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 shortest distance between the skew lines $\mathbf{r}=(2 \hat{\mathbf{i}}-\hat{\mathbf{j}})+t(\hat{\mathbf{i}}+2 \hat{\mathbf{k}})$ and $\mathbf{r}=(-2 \hat{\mathbf{i}}+\hat{\mathbf{k}})+s(\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k...

2Three Dimensional Geometry

Angle between the planes, $\mathbf{r} \cdot(12 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})=5$ and, $\mathbf{r} \cdot(5 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}})=7$ is

3Three Dimensional Geometry

If the line with direction ratios $(1, \alpha, \beta)$ is perpendicular to the line with direction ratios $(-1,2,1)$ and parallel to the line with direction ratios $(\alpha, 1, \beta)$ then $(\alpha, \beta)$ is

4Three Dimensional Geometry

If the plane $x-y+z+4=0$ divides the line joining the points $P(2,3,-1)$ and $Q(1,4,-2)$ in the ratio $l: m$, then $l+m$ is

5Three Dimensional Geometry

Let $O(\mathbf{O}), A(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}), B(-2 \hat{\mathbf{i}}+3 \hat{\mathbf{k}}), C(2 \hat{\mathbf{i}}+\hat{\mathbf{j}})$ and $D(4 \hat{\mathbf{k}})$ are position vectors of the points $O, A, B, C$ and ...

6Three Dimensional Geometry

The distance of the point $O(\mathbf{O})$ from the plane $\mathbf{r}$. $(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})=5$ measured parallel to $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-6 \hat{\mathbf{k}}$ is

7Three Dimensional Geometry

If $A(1,0,2), B(2,1,0), C(2,-5,3)$ and $D(0,3,2)$ are four points and the point of intersection of the lines $A B$ and $C D$ is $P(a, b, c)$, then $a+b+c=$

8Three Dimensional Geometry

The direction cosines of two lines are connected by the relations $l+m-n=0$ and $l m-2 m n+n l=0$. If $\theta$ is the acute angle between those lines, then $\cos \theta=$

9Three Dimensional Geometry

The distance from a point $(1,1,1)$ to a variable plane $\pi$ is 12 units and the points of intersections of the plane $\pi$ and $X, Y, Z$ - axes are $A, B, C$ respectively, If the point of intersection of the planes through the points $A, ...

10Three Dimensional Geometry

The angle between the line with the direction ratios $(2,5,1)$ and the plane $8 x+2 y-z=14$ is

11Three Dimensional Geometry

The direction cosines of the line of intersection of the planes $x+2 y+z-4=0$ and $2 x-y+z-3=0$ are

12Three Dimensional Geometry

If $L_1$ and $L_2$ are two lines which pass through origin and having direction ratios $(3,1,-5)$ and $(2,3,-1)$ respectively, then equation of the plane containing $L_1$ and $L_2$ is

13Three Dimensional Geometry

The perpendicular distance from the point $(-1,1,0)$ to the line joining the points $(0,2,4)$ and $(3,0,1)$ is

14Three Dimensional Geometry

If $A(1,2,0), B(2,0,1), C(-3,0,2)$ are the vertices of $\triangle A B C$, then the length of the internal bisector of $\angle B A C$ is

15Three Dimensional Geometry

The shortest distance between the skew lines $\mathbf{r}=(-\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})+t(3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})$ and $\mathbf{r}=(7 \hat{\mathbf{i}}+4 \hat{\mathbf{k}})+s(\hat{\...

16Three Dimensional Geometry

A line $L$ passes through the points $(1,2,-3)$ and $(\beta, 3,1)$ and a plane $\pi$ passes through the points $(2,1,-2)$, $(-2,-3,6),(0,2,-1)$. If $\theta$ is the angle between the line $L$ and plane $\pi$, then $27 \cos ^2 \theta=$

17Three Dimensional Geometry

If the distance between the planes $2 x+y+z+1=0$ and $2 x+y+z+\alpha=0$ is 3 units, then product of all possible values of $\alpha$ is

18Three Dimensional Geometry

Let $P(\alpha, 4,7)$ and $Q(\beta, \beta, 8)$ be two points. If $Y Z$-plane divides the join of the points $P$ and $Q$ in the ratio $2: 3$ and $Z X$-plane divides the join of $P$ and $Q$ in the ratio $4: 5$, then length of line segment $P Q...

19Three Dimensional Geometry

The equation $a x y+b y=c y$ represents the locus of the points which lie on

20Three Dimensional Geometry

If the points with position vectors $(\alpha \hat{\mathbf{i}}+10 \hat{\mathbf{j}}+13 \hat{\mathbf{k}}),(6 \hat{\mathbf{i}}+11 \hat{\mathbf{j}}+11 \hat{\mathbf{k}}),\left(\frac{9}{2} \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}-8 \hat{\mathbf{k}}...

21Three Dimensional Geometry

The distance between two parallel planes $a x+b y+c z+d_1=0, a x+b y+c z+d_2=0$ is given by $\frac{\left|d_1-d_2\right|}{\sqrt{a^2+b^2+c^2}}$. If the plane $2 x-y+2 z+3=0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the pla...

22Three Dimensional Geometry

Let $\pi_1$ be the plane determined by the vectors $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ and $3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$. Let $\pi_2$ be the plane determined by the vectors $\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $3 \hat{\math...

23Three Dimensional Geometry

$\triangle A B C$ is formed by $A(1,8,4), B(0,-11,4)$ and $C(2,-3,1)$. If $D$ is the foot of the perpendicular drawn from $A$ to $B C$, then the coordinates of $D$ are

24Three Dimensional Geometry

If the line passing through the points $(5,1, a)$ and $(3, b, 1)$ crosses the $Y Z$-plane at the point $\left(0, \frac{17}{2}, \frac{-13}{2}\right)$, then $a+b=$

25Three Dimensional Geometry

If $A(3,-1,11), B(0,2,3)$ and $C(4,8,11)$ are three points, then the coordinates of the foot of the perpendicular drawn from the point $A$ to the line joining the points $B$ and $C$ is

26Three Dimensional Geometry

The distance of a point $\mathbf{a}$ from the plane $\mathbf{r} \cdot \mathbf{m}=q$ is given by $\frac{|\mathbf{a} \cdot \mathbf{m}-9|}{|\mathbf{m}|}$. If the distance of the point $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ fr...

27Three Dimensional Geometry

If $S$ is the set of all real values of ' $a$ ' such that a plane passing through the points $\left(-a^2, 1,1\right),\left(1,-a^2, 1\right)$ and $\left(1,1,-a^2\right)$ also passes through the point $(-1,-1,1)$, then $S=$

28Three Dimensional Geometry

Let $A B C D$ be a tetrahedron in which the coordinates of each of its vertices are in arithmetic progression with same common difference. If the centroid $G$ of the tetrahedron is $(2,3, k)$, then the distance of $G$ from the origin is

29Three Dimensional Geometry

Let the plane \(\pi\) pass through the point (1, 0, 1) and perpendicular to the planes \(2x + 3y - z = 2\) and \(x - y + 2z = 1\). Let the equation of the plane passing through the point (11, 7, 5) and parallel to the plane \(\pi\) be $$ax ...

30Three Dimensional Geometry

If the direction cosines of a line are \(\left(\frac{a}{\sqrt{83}}, \frac{5}{\sqrt{83}}, \frac{c}{\sqrt{83}}\right)\) and \(c-a=4\), then \(ca=\)

31Three Dimensional Geometry

If P divides the line segment joining the points \(A(1,2,-1)\) and \(B(-1,0,1)\) externally in the ratio 1 : 2 and \(Q=(1,3,-1)\), then \(PQ=\)

32Three Dimensional Geometry

The \(x\)-intercept of a plane \(\pi\) passing through the point \((1,1,1)\) is \(\frac{5}{2}\) and the perpendicular distance from the origin to the plane \(\pi\) is \(\frac{5}{7}\). If the \(y\)-intercept of the plane \(\pi\) is negative ...

33Three Dimensional Geometry

If the point \((a, 8,-2)\) divides the line segment joining the points \((1,4,6)\) and \((5,2,10)\) in the ratio \(m: n\), then \(\frac{2 m}{n}-\frac{a}{3}=\)

34Three Dimensional Geometry

If \((a, b, c)\) are the direction ratios of a line joining the points \((4,3,-5)\) and \((-2,1,-8)\), then the point \(P(a, 3 b, 2 c)\) lies on the plane

35Three Dimensional Geometry

The point of intersection of the lines \(\mathbf{r}=2 \mathbf{b}+t(6 \mathbf{c}-\mathbf{a})\) and \(\mathbf{r}=\mathbf{a}+s(\mathbf{b}-3 \mathbf{c})\) is

36Three Dimensional Geometry

\(D, E, F\) are respectively the points on the sides \(B C, C A\) and \(A B\) of a \(\triangle A B C\) dividing them in the ratio \(2: 3,1: 2,3: 1\) internally. The lines \(\mathbf{B E}\) and \(\mathbf{C F}\) intersect on the line $$\mathbf...

37Three Dimensional Geometry

If \((2,3, c)\) are the direction ratios of a ray passing through the point \(C(5, q, 1)\) and also the mid-point of the line segment joining the points \(A(p,-4,2)\) and \(B(3,2,-4)\), then \(c \cdot(p+7 q)=\)

38Three Dimensional Geometry

If \(x\)-coordinate of a point \(P\) on the line joining the points \(Q(2,2,1)\) and \(R(5,2,-2)\) is 4, then the \(y\)-coordinate of \(P=\)

39Three Dimensional Geometry

If the equation of the plane passing through the point \(A(-2,1,3)\) and perpendicular to the vector \(3 \hat{i}+\hat{j}+5 \hat{k}\) is \(a x+b y+c z+d=0\), then \(\frac{a+b}{c+d}=\)

40Three Dimensional Geometry

If the equation of the plane which is at a distance of \(1 / 3\) units from the origin and perpendicular to a line whose directional ratios are \((1,2,2)\) is \(x+p y+q z+r=0\), then \(\sqrt{p^2+q^2+r^2}=\)

41Three Dimensional Geometry

If the direction cosines of two lines are \(\left( {{2 \over 3},{2 \over 3},{1 \over 3}} \right)\) and \(\left( {{5 \over {13}},{{12} \over {13}},0} \right)\), then identify the direction ratios of a line which is bisecting one o the angle ...

42Three Dimensional Geometry

A(2, 3, 4), B(4, 5, 7), C(2, \(-\)6, 3) and D(4, \(-\)4, k) are four points. If the line AB is parallel to CD, then k is equal to

43Three Dimensional Geometry

If the vertices of the triangles are (1, 2, 3), (2, 3, 1), (3, 1, 2) and if H, G, S and I respectively denote its orthocentre, centroid, circumcentre and incentre, then H + G + S + I is equal to

44Three Dimensional Geometry

The line passing through \((1,1,-1)\) and parallel to the vector \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) meets the line \(\frac{x-3}{-1}=\frac{y+2}{5}=\frac{z-2}{-4}\) at \(A\) and the plane \(2 x-y+2 z+7=0\) at \(B\). Then...

45Three Dimensional Geometry

If the lines, \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{\lambda}\) and \(\frac{x-2}{3}=\frac{y-3}{2}=\frac{z-2}{3}\) are coplanar, then \(\sin ^{-1}(\sin \lambda)+\cos ^{-1}(\cos \lambda)\) is equal to

46Three Dimensional Geometry

The sum of intercepts of the plane \(4 x+3 y+2 z=2\) on the coordinate axes is

47Three Dimensional Geometry

The points (2, 3, 4), (\(-\)1, \(-\)2, 1) and (5, 8, 7) are

48Three Dimensional Geometry

The direction cosines of the line joining the points \((-2,4,-5)\) and \((1,2,3)\) are

49Three Dimensional Geometry

The equation of the plane passing through \(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}\) and parallel to the vectors \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and $$\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf...

50Three Dimensional Geometry

Let \(O\) be the origin and \(P\) be a point which is at a distance of 3 units from the origin. If the direction ratios of \(\overline{O P}\) are \((1,-2,-2)\), then the coordinates of \(P\) are