Three Dimensional Geometry · Mathematics · MHT CET

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MCQ (Single Correct Answer)

1

The equation of the plane passing through the point of intersection of the planes $2 x-y+z-3=0$ and $4 x-3 y+5 z+9=0$ and parallel to the line $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z-3}{5}$ is $\alpha x+\beta y+\gamma z+d=0$ Then $\alpha+\beta+\gamma+d=$

MHT CET 2025 20th April Evening Shift
2

The distance of the point $(2,4,0)$ from the point of intersection of the lines $\frac{x+6}{3}=\frac{y}{2}=\frac{z+1}{1}$ and $\frac{x-7}{4}=\frac{y-9}{3}=\frac{z-4}{2}$ is

MHT CET 2025 20th April Evening Shift
3

The co-ordinates of the point where the line joining the points $(2,-3,1)$ and $(3,-4,-5)$ and intersects the plane $2 x+y+z=7$ are

MHT CET 2025 20th April Evening Shift
4

If the line $\frac{x+1}{3}=\frac{y-k}{7}=\frac{z-4}{8}$ lies in the plane $2 x+\mathrm{p} y+7 z-41=0$ which is perpendicular to the plane $x+4 y-2 z+13=0$ then $\mathrm{k}=$

MHT CET 2025 20th April Evening Shift
5

The direction cosines of the line $x-y+2 z=5$ and $3 x+y+z=6$ are

MHT CET 2025 20th April Morning Shift
6

If the angle between the planes $x-2 y+3 z-5=0$ and $x+\alpha y+2 z+7=0$ is $\cos ^{-1}\left(\frac{1}{14}\right)$ then the difference between the values of $\alpha$ is

MHT CET 2025 20th April Morning Shift
7

If the shortest distance between the lines $\frac{x-\mathrm{k}}{2}=\frac{y-4}{3}=\frac{\mathrm{z}-3}{4}$ and $\frac{x-2}{4}=\frac{y-4}{6}=\frac{\mathrm{z}-7}{8}$ is $\frac{13}{\sqrt{29}}$, then $\mathrm{k}=$

MHT CET 2025 20th April Morning Shift
8

The acute angle between the lines $x=-2+2 \mathrm{t}, y=3-4 \mathrm{t}, \mathrm{z}=-4+\mathrm{t}$ and $x=-2-\mathrm{t}, y=3+2 \mathrm{t}, \mathrm{z}=-4+3 \mathrm{t}$ is

MHT CET 2025 20th April Morning Shift
9

If the plane $\frac{x}{3}+\frac{y}{2}-\frac{z}{4}=1$ cuts the co-ordinate axes at points $\mathrm{A}, \mathrm{B}$ and C , then the area of the triangle ABC is

MHT CET 2025 20th April Morning Shift
10

A plane passes through $(2,1,2)$ and $(1,2,1)$ and parallel to the line $2 x=3 y$ and $\mathrm{z}=1$, then the plane also passes through the point

MHT CET 2025 19th April Evening Shift
11
The equation of the plane passing through the line of intersection of the planes $x+y+z=1$ and $3 x+4 y+5 z=2$ and perpendicular to the XY- plane is
MHT CET 2025 19th April Evening Shift
12

The coordinates of the foot of the perpendicular drawn from a point $\mathrm{P}(-1,1,2)$ to the plane $2 x-3 y+z-11=0$ are

MHT CET 2025 19th April Evening Shift
13

The lines $\frac{x-3}{1}=\frac{y-2}{1}=\frac{z-5}{-k}$ and $\frac{x-4}{\mathrm{k}}=\frac{y-3}{1}=\frac{\mathrm{z}-3}{2}$ are coplanar, hence $\mathrm{k}=$

MHT CET 2025 19th April Evening Shift
14

If the shortest distance between the lines $\bar{r}_1=\alpha \hat{i}+2 \hat{j}+2 \hat{k}+\lambda(\hat{i}-2 \hat{j}+2 \hat{k}), \lambda \in \mathbb{R}, \alpha>0 \quad$ and $\bar{r}_2=-4 \hat{i}-\hat{k}+\mu(3 \hat{i}-2 \hat{j}-2 \hat{k}), \mu \in R$, is 9 , then the value of $\alpha$ is

MHT CET 2025 19th April Evening Shift
15
The Cartesian equation of plane through $\mathrm{A}(7,8,6)$ and parallel to the XY plane is
MHT CET 2025 19th April Morning Shift
16
The distance of the point $(-3,2,3)$ from the line passing through $(4,6,-2)$ and having direction ratios $-1,2,3$ is $\qquad$ units.
MHT CET 2025 19th April Morning Shift
17
A plane passes through $(1,-2,1)$ and is perpendicular to the planes $2 x-2 y+z=0$ and $x-y+2 z=4$. The distance of the point $(1,2,2)$ from this plane is ________ units.
MHT CET 2025 19th April Morning Shift
18
If the sum of the squares of the distance of the point $\mathrm{P}(x, y, \mathrm{z})$ from the co-ordinate axes is 242 , then the distance of the point P from the origin is units.
MHT CET 2025 19th April Morning Shift
19
If the points $\mathrm{A}(2-x, 2,2), \mathrm{B}(2,2-y, 2)$, $\mathrm{C}(2,2,2-\mathrm{z})$ and $\mathrm{D}(1,1,1)$ are coplanar, then the locus of point $\mathrm{P}(x, y, \mathrm{z})$ is
MHT CET 2025 19th April Morning Shift
20
If the lines $\frac{3-x}{2}=\frac{5 y-2}{3 \lambda+1}=5-\mathrm{z}$ and $\frac{x+2}{-1}=\frac{1-3 y}{7}=\frac{4-z}{2 \mu}$ are at right angles, then $7 \lambda-10 \mu=$
MHT CET 2025 19th April Morning Shift
21
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 $\sin \theta=\frac{1}{3}$, then $\lambda+1=$
MHT CET 2025 19th April Morning Shift
22
If the directed line makes an angle $45^{\circ}$ and $60^{\circ}$ with the X and Y -axes respectively, then the obtuse angle $\theta$ made by the line with the Z -axis is
MHT CET 2025 19th April Morning Shift
23

The co-ordinates of the foot of the perpendicular from the point $(0,2,3)$ on the line $\frac{x+3}{5}=\frac{y+1}{2}=\frac{z+4}{3}$ is

MHT CET 2024 16th May Evening Shift
24

A line having direction ratios $1,-4,2$ intersects the lines $\frac{x-7}{3}=\frac{y-1}{-1}=\frac{z+2}{1}$ and $\frac{x}{2}=\frac{y-7}{3}=\frac{z}{1}$ at the points $A$ and $B$ resp., then co-ordinates of points A and B are

MHT CET 2024 16th May Evening Shift
25

A plane makes positive intercepts of unit length on each of $X$ and $Y$ axis. If it passes through the point $(-1,1,2)$ and makes angle $\theta$ with the X -axis, then $\theta$ is

MHT CET 2024 16th May Evening Shift
26

The equation of plane through the point $(2,-1,-3)$ and parallel to lines $\frac{x-1}{3}=\frac{y+2}{2}=\frac{z}{-4}$ and $\frac{x}{2}=\frac{y-1}{-3}=\frac{z-2}{2}$ is

MHT CET 2024 16th May Evening Shift
27

The equation of the plane, passing through the intersection of the planes $x+y+z=1$ and $2 x+3 y-z+4=0$ and parallel to $Y$-axis is

MHT CET 2024 16th May Morning Shift
28

A line with positive direction cosines passes through the point $\mathrm{P}(2,-1,2)$ and makes equal angles with co-ordinate axes. The line meets the plane $2 x+y+z=9$ at point Q. Then the length of the line segment PQ equals

MHT CET 2024 16th May Morning Shift
29

If the distance between the plane Ax-2y+z $=\mathrm{d}$ and the plane containing the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}$ is $\sqrt{6}$ units, then $|d|$ is

MHT CET 2024 16th May Morning Shift
30

The length of the projection of the line segment joining the points $(5,-1,4)$ and $(4,-1,3)$ on the plane $x+y+z=7$ is

MHT CET 2024 16th May Morning Shift
31

A line makes $45^{\circ}$ angle with positive X -axis and makes equal angles with positive Y -axis ad Z-axis respectively, then the sum of the three angles which the line makes with positive X -axis, Y -axis and Z -axis is

MHT CET 2024 15th May Evening Shift
32

If the lines $\frac{x-1}{2}=\frac{y+2}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-\mathrm{k}}{2}=\frac{\mathrm{z}}{1}$ intersect, then k has the value

MHT CET 2024 15th May Evening Shift
33

The vector equation of the plane through the line of intersection of the planes $x+y+z=1$ and $2 x+3 y+4 z=5$, which is perpendicular to the plane $x-y+z=0$, is

MHT CET 2024 15th May Evening Shift
34

The equation of a line passing through the point $(2,-1,1)$ and parallel to the line joining the points $\hat{i}+2 \hat{j}+2 \hat{k}$ and $-\hat{i}+4 \hat{j}+\hat{k}$ is

MHT CET 2024 15th May Evening Shift
35

The foot of the perpendicular drawn from origin to a plane is $\mathrm{M}(2,1,-2)$, then vector equation of the plane is

MHT CET 2024 15th May Evening Shift
36

Let $\mathrm{L}_1: \frac{x+2}{5}=\frac{y-3}{2}=\frac{\mathrm{z}-6}{1}$ and $\mathrm{L}_2: \frac{x-3}{4}=\frac{y+2}{3}=\frac{z-3}{5}$ be the given lines. Then the unit vector perpendicular to both $\mathrm{L}_1$ and $\mathrm{L}_2$ is

MHT CET 2024 15th May Morning Shift
37

The perpendicular distance from the origin to the plane containing the two lines $\frac{x+2}{3}=\frac{y-2}{5}=\frac{z+5}{7}$ and $\frac{x-1}{1}=\frac{y-4}{4}=\frac{z+4}{7}$, is

MHT CET 2024 15th May Morning Shift
38

Let $P(2,1,5)$ be a point in space and $Q$ be a point on the line $\bar{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(-3 \hat{i}+\hat{j}+5 \hat{k})$. Then the value of $\mu$ for which the vector $\overline{\mathrm{PQ}}$ is parallel to the plane $3 x-y+4 z=1$ is

MHT CET 2024 15th May Morning Shift
39

The centroid of tetrahedron with vertices $\mathrm{P}(5,-7,0), \mathrm{Q}(\mathrm{a}, 5,3), \mathrm{R}(4,-6, b)$ and $\mathrm{S}(6, \mathrm{c}, 2)$ is $(4,-3,2)$, then the value of $2 a+3 b+c$ is equal to

MHT CET 2024 15th May Morning Shift
40

A variable plane passes through the fixed point $(3,2,1)$ and meets $X, Y$ and $Z$ axes at points $A$, B and C respectively. A plane is drawn parallel to YZ - plane through A , a second plane is drawn parallel to ZX -plan through B , a third plane is drawn parallel to XY - plane through C . Then locus of the point of intersection of these three planes, is

MHT CET 2024 11th May Evening Shift
41

The distance of the point $(1,-5,9)$ from the plane $x-y+z=5$ measured along the line $x=y=\mathrm{z}$ is __________ units.

MHT CET 2024 11th May Evening Shift
42

If for some $\alpha \in \mathbb{R}$, the lines $\mathrm{L}_1: \frac{x+1}{2}=\frac{y-2}{-1}=\frac{z-1}{1}$ and $\mathrm{L}_2: \frac{x+2}{\alpha}=\frac{y+1}{5-\alpha}=\frac{z+1}{1}$ are coplanar, then the line $L_2$ passes through the point

MHT CET 2024 11th May Evening Shift
43

Let $P(3,2,6)$ be a point in space and $Q$ be a point on the line $\bar{r}=\hat{i}-\hat{j}+2 \hat{k}+\mu(-3 \hat{i}+\hat{j}+5 \hat{k})$. Then the value of $\mu$ for which the vector $\overline{\mathrm{PQ}}$ is parallel to the plane $x-4 y+3 z=1$ is

MHT CET 2024 11th May Evening Shift
44

The perpendicular distance of the origin from the plane $2 x+y-2 z-18=0$ is

MHT CET 2024 11th May Morning Shift
45

The plane $2 x+3 y+4 z=1$ meets $X$-axis in $A$, Y -axis in B and Z -axis in C . Then the centroid of $\triangle A B C$ is

MHT CET 2024 11th May Morning Shift
46

If the lines $\frac{x+1}{-10}=\frac{y+k}{-1}=\frac{z-4}{1} \quad$ and $\frac{x+10}{-1}=\frac{y+1}{-3}=\frac{z-1}{4}$ intersect each other, then the value of $k$ is

MHT CET 2024 11th May Morning Shift
47

The equation of the line passing through the point $(3,1,2)$ and perpendicular to the lines $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x}{-3}=\frac{y}{2}=\frac{z}{5}$ is

MHT CET 2024 11th May Morning Shift
48

The area of the triangle with vertices $(1,2,0)$, $(1,0,2)$ and $(0,3,1)$ is

MHT CET 2024 11th May Morning Shift
49

If the volume of tetrahedron whose vertices are $A \equiv(1,-6,10), B \equiv(-1,-3,7), C \equiv(5,-1, k)$ and $D \equiv(7,-4,7)$ is 11 cu . units, then the value of $k$ is

MHT CET 2024 11th May Morning Shift
50

The vector equation of the plane passing through the point $\mathrm{A}(1,2,-1)$ and parallel to the vectors $2 \hat{i}+\hat{j}-\hat{k}$ and $\hat{i}-\hat{j}+3 \hat{k}$ is

MHT CET 2024 10th May Evening Shift
51

The shortest distance between lines $\bar{r}=(\hat{i}+2 \hat{j}-\hat{k})+\lambda(2 \hat{i}+\hat{j}-3 \hat{k})$ and $\bar{r}=(2 \hat{i}-\hat{j}+2 \hat{k})+\mu(\hat{i}-\hat{j}+\hat{k})$ is

MHT CET 2024 10th May Evening Shift
52

If the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-\mathrm{k}}{2}=\frac{\mathrm{z}}{1}$ intersect, then the value of k is

MHT CET 2024 10th May Evening Shift
53

The projection of $\overline{\mathrm{AB}}$ on $\overline{\mathrm{CD}}$, where $A \equiv(2,-3,0), B \equiv(1,-4,-2), C \equiv(4,6,8)$ and $\mathrm{D} \equiv(7,0,10)$ is

MHT CET 2024 10th May Evening Shift
54

The equation of the plane through the point $(2,-1,-3)$ and parallel to the lines $\frac{x-1}{3}=\frac{y+2}{2}=\frac{z}{-4}$ and $\frac{x}{2}=\frac{y-1}{-3}=\frac{z-2}{2}$ is

MHT CET 2024 10th May Evening Shift
55

Equation of the plane, through the points $(-1,2,-2)$ and $(-1,3,2)$ and perpendicular to $y \mathrm{z}$ - plane, is

MHT CET 2024 10th May Morning Shift
56

If the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-1}{4}$ and $\frac{x-3}{-1}=\frac{y-\mathrm{k}}{2}=\frac{\mathrm{z}}{1}$ intersect, then k is equal to

MHT CET 2024 10th May Morning Shift
57

If the line, $\frac{x-3}{2}=\frac{y+2}{1}=\frac{z+4}{3}$ lies in the plane, $\ell x+m y-z=9$, then $\ell^2+m^2$ is equal to

MHT CET 2024 10th May Morning Shift
58

If the line $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$ lies in the plane $x+3 y-\alpha z+\beta=0$, then $(\alpha, \beta)=$

MHT CET 2024 10th May Morning Shift
59

A plane which is perpendicular to two planes $2 x-2 y+z=0$ and $x-y+2 z=4$, passes through $(1,2,1)$. The distance of the plane from the point $(2,3,4)$ is

MHT CET 2024 9th May Evening Shift
60

The value of m such that $\frac{x-4}{1}=\frac{y-2}{1}=\frac{z+m}{2}$ lies in the plane $2 x-4 y+z=7$ is

MHT CET 2024 9th May Evening Shift
61

A line with positive direction cosines passes through the point $\mathrm{P}(2,1,2)$ and makes equal angles with the coordinate axes. The line meets the plane $2 x+y+\mathrm{z}=9$ at point Q . The length of the line segment PQ equals $\qquad$ units.

MHT CET 2024 9th May Evening Shift
62

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

MHT CET 2024 9th May Evening Shift
63

The equation of the plane, passing through the point $(1,1,1)$ and perpendicular to the planes $2 x+y-2 z=5$ and $3 x-6 y-2 z=7$, is

MHT CET 2024 9th May Morning Shift
64

The distance of the point $(1,3,-7)$ from the plane passing through the point $(1,-1,-1)$ having normal perpendicular to both the lines $\frac{x-1}{1}=\frac{y+2}{-2}=\frac{z-4}{3}$ and $\frac{x-2}{2}=\frac{y+1}{-1}=\frac{z+7}{-1}$ is

MHT CET 2024 9th May Morning Shift
65

The value of m , such that $\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-m}{2}$ lies in the plane $2 x-4 y+z=7$, is

MHT CET 2024 9th May Morning Shift
66

The length of the perpendicular from the point $\mathrm{A}(1,-2,-3)$ on the line $\frac{x-1}{2}=\frac{y+3}{-1}=\frac{z+1}{-2}$ is

MHT CET 2024 9th May Morning Shift
67

If the points $(1,-1, \lambda)$ and $(-3,0,1)$ are equidistant from the plane $3 x-4 y-12 z+13=0$, then the sum of all possible values of $\lambda$ is

MHT CET 2024 4th May Evening Shift
68

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

MHT CET 2024 4th May Evening Shift
69

The equation of the plane, passing through the point $(-1,2,-3)$ and parallel to the lines $\frac{x-1}{3}=\frac{y-2}{2}=\frac{z}{-4}$ and $\frac{x}{2}=\frac{y-1}{-3}=\frac{z-2}{2}$, is

MHT CET 2024 4th May Evening Shift
70

The co-ordinates of the point where the line through $\mathrm{A}(3,4,1)$ and $\mathrm{B}(5,1,6)$ crosses the $x y$-plane are

MHT CET 2024 4th May Evening Shift
71

The Cartesian equation of a line is $2 x-2=3 y+1=6 z-2$, then the vector equation of the line is

MHT CET 2024 4th May Evening Shift
72

The lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{-k} \quad$ and $\frac{x-1}{\mathrm{k}}=\frac{y-4}{2}=\frac{\mathrm{z}-5}{1}$ are coplanar if

MHT CET 2024 4th May Morning Shift
73

Let $\mathrm{L}_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}$ and $\mathrm{L}_2: \frac{x-2}{1}=\frac{y+2}{2}=\frac{z-3}{3}$ be two given lines. Then the unit vector perpendicular to $L_1$ and $L_2$ is

MHT CET 2024 4th May Morning Shift
74

Let $a, b \in R$. If the mirror image of the point $\mathrm{p}(\mathrm{a}, 6,9)$ w.r.t. line $\frac{x-3}{7}=\frac{y-2}{5}=\frac{z-1}{-9}$ is $(20, b,-a-9)$, then $|a+b|$ is equal to

MHT CET 2024 4th May Morning Shift
75

A plane which is perpendicular to two planes $2 x-2 y+z=0$ and $x-y+2 z=4$, passes through $(1,-2,1)$. The distance of the plane from the point $(1,2,2)$ is

MHT CET 2024 4th May Morning Shift
76

Let $\mathrm{L}_1$ $\frac{x+1}{3}=\frac{y+2}{2}=\frac{z+1}{1}$ and $\mathrm{L}_2: \frac{x-2}{2}=\frac{y+2}{1}=\frac{z-3}{3}$ be the given lines. Then the unit vector perpendicular to $L_1$ and $L_2$ is

MHT CET 2024 3rd May Evening Shift
77

The equation of the plane passing through the point $(1,1,1)$ and perpendicular to the planes $2 x-y-2 z=5$ and $3 x-6 y+2 z=7$ is

MHT CET 2024 3rd May Evening Shift
78

Equation of the plane containing the straight line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$ and $\frac{x}{4}=\frac{y}{2}=\frac{z}{3}$ is

MHT CET 2024 3rd May Evening Shift
79

If $A(-4,5, P), B(3,1,4)$ and $C(-2,0, q)$ are the vertices of a triangle $A B C$ and $G(r, q, 1)$ is its centroid, then the value of $2 p+q-r$ is equal to

MHT CET 2024 3rd May Evening Shift
80

On which of the following lines lies the point of intersection of the line, $\frac{x-4}{2}=\frac{y-5}{2}=\frac{z-3}{1}$ and the plane $x+y+z=2$ ?

MHT CET 2024 3rd May Evening Shift
81

Equation of the plane containing the straight line $\frac{x}{3}=\frac{y}{2}=\frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{4}=\frac{y}{3}=\frac{z}{2}$ and $\frac{x}{2}=\frac{y}{-4}=\frac{z}{3}$ is

MHT CET 2024 3rd May Morning Shift
82

The value of $m$, such that $\frac{x-4}{1}=\frac{y-2}{1}=\frac{2 z-m}{3}$ lies in the plane $2 x-5 y+2 z=7$, is

MHT CET 2024 3rd May Morning Shift
83

The image of the line $\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-5}$ in the plane $2 x-y+z+3=0$ is the line

MHT CET 2024 3rd May Morning Shift
84

Let $\mathrm{P}(2,3,6)$ be a point in space and Q be a point on the line $\bar{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(-3 \hat{i}+\hat{j}+5 \hat{k})$. Then the value of $\mu$ for which vector $\overline{\mathrm{PQ}}$ is parallel to the plane $x-4 y+4 z=1$ is

MHT CET 2024 3rd May Morning Shift
85

Distance between the parallel lines $\frac{x}{3}=\frac{y-1}{-2}=\frac{z}{1}$ and $\frac{x+4}{3}=\frac{y-3}{-2}=\frac{z+2}{1}$ is

MHT CET 2024 2nd May Evening Shift
86

The equation of the plane, passing through the mid point of the line segment of join of the points $\mathrm{P}(1,2,5)$ and $\mathrm{Q}(3,4,3)$ and perpendicular to it, is

MHT CET 2024 2nd May Evening Shift
87

The area of the triangle, whose vertices are $A \equiv(1,-1,2), B \equiv(2,1,-1)$ and $C \equiv(3,-1,2)$, is

MHT CET 2024 2nd May Evening Shift
88

The equation of the line, through $\mathrm{A}(1,2,3)$ and perpendicular to the vector $2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\hat{i}+3 \hat{j}+2 \hat{k}$, is

MHT CET 2024 2nd May Evening Shift
89

Let $P$ be the image of the point $(3,1,7)$ with respect to the plane $x-y+z=3$. Then the equation of the plane passing through $P$ and containing the straight line $\frac{x}{1}=\frac{y}{2}=\frac{z}{1}$ is

MHT CET 2024 2nd May Evening Shift
90

The incentre of the triangle whose vertices are $P(0,3,0), Q(0,0,4)$ and $R(0,3,4)$ is

MHT CET 2024 2nd May Evening Shift
91

The vector equation of a line whose Cartesian equations are $y=2,4 x-3 z+5=0$ is

MHT CET 2024 2nd May Morning Shift
92

The Cartesian equation of the plane, passing through the points $(3,1,1),(1,2,3)$ and $(-1,4,2)$, is

MHT CET 2024 2nd May Morning Shift
93

The equation of the line passing through the point $(-1,3,-2)$ and perpendicular to each of the lines $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ and $\frac{x+2}{-3}=\frac{y-1}{2}=\frac{z+1}{5}$, is

MHT CET 2024 2nd May Morning Shift
94

If the line $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z+4}{3}$ lies in the plane $\ell x+m y-z=9$, then $\ell^2+m^2$ is

MHT CET 2024 2nd May Morning Shift
95

The mirror image of $$\mathrm{P}(2,4,-1)$$ in the plane $$x-y+2 z-2=0$$ is $$(\mathrm{a}, \mathrm{b}, \mathrm{c})$$, then the value of $$a+b+c$$ is

MHT CET 2023 14th May Evening Shift
96

If the lines $$\frac{x-\mathrm{k}}{2}=\frac{y+1}{3}=\frac{\mathrm{z}-1}{4}$$ and $$\frac{x-3}{1}=\frac{y-\frac{9}{2}}{2}=\frac{\mathrm{z}}{1}$$ intersect, then the value of $$\mathrm{k}$$ is

MHT CET 2023 14th May Evening Shift
97

A vector parallel to the line of intersection of the planes $$\bar{r} \cdot(3 \hat{i}-\hat{j}+\hat{k})=1$$ and $$\bar{r} \cdot(\hat{i}+4 \hat{j}-2 \hat{k})=2$$ is

MHT CET 2023 14th May Evening Shift
98

The length of the perpendicular drawn from the point $$(1,2,3)$$ to the line $$\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$$ is

MHT CET 2023 14th May Evening Shift
99

If $$\triangle \mathrm{ABC}$$ is right angled at $$\mathrm{A}$$, where $$A \equiv(4,2, x), \mathrm{B} \equiv(3,1,8)$$ and $$C \equiv(2,-1,2)$$, then the value of $$x$$ is

MHT CET 2023 14th May Morning Shift
100

The angle between the lines, whose direction cosines $$l, \mathrm{~m}, \mathrm{n}$$ satisfy the equations $$l+\mathrm{m}+\mathrm{n}=0$$ and $$2 l^2+2 \mathrm{~m}^2-\mathrm{n}^2=0$$, is

MHT CET 2023 14th May Morning Shift
101

Equation of the plane passing through $$(1,-1,2)$$ and perpendicular to the planes $$x+2 y-2 z=4$$ and $$3 x+2 y+z=6$$ is

MHT CET 2023 14th May Morning Shift
102

A line with positive direction cosines passes through the point $$\mathrm{P}(2,-1,2)$$ and makes equal angles with the co-ordinate axes. The line meets the plane $$2 x+y+z=9$$ at point $$\mathrm{Q}$$. The length of the line segment $$P Q$$ equals

MHT CET 2023 14th May Morning Shift
103

If the shortest distance between the lines $$\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{\lambda}$$ and $$\frac{x-2}{1}=\frac{y-4}{4}=\frac{z-5}{5}$$ is $$\frac{1}{\sqrt{3}}$$, then sum of possible values of $$\lambda$$ is

MHT CET 2023 14th May Morning Shift
104

Consider the lines $$\mathrm{L}_1: \frac{x+1}{3}=\frac{y+2}{1}=\frac{\mathrm{z}+1}{2}$$

$$\mathrm{L}_2: \frac{x-2}{1}=\frac{y+2}{2}=\frac{\mathrm{z}-3}{3}$$, then the unit vector perpendicular to both $$\mathrm{L}_1$$ and $$\mathrm{L}_2$$ is

MHT CET 2023 14th May Morning Shift
105

A tetrahedron has vertices at $$P(2,1,3), Q(-1,1,2), R(1,2,1)$$ and $$O(0,0,0)$$, then angle between the faces $$O P Q$$ and $$P Q R$$ is

MHT CET 2023 13th May Evening Shift
106

A plane is parallel to two lines whose direction ratios are $$2,0,-2$$ and $$-2,2,0$$ and it contains the point $$(2,2,2)$$. If it cuts coordinate axes at $$A, B, C$$, then the volume of the tetrahedron $$O A B C$$ (in cubic units) is

MHT CET 2023 13th May Evening Shift
107

The incentre of the $$\triangle A B C$$, whose vertices are $$A(0,2,1), B(-2,0,0)$$ and $$C(-2,0,2)$$, is

MHT CET 2023 13th May Evening Shift
108

The acute angle between the line joining the points $$(2,1,-3),(-3,1,7)$$ and a line parallel to $$\frac{x-1}{3}=\frac{y}{4}=\frac{z+3}{5}$$ through the point $$(-1,0,4)$$ is

MHT CET 2023 13th May Evening Shift
109

The foot of the perpendicular from the point $$(1,2,3)$$ on the line $$\mathbf{r}=(6 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}+7 \hat{\mathbf{k}})+\lambda(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})$$ has the coordinates

MHT CET 2023 13th May Evening Shift
110

The distance of the point $$(1,6,2)$$ from the point of intersection of the line $$\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}$$ and the plane $$x-y+z=16$$ is

MHT CET 2023 13th May Evening Shift
111

A line drawn from the point $$\mathrm{A}(1,3,2)$$ parallel to the line $$\frac{x}{2}=\frac{y}{4}=\frac{z}{1}$$, intersects the plane $$3 x+y+2 z=5$$ in point $$\mathrm{B}$$, then co-ordinates of point $$\mathrm{B}$$ are

MHT CET 2023 13th May Morning Shift
112

A line $$\mathrm{L}_1$$ passes through the point, whose p. v. (position vector) $$3 \hat{i}$$, is parallel to the vector $$-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$$. Another line $$\mathrm{L}_2$$ passes through the point having p.v. $$\hat{i}+\hat{j}$$ is parallel to vector $$\hat{i}+\hat{k}$$, then the point of intersection of lines $$L_1$$ and $$L_2$$ has p.v.

MHT CET 2023 13th May Morning Shift
113

The equation of the line passing through the point $$(-1,3,-2)$$ and perpendicular to each of the lines $$\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$$ and $$\frac{x+2}{-3}=\frac{y-1}{2}=\frac{z+1}{5}$$ is

MHT CET 2023 13th May Morning Shift
114

If $$A(1,4,2)$$ and $$C(5,-7,1)$$ are two vertices of triangle $$A B C$$ and $$G\left(\frac{4}{3}, 0, \frac{-2}{3}\right)$$ is centroid of the triangle $$A B C$$, then the mid point of side $$B C$$ is

MHT CET 2023 13th May Morning Shift
115

The distance of the point $$(-1,-5,-10)$$ from the point of intersection of the line $$\frac{x-2}{3}=\frac{y+1}{4}=\frac{z-2}{12}$$ and the plane $$x-y+z=5$$ is

MHT CET 2023 13th May Morning Shift
116

The equation of the line, passing through $$(1,2,3)$$ and parallel to planes $$x-y+2 z=5$$ and $$3 x+y+z=6$$, is

MHT CET 2023 12th May Evening Shift
117

The shortest distance (in units) between the lines $$\frac{x+1}{3}=\frac{y+2}{1}=\frac{z+1}{2}$$ and $$\bar{r}=(2 \hat{i}-2 \hat{j}+3 \hat{k})+\lambda(\hat{i}+2 \hat{j})$$ is

MHT CET 2023 12th May Evening Shift
118

The length (in units) of the projection of the line segment, joining the points $$(5,-1,4)$$ and $$(4,-1,3)$$, on the plane $$x+y+z=7$$ is

MHT CET 2023 12th May Evening Shift
119

If the volume of tetrahedron, whose vertices are $$\mathrm{A}(1,2,3), \mathrm{B}(-3,-1,1), \mathrm{C}(2,1,3)$$ and $$D(-1,2, x)$$ is $$\frac{11}{6}$$ cubic units, then the value of $$x$$ is

MHT CET 2023 12th May Evening Shift
120

Equation of plane containing the line $$\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$$ and perpendicular to the plane containing the lines $$\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$$ and $$\frac{x}{4}=\frac{y}{2}=\frac{z}{3}$$ is

MHT CET 2023 12th May Evening Shift
121

The centroid of tetrahedron with vertices at $$\mathrm{A}(-1,2,3), \mathrm{B}(3,-2,1), \mathrm{C}(2,1,3)$$ and $$\mathrm{D}(-1,-2,4)$$ is

MHT CET 2023 12th May Morning Shift
122

A plane is parallel to two lines whose direction ratios are $$1,0,-1$$ and $$-1,1,0$$ and it contains the point $$(1,1,1)$$. If it cuts the co-ordinate axes at $$\mathrm{A}, \mathrm{B}, \mathrm{C}$$, then the volume of the tetrahedron $$\mathrm{OABC}$$ (in cubic units) is

MHT CET 2023 12th May Morning Shift
123

The equation of the plane through $$(-1,1,2)$$ whose normal makes equal acute angles with co-ordinate axes is

MHT CET 2023 12th May Morning Shift
124

The distance of the point $$\mathrm{P}(-2,4,-5)$$ from the line $$\frac{x+3}{3}=\frac{y-4}{5}=\frac{z+8}{6}$$ is

MHT CET 2023 12th May Morning Shift
125

If the line $$\frac{1-x}{3}=\frac{7 y-14}{2 p}=\frac{z-3}{2}$$ and $$\frac{7-7 x}{3 \mathrm{p}}=\frac{y-5}{1}=\frac{6-\mathrm{z}}{5}$$ are at right angles, then $$\mathrm{p}=$$

MHT CET 2023 12th May Morning Shift
126
If the lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $x-3=\frac{y-\mathrm{k}}{2}=\mathrm{z}$ intersect, then the value of $\mathrm{k}$ is
MHT CET 2023 11th May Evening Shift
127

If the line $$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}$$ meets the plane $$x+2 y+3 z=15$$ at the point $$P$$, then the distance of $$\mathrm{P}$$ from the origin is

MHT CET 2023 11th May Evening Shift
128

The equation of line passing through the point $$(1,2,3)$$ and perpendicular to the lines $$\frac{x-2}{3}=\frac{y-1}{2}=\frac{z+1}{-2}$$ and $$\frac{x}{2}=\frac{y}{-3}=\frac{z}{1}$$ is

MHT CET 2023 11th May Evening Shift
129

The angle between the line $$\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}$$ and plane $$x-2 y-\lambda z=3$$ is $$\cos ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)$$, then value of $$\lambda$$ is

MHT CET 2023 11th May Evening Shift
130

If the direction cosines $$l, \mathrm{~m}, \mathrm{n}$$ of two lines are connected by relations $$l-5 \mathrm{~m}+3 \mathrm{n}=0$$ and $$7 l^2+5 \mathrm{~m}^2-3 \mathrm{n}^2=0$$, then value of $$l+\mathrm{m}+\mathrm{n}$$ is

MHT CET 2023 11th May Morning Shift
131

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

MHT CET 2023 11th May Morning Shift
132

A plane is parallel to two lines, whose direction ratios are $$1,0,-1$$ and $$-1,1,0$$ and it contains the point $$(1,1,1)$$. If it cuts co-ordinate axes $$(\mathrm{X}, \mathrm{Y}, \mathrm{Z}$$ - axes resp.) at $$\mathrm{A}, \mathrm{B}, \mathrm{C}$$, then the volume of the tetrahedron $$\mathrm{OABC}$$ is _________ cu. units.

MHT CET 2023 11th May Morning Shift
133

The lines $$\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5} \quad$$ and $$\frac{x+2}{4}=\frac{y-1}{3}=\frac{z+1}{2}$$

MHT CET 2023 11th May Morning Shift
134

The vector equation of the line $$2 x+4=3 y+1=6 z-3$$ is

MHT CET 2023 11th May Morning Shift
135

The plane through the intersection of planes $$x+y+z=1$$ and $$2 x+3 y-z+4=0$$ and parallel to $$\mathrm{Y}$$-axis also passes through the point

MHT CET 2023 10th May Evening Shift
136

The perpendicular distance of the origin from the plane $$x-3 y+4 z-6=0$$ is

MHT CET 2023 10th May Evening Shift
137

Two lines $$\frac{x-3}{1}=\frac{y+1}{3}=\frac{z-6}{-1}$$ and $$\frac{x+5}{7}=\frac{y-2}{-6}=\frac{z-3}{4} \quad$$ intersect at the point R. Then reflection of $$\mathrm{R}$$ in the $$x y$$-plane has co-ordinates

MHT CET 2023 10th May Evening Shift
138

The line $$\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$$ lies in the plane $$x+3 y-\alpha z+\beta=0$$, then the value of $$\alpha^2+\alpha \beta+\beta^2$$ is

MHT CET 2023 10th May Morning Shift
139

Let $$\mathrm{P}$$ be a plane passing through the points $$(2,1,0),(4,1,1)$$ and $$(5,0,1)$$ and $$R$$ be the point $$(2,1,6)$$. Then image of $$R$$ in the plane $$P$$ is

MHT CET 2023 10th May Morning Shift
140

The co-ordinates of the point, where the line through $$A(3,4,1)$$ and $$B(5,1,6)$$ crosses the $$\mathrm{XZ}$$-plane, are

MHT CET 2023 10th May Morning Shift
141

$$\mathrm{ABC}$$ is a triangle in a plane with vertices $$\mathrm{A}(2,3,5), \mathrm{B}(-1,3,2)$$ and $$\mathrm{C}(\lambda, 5, \mu)$$. If median through $$\mathrm{A}$$ is equally inclined to the co-ordinate axes, then value of $$\lambda+\mu$$ is

MHT CET 2023 10th May Morning Shift
142

If a line $$\mathrm{L}$$ is the line of intersection of the planes $$2 x+3 y+z=1$$ and $$x+3 y+2 z=2$$. If line $$\mathrm{L}$$ makes an angle $$\alpha$$ with the positive $$\mathrm{X}$$-axis, then the value of $$\sec \alpha$$ is

MHT CET 2023 9th May Evening Shift
143

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

MHT CET 2023 9th May Evening Shift
144

The co-ordinates of the point, where the line $$\frac{x-1}{2}=\frac{y-2}{-3}=\frac{z+5}{4}$$ meets the plane $$2 x+4 y-\mathrm{z}=3$$, are

MHT CET 2023 9th May Evening Shift
145

The equation of a plane, containing the line of intersection of the planes $$2 x-y-4=0$$ and $$y+2 z-4=0$$ and passing through the point $$(2,1,0)$$, is

MHT CET 2023 9th May Evening Shift
146

The foot of the perpendicular drawn from the origin to the plane is $$(4,-2,5)$$, then the Cartesian equation of the plane is

MHT CET 2023 9th May Morning Shift
147

A vector $$\overrightarrow{\mathrm{n}}$$ is inclined to $$\mathrm{X}$$-axis at $$45^{\circ}$$, $$\mathrm{Y}$$-axis at $$60^{\circ}$$ and at an acute angle to Z-axis If $$\overrightarrow{\mathrm{n}}$$ is normal to a plane passing through the point $$(-\sqrt{2}, 1,1)$$, then equation of the plane is

MHT CET 2023 9th May Morning Shift
148

If the Cartesian equation of a line is $$6 x-2=3 y+1=2 z-2$$, then the vector equation of the line is

MHT CET 2023 9th May Morning Shift
149

The distance between parallel lines

$$\frac{x-1}{2}=\frac{y-2}{-2}=\frac{z-3}{1}$$ and

$$\frac{x}{2}=\frac{y}{-2}=\frac{z}{1}$$ is :

MHT CET 2022 11th August Evening Shift
150

A line makes the same angle '$$\alpha$$' with each of the $$x$$ and $$y$$ axes. If the angle '$$\theta$$', which it makes with the $$z$$-axis, is such that $$\sin ^2 \theta=2 \sin ^2 \alpha$$, then the angle $$\alpha$$ is

MHT CET 2022 11th August Evening Shift
151

A tetrahedron has verticles $$P(1,2,1), Q(2,1,3), R(-1,1,2)$$ and $$O(0,0,0)$$. Then the angle between the faces $$O P Q$$ and $$P Q R$$ is

MHT CET 2022 11th August Evening Shift
152

The Cartesian equation of a line passing through $$(1,2,3)$$ and parallel to $$x-y+2 z=5$$ and $$3 x+y+z=6$$ is

MHT CET 2022 11th August Evening Shift
153

The equation of the plane passing through the points $$(2,3,1),(4,-5,3)$$ and parallel to $$X$$-axis is

MHT CET 2022 11th August Evening Shift
154

The equation of the plane which passes through (2, $$-$$3, 1) and is normal to the line joining the points (3, 4, $$-$$1) and (2, $$-$$1, 5) is given by

MHT CET 2021 24th September Evening Shift
155

If $$G(3,-5, r)$$ is the centroid of $$\triangle A B C$$, where $$A \equiv(7,-8,1), B \equiv(p, q, 5), C \equiv(q+1,5 p, 0)$$ are vertices of the triangle $$A B C$$, then the values of $$p, q, r$$ are respectively

MHT CET 2021 24th September Evening Shift
156

If the lines $$\frac{2 x-4}{\lambda}=\frac{y-1}{2}=\frac{z-3}{1}$$ and $$\frac{x-1}{1}=\frac{3 y-1}{\lambda}=\frac{z-2}{1}$$ are perpendicular to each other, then $$\lambda=$$

MHT CET 2021 24th September Evening Shift
157

The co-ordinates of the points on the line $$\frac{x+2}{1}=\frac{y-1}{2}=\frac{z+1}{-2}$$ at a distance of 12 units from the point A($$-$$2, 1, $$-$$1) are

MHT CET 2021 24th September Evening Shift
158

If the vector equation of the plane $$\bar{r}=(2 \hat{i}+\hat{k})+\lambda \hat{i}+\mu(\hat{i}+2 \hat{j}-3 \hat{k})$$ in scalar product form is given by $$\overline{\mathrm{r}} \cdot(3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})=\alpha$$ then $$\alpha=$$

MHT CET 2021 24th September Evening Shift
159

If the lines $$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$$ and $$\frac{x-2}{1}=\frac{y+m}{2}=\frac{z-2}{1}$$ intersect each other, then value of m is

MHT CET 2021 24th September Morning Shift
160

The length of perpendicular drawn from the point $$2 \hat{i}-\hat{j}+5 \hat{k}$$ to the line $$\overline{\mathrm{r}}=(11 \hat{i}-2 \hat{j}-8 \hat{k})+\lambda(10 \hat{i}-4 \hat{j}-11 \hat{k})$$ is

MHT CET 2021 24th September Morning Shift
161

Equation of the plane passing through the point $$(1,2,3)$$ and parallel to the plane $$2 x+3 y-4 z=0 $$

MHT CET 2021 24th September Morning Shift
162

If $$\mathrm{A}$$ and $$\mathrm{B}$$ are the foot of the perpendicular drawn from the point $$\mathrm{Q}(\mathrm{a}, \mathrm{b}, \mathrm{c})$$ to the planes $$\mathrm{YZ}$$ and $$\mathrm{ZX}$$ respectively, then the equation of the plane through the points $$\mathrm{A}, \mathrm{B}$$, and $$\mathrm{O}$$ is (where $$\mathrm{O}$$ is the origin)

MHT CET 2021 24th September Morning Shift
163

If $$\mathrm{A}=(-2,2,3), \mathrm{B}=(3,2,2), \mathrm{C}=(4,-3,5)$$ and $$\mathrm{D}=(7,-5,-1)$$ Then the projection of $$\overline{\mathrm{AB}}$$ on $$\overline{\mathrm{CD}}$$ is

MHT CET 2021 24th September Morning Shift
164

The Cartesian equation of a plane which passes through the points $$\mathrm{A}(2,2,2)$$ and making equal nonzero intercepts on the co-ordinate axes is

MHT CET 2021 23rd September Evening Shift
165

The co-ordinates of the foot of the perpendicular drawn from the point $$2 \hat{i}-\hat{j}+5 \hat{k}$$ to the line $$\vec{r}=(11 \hat{i}-2 \hat{j}-8 \hat{k})+\lambda(10 \hat{i}-4 \hat{j}-11 \hat{k})$$ are

MHT CET 2021 23rd September Evening Shift
166

If A(3, 2, $$-$$1), B($$-$$2, 2, $$-$$3) and D($$-$$2, 5, $$-$$4) are the vertices of a parallelogram, then the area of the parallelogram is

MHT CET 2021 23th September Morning Shift
167

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

MHT CET 2021 23th September Morning Shift
168

The coordinates of the foot of the perpendicular drawn from the origin to the plane $$2 x+y-2 z=18$$ are

MHT CET 2021 23th September Morning Shift
169

The vector equation of the line passing through $$\mathrm{P}(1,2,3)$$ and $$\mathrm{Q}(2,3,4)$$ is

MHT CET 2021 23th September Morning Shift
170

Equation of planes parallel to the plane $$x-2y+2z+4=0$$ which are at a distance of one unit from the point (1, 2, 3) are

MHT CET 2021 23th September Morning Shift
171

The area of triangle with vertices $$(1,2,0),(1,0, a)$$ and $$(0,3,1)$$ is $$\sqrt{6}$$ sq. units, then the values of '$$a$$' are

MHT CET 2021 22th September Evening Shift
172

If $$\mathrm{G}(4,3,3)$$ is the centroid of the triangle $$\mathrm{ABC}$$ whose vertices are $$\mathrm{A}(\mathrm{a}, 3,1), \mathrm{B}(4,5, \mathrm{~b})$$ and $$C(6, c, 5)$$, then the values of $$a, b, c$$ are

MHT CET 2021 22th September Evening Shift
173

The d.r.s. of the normal to the plane passing through the origin and the line of intersection of the planes $$x+2 y+3 z=4$$ and $$4 x+3 y+2 z=1$$ are

MHT CET 2021 22th September Evening Shift
174

The line $$\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$$ lies in the plane $$x+3 y-\alpha z+\beta=0$$, then value of $$\alpha \beta$$ is

MHT CET 2021 22th September Evening Shift
175

If the points $$P(4,5, x), Q(3, y, 4)$$ and $$R(5,8,0)$$ are collinear, then the value of $$x+y$$ is

MHT CET 2021 22th September Evening Shift
176

A line drawn from a point $$A(-2,-2,3)$$ and parallel to the line $$\frac{x}{-2}=\frac{y}{2}=\frac{z}{-1}$$ meets the $$\mathrm{YOZ}$$ plane in point $$\mathrm{P}$$, then the co-ordinates of the point $$\mathrm{P}$$ are

MHT CET 2021 22th September Evening Shift
177

The Cartesian equation of a line is $$3 x+1=6 y-2=1-z$$, then its vector equation is

MHT CET 2021 22th September Morning Shift
178

The plane $$\frac{x}{2}+\frac{y}{3}+\frac{z}{4}=1$$ cuts the $$X$$-axis at A, Y-axis at B and Z-axis at C, then the area of $$\triangle \mathrm{ABC}=$$

MHT CET 2021 22th September Morning Shift
179

If a plane meets the axes $$\mathrm{X}, \mathrm{Y}, \mathrm{Z}$$ in $$\mathrm{A}, \mathrm{B}, \mathrm{C}$$ respectively such that centroid of $$\triangle \mathrm{ABC}$$ is $$(1,2,3)$$, then the equation of the plane is

MHT CET 2021 22th September Morning Shift
180

The shortest distance between lines $$\bar{r}=(2 \hat{i}-\hat{j})+\lambda(2 \hat{i}+\hat{j}-3 \hat{k})$$ and $$\bar{r}=(\hat{r}-\hat{j}+2 \hat{k})+\mu(2 \hat{i}+\hat{j}-5 \hat{k})$$ is

MHT CET 2021 22th September Morning Shift
181

The direction cosines $$\ell, \mathrm{m}, \mathrm{n}$$ of the line $$\frac{\mathrm{x}+2}{2}=\frac{2 \mathrm{y}-5}{3} ; \mathrm{z}=-1$$ are

MHT CET 2021 21th September Evening Shift
182

Equation of the plane passing through the point (2, 0, 5) and parallel to the vectors $$\widehat i - \widehat j + \widehat k$$ and $$3\widehat i + 2\widehat j - \widehat k$$ is

MHT CET 2021 21th September Evening Shift
183

The co-ordinates of the point $$\mathrm{P} \equiv(1,2,3)$$ and $$\mathrm{O} \equiv(0,0,0)$$, then the direction cosines of $$\overline{\mathrm{OP}}$$ are

MHT CET 2021 21th September Evening Shift
184

The equation of the plane containing the line $$\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$$ and the point $$(0,7,-7)$$ is

MHT CET 2021 21th September Evening Shift
185

The equation of a line passing through $$(3,-1,2)$$ and perpendicular to the lines $$\bar{r}=(\hat{i}+\hat{j}-\hat{k})+\lambda(2 \hat{i}-2 \hat{j}+\hat{k})$$ and $$\bar{r}=(2 \hat{i}+\hat{j}-3 \hat{k})+\mu(\hat{i}-2 \hat{j}+2 \hat{k})$$ is

MHT CET 2021 21th September Evening Shift
186

The area of the parallelogram with vertices A(1, 2, 3), B(1, 3, a), C(3, 8, 6) and D(3, 7, 3) is $$\sqrt{265}$$ sq. units, then a =

MHT CET 2021 21th September Evening Shift
187

If the lines $\frac{1-x}{3}=\frac{7 y-14}{2 \lambda}=\frac{z-3}{2}$ and $\frac{7-7 x}{3 \lambda}=\frac{y-5}{1}=\frac{6-z}{5}$ are at right angles, then $\lambda=$

MHT CET 2021 21th September Morning Shift
188

The Cartesian equation of the plane passing through the point A(7, 8, 6) and parallel to the XY plane is

MHT CET 2021 21th September Morning Shift
189

The equation of the plane passing through $$(-2,2,2)$$ and $$(2,-2,-2)$$ and perpendicular to the plane $$9 x-13 y-3 z=0$$ is

MHT CET 2021 21th September Morning Shift
190

The Cartesian equation of the line passing through the points A(2, 2, 1) and B(1, 3, 0) is

MHT CET 2021 20th September Evening Shift
191

The Cartesian equation of the plane $$\overline{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}})+\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})+\mu(\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$$ is

MHT CET 2021 20th September Evening Shift
192

The equation of the plane that contains the line of intersection of the planes. $$x+2 y+3 z-4=0$$ and $$2 x+y-z+5=0$$ and is perpendicular to the plane $$5 x+3 y-6 z+8=0$$ is

MHT CET 2021 20th September Evening Shift
193

The vector equation of the line whose Cartesian equations are y = 2 and 4x $$-$$ 3z + 5 = 0 is

MHT CET 2021 20th September Evening Shift
194

The Cartesian equation of the plane passing through the point $$(0,7,-7)$$ and containing the line $$\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$$ is

MHT CET 2021 20th September Morning Shift
195

If the lines $$\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$$ and $$\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$$ intersect, then the values of $$k$$ is

MHT CET 2021 20th September Morning Shift
196

The parametric equations of a line passing through the points $$\mathrm{A}(3,4,-7)$$ and $$\mathrm{B}(1,-1,6)$$ are

MHT CET 2021 20th September Morning Shift
197

The angle between a line with direction ratios 2, 2, 1 and a line joining (3, 1, 4) and (7, 2, 12) is

MHT CET 2021 20th September Morning Shift
198

If the line $$\frac{x+1}{2}=\frac{y-m}{3}=\frac{z-4}{6}$$ lies in the plane $$3 x-14 y+6 z+49=0$$, then the value of $$m$$ is

MHT CET 2021 20th September Morning Shift
199

The point $P$ lies on the line $A, B$ where $A=(2,4,5)$ and $B \equiv(1,2,3)$. If $z$ co-ordinate of point $P$ is 3 , the its $y$ co-ordinate is

MHT CET 2020 19th October Evening Shift
200

A line makes angles $\alpha, \beta, \gamma$ with the co-ordinate axes and $\alpha+\beta=90^{\circ}$, then $\gamma=$

MHT CET 2020 19th October Evening Shift
201

The equations of planes parallel to the plane $x+2 y+2 z+8=0$, which are at a distance of 2 units from the point $(1,1,2)$ are

MHT CET 2020 19th October Evening Shift
202

The equation of a plane containing the point $(1,-1,2)$ and perpendicular to the planes $2 x+3 y-2 z=5$ and $x+2 y-3 z=8$ is

MHT CET 2020 19th October Evening Shift
203

The equation of the line passing through $(1,2,3)$ and perpendicular to the lines $x-1=\frac{y+2}{2}=\frac{z+4}{4}$ and $\frac{x-1}{2}=\frac{y-2}{2}=z+3$ is

MHT CET 2020 19th October Evening Shift
204

If the plane $$2 x+3 y+5 z=1$$ intersects the co-ordinate axes at the points $$A, B, C$$, then the centroid of $$\triangle A B C$$ is

MHT CET 2020 16th October Evening Shift
205

The direction co-sines of the line which bisects the angle between positive direction of $$Y$$ and $$Z$$ axes are

MHT CET 2020 16th October Evening Shift
206

The angle between the lines $$\frac{x-1}{4}=\frac{y-3}{1}=\frac{z}{8}$$ and $$\frac{x-2}{2}=\frac{y+1}{2}=\frac{z-4}{1}$$ is

MHT CET 2020 16th October Evening Shift
207

If the line $$r=(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\lambda(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})$$ is parallel to the plane $$r \cdot(3 \hat{i}-2 \hat{\mathbf{j}}+m \hat{\mathbf{k}})=10$$, then the value of $$m$$ is

MHT CET 2020 16th October Evening Shift
208

The points $$A(-a,-b), B(0,0), C(a, b)$$ and $$D\left(a^2, a b\right)$$ are

MHT CET 2020 16th October Evening Shift
209

The cosine of the angle included between the lines $$\mathbf{r}=(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}})$$ and $$\mathbf{r}=(\hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\mu(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-6 \hat{\mathbf{k}})$$ where $$\lambda, \mu \in R$$ is.

MHT CET 2020 16th October Evening Shift
210

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

MHT CET 2020 16th October Morning Shift
211

The angle between the line $$r =(\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(3 \hat{\mathbf{i}}+\hat{\mathbf{j}})$$ and the plane $$\mathbf{r} \cdot(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})=8$$ is

MHT CET 2020 16th October Morning Shift
212

The direction cosines of a line which is perpendicular to lines whose direction ratios are $$3,-2,4$$ and $$1,3,-2$$ are

MHT CET 2020 16th October Morning Shift
213

If the lines given by $$\frac{x-1}{2 \lambda}=\frac{y-1}{-5}=\frac{z-1}{2}$$ and $$\frac{x+2}{\lambda}=\frac{y+3}{\lambda}=\frac{z+5}{1}$$ are parallel, then the value of $$\lambda$$ is

MHT CET 2020 16th October Morning Shift
214

The vector equation of the plane $\mathbf{r}=(2 \hat{\mathbf{i}}+\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}})+\mu(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})$ in scalar product form is $\mathbf{r} \cdot(3 \hat{\mathbf{i}}+2 \hat{\mathbf{k}})=\alpha$, then $\alpha=\ldots$

MHT CET 2019 3rd May Morning Shift
215

The direction ratios of the normal to the plane passing through origin and the line of intersection of the planes $x+2 y+3 z=4$ and $4 x+3 y+2 z=1$ are $\ldots \ldots$

MHT CET 2019 3rd May Morning Shift
216

If line $\frac{2 x-4}{\lambda}=\frac{y-1}{2}=\frac{z-3}{1}$ and $\frac{x-1}{1}=\frac{3 y-1}{\lambda}=\frac{z-2}{1}$ are perpendicular to each other then $\lambda=$ ............

MHT CET 2019 3rd May Morning Shift
217

Which of the following can not be the direction cosines of a line?

MHT CET 2019 3rd May Morning Shift
218

If lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-\lambda}{2}=\frac{z}{1}$ intersect each other, then $\lambda=\ldots \ldots$

MHT CET 2019 2nd May Evening Shift
219

Equations of planes parallel to the plane $x-2 y+2 z+4=0$ which are at a distance of one unit from the point $(1,2,3)$ are ............

MHT CET 2019 2nd May Evening Shift
220

If $P(6,10,10), Q(1,0,-5), R(6,-10, \lambda)$ are vertices of a triangle right angled at $Q$, then value of $\lambda$ is ............

MHT CET 2019 2nd May Evening Shift
221

If the foot of the perpendicular drawn from the point $(0,0,0)$ to the plane is $(4,-2,-5)$ then the equation of the plane is .............

MHT CET 2019 2nd May Evening Shift
222

If $G(3,-5, r)$ is centroid of triangle $A B C$ where $A(7,-8,1), B(p, q, 5)$ and $C(q+1,5 p, 0)$ are vertices of a triangle then values of $p, q, r$ are respectively ......

MHT CET 2019 2nd May Morning Shift
223

The angle between lines $\frac{x-2}{2}=\frac{y-3}{-2}=\frac{z-5}{1}$ and $\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-5}{2}$ is ............

MHT CET 2019 2nd May Morning Shift
224

If the line passes through the points $P(6,-1,2), Q(8,-7,2 \lambda)$ and $R(5,2,4)$ then value of $\lambda$ is ...........

MHT CET 2019 2nd May Morning Shift
225

The co-ordinates of the foot of perpendicular drawn from origin to the plane $2 x-y+5 z-3=0$ are $\ldots \ldots$

MHT CET 2019 2nd May Morning Shift
226

The equation of the plane passing through the point $(-1,2,1)$ and perpendicular to the line joining the points $(-3,1,2)$ and $(2,3,4)$ is

MHT CET 2019 2nd May Morning Shift
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