12th Maths Weekly Test -1 (Applications of Vector Algebra)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test -1 (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

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

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

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

\( { \pi }\)
\(\vec { a } .\vec { b } =\vec { b } .\vec { c } =\vec { c } .\vec { a } =0\) , 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

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)

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.

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

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.

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
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

Part - D

Generic 5 - Answer All Question

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.

Find the vector and Cartesian equation of the plane passing through the point (1,1, -1) and perpendicular to the planes x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0

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

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12th Maths Weekly Test -1 (Applications of Vector Algebra)

St. Britto Hr. Sec. School - Madurai

12th Maths Weekly Test -1 (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

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

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

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

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.

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.

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

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.

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

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

Part - D

Generic 5 - Answer All Question

Find the vector and Cartesian equation of the plane passing through the point (1,1, -1) and perpendicular to the planes x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0

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

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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