#### Applications of Vector Algebra Two Marks Questions

12th Standard EM

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Maths

Time : 00:45:00 Hrs
Total Marks : 30
15 x 2 = 30
1. Find the parametric form of vector equation and Cartesian equations of the straight line passing through the point (−2,3, 4) and parallel to the straight line $\frac { x-1 }{ -4 } =\frac { y-3 }{ 5 } =\frac { z-8 }{ 6 }$

2. Find the vector and Cartesian equations of the plane passing through the point with position vector $4\hat { i } +2\hat { j } -3\hat { k }$ and normal to vector $2\hat { i } -\hat { j } +\hat { k }$

3. A variable plane moves in such a way that the sum of the reciprocals of its intercepts on the coordinate axes is a constant. Show that the plane passes through a fixed point

4. Find the vector and Cartesian equations of the plane passing through the point with position vector $2\hat { i } +6\hat { j } +3\hat { k }$ and normal to the vector $\hat { i } +3\hat { j } +5\hat { k }$

5. A plane passes through the point (−1,1, 2) and the normal to the plane of magnitude $3\sqrt { 3 }$ makes equal acute angles with the coordinate axes. Find the equation of the plane.

6. Find the angle between the line $\vec { r } =(2\hat { i } -\hat { j } +\hat { k } )+t(\hat { i } +2\hat { j } -2\hat { k } )$ and the plane $\vec { r } =(6\hat { i } +3\hat { j } +2\hat { k } )=8$

7. Find the angle between the planes $\vec { r } .(\hat { i } +\hat { j } -2\hat { k } )$ = 3 and 2x - 2y + z =2

8. Find the length of the perpendicular from the point (1, -2, 3) to the plane x - y + z =5.

9. Find the acute angle between the planes $\vec { r } .(2\hat { i } +2\hat { j } +2\hat { k } )$ and 4x-2y+2z=15.

10. Find the distance of a point (2,5, −3) from the plane $\vec { r } .(6\hat { i } -3\hat { j } +2\hat { k } )$=5

11. Find the distance between the parallel planes x+2y-2z=0 and 2x+4y-4z+5=0

12. If $\overset { \rightarrow }{ a } =\overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k }$$\overset { \rightarrow }{ b } =-\overset { \wedge }{ i } +2\overset { \wedge }{ j } +\overset { \wedge }{ k }$ and $\overset { \rightarrow }{ c } =3\overset { \wedge }{ i } +\overset { \wedge }{ j }$ find $\frac { \lambda }{ c }$ such that $\overset { \rightarrow }{ a } +\lambda \overset { \rightarrow }{ b }$ is perpendicular to $\overset { \rightarrow }{ c }$

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-a

13. Find the Cartesian equation of a.line passing through the pointsA(2, -1, 3) and B(4, 2, 1)

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-1

14. Find the parametric form of vector equation of the plane passing through the point (1, -1, 2) having 2, 3, 3.as direction ratios of normal to the plane.

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2

15. Let $\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c }$ be unit vectors such $\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } =\overset { \rightarrow }{ a } .\overset { \rightarrow }{ c } =0$ and the angle between $\overset { \rightarrow }{ b }$ and $\overset { \rightarrow }{ c }$ is $\frac { \pi }{ 6 }$Prove that $\overset { \rightarrow }{ a } =\pm 2\left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right)$

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Type I even degree reciprocal equation