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11th Maths Monthly Test -2( Vector Algebra )

St. Britto Hr. Sec. School - Madurai

11th Maths Monthly Test -2( Vector Algebra )-Aug 2020

Time : 2.30 Hours

Part - A

Multiple Choice Question - Answer All Question

The number of vectors of unit length perpendicular to the vectors \(\overrightarrow { a } =\hat { i } +\hat { j } +\hat { k } \) and \(\overrightarrow { b } =2\hat { i } +\hat { j } +6\hat { k } \) is

1
2
3
infinite

If \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are two vectors of magnitude 2 and inclined at an angle 60° , then the angle between \(\overrightarrow{a}\) and \(\overrightarrow{a}+\overrightarrow{b}\) is

30°
60°
45°
90°

A vector \(\overrightarrow{OP}\) makes 60° and 45° with the positive direction of the x and y axes respectively. Then the angle between \(\overrightarrow{OP}\)and the z-axis is

45°
60°
90°
30°

If m \(\left( \overset { \rightarrow }{ 2 } +\overset { \rightarrow }{ j } +\overset { \rightarrow }{ k } \right) \) is a unit vector then the value of m is

\(\pm \frac { 1 }{ \sqrt { 3 } } \)
\(\pm \frac { 1 }{ \sqrt { 5 } } \)
\(\pm \frac { 1 }{ \sqrt { 6 } } \)
\(\pm \frac { 1 }{ {2 } } \)

If \(\overrightarrow{a}\) and \(\overrightarrow{b}\) having same magnitude and angle between them is 60° and their scalar product is \({1\over2}\) then \(|\overrightarrow{a}|\) is

2
3
7
1

A vector makes equal angle with the positive direction of the coordinate axes. Then each angle is equal to

\(cos^{-1}({1\over 3})\)
\(cos^{-1}({2\over 3})\)
\(cos^{-1}({1\over\sqrt 3})\)
\(cos^{-1}({2\over\sqrt 3})\)

The value of \(\theta \in (0,{\pi\over 2})\) for which the vectors \(\overrightarrow{a}=(sin \theta)\hat{i}+(cos\theta)\hat{j}\) and \(\overrightarrow{b}=\hat{i}-\sqrt{3}\hat{j}+2\hat{k}\) are perpendicular, is equal to

\({\pi\over 3}\)
\({\pi\over 6}\)
\({\pi\over 4}\)
\({\pi\over 2}\)

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

The vectors \(\overrightarrow{a}-\overrightarrow{b},\overrightarrow{b}-\overrightarrow{c},\overrightarrow{c}-\overrightarrow{a}\) are

parallel to each other
unit vectors
mutually perpendicular vectors
coplanar vectors.

Find the odd one out of the following

matrix multiplication
vector cross product
Subtraction
Matrix Addition

One of the diagonals of parallelogram ABCD with \(\overrightarrow{a}\) and \(\overrightarrow{b}\) as adjacent sides is \(\overrightarrow{a}+\overrightarrow{b}\) . The other diagonal \(\overrightarrow{BD}\) is

\(\overrightarrow{a}-\overrightarrow{b}\)
\(\overrightarrow{b}-\overrightarrow{a}\)
\(\overrightarrow{a}+\overrightarrow{b}\)
\(\overrightarrow{a}+\overrightarrow{b}\over 3\)

If \(\overrightarrow{a},\overrightarrow{b}\) are the position vectors A and B, then which one of the following points whose position vector lies on AB, is

\(\overrightarrow{a}+\overrightarrow{b}\)
\({2\overrightarrow{a}-\overrightarrow{b}\over 2}\)
\({2\overrightarrow{a}+\overrightarrow{b}\over 2}\)
\({\overrightarrow{a}-\overrightarrow{b}\over 3}\)

If the projection of \(5\hat{i}-\hat{j}-3\hat{k}\) on the vector \(\hat{i}+3\hat{j}-\lambda\hat{k}\) is same as the projection of \(\hat{i}+3\hat{j}+\lambda\hat{k}\) on \(5\hat{i}-\hat{j}-3\hat{k}\),then \(\lambda\) is equal to

\(\pm 4\)
\(\pm 3\)
\(\pm 5\)
\(\pm 1\)

Match List - I with List II

List I List II

1 \(\hat{i}.\hat{i}\) a 0

2 \(\hat{i}.\hat{j}\) b \(\hat{k}\)

3 \(\hat{i}\times\hat{i}\) c 1

4 \(\hat{i}\times\hat{j}\) d 0

The Correct match is

i ii iii iv

b c d a
i ii iii iv

c a d b
i ii iii iv

d b a c
i ii iii iv

d c b a

If (1, 2, 4) and (2, - 3\(\lambda\) - 3) are the initial and terminal points of the vector \(\hat{i}+\hat{j}-7\hat{k}\) ,then the value of \(\lambda\)is equal to

\(7\over 3\)
-\(7\over 3\)
-\(5\over 3\)
\(5\over 3\)

Part - B

Generic 2 - Answer All Question

If \(\overrightarrow{a}\) and \(\overrightarrow{b}\)are two vectors such that | \(\overrightarrow{a}\) |=10,| \(\overrightarrow{b}\) |=15 and \(\overrightarrow{a}\).\(\overrightarrow{b}\) = 75 \(\sqrt{2}\), find the angle between \(\overrightarrow{a}\)and \(\overrightarrow{b}\).

Find the angle between the vectors \(2\hat{i}+3\hat{j}-6\hat{k}\) and \(6\hat{i}-3\hat{j}+2\hat{k}\)

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 \(\overrightarrow{a}=-3\hat{i}+4\hat{j}-7\hat{k}\) and \(\overrightarrow{b}=6\hat{i}+2\hat{j}-3\hat{k},\) verify \(\overrightarrow{b}\)are\(\overrightarrow{a}\times \overrightarrow{b}\) perpendicular to each other

Find the projection of the vector \(\hat{i}-\hat{j}\) on the vector \(\hat{i}+\hat{j}\).

Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (-5, 7)

Part - C

Generic 3 - Answer All Question

Find the cosine and sine angle between the vectors \(\overrightarrow{a}=2\hat{i}+\hat{j}+3\hat{k}\) and \(\overrightarrow{b}=4\hat{i}-2\hat{j}+2\hat{k}\).

Find the area of the triangle whose vertices are A(3, - 1, 2), B(1, - 1, - 3) and C(4, - 3, 1).

Find the angle A of the triangle whose vertices are A(0, -1, 2), B(3, 1, 4) and C(5, 7, 1).

Show that the points whose position vectors 4\(\hat{i}\) +5\(\hat{j}\) + \(\hat{k}\) , -\(\hat{j}\) -\(\hat{k}\) , 3\(\hat{i}\)+9\(\hat{j}\) +4 \(\hat{k}\) and -4\(\hat{i}\) +4\(\hat{j}\) +4\(\hat{k}\) are coplanar.

If ABCDE is a pentagaon then prove that \(\overrightarrow { AB } +\overrightarrow { AE } +\overrightarrow { BC } +\overrightarrow { DC } +\overrightarrow { ED } +\overrightarrow { AC } =3\overrightarrow { AC } \)

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

Part - D

Generic 5 - Answer All Question

If \(\left| \vec { a } \right| =\left| \vec { b } \right| =\left| \vec { a } +\vec { b } \right| \)=1 then prove that \({ \left| \vec { a } -\vec { b } \right| }=\sqrt { 3 } \)

The position vectors \(\overrightarrow{a}\),\(\overrightarrow{b}\), \(\overrightarrow{c}\) of three points satisfy the relation 2\(\overrightarrow{a}\)-7\(\overrightarrow{b}\) 5\(\overrightarrow{c}\) =\(\overrightarrow{0}\) .Are these points collinear?

Find the projection of:
(i) \(\hat { i } -\hat { j } \) on Z-axis
(ii) \(\hat { i } +2\hat { j } -2\hat { k } \) on \(2\hat { i } -\hat { \quad j } -2\hat { k } \)
(iii) \(3\hat { i } +\hat { j } -\hat { k } \) on \(4\hat { i } -\hat { j } +2\hat { k } \)

Let\(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\)be three vectors such that | \(\overrightarrow{a}\) |= 3, | \(\overrightarrow{b}\) |= 4, | \(\overrightarrow{c}\) |= 5 and each one of them being perpendicular to the sum of the other two, find | \(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}\) |.

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

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

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