97=0$ …
  19. If $(1,5,35)$, $(7,5,5)$, $(1,\lambda,7)$ and $(2\lambda,1,2)$ are coplanar, then the sum of all possible values of $\lambda$ is :
  20. Let the image of the point $P(1,2,6)$ in the plane passing through the points $A(1,2,0)$, $B(1,4,1)$ and $C(0,5,1)$ be $Q(\alpha,\beta,\gamma)$. Then …
  21. The angle between the straight lines, whose direction cosines are given by the equations $2l+2m-n=0$ and $mn+nl+lm=0$, is :
  22. Let $L_1$ and $L_2$ be two lines passing through the points $P(b-c,c-a,a-b)$ and $Q\left(\frac{1}{l},\frac{1}{m},\frac{1}{n}\right)$ respectively, where …
  23. Let the foot of perpendicular from a point $P(1,2,-1)$ to the straight line $L:\frac{x}{1}=\frac{y}{0}=\frac{z}{-1}$ be $N$. Let a line be drawn from $P$ parallel to the …
  24. The equation of the plane passing through the line of intersection of the planes $\vec{r}\cdot\left(\hat{i}+\hat{j}+\hat{k}\right)=1$ and …
  25. The distance of the point $(1,-2,3)$ from the plane $x-y+z=5$ measured parallel to a line, whose direction ratios are $2,3,-6$ is :
  26. A hall has a square floor of dimension $10\text{ m}\times10\text{ m}$ (see the figure) and vertical walls. If the angle $GPH$ between the diagonals $AG$ and $BH$ is …
  27. If the shortest distance between the straight lines $3(x-1)=6(y-2)=2(z-1)$ and $4(x-2)=2(y-\lambda)=(z-3)$, $\lambda\in\mathbf{R}$ is $\frac{1}{\sqrt{38}}$, then the …
  28. Equation of a plane at a distance $\sqrt{\frac{2}{21}}$ from the origin, which contains the line of intersection of the planes $x-y-z-1=0$ and $2x+y-3z+4=0$, is :
  29. Let the equation of the plane, that passes through the point $(1,4,-3)$ and contains the line of intersection of the planes $3x-2y+4z-7=0$ and $x+5y-2z+9=0$, be …
  30. Let the lines $l_1:\frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha}{-2}$ and $l_2:3x+2y+z-2=0=x-3y+2z-13$ be coplanar. If the point $P(a,b,c)$ on $l_1$ is nearest to the …
  31. A plane $P$ contains the line of intersection of the plane $\vec{r}\cdot\left(\hat{i}+\hat{j}+\hat{k}\right)=6$ and …
  32. The distance of the origin from the point of intersection of the line, $\frac{x}{2}=\frac{y+1}{3}=\frac{z-1}{-1}$ and the plane, $2x+y+6z+1=0$ is
  33. Let $L$ be a line obtained from the intersection of two planes $x+2y+z=6$ and $y+2z=4$. If point $P(\alpha,\beta,\gamma)$ is the foot of perpendicular from $(3,2,1)$ on …
  34. If the mirror image of the point $(1,3,5)$ with respect to the plane $4x-5y+2z=8$ is $(\alpha,\beta,\gamma)$, then $5(\alpha+\beta+\gamma)$ equals :
  35. Let $P$ be the plane passing through the point $(1,2,3)$ and the line of intersection of the planes $\vec{r}\cdot\left(\hat{i}+\hat{j}+4\hat{k}\right)=16$ and …
  36. The equation of the plane which contains the $y$-axis and passes through the point $(1,2,3)$ is :
  37. A plane passes through the points $A(1,2,3)$, $B(3,5,2)$ and $C(2,1,5)$. If $P(a,b,c)$ be a point such that $a(b+c)=bc$, then the acute angle which the line $OP$ ($O$ is …
  38. If the equation of plane passing through the mirror image of a point $(2,3,1)$ with respect to line $\frac{x+1}{2}=\frac{y-3}{1}=\frac{z+2}{-1}$ and containing the line …
  39. A plane $P$ contains the line $x+2y+3z+1=0=x-y-z-6$, and is perpendicular to the plane $-2x+y+z+8=0$. Then which of the following points lies on $P$ ?
  40. Let the plane $P:4x-y+z=10$ be rotated by an angle $\frac{\pi}{2}$ about its line of intersection with the plane $x+y-z=4$. If $\alpha$ is the distance of the point …
  41. The plane which is parallel to $\vec{r}\cdot\left(\hat{i}+2\hat{j}+3\hat{k}\right)=6$ and passes through $(1,2,3)$, also passes through the point
  42. For $a,b\in\mathbb{Z}$ and $|a-b|\le10$, let the angle between the plane $P:ax+y-z=b$ and the line $l:x-1=a-y=z+1$ be $\cos^{-1}\left(\frac{1}{3}\right)$. If the …
  43. If $(x,y,z)$ be an arbitrary point lying on a plane $P$ which passes through the points $(42,0,0)$, $(0,42,0)$ and $(0,0,42)$, then the value of the expression …
  44. Consider the three planes $P_1:3x+15y+21z=9$, $P_2:x-3y-z=5$, and $P_3:2x+10y+14z=5$ Then, which one of the following is true ?
  45. Let two vertices of a triangle $ABC$ be $(2,4,6)$ and $(0,-2,-5)$, and its centroid be $(2,1,-1)$. If the image of the third vertex in the plane $x+2y+4z=11$ is …
  46. Let $P$ be the plane passing through the points $(5,3,0)$, $(13,3,-2)$ and $(1,6,2)$. For $\alpha\in\mathbb{N}$, if the distances of the points $A(3,4,\alpha)$ and …
  47. Let the line $L$ pass through the point $(0,1,2)$, intersect the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and be parallel to the plane $2x+y-3z=4$. Then the …
  48. If the equation of the plane containing the line $x+2y+3z-4=0=2x+y-z+5$ and perpendicular to the plane …
  49. A plane intersects the $yz$-plane at $x=0$, $2y-3z=5$ and it intersects the $xy$-plane at $z=0$, $7x+4y=10$. Then the distance of the point $(1,-2,-1)$ from this plane is
  50. The distance of the point $(-1,2,-2)$ from the line of intersection of the planes $2x+3y+2z=0$ and $x-2y+z=0$ is :

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