Line and Plane - Maharashtra Board STD 12 Science Maths Questions

Q 1Answer the following questions in short.1 Mark

The acute angle between the two planes $x+y+2 z=3$ and $3 x-2 y+2 z=7$ is ______.

A
$\sin ^{-1}\left(\frac{5}{\sqrt{102}}\right)$
B
$\cos ^{-1}\left(\frac{5}{\sqrt{102}}\right)$
C
$\sin ^{-1}\left(\frac{15}{\sqrt{102}}\right)$
D
$\cos ^{-1}\left(\frac{15}{\sqrt{102}}\right)$

Q 2Solve the Following Question.(2 Marks)2 Marks

Find the Cartesian equations of the line passing through $A(2,2,1)$ and $B(1,3,0)$.

Q 3Solve the Following Question.(2 Marks)2 Marks

The direction ratios of the line which is perpendicular to the two lines $\frac{x-7}{2}=\frac{y+17}{-3}=\frac{z-6}{1}$ and $\frac{x+5}{1}=\frac{y+3}{2}=\frac{z-6}{-2}$ are _____.

A
$4,5,7$
B
$4,-5,7$
C
$4,-5,-7$
D
$-4,5,8$

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Q 4Solve the Following Question.(2 Marks)2 Marks

Show that the points $(1,1,1)$ and $(-3,0,1)$ are equidistant from the plane $\bar{L}(3 \bar{i}+4 \bar{\jmath}-12 \bar{k})+13=0$

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Q 5Solve the Following Question.(2 Marks)2 Marks

Find the direction cosines of the line $\frac{x+2}{2}=\frac{2 y-5}{3} ; z=-1$

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Q 6Solve the Following Question.(2 Marks)2 Marks

Find the vector equation of the plane which is at a distance of 5 units from the origin and which is normal to the vector $2 \hat{i}+\hat{j}+2 \hat{k}$

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Q 7Solve the Following Question.(3 Marks)3 Marks

Find the co-ordinates of the foot of the perpendicular drawn from the origin to the plane $2 x+y-2 z=18$.

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Q 8Solve the Following Question.(3 Marks)3 Marks

Find the shortest distance between the lines
$\bar{r}=(4 \hat{i}-\hat{j})+\lambda(\hat{i}+2 \hat{j}-3 \hat{k}) \text { and } \bar{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(\hat{i}+4 \hat{j}-5 \hat{k})$

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Q 9Solve the Following Question.(3 Marks)3 Marks

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

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Q 10Solve the Following Question.(3 Marks)3 Marks

Find the shortest distance between the lines $\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$ and $\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z+7}{1}$.

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Q 11Solve the Following Question.(3 Marks)3 Marks

Find the vector equation of the plane passing through the point $A(-1,2,-5)$ and parallel to the vectors $4 \hat{i}-\hat{j}+3 \hat{k}$ and $\hat{i}+\hat{\jmath}-\hat{k}$.

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Q 12Solve the Following Question.(4 Marks)4 Marks

Show that the lines $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$ and $\frac{x}{1}=\frac{y-7}{-3}=\frac{z+7}{2}$ are coplanar. Find the equation of the plane containing them.

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Q 13Solve the Following Question.(4 Marks)4 Marks

Parametric form of the equation of the plane is $\bar{r}=(2 \bar{i}+4 \bar{k})-\lambda \bar{i}+\mu(\bar{i}+2 \bar{j}-3 \bar{k}) \lambda$ and $\mu$ are parameters. Find normal to the plane and hence equation of the plane in normal form. Write its Cartestan form.

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Q 14Solve the Following Question.(4 Marks)4 Marks

Find the angle between the line $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z+2}{4}$ and the plane $2 x+y-3 z+4=0$

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Q 15Solve the Following Question.(4 Marks)4 Marks

Find the vector equation of the plane passing through the points $A (1,0,1), B (1,-1,1)$ and $C (4,-3,2)$

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Q 16Solve the Following Question.(4 Marks)4 Marks

Find the vector and Cartesian equations of the plane passing through the points $A (1,1,-2), B (1,2,1)$ and $C (2,-1,1)$

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