Three Dimensional Geometry PYQs - Last 5 Years

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All Three Dimensional Geometry PYQs Last 10 Years Create chapter test

Practice 108 AP EAPCET Mathematics questions from Three Dimensional Geometry. Use the year-wise and type-wise breakdown to prioritize recent PYQs, then continue into the question list below.

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Last 5 Years Three Dimensional Geometry Questions

Showing 50 of 108 filtered questions.

1Three Dimensional Geometry

The circumradius of the triangle formed by the points $(2,-1,1),(1,-3,-5)$ and $(3,-4,-4)$ is

2Three Dimensional Geometry

Let $A(2,3,5), B(-1,3,2)$ and $C(\lambda, 5, \mu)$ be the vertices of $\triangle A B C$. If the median through the vertex $A$ is equally inclined to the coordinate axes, then

3Three Dimensional Geometry

Equation of the plane passing through the origin and perpendicular to the planes $x+2 y-z=1$ and $3 x-4 y+z=5$ is

4Three Dimensional Geometry

The points $A(-1,2,3), B(2,-3,1)$ and $C(3,1,-2)$

5Three Dimensional Geometry

The directions cosines of the line making angles $\frac{\pi}{4}, \frac{\pi}{3}$ and $\theta\left(0<\theta<\frac{\pi}{2}\right)$ respectively with $X, Y$ and $Z$ axes are

6Three Dimensional Geometry

If the equation of the plane passing through the point $(3,2,5)$ and perpendicular to the planes $2 x-3 y+5 z=7$ and $5 x+2 y-3 z=11$ is $x+b y+c z+d=0$, then $2 b+3 c+d=$

7Three Dimensional Geometry

The vector equation of a plane passing through the line of intersection of the planes $\mathbf{r} \cdot(\hat{\mathbf{i}}-2 \hat{\mathbf{k}})=3, \mathbf{r} \cdot(2 \hat{\mathbf{j}}+\hat{\mathbf{k}})=5$ and the point $\hat{\mathbf{i}}+2 \hat{...

8Three Dimensional Geometry

If the line joining the points $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ and $\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ intersects the plane passing through the points $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}, 2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and ...

9Three Dimensional Geometry

If the image of the point $A(1,1,1)$ with respect to the plane $4 x+2 y+4 z+1=0$ is $B(\alpha, \beta, \gamma)$, then $\alpha+\beta+\gamma=$

10Three Dimensional Geometry

The point in the $X Y$ - plane which is equidistant from the points $A(2,0,3), B(0,3,2)$ and $C(0,0,1)$ has the coordinates

11Three Dimensional Geometry

Line $L_1$ passes through the point $\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and $\hat{\mathbf{k}}-\hat{\mathbf{i}}$. Line $L_2$ passes through the point $\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and is parallel to the vector $\hat{\mathbf{i}}+\hat...

12Three Dimensional Geometry

If the direction ratio of two lines $L_1$ and $L_2$ are given by $(1,-2,2)$ and $(-2,3,-6)$ respectively, then the direction ratios of the line which is perpendicular to the linesh and $L_2$ are

13Three Dimensional Geometry

Assertion (A) For the lines $\mathbf{r}=\mathbf{a}+t \mathbf{b}$ and $\mathbf{r}=\mathbf{p}+s \mathbf{q}$, if $(\mathbf{a}-\mathbf{p}) \cdot(\mathbf{b} \times \mathbf{q}) \neq 0$, then the two lines are coplanar.
Reason $(\mathrm{R})|(\math...

14Three Dimensional Geometry

Let $A=(2,0,-1), B=(1,-2,0), C=(1,2,-1)$ and $D=(0,-1,-2)$ be four points.
If $\theta$ is the acute angle between the plane determined by $A, B, C$ and the plane determined by $A, C, D$, then $\tan \theta=$

15Three Dimensional Geometry

The locus of a point at which the line joining the points $(-3,1,2),(1,-2,4)$ subtends a right angle, is

16Three Dimensional Geometry

If $A(1,2,3), B(2,3,-1), C(3,-1,-2)$ are the vertices of a $\triangle A B C$, then the direction ratios of the bisector of $\angle A B C$ are

17Three Dimensional Geometry

If $A(2,-1,1), B(2,5,1)$ and $C(0,-2,3)$ are the vertices of a triangle. If $D$ is the point of intersection of the side $B C$ and the internal angular bisector of angle $A$, then $A D=$

18Three Dimensional Geometry

A plane $\pi$ given by $a x+b y+11 z+d=0$ is perpendicular to the planes $2 x-3 y+z=4$, $3 x+y-z=5$ and the perpendicular distance from the origin to the plane $\pi$ is $\sqrt{6}$ units. If all the intercepts made by the plane $\pi$ on the ...

19Three Dimensional Geometry

If the direction cosines of two lines satisfy the equations $l-2 m+n=0, l m+10 m n-2 n l=0$ and $\theta$ is the angle between the lines, then $\cos \theta=$

20Three Dimensional Geometry

If $A(0,1,2), B(2,-1,3)$ and $C(1,-3,1)$ are the vertices of a triangle, then the distance between its circumcentre and orthocentre is

21Three Dimensional Geometry

If $(2,-1,3)$ is the foot of the perpendicular drawn from the origin $(0,0,0)$ to a plane, then the equation of that plane is

22Three Dimensional Geometry

Angle between a diagonal of a cube and a diagonal of its face which are coterminus is

23Three Dimensional Geometry

A plane $\pi$ is passing through the points $A(1,-2,3)$ and $B(6,4,5)$. If the plane $\pi$ is perpendicular the plane $3 x-y+z=2$, then the perpendicular distance from $(0,0,0)$ to the plane $\pi$ is

24Three Dimensional Geometry

$\hat{\mathbf{i}}-2 \hat{\mathbf{j}}$ is a point on the line parallel to the vector $2 \hat{\mathbf{i}}+\hat{\mathbf{k}}$. If $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ is a point on the plane parallel to the vectors $2 \hat{\mathbf{j}}-\hat{\ma...

25Three Dimensional Geometry

For a positive real number $p$, if the perpendicular distance from a point $-\hat{\mathbf{i}}+p \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ to the plane $\mathbf{r} \cdot(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})=7$ is 6 units, the...

26Three Dimensional Geometry

If the line of intersection of the planes $2 x+3 y+z=1$ and $x+3 y+2 z=2$ makes an angle $\alpha$ with the positive $X$-axis, then $\cos \alpha=$

27Three Dimensional Geometry

On a line with direction cosines $l, m, n, A\left(x_1, y_1, z_1\right)$ is a fixed point. If $B=\left(x_1+4 k l, y_1+4 k m, z_1+4 k n\right)$ and $C=\left(x_1+k l, y_1+k m, z_1+k n\right)(k>0)$, then the ratio in which the point $B$ divides...

28Three Dimensional Geometry

If $Q(\alpha, \beta, \gamma)$ is the harmonic conjugate of the point $P(0,-7,1)$ with respect to the line segment joining the points $(2,-5,3)$ and $(-1,-8,0)$, then $\alpha-\beta+\gamma=$

29Three Dimensional Geometry

If the four points $(6,2,4),(1,3,5),(1,-2,3)$ and $(6, k, 2)$ are coplanar, then $k=$

30Three Dimensional Geometry

The point of intersection of the lines represented by $\mathbf{r}=(\hat{\mathbf{i}}-6 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})+\mathbf{t}(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}})$ and $\mathbf{r}=(4 \hat{\mathbf{j}}+\hat{\mathbf{k}...

31Three Dimensional Geometry

$G(1,0,1)$ is the centroid of the $\triangle A B C$. If $A=(1,-4,2)$ and $B=(3,1,0)$, then $A G^2+C G^2=$

32Three Dimensional Geometry

If the sum of the distances of the point $(3,4, \alpha), \alpha \in R$ from $X$-axis, $Y$-axis and $Z$-axis is minimum, then $\sec \alpha=$

33Three Dimensional Geometry

If the equation of the plane passing through the point $(2,-1,3)$ and perpendicular to each of the planes $3 x-2 y+z=8$ and $x+y+z=6$ is $l x+m y+n z=1$, then $4 m+2 n-3 l=$

34Three Dimensional Geometry

The length of the internal bisector of angle $A$ in $\triangle A B C$ with vertices $A(4,7,8), B(2,3,4)$ and $C(2,5,7)$ is

35Three Dimensional Geometry

If the angle $\theta$ between the line $\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}$ and the plane $2 x-y+\sqrt{\lambda} z+4=0$ is such that $\sin \theta=\frac{1}{3^{\prime}}$ then the value of $\lambda=$

36Three Dimensional Geometry

If the direction cosines of lines are given by $l+m+n=0$ and $m n-2 l m-2 n l=0$, then the acute angle between those lines is

37Three Dimensional Geometry

If the perpendicular distance from $(1,2,4)$ to the plane $2 x+2 y-z+k=0$ is 3 , then $k=$

38Three Dimensional Geometry

If $P=(0,1,2), Q=(4,-2,1)$, and $O=(0,0,0)$, then $\angle P O Q=$

39Three Dimensional Geometry

The orthocentre of triangle fromed by points $(2,1,5)$ $(3,2,3)$ and $(4,0,4)$ is

40Three Dimensional Geometry

The distance of a point $(2,3,-5)$ from the plane $\hat{\mathbf{r}} \cdot(4 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}})=4$ is

41Three Dimensional Geometry

If $A=(1,2,3), B=(3,4,7)$ and $C=(-3,-2,-5)$ are three points, then the ratio in which the point $C$ divides $A B$ externally is

42Three Dimensional Geometry

The foot of the perpendicular drawn from a point $A(1,1,1)$ on to a plane $\pi$ is $P(-3,3,5)$.If the equation of the plane parallel to the plane of $\pi$ and passing through the mid-point of $A P$ is $a x-y+c z+d=0$, then $a+c-d$ is equal ...

43Three Dimensional Geometry

If $l, m$ and $n$ are the direction cosines of a line that is perpendicular to the lines having the direction ratios $(1,2,-1)$ and $(1,-2,1)$, then $(l+m+n)^2$ is equal to

44Three Dimensional Geometry

If a line $L$ makes angles $\frac{\pi}{3}$ and $\frac{\pi}{4}$ with $Y$-axis and $Z$-axis respectively, then the angle between $L$ and another line having direction ratio $1,1,1$ is

45Three Dimensional Geometry

If $\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}$ are the vertices of a tetrahedron, t...

46Three Dimensional Geometry

Let $\pi$ be the plane that passes through the point $(-2,1,-1)$ and parallel to the plane $2 x-y+2 z=0$. Then the foot of perpendicular drawn from the point $(1,2,1)$ to the plane $\pi$ is

47Three Dimensional Geometry

$A(1,2,1), B(2,3,2), C(3,1,3)$ and $D(2,1,3)$ are the vertices of a tetrahedron. If $\theta$ is the angle between the faces $A B C$ and $A B D$, then $\cos \theta=$

48Three Dimensional Geometry

If $P(2, \beta, \alpha)$ lies on the plane $x+2 y-z-2=0$ and $Q(\alpha,-1, \beta)$ lies on the plane $2 x-y+3 z+6=0$, then the direction cosines of the $P Q$ are

49Three Dimensional Geometry

Consider the tetrahedron with the vertices $A(3,2,4)$, $B\left(x_1, y_1, 0\right), C\left(x_2, y_2, 0\right)$ and $D\left(x_3, y_3, 0\right)$.If the $\triangle B C D$ is formed by the lines $y=x, x+y=6$ and $y=1$, then the centroid of the t...

50Three Dimensional Geometry

Let $P\left(x_1, y_1, z_1\right)$ be the foot of perpendicular drawn from the point $Q(2,-2,1)$ to the plane $x-2 y+z=1$. If $d$ is the perpendicular from the point $Q$ to the plane and $l=x_1+y_1+z_1$, then $l+3 d^2$ is