12th Maths Monthly Test - 3 ( Applications of Vector Algebra )

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 3 ( Applications of Vector Algebra )-Aug 2020

Time : 2.00 Hours

Part - A

Generic 2 - Answer All Question

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.

If \(\vec { a } =\hat { i } -2\hat { j } +3\hat { k } ,\vec { b } =2\hat { i } +\hat { j } -2\hat { k } ,\vec { c } =3\hat { i } +2\hat { j } +\hat { k } \) find
(i) \((\vec { a } \times \vec { b } )\times \vec { c } \)
(ii) \(\vec { a } \times (\vec { b } \times \vec { c } )\)

Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and \(\frac { 1 }{ 2 } \left| \vec { AC } \times \vec { BC } \right| \).

If the straight lines \(\frac { x-1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ 2 } \) and \(\frac { x+1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ \lambda } \) are coplanar, find λ and equations of the planes containing these two lines.

Find the distance between the planes \(\vec { r } .(2\hat { i } -\hat { j } -\hat { k } )\)=6 and \(\vec { r } .(6\hat { i } -\hat { j } -2\hat { k } )\)= 27

Find the equation of the plane which passes through the point (3, 4, -1) and is parallel to the plane 2x - 3y + 5z + 7 = 0. Also, find the distance between the two planes.

If G is the centroid of a ΔABC, prove that (area of ΔGAB) = (area of ΔGBC) = (area of ΔGCA) = \(\frac{1}{3}\) (area of ΔABC)

A force of magnitude 6 units acting parallel to \(2\hat i - 2\hat j + \hat k\) displaces the point of application from (1, 2, 3) to (5, 3, 7). Find the work done.

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.

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

If \(\vec { a } =2\hat { i } +3\hat { j } -\hat { k } ,\vec { b } =-\hat { i } +2\hat { j } -4\hat { k } ,\vec { c } =\hat { i } +\hat { j } +\hat { k } \) then find the value of \((\vec { a } \times \vec { b } )\times (\vec { a } \times \vec { c } )\).

Show that the lines \(\frac { x-2 }{ 1 } =\frac { y-3 }{ 1 } =\frac { z-4 }{ 3 } \) and \(\frac { x-2 }{ 1 } =\frac { y-4 }{ 1 } =\frac { z-5 }{ 3 } \) coplanar.Also, find the plane containing these lines.
Find the parametric form of vector equation of the straight line passing through (−1, 2,1) and parallel to the straight line \(\vec { r } =(2\hat { i } +3\hat { j } -\hat { k } )+t(\hat { i } -2\hat { j } +\hat { k } )\) and hence find the shortest distance
between the lines.

Find the intercepts cut off by the plane \(\vec { r } .(6\hat { i } +4\hat { j } -3\hat { k } )\)=12 on the coordinate axes.
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.

If the straight line joining the points (2, 1, 4) and (a−1, 4, −1) is parallel to the line joining the points (0, 2, b −1) and (5, 3, −2) , find the values of a and b.

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

Part - B

Generic 3 - Answer All Question

Find the distance of the point (5,-5,-10) from the point of intersection of a straight line passing through the points A(4,1,2) and B(7,5,4) with the plane x-y+z=5

With usual notations, in any triangle ABC, prove the following by vector method.
(i) a²=b²+c²−2bc cos A
(ii) b²=c²+a²−2ca cos B
(iii) c²= a²+b²−2ab cos C

Prove that \(\left[ \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } \right] \)=\(\left[ \overset { \rightarrow }{ a }\ \overset { \rightarrow }{ b }\ \overset { \rightarrow }{ c } \right] \)

Find the angle between the lines \(\vec { r } =(\hat { i } +2\hat { j } +4\hat { k } )+t(2\hat { i } +2\hat { j } +\hat { k } )\) and the straight line passing through the points (5, 1, 4) and (9, 2, 12)

Find the shortest distance between the two given straight lines \(\vec { r } =(2\hat { i } +3\hat { j } +4\hat { k } )+t(-2\hat { i } +\hat { j } -2\hat { k } )\) and \(\frac { x-3 }{ 2 } =\frac { y }{ -1 } =\frac { z+2 }{ 2 } \)

Show that the four points (6, -7, 0), (16, -19, -4), (0, 3, -6), (2, -5, 10) lie on a same plane.
The vector equation in parametric form of a line is \(\vec { r } =(3\hat { i } -2\hat { j } +6\hat { k } )+t(2\hat { i } -\hat { j } +3\hat { k } )\). Find
(i) the direction cosines of the straight line
(ii) vector equation in non-parametric form of the line
(iii) Cartesian equations of the line.

Find the non-parametric form of vector equation, and Cartesian equation of the plane passing through the point (0, 1, -5) and parallel to the straight lines \(\vec { r } =(\hat { i } +2\hat { j } -4\hat { k } )+s(\hat { i } +3\hat { j } +6\hat { k } )\) and \(\hat { r } =(\hat { i } -3\hat { j } +5\hat { k } )+t(\hat { i } +\hat { j } -\hat { k } )\)

A particle acted upon by constant forces \(\hat { 2j } +\hat { 5j } +\hat { 6k } \) and \(-\hat { i } -\hat { 2j } -\hat { k } \) is displaced from the point (4, −3, −2) to the point (6, 1, −3) . Find the total work done by the forces.

Prove that \([\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } ]\) = \([{ \vec { a } ,\vec { b } ,\vec { c } }]^{ 2 }\)

For any four vectors \(\vec { a } ,\vec { b } ,\vec { c } ,\vec { d } \) we have \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )=[\vec { a } ,\vec { b } ,\vec { d } ]\vec { c } -[\vec { a } ,\vec { b } ,\vec { c } ]\vec { d } =[\vec { a } ,\vec { c } ,\vec { d } ]\vec { b } -[\vec { b } ,\vec { c } ,\vec { d } ]\vec { a } \)

Part - C

Generic 5 - Answer All Question

If the Cartesian equation of a plane is 3x - 4y + 3z = -8, find the vector equation of the plane in the standard form.

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

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.

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

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 \(\frac { x-1 }{ 1 } =\frac { 2y+1 }{ 2 } =\frac { z+1 }{ -1 } \)
Prove by vector method that sin(α −β )=sinα cosβ −cosα sinβ

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.

If \(\vec { a } =\vec { i } -\vec { j } ,\vec { b } =\hat { i } -\hat { j } -4\hat { k } ,\vec { c } =3\hat { j } -\hat { k } \) and \(\vec { d } =2\hat { i } +5\hat { j } +\hat { k } \)
(i) \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )=[\vec { a } ,\vec { b } ,\vec { d } ]\vec { c } -[\vec { a } ,\vec { b } ,\vec { c } ]\vec { d } \)
(ii) \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )=[\vec { a } ,\vec { c } ,\vec { d } ]\vec { b } -[\vec { b } ,\vec { c } ,\vec { d } ]\vec { a } \)

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

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12th Maths Monthly Test - 3 ( Applications of Vector Algebra )

St. Britto Hr. Sec. School - Madurai

12th Maths Monthly Test - 3 ( Applications of Vector Algebra )-Aug 2020

Part - A

Generic 2 - Answer All Question

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.

If \(\vec { a } =\hat { i } -2\hat { j } +3\hat { k } ,\vec { b } =2\hat { i } +\hat { j } -2\hat { k } ,\vec { c } =3\hat { i } +2\hat { j } +\hat { k } \) find (i) \((\vec { a } \times \vec { b } )\times \vec { c } \) (ii) \(\vec { a } \times (\vec { b } \times \vec { c } )\)

Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and \(\frac { 1 }{ 2 } \left| \vec { AC } \times \vec { BC } \right| \).

If the straight lines \(\frac { x-1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ 2 } \) and \(\frac { x+1 }{ 2 } =\frac { y+1 }{ \lambda } =\frac { z }{ \lambda } \) are coplanar, find λ and equations of the planes containing these two lines.

Find the distance between the planes \(\vec { r } .(2\hat { i } -\hat { j } -\hat { k } )\)=6 and \(\vec { r } .(6\hat { i } -\hat { j } -2\hat { k } )\)= 27

Find the equation of the plane which passes through the point (3, 4, -1) and is parallel to the plane 2x - 3y + 5z + 7 = 0. Also, find the distance between the two planes.

If G is the centroid of a ΔABC, prove that (area of ΔGAB) = (area of ΔGBC) = (area of ΔGCA) = \(\frac{1}{3}\) (area of ΔABC)

A force of magnitude 6 units acting parallel to \(2\hat i - 2\hat j + \hat k\) displaces the point of application from (1, 2, 3) to (5, 3, 7). Find the work done.

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.

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

If \(\vec { a } =2\hat { i } +3\hat { j } -\hat { k } ,\vec { b } =-\hat { i } +2\hat { j } -4\hat { k } ,\vec { c } =\hat { i } +\hat { j } +\hat { k } \) then find the value of \((\vec { a } \times \vec { b } )\times (\vec { a } \times \vec { c } )\).

Show that the lines \(\frac { x-2 }{ 1 } =\frac { y-3 }{ 1 } =\frac { z-4 }{ 3 } \) and \(\frac { x-2 }{ 1 } =\frac { y-4 }{ 1 } =\frac { z-5 }{ 3 } \) coplanar.Also, find the plane containing these lines.

Find the parametric form of vector equation of the straight line passing through (−1, 2,1) and parallel to the straight line \(\vec { r } =(2\hat { i } +3\hat { j } -\hat { k } )+t(\hat { i } -2\hat { j } +\hat { k } )\) and hence find the shortest distance between the lines.

Find the intercepts cut off by the plane \(\vec { r } .(6\hat { i } +4\hat { j } -3\hat { k } )\)=12 on the coordinate axes.

If the straight line joining the points (2, 1, 4) and (a−1, 4, −1) is parallel to the line joining the points (0, 2, b −1) and (5, 3, −2) , find the values of a and b.

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

Part - B

Generic 3 - Answer All Question

Find the distance of the point (5,-5,-10) from the point of intersection of a straight line passing through the points A(4,1,2) and B(7,5,4) with the plane x-y+z=5

With usual notations, in any triangle ABC, prove the following by vector method. (i) a2=b2+c2−2bc cos A (ii) b2=c2+a2−2ca cos B (iii) c2= a2+b2−2ab cos C

Find the angle between the lines \(\vec { r } =(\hat { i } +2\hat { j } +4\hat { k } )+t(2\hat { i } +2\hat { j } +\hat { k } )\) and the straight line passing through the points (5, 1, 4) and (9, 2, 12)

Find the shortest distance between the two given straight lines \(\vec { r } =(2\hat { i } +3\hat { j } +4\hat { k } )+t(-2\hat { i } +\hat { j } -2\hat { k } )\) and \(\frac { x-3 }{ 2 } =\frac { y }{ -1 } =\frac { z+2 }{ 2 } \)

Show that the four points (6, -7, 0), (16, -19, -4), (0, 3, -6), (2, -5, 10) lie on a same plane.

The vector equation in parametric form of a line is \(\vec { r } =(3\hat { i } -2\hat { j } +6\hat { k } )+t(2\hat { i } -\hat { j } +3\hat { k } )\). Find (i) the direction cosines of the straight line (ii) vector equation in non-parametric form of the line (iii) Cartesian equations of the line.

A particle acted upon by constant forces \(\hat { 2j } +\hat { 5j } +\hat { 6k } \) and \(-\hat { i } -\hat { 2j } -\hat { k } \) is displaced from the point (4, −3, −2) to the point (6, 1, −3) . Find the total work done by the forces.

Prove that \([\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } ]\) = \([{ \vec { a } ,\vec { b } ,\vec { c } }]^{ 2 }\)

Part - C

Generic 5 - Answer All Question

If the Cartesian equation of a plane is 3x - 4y + 3z = -8, find the vector equation of the plane in the standard form.

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

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

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 \(\frac { x-1 }{ 1 } =\frac { 2y+1 }{ 2 } =\frac { z+1 }{ -1 } \)

Prove by vector method that sin(α −β )=sinα cosβ −cosα sinβ

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

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If \(\vec { a } =\hat { i } -2\hat { j } +3\hat { k } ,\vec { b } =2\hat { i } +\hat { j } -2\hat { k } ,\vec { c } =3\hat { i } +2\hat { j } +\hat { k } \) find
(i) \((\vec { a } \times \vec { b } )\times \vec { c } \)
(ii) \(\vec { a } \times (\vec { b } \times \vec { c } )\)

Find the parametric form of vector equation of the straight line passing through (−1, 2,1) and parallel to the straight line \(\vec { r } =(2\hat { i } +3\hat { j } -\hat { k } )+t(\hat { i } -2\hat { j } +\hat { k } )\) and hence find the shortest distance
between the lines.

With usual notations, in any triangle ABC, prove the following by vector method.
(i) a²=b²+c²−2bc cos A
(ii) b²=c²+a²−2ca cos B
(iii) c²= a²+b²−2ab cos C

The vector equation in parametric form of a line is \(\vec { r } =(3\hat { i } -2\hat { j } +6\hat { k } )+t(2\hat { i } -\hat { j } +3\hat { k } )\). Find
(i) the direction cosines of the straight line
(ii) vector equation in non-parametric form of the line
(iii) Cartesian equations of the line.