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12th Maths Monthly Test - 2 ( Applications of Vector Algebra )

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 2 ( Applications of Vector Algebra )-Aug 2020

Time : 2.30 Hours

Part - A

Generic 2 - Answer All Question

Verify whether the line \(\frac { x-3 }{ -4 } =\frac { y-4 }{ -7 } =\frac { z+3 }{ 12 } \) lies in the plane 5x-y+z=8.

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three vectors, prove that \([\vec { a } +\vec { c } ,\vec { a } +\vec { b } ,\vec { a } +\vec { b } +\vec { c } ]\) = \([\vec { a } ,\vec { b } ,\vec { c } ]\)

The volume of the parallelepiped whose coterminus edges are \(7\hat { i } +\lambda \hat { j } -3\hat { k } ,\hat { i } +2\hat { j } -\hat { k } \), \(-3\hat { i } +7\hat { j } +5\hat { k } \) is 90 cubic units. Find the value of λ.

Show that the points (2, 3, 4),(−1, 4, 5) and (8,1, 2) are collinear.

Find the angle between the following lines.
\(\vec { r } =(4\hat { i } -\hat { j } )+t(\hat { i } +2\hat { j } -2\hat { k } )\), \(\hat{r}=(\hat { i } -2\hat { j } +4 \hat { k } )+s(\hat {- i } -2\hat { j } +2\hat { k } )\)

Find the coordinates of the foot of the perpendicular and length of the perpendicular from the point ( 4,3,2) to the plane x + 2y + 3z = 2

Find a parametric form of vector equation of a plane which is at a distance of 7 units from the origin having 3, −4,5 as direction ratios of a normal to it.
Prove by vector method that the parallelograms on the same base and between the same parallels are equal in area.

Find the point of intersection of the line x - 1 = \(\frac { y }{ 2 } \) = z + 1 with the plane 2x - y + 2z = 2. Also, find the angle between the line and the plane.

Show that the lines \(\frac { x-1 }{ 4 } =\frac { 2-y }{ 6 } =\frac { z-4 }{ 12 } \) and \(\frac { x-3 }{ -2 } =\frac { y-3 }{ 3 } =\frac { 5-z }{ 6 } \) are parallel.

Forces of magnit \(5\sqrt { 2 } \) and \(5\sqrt { 2 } \) units acting in the directions \(\hat { 3i } +\hat { 4j } +\hat { 5k } \) and \(\hat { 10i } +\hat { 6j } -\hat { 8k } \) respectively, act on a particle which is displaced from the point with position vector \(\hat { 4i } +\hat { 3j } -\hat { 2k } \) to the point with position vector \(\hat { 6i } +\hat { j } -\hat { 3k } \). Find the work done by the forces.

Find the direction cosines of the straight line passing through the points (5,6,7) and (7,9,13) . Also, find the parametric form of vector equation and Cartesian equations of the straight line passing through two given points.

Find the parametric form of vector equation, and Cartesian equations of the plane passing through the points (2, 2,1), (9,3,6) and perpendicular to the plane 2x + 6y + 6z = 9

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

Find the direction cosines and length of the perpendicular from the origin to the plane \(\vec { r } .(3\hat { i } -4\hat { j } +12\hat { k } )=5\)

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 \(\hat { i } +3\hat { j } +5\hat { k } \)

Find the parametric form of vector equation of a line passing through a point (2, -1, 3) and parallel to line \({ \overset { \rightarrow }{ r } }=\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) +t\left( 2\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \right) \)

If \(\hat { 2i } -\hat { j } +\hat { 3k } ,\hat { 3i } +\hat { 2j } +\hat { k } ,\hat { i } +\hat { mj } +\hat { 4k } \) are coplanar, find the value of m.

Prove by vector method that if a line is drawn from the centre of a circle to the midpoint of a chord, then the line is perpendicular to the chord.

If the straight lines \(\frac { x-1 }{ 1 } =\frac { y-2 }{ 1 } =\frac { z-3 }{ { m }^{ 2 } } \) and \(\frac { x-3 }{ 1 } =\frac { y-2 }{ { m }^{ 2 } } =\frac { z-1 }{ 2 } \) are coplanar, find the distinct real values of m.

If the straight lines \(\frac { x-5 }{ 5m+2 } =\frac { 2-y }{ 5 } =\frac { 1-z }{ -1 } \) and \(x=\frac { 2y+1 }{ 4m } =\frac { 1-z }{ -3 } \) are perpendicular to each other, find the value of m.

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 \(2\hat { i } -\hat { j } +\hat { k } \)

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 \(\vec { r } =(6\hat { i } +3\hat { j } +2\hat { k } )=8\)
Find the angle between the following lines.
\(\frac { x+4 }{ 3 } =\frac { y-7 }{ 4 } =\frac { z+5 }{ 5 } \), \(\vec { r } =4\hat { k } +t(2\hat { i } +\hat { j } +\hat { k } )\)

Find the direction cosines of the normal to the plane 12x + 3y − 4z = 65 . Also, find the non-parametric form of vector equation of a plane and the length of the perpendicular to the plane from the origin.

Find the volume of the parallelepiped whose coterminous edges are represented by the vectors \(-6\hat { i } +14\hat { j } +10\hat { k } ,14\hat { i } -10\hat { j } -6\hat { k } \) and \(2\hat { i } +4\hat { j } -2\hat { k } \)

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

Part - B

Generic 5 - Answer All Question

Find the equation of a straight line passing through the point of intersection of the straight lines \(\vec { r } =(\hat { i } +\hat { 3j } -\hat { k } )+t(2\hat { i } +3\hat { j } +2\hat { k } )\quad \) and \(\frac { x-2 }{ 1 } =\frac { y-4 }{ 2 } =\frac { z+3 }{ 4 } \) and perpendicular to both straight lines.

If \(\vec { a } =-2\hat { i } +3\hat { j } -2\hat { k } ,\vec { b } =3\hat { i } -\hat { j } +3\hat { k } ,\vec { c } =2\hat { i } -5\hat { j } +\hat { k } \) find \((\vec { a } \times \vec { b } )\times \vec { c } \) and \((\vec { a } \times \vec { b } )\times \vec { c } \). State whether they are equal.

Find the point of intersection of the lines \(\frac { x-1 }{ 2 } =\frac { y-2 }{ 3 } =\frac { z-3 }{ 4 } \) and \(\frac { x-4 }{ 5 } =\frac { y-1 }{ 2 } =z\)
Find the image of the point whose position vector is \(\hat { i } +2\hat { j } +3\hat { k } \) in the plane \(\vec { r } .(\hat { i } +2\hat { j } +4\hat { k } )\) = 38

By vector method, prove that cos(α + β) = cos α cos β - sin α sin β

If \(\left| \overset { \rightarrow }{ A } \right| =\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \wedge }{ i } =\overset { \wedge }{ j } -\overset { \wedge }{ k } \) are two given vector, then find a vector B satisfying the equations \(\overset { \rightarrow }{ A } \times \overset { \rightarrow }{ B } \)= \(\overset { \rightarrow }{ C } \) and \(\overset { \rightarrow }{ A } \).\(\overset { \rightarrow }{ B } \)=3

Find the vector parametric, vector non-parametric and Cartesian form of the equation of the plane passing through the points (-1, 2, 0), (2, 2, -1)and parallel to the straight line \(\frac { x-1 }{ 1 } =\frac { 2y+1 }{ 2 } =\frac { z+1 }{ -1 } \)

With usual notations, in any triangle ABC, prove by vector method that \(\frac { a }{ sinA } =\frac { b }{ sinB }=\frac { c }{ sinc }\)

Prove by vector method that the perpendiculars (attitudes) from the vertices to the opposite sides of a triangle are concurrent.

Find the vector equation in parametric form and Cartesian equations of a straight passing through the points (-5, 7, 14) and (13, -5, 2). Find the point where the straight line crosses the xy - plane.

Show that the points A, B, C with position vector \(2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\overset { \wedge }{ k } ,\overset { \wedge }{ i } -3\overset { \wedge }{ j } -5\overset { \wedge }{ k } \) and \(3\overset { \wedge }{ i } -4\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) respectively are the vector of a right angled, triangle. Also, find the remaining angles of the triangle.

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