11th Maths Monthly Test- 1 (Vector Algebra )

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test- 1 (Vector Algebra ) -Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

If \(\left| \vec { a } \right| \)=4 and \(-3\le \lambda \le 2\) then the range of \(\left| \lambda \vec { a } \right| \)is

[0, 8]
[-12, 8]
[0, 12]
[8, 12]

If \(|\overrightarrow{a}+\overrightarrow{b}|=60,\)\(|\overrightarrow{a}+\overrightarrow{b}|=40\) and \(|\overrightarrow{b}|=46\) , then \(|\overrightarrow{a}|\) is

The unit vector parallel to the resultant of the vectors \(\hat{i}+\hat{j}-\hat{k}\) and\(\hat{i}-2\hat{j}+\hat{k}\) is

\({\hat{i}-\hat{j}+\hat{k}\over\sqrt{5}}\)
\({2\hat{i}+\hat{j}\over\sqrt{5}}\)
\({2\hat{i}-\hat{j}+\hat{k}\over\sqrt{5}}\)
\({2\hat{i}-\hat{j}\over\sqrt{5}}\)

The value of the expression \({ \left| \overrightarrow { a } \times \overrightarrow { b } \right| }^{ 2 }+{ (\overrightarrow { a } .\overrightarrow { b } ) }^{ 2 }\) is

cos² \(\theta\)
sin² \(\theta\)
\({ |\overrightarrow { a } | }^{ 2 }|{ \overrightarrow { b } | }^{ 2 }\)
\({ (|\overrightarrow { a } |+|\overrightarrow { b } |) }^{ 2 }\)

If \(|\overrightarrow { a } |=|\overrightarrow { b } |\) then

\(\overrightarrow { a } =\overrightarrow { b } \)
\(\overrightarrow { a } =\overrightarrow { -b } \)
\(\overrightarrow { a } =\pm \overrightarrow { b } \)
both are null vectors

The projection of \(\overrightarrow { b } \) on \(\overrightarrow { a } \) is

\(\left( \frac { \overrightarrow { a } .\overrightarrow { b } }{ |\overrightarrow { b } | } \right) \overrightarrow { b } \)
\(\frac { \overrightarrow { a } .\overrightarrow { b } }{ |\overrightarrow { b } | } \)
\(\frac { \overrightarrow { a } .\overrightarrow { b } }{ |\overrightarrow { a } | } \)
\(\left( \frac { \overrightarrow { a } .\overrightarrow { b } }{ |\overrightarrow { a } | } \right) \)

The angle between two vectors\(\vec{a}\) and\(\vec{b}\) with magnitudes\(\sqrt{3}\) and 4 respectively and \(\vec{a}.\vec{b}=2\sqrt{3}\) is

\(\frac{\pi}{6}\)
\(\frac { \pi }{ 3 } \)
\(\frac { \pi }{ 2 } \)
\(\frac { 5\pi }{ 2 } \)

If \(\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+x\hat{j}+\hat{k},\overrightarrow{c}=\hat{i}-\hat{j}+4\hat{k}\) and \(\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})=70,\) then x is equal to

Find the odd one out of the following

\(\hat { i } +2\hat { j } +3\hat { k } \)
\(2\hat { i } +4\hat { j } +6\hat { k } \)
\(7\hat { i } +14\hat { j } +21\hat { k } \)
\(\hat { i } +3\hat { j } +2\hat { k } \)

Assertion (A) : \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) are the position vector three collinear points then 2 \(\overset { \rightarrow }{ a }=\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \)
Reason (R): Collinear points, have same direction

Both A and R are true and R is the correct explanation of A
Both A and R are true and R is not a correct explantion of A
A is true but R is false
A is false but R is true

Part - B

Generic 2 - Answer All Question

Find the vectors of magnitude \(10\sqrt{3}\) that are perpendicular to the plane which contains \(\hat{i}+2\hat{j}+\hat{k}\) and\(\hat{i}+3\hat{j}+4\hat{k}\)

If \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\) , \(\vec{b}=-\hat{i}+2\hat{j}+\hat{k}\)and\(\vec{c}=3\hat{i}+\hat{j}\) .Find \(\lambda\)so that \(\vec{a}+\lambda \vec{b}\) is perpendicular to \(\vec{c}\).

Represent graphically the displacement of 60 km 50° south of east.

Find the area of the parallelogram whose adjacent sides are \(\overrightarrow{a}=3\hat{i}+\hat{j}+4\hat{k}\) and \(\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}\) .
Find the value \(\lambda\) for which the vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are perpendicular, where \(\overrightarrow{a}=2\hat{i}+\lambda\hat{j}+\hat{k}\) and \(\overrightarrow{b}=\hat{i}-2\hat{j}+3\hat{k}\)

If \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\) , \(\vec{b}=-\hat{j}+4\hat{k}\)and \(\vec{c}=\hat{i}-2\hat{j}+\hat{k}\). Find\((\vec{a}+\vec{b}).\vec{c}\) and \(\vec{c}.(\vec{a}+\vec{b})\) Are they equal?

Find the direction cosines of a vector whose direction ratios are 1,2,3

Verify whether the following ratios are direction cosines of some vector or not\({4\over 3}.0,{3\over 4}\)

Find the direction cosines and direction ratios for the following vectors.3\(\hat{i}\)-3\(\hat{k}\)+4\(\hat{j}\)

Find the direction cosines of a vector whose direction ratios are 3,-1,3

Part - C

Generic 3 - Answer All Question

Examine whether the vectors \(\hat { i } +3\hat { j } +\hat { k } ,2\hat { i } -\hat { j } -\hat { k } \) and \(7\hat { j } +5\hat { k } \) are coplanar

If \(|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|\) prove that \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are perpendicular.

Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.

Find a point whose position vector has magnitude 5 and parallel to the vector 4\(\hat{i}\)- 3\(\hat{j}\)+10\(\hat{k}\).

If \(|\overrightarrow{a}|=5,|\overrightarrow{b}|=6,|\overrightarrow{c}|=7\) and \(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c} =\overrightarrow{0}\) ,find \(\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}\).
Show that the points whose positions vectors \(4\hat { i } -3\hat { j } +\hat { k } \) ,\(2\hat { i } -4\hat { j } +5\hat { k } \) ,\(\hat { i } -\hat { j } \) from a right angled triangle.

If \(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\)are position vectors of the vertices A, B, C of a triangle ABC, show that the area of the triangle ABC is \({1\over 2}|\overrightarrow{a}\times \overrightarrow{b}+ \overrightarrow{b}+\overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}|\) .Also deduce the condition for collinearity of the points A, B, and C.

Find the area of a triangle having the points A(1,0,0),B(0,1,0), and C(0,0,1) A BC as its vertices.

Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side whose length is half of the length of the third side.

Find the unit vectors parallel to the sum of \(3\vec { i } -5\vec { j } +8\vec { k } \) and \(-2\vec { i } -2\vec { k } \)

Part - D

Generic 5 - Answer All Question

If \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are vectors represented by two adjacent sides of a regular hexagon, then find the vectors represented by other sides.

If D is the midpoint of the side BC of a triangle ABC, prove that \(\overrightarrow{AB}\) + \(\overrightarrow{AC}\) = 2\(\overrightarrow{AD}\) .

Show that the points (2, - 1, 3), (4, 3, 1) and (3, 1, 2) are collinear.

If \(\overrightarrow{a},\overrightarrow{b}\)are unit vectors and \(\theta\) is the angle between them, show that \(tan {\theta \over 2}={|\overrightarrow{a}-\overrightarrow{b}|\over|\overrightarrow{a}+\overrightarrow{b}|}\)

Prove that the points whose position vectors \(2\hat{i}+4\hat{j}+3\hat{k},4\hat{i}+\hat{j}+9\hat{k}\) and \(10\hat{i}-\hat{j}+6\hat{k}\) form a right angled triangle.
Find the value or values of m for which m (\(\hat{i}+\hat{j}+\hat{k}\)) is a unit vector

The position vectors of the points P, Q, R, S are \(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\),2 \(\hat{i}\)+5\(\hat{j}\),3\(\hat{i}\)+2\(\hat{j}\)-3\(\hat{k}\), and \(\hat{i}\)-6\(\hat{j}\)-\(\hat{k}\), respectively. Prove that the line PQ and RS are parallel.

If G is the centroid of a triangle ABC, prove that \(\overrightarrow{GA}\)+\(\overrightarrow{GB}\) +\(\overrightarrow{GC}\) = \(\overrightarrow{0}\).

Let ABC be a triangle,\(\overrightarrow{BC}=\overrightarrow{a},\overrightarrow{CA}=\overrightarrow{b}\) and \(\overrightarrow{AB}=\overrightarrow{c}\). Then prove that \(\overrightarrow {a}\times \overrightarrow {b}=\overrightarrow {b}\times \overrightarrow {c}=\overrightarrow {c}\times \overrightarrow {a}.\)

Let A, Band C represent the angles of a \(\triangle\)ABC and a, b, c represent the lengths of the sides opposite to them, then prove that a = b cos C + c cos B (Projection formula)

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11th Maths Monthly Test- 1 (Vector Algebra )

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test- 1 (Vector Algebra ) -Aug 2020

Part - A

Multiple Choice Question - Answer All Question

If \(\left| \vec { a } \right| \)=4 and \(-3\le \lambda \le 2\) then the range of \(\left| \lambda \vec { a } \right| \)is

If \(|\overrightarrow{a}+\overrightarrow{b}|=60,\)\(|\overrightarrow{a}+\overrightarrow{b}|=40\) and \(|\overrightarrow{b}|=46\) , then \(|\overrightarrow{a}|\) is

The unit vector parallel to the resultant of the vectors \(\hat{i}+\hat{j}-\hat{k}\) and\(\hat{i}-2\hat{j}+\hat{k}\) is

The value of the expression \({ \left| \overrightarrow { a } \times \overrightarrow { b } \right| }^{ 2 }+{ (\overrightarrow { a } .\overrightarrow { b } ) }^{ 2 }\) is

If \(|\overrightarrow { a } |=|\overrightarrow { b } |\) then

The projection of \(\overrightarrow { b } \) on \(\overrightarrow { a } \) is

The angle between two vectors\(\vec{a}\) and\(\vec{b}\) with magnitudes\(\sqrt{3}\) and 4 respectively and \(\vec{a}.\vec{b}=2\sqrt{3}\) is

If \(\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+x\hat{j}+\hat{k},\overrightarrow{c}=\hat{i}-\hat{j}+4\hat{k}\) and \(\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})=70,\) then x is equal to

Find the odd one out of the following

Part - B

Generic 2 - Answer All Question

Find the vectors of magnitude \(10\sqrt{3}\) that are perpendicular to the plane which contains \(\hat{i}+2\hat{j}+\hat{k}\) and\(\hat{i}+3\hat{j}+4\hat{k}\)

If \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\) , \(\vec{b}=-\hat{i}+2\hat{j}+\hat{k}\)and\(\vec{c}=3\hat{i}+\hat{j}\) .Find \(\lambda\)so that \(\vec{a}+\lambda \vec{b}\) is perpendicular to \(\vec{c}\).

Represent graphically the displacement of 60 km 50° south of east.

Find the area of the parallelogram whose adjacent sides are \(\overrightarrow{a}=3\hat{i}+\hat{j}+4\hat{k}\) and \(\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}\) .

Find the value \(\lambda\) for which the vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are perpendicular, where \(\overrightarrow{a}=2\hat{i}+\lambda\hat{j}+\hat{k}\) and \(\overrightarrow{b}=\hat{i}-2\hat{j}+3\hat{k}\)

If \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\) , \(\vec{b}=-\hat{j}+4\hat{k}\)and \(\vec{c}=\hat{i}-2\hat{j}+\hat{k}\). Find\((\vec{a}+\vec{b}).\vec{c}\) and \(\vec{c}.(\vec{a}+\vec{b})\) Are they equal?

Find the direction cosines of a vector whose direction ratios are 1,2,3

Verify whether the following ratios are direction cosines of some vector or not\({4\over 3}.0,{3\over 4}\)

Find the direction cosines and direction ratios for the following vectors.3\(\hat{i}\)-3\(\hat{k}\)+4\(\hat{j}\)

Find the direction cosines of a vector whose direction ratios are 3,-1,3

Part - C

Generic 3 - Answer All Question

Examine whether the vectors \(\hat { i } +3\hat { j } +\hat { k } ,2\hat { i } -\hat { j } -\hat { k } \) and \(7\hat { j } +5\hat { k } \) are coplanar

If \(|\overrightarrow{a}+\overrightarrow{b}|=|\overrightarrow{a}-\overrightarrow{b}|\) prove that \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are perpendicular.

Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram.

Find a point whose position vector has magnitude 5 and parallel to the vector 4\(\hat{i}\)- 3\(\hat{j}\)+10\(\hat{k}\).

If \(|\overrightarrow{a}|=5,|\overrightarrow{b}|=6,|\overrightarrow{c}|=7\) and \(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c} =\overrightarrow{0}\) ,find \(\overrightarrow{a}.\overrightarrow{b}+\overrightarrow{b}.\overrightarrow{c}+\overrightarrow{c}.\overrightarrow{a}\).

Show that the points whose positions vectors \(4\hat { i } -3\hat { j } +\hat { k } \) ,\(2\hat { i } -4\hat { j } +5\hat { k } \) ,\(\hat { i } -\hat { j } \) from a right angled triangle.

Find the area of a triangle having the points A(1,0,0),B(0,1,0), and C(0,0,1) A BC as its vertices.

Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side whose length is half of the length of the third side.

Find the unit vectors parallel to the sum of \(3\vec { i } -5\vec { j } +8\vec { k } \) and \(-2\vec { i } -2\vec { k } \)

Part - D

Generic 5 - Answer All Question

If \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are vectors represented by two adjacent sides of a regular hexagon, then find the vectors represented by other sides.

If D is the midpoint of the side BC of a triangle ABC, prove that \(\overrightarrow{AB}\) + \(\overrightarrow{AC}\) = 2\(\overrightarrow{AD}\) .

Show that the points (2, - 1, 3), (4, 3, 1) and (3, 1, 2) are collinear.

If \(\overrightarrow{a},\overrightarrow{b}\)are unit vectors and \(\theta\) is the angle between them, show that \(tan {\theta \over 2}={|\overrightarrow{a}-\overrightarrow{b}|\over|\overrightarrow{a}+\overrightarrow{b}|}\)

Prove that the points whose position vectors \(2\hat{i}+4\hat{j}+3\hat{k},4\hat{i}+\hat{j}+9\hat{k}\) and \(10\hat{i}-\hat{j}+6\hat{k}\) form a right angled triangle.

Find the value or values of m for which m (\(\hat{i}+\hat{j}+\hat{k}\)) is a unit vector

The position vectors of the points P, Q, R, S are \(\hat{i}\) + \(\hat{j}\) + \(\hat{k}\),2 \(\hat{i}\)+5\(\hat{j}\),3\(\hat{i}\)+2\(\hat{j}\)-3\(\hat{k}\), and \(\hat{i}\)-6\(\hat{j}\)-\(\hat{k}\), respectively. Prove that the line PQ and RS are parallel.

If G is the centroid of a triangle ABC, prove that \(\overrightarrow{GA}\)+\(\overrightarrow{GB}\) +\(\overrightarrow{GC}\) = \(\overrightarrow{0}\).

Let A, Band C represent the angles of a \(\triangle\)ABC and a, b, c represent the lengths of the sides opposite to them, then prove that a = b cos C + c cos B (Projection formula)

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