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

St. Britto Hr. Sec. School - Madurai

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

Time : 1.30 Hours

Part - A

Multiple Choice Question - Answer All Question

If \(\vec{a}\) and \(\vec{b}\) are parallel vectors, then \([\vec { a } ,\vec { c } ,\vec { b } ]\) is equal to

2
-1
1
0

\(\vec { a } .\vec { b } =\vec { b } .\vec { c } =\vec { c } .\vec { a } =0\) , then the value of \([\vec { a } ,\vec { b } ,\vec { c } ]\) is

\(\left| \vec { a } \right| \left| \vec { b } \right| \left| \vec { c } \right| \)
\(\frac{1}{3}\)\(\left| \vec { a } \right| \left| \vec { b } \right| \left| \vec { c } \right| \)
1
-1

The volume of the parallelepiped with its edges represented by the vectors \(\hat { i } +\hat { j } ,\hat { i } +2\hat { j } ,\hat { i } +\hat { j } +\pi \hat { k } \) is

\(\frac { \pi }{ 2 } \)
\(\frac { \pi }{ 3} \)
\( { \pi }\)
\(\frac { \pi }{ 4 } \)

If \(\vec { a } =\hat { i } +\hat { j } +\hat { k } \), \(\vec { b } =\hat { i } +\hat { j } \), \(\vec { c } =\hat { i } \) and \((\vec { a } \times \vec { b } )\times\vec { c } \) = \(\lambda \vec { a } +\mu \vec { b } \) then the value of \(\lambda +\mu \) is

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors such that \(\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } } \), then the angle between

\(\frac { \pi }{ 2 } \)
\(\frac { 3\pi }{ 4 } \)
\(\frac { \pi }{ 4 } \)
\( { \pi }\)

Consider the vectors \(\vec { a } ,\vec { b } ,\vec { c } ,\vec { d } \) such that \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )\) = \(\vec { 0 } \) Let \({ P }_{ 1 }\) and \({ P }_{ 2 }\) be the planes determined by the pairs of vectors \(\vec { a } ,\vec { b } \) and \(\vec { c } ,\vec { d } \) respectively. Then the angle between \({ P }_{ 1 }\) and \({ P }_{ 2 }\) is

0°
45°
60°
90°

If \(\vec { a } =2\hat { i } +3\hat { j } -\hat { k } ,\vec { b } =\hat { i } +2\hat { j } +5\hat { k } ,\vec { c } =3\hat { i } +5\hat { j } -\hat { k } \) then a vector perpendicular to \(\vec { a } \) and lies in the plane containing \(\vec { b } \) and \(\vec { c } \) is

\(-17\hat { i } +21\hat { j } -\hat { 97k } \)
\(17\hat { i } +21\hat { j } -\hat { 123k } \)
\(-17\hat { i } -21\hat { j } +\hat { 197k } \)
\(-17\hat { i } -21\hat { j } -\hat { 97k } \)

If the line \(\frac { x-2 }{ 3 } =\frac { y-1 }{ -5 }= \frac { z+2 }{ 2 } \) lies in the plane x + 3y + - αz + β = 0, then (α, β) is

(-5, 5)
(-6, 7)
(5, 5)
(6, -7)

The coordinates of the point where the line \(\vec { r } =(6\hat { i } -\hat { j } -3\hat { k } )+t(\hat {- i } +4\hat { j } )\) meets the plane \(\vec { r } =(\hat { i } +\hat { j } -\hat { k } )\) = 3 are

(2,1,0)
(7,1,7)
(1,2,6)
(5,-1,1)

The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0

\(\frac { \sqrt { 7 } }{ 2\sqrt { 2 } } \)
\(\frac{7}{2}\)
\(\frac { \sqrt { 7 } }{ 2 } \)
\(\frac { 7 }{ 2\sqrt { 2 } } \)

The vector equation \(\vec { r } =(\hat { i } -2\hat { j } -\hat { k } )+t(6\hat { i } -\hat { k) } \) represents a straight line passing through the points

(0,6,1)− and (1,2,1)
(0,6,-1) and (1,4,2)
(1,-2,-1) and (1,4,-2)
(1,-2,-1) and (0,-6,1)

If the planes \(\vec { r } =(2\hat { i } -\lambda \hat { j } +\hat { k } )=3\) and \(\vec { r } =(4+\hat { j } -\mu \hat { k } )=5\) are parallel, then the value of λ and μ are

\(\frac { 1 }{ 2 } ,-2\)
\(-\frac { 1 }{ 2 } ,2\)
\(-\frac { 1 }{ 2 } ,-2\)
\(\frac { 1 }{ 2 } ,2\)

The vector, d\(\overset { \wedge }{ i } +\overset { \wedge }{ j } +2\overset { \wedge }{ k } ,\overset { \wedge }{ i } +\lambda \overset { \wedge }{ j } -\overset { \wedge }{ k } \) t and \(2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\lambda \overset { \wedge }{ k } \) are co-planar if

λ = -2
λ = 1+\(\sqrt { 3 } \)
λ = 1 - \(\sqrt { 3 } \)
λ = -2,1土 \(\sqrt { 3 } \)

The number of vectors of unit length perpendicular to the vectors \(\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) \) and \(\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right) \)is

If \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are two unit vectors, then the vectors \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) \times \left( \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } \right) \) is parallel to the vector

\(\overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \)
\(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \)
2\(\overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \)
2\(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \)

If \(\overset { \rightarrow }{ a } \),\(\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \) are any three vectors, then \(\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) \) if and only if

\(\overset { \rightarrow }{ b } \), \(\overset { \rightarrow }{ c } \) are collinear
\(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ c } \) are collinear
\(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are collinear
none

If \(\left| \overset { \rightarrow }{ a } \right| =\left| \overset { \rightarrow }{ b } \right| =1\)such that \(\overset { \rightarrow }{ a } +2\overset { \rightarrow }{ b } \) and \(5\overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \) are perpendicular to each other, then the angle between \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) is

45^o
60^o
cos^-1 \(\left( \frac { 1 }{ 3 } \right) \)
cos^-1 \(\left( \frac { 2 }{ 7 } \right) \)

The p.v, OP of a point P make angles 60^o and 45^owith X and Y axis respectively. The angle of inclination of \(\overset { \rightarrow }{ OP } \)with z-axis is

75^o
60^o
45^o
3

If θ is the angle between the vectors \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \), then sinθ is

\(\frac { \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } }{ \left| \overset { \rightarrow }{ a } \right| \left| \overset { \rightarrow }{ b } \right| } \)
\(\frac { \left| \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } \right| }{ \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } } \)
\(\sqrt { 1-{ \left( \frac { \overset { \rightarrow }{ a. } \overset { \rightarrow }{ b } }{ \left| \overset { \rightarrow }{ a } \right| \left| \overset { \rightarrow }{ b } \right| } \right) }^{ 2 } } \)
0

If \(\overset { \rightarrow }{ a } =\left| \overset { \rightarrow }{ a } \right| \overset { \rightarrow }{ e } \) then \(\overset { \rightarrow }{ e } .\overset { \rightarrow }{ e } \)

0
e
1
\(\overset { \rightarrow }{ 0 } \)

Part - B

Generic 2 - Answer All Question

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.

Prove by vector method that an angle in a semi-circle is a right angle.

Using vector method, prove that if the diagonals of a parallelogram are equal, then it is a rectangle

Prove by vector method that the parallelograms on the same base and between the same parallels are equal in area.

Using vector method, prove that cos(α − β )=cos α cos β +sin α sin β

A particle acted on by constant forces \(8\hat i+2\hat j-6\hat k\) and \(6\hat i+2\hat j -2\hat k\) is displaced from the point (1, 2, 3) to the point (5, 4, 1). Find the total work done by the forces.

Find the magnitude and direction cosines of the torque of a force represented by \(3\hat { i } +4\hat { j } -5\hat { k } \) about the point with position vector \(2\hat { i } -3\hat { j } +4\hat { k } \) acting through a point whose position vector is \(4\hat { i } +2\hat { j } -3\hat { k } \).

If \(\vec { a } =\hat { i } -2\hat { j } +3\hat { k }, \vec { 6 } =2\hat { i } +\hat { j } -2\hat { k }, \vec { c } =3\hat { i } +2\hat { j } +\hat { k } \) find \(\vec { a } .(\vec { b } \times \vec { c } )\).

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.

Find the altitude of a parallelepiped determined by the vectors \(\vec { a } =-2\hat { i } +5\hat { j } +3\hat { k } \), \(\hat { b } =\hat { i } +3\hat { j } -2\hat { k } \) and \(\vec { c } =-3\vec { i } +\vec { j } +4\vec { k } \) if the base is taken as the parallelogram determined by \(\vec { b } \) and \(\vec { c } \)

Part - C

Generic 3 - Answer All Question

Dot product of a vector with vector \(\overset { \wedge }{ 3i } -5\overset { \wedge }{ k } \), \(2\overset { \wedge }{ i } +7\overset { \wedge }{ j } \) and \(\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) are respectively -1, 6 and 5. Find the vector.

Find the equation of the plane through the intersection of the planes 2x-3y+ z-4 -0 and x - y + Z + 1 - 0 and perpendicular to the plane x + 2y - 3z + 6 = 0

If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } -\overset { \wedge }{ j } ,\overset { \rightarrow }{ b } =\overset { \wedge }{ j } -\overset { \wedge }{ k } ,\overset { \rightarrow }{ c } =\overset { \wedge }{ k } -\overset { \wedge }{ i } \) then find \(\left[ \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ b } -\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } -\overset { \rightarrow }{ a } \right] \)

Prove by vector method, that in a right angled triangle the square of the hypotenuse is equal to the sum of the square of the other two sides.

Show that the four points whose position vectors are \(6\overset { \wedge }{ i } -7\overset { \wedge }{ j } ,16\overset { \wedge }{ i } -29\overset { \wedge }{ j } -4\overset { \wedge }{ k } ,3\overset { \wedge }{ i } -6\overset { \wedge }{ j } \) are co-planar

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

If D is the midpoint of the side BC of a triangle ABC, then show by vector method that \({ \left| \vec { AB } \right| }^{ 2 }+{ \left| \vec { AC } \right| }^{ 2 }=2({ \left| \vec { AD} \right| }^{ 2 }+{ \left| \vec { BD } \right| }^{ 2 })\)

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.

Find the magnitude and the direction cosines of the torque about the point (2, 0, -1) of a force \((\hat { 2i } +\hat { j } -\hat { k } )\), whose line of action passes through the origin

If the vectors \(\vec { a } ,\vec { b } ,\vec { c } \) are coplanar, then prove that the vectors \(\vec { a } +\vec { b } ,\vec { b } +\vec { c } ,\vec { c } +\vec { a } \) are also coplanar

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

St. Britto Hr. Sec. School - Madurai

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

Part - A

Multiple Choice Question - Answer All Question

If \(\vec{a}\) and \(\vec{b}\) are parallel vectors, then \([\vec { a } ,\vec { c } ,\vec { b } ]\) is equal to

\(\vec { a } .\vec { b } =\vec { b } .\vec { c } =\vec { c } .\vec { a } =0\) , then the value of \([\vec { a } ,\vec { b } ,\vec { c } ]\) is

The volume of the parallelepiped with its edges represented by the vectors \(\hat { i } +\hat { j } ,\hat { i } +2\hat { j } ,\hat { i } +\hat { j } +\pi \hat { k } \) is

If \(\vec { a } =\hat { i } +\hat { j } +\hat { k } \), \(\vec { b } =\hat { i } +\hat { j } \), \(\vec { c } =\hat { i } \) and \((\vec { a } \times \vec { b } )\times\vec { c } \) = \(\lambda \vec { a } +\mu \vec { b } \) then the value of \(\lambda +\mu \) is

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors such that \(\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } } \), then the angle between

If \(\vec { a } =2\hat { i } +3\hat { j } -\hat { k } ,\vec { b } =\hat { i } +2\hat { j } +5\hat { k } ,\vec { c } =3\hat { i } +5\hat { j } -\hat { k } \) then a vector perpendicular to \(\vec { a } \) and lies in the plane containing \(\vec { b } \) and \(\vec { c } \) is

If the line \(\frac { x-2 }{ 3 } =\frac { y-1 }{ -5 }= \frac { z+2 }{ 2 } \) lies in the plane x + 3y + - αz + β = 0, then (α, β) is

The coordinates of the point where the line \(\vec { r } =(6\hat { i } -\hat { j } -3\hat { k } )+t(\hat {- i } +4\hat { j } )\) meets the plane \(\vec { r } =(\hat { i } +\hat { j } -\hat { k } )\) = 3 are

The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0

The vector equation \(\vec { r } =(\hat { i } -2\hat { j } -\hat { k } )+t(6\hat { i } -\hat { k) } \) represents a straight line passing through the points

If the planes \(\vec { r } =(2\hat { i } -\lambda \hat { j } +\hat { k } )=3\) and \(\vec { r } =(4+\hat { j } -\mu \hat { k } )=5\) are parallel, then the value of λ and μ are

The vector, d\(\overset { \wedge }{ i } +\overset { \wedge }{ j } +2\overset { \wedge }{ k } ,\overset { \wedge }{ i } +\lambda \overset { \wedge }{ j } -\overset { \wedge }{ k } \) t and \(2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\lambda \overset { \wedge }{ k } \) are co-planar if

The number of vectors of unit length perpendicular to the vectors \(\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) \) and \(\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right) \)is

The p.v, OP of a point P make angles 60o and 45o with X and Y axis respectively. The angle of inclination of \(\overset { \rightarrow }{ OP } \)with z-axis is

If θ is the angle between the vectors \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \), then sinθ is

If \(\overset { \rightarrow }{ a } =\left| \overset { \rightarrow }{ a } \right| \overset { \rightarrow }{ e } \) then \(\overset { \rightarrow }{ e } .\overset { \rightarrow }{ e } \)

Part - B

Generic 2 - Answer All Question

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.

Prove by vector method that an angle in a semi-circle is a right angle.

Using vector method, prove that if the diagonals of a parallelogram are equal, then it is a rectangle

Prove by vector method that the parallelograms on the same base and between the same parallels are equal in area.

Using vector method, prove that cos(α − β )=cos α cos β +sin α sin β

A particle acted on by constant forces \(8\hat i+2\hat j-6\hat k\) and \(6\hat i+2\hat j -2\hat k\) is displaced from the point (1, 2, 3) to the point (5, 4, 1). Find the total work done by the forces.

Find the magnitude and direction cosines of the torque of a force represented by \(3\hat { i } +4\hat { j } -5\hat { k } \) about the point with position vector \(2\hat { i } -3\hat { j } +4\hat { k } \) acting through a point whose position vector is \(4\hat { i } +2\hat { j } -3\hat { k } \).

If \(\vec { a } =\hat { i } -2\hat { j } +3\hat { k }, \vec { 6 } =2\hat { i } +\hat { j } -2\hat { k }, \vec { c } =3\hat { i } +2\hat { j } +\hat { k } \) find \(\vec { a } .(\vec { b } \times \vec { c } )\).

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.

Part - C

Generic 3 - Answer All Question

Dot product of a vector with vector \(\overset { \wedge }{ 3i } -5\overset { \wedge }{ k } \), \(2\overset { \wedge }{ i } +7\overset { \wedge }{ j } \) and \(\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) are respectively -1, 6 and 5. Find the vector.

Find the equation of the plane through the intersection of the planes 2x-3y+ z-4 -0 and x - y + Z + 1 - 0 and perpendicular to the plane x + 2y - 3z + 6 = 0

Prove by vector method, that in a right angled triangle the square of the hypotenuse is equal to the sum of the square of the other two sides.

Show that the four points whose position vectors are \(6\overset { \wedge }{ i } -7\overset { \wedge }{ j } ,16\overset { \wedge }{ i } -29\overset { \wedge }{ j } -4\overset { \wedge }{ k } ,3\overset { \wedge }{ i } -6\overset { \wedge }{ j } \) are co-planar

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

If D is the midpoint of the side BC of a triangle ABC, then show by vector method that \({ \left| \vec { AB } \right| }^{ 2 }+{ \left| \vec { AC } \right| }^{ 2 }=2({ \left| \vec { AD} \right| }^{ 2 }+{ \left| \vec { BD } \right| }^{ 2 })\)

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.

Find the magnitude and the direction cosines of the torque about the point (2, 0, -1) of a force \((\hat { 2i } +\hat { j } -\hat { k } )\), whose line of action passes through the origin

If the vectors \(\vec { a } ,\vec { b } ,\vec { c } \) are coplanar, then prove that the vectors \(\vec { a } +\vec { b } ,\vec { b } +\vec { c } ,\vec { c } +\vec { a } \) are also coplanar

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The p.v, OP of a point P make angles 60^o and 45^owith X and Y axis respectively. The angle of inclination of \(\overset { \rightarrow }{ OP } \)with z-axis is

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