Assertion (A) & Reason (B) MCQ - Maths STD 12 Science Questions - Vidyadip

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Assertion (A) & Reason (B) MCQ

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23 questions · 9 auto-graded MCQ + 14 self-marked written.

MCQ 11 Mark

Assertion $(A)$ : The vectors :
\[\vec{a}=6 \hat{i}+2 \hat{j}-8 \hat{k}, \vec{b}=10 \hat{i}-2 \hat{j}-6 \hat{k}, \vec{c}=4 \hat{i}-4 \hat{j}+2 \hat{k}\]
represent the sides of a right angled triangle.
Reason $(R)$ : Three non-zero vectors of which none of two are collinear forms a triangle if their resultant is zero vector or sum of any two vectors is equal to the third.

A
Both Assertion (A) and Reason (R) are true and the Reason $(R)$ is the correct explanation of the Assertion (A).
B
Both Assertion (A) and Reason $(R)$ are true but Reason $(R)$ is not the correct explanation of the Assertion (A).
C
Assertion (A) is true but Reason (R) is false.
D
Assertion (A) is false but Reason $( R )$ is true.

Answer

$
\begin{array}{l}\text {Given, } \vec{a}=6 \hat{i}+2 \hat{j}-8 \hat{k}, \vec{b}=10 \hat{i}-2 \hat{j}-6 \hat{k} \text { and } \vec{c}=4 \hat{i}-4 \hat{j}+2 \hat{k} \\
\begin{aligned}
\Rightarrow \quad \vec{a}+\vec{c} & =(6 \hat{i}+2 \hat{j}-8 \hat{k})+(4 \hat{i}-4 \hat{j}+2 \hat{k}) \\
& =10 \hat{i}-2 \hat{j}-6 \hat{k}=\vec{b}
\end{aligned}
\end{array}
$Hence, $\vec{a}, \vec{b}$ and $\vec{c}$ are the sides of a triangle.
Let $\theta$ be the angle between $\vec{a}$ and $\vec{c}$, then
$
\begin{array}{l}
\cos \theta=\frac{\vec{a} \cdot \vec{c}}{|\vec{a}| \cdot|\vec{c}|}=\frac{6 \cdot 4+2(-4)+(-8)(2)}{\sqrt{6^2+2^2+(-8)^2} \sqrt{4^2+(-4)^2+2^2}} \\
=\frac{24-8-16}{\sqrt{104} \sqrt{36}}=0 \\
\Rightarrow \quad \cos \theta=\cos 90^{\circ} \Rightarrow \theta=90^{\circ} \\
\therefore \quad \text { Assertion (A) is true. }
\end{array}
$$\therefore \quad$ Assertion $( A )$ is true.
Hence, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

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MCQ 21 Mark

Assertion (A): For two hon-zero vectors $\vec{a}$ and $b , \vec{a} \cdot b = b \cdot \vec{a}$.
Reason (R): For two non-zero vectors $\vec{a}$ and $\vec{b}, \vec{a} \times \vec{b}=\vec{b} \times \vec{a}$.

A
Both Assertion (A) and Reason (R) are true and Reason $(R)$ is the correct explanation of the Assertion (A).
B
Both Assertion (A) and Reason (R) are true, but Reason $(R)$ is not the correct explanation of the Assertion (A).
C
Assertion (A) is true but Reason (R) is false.
D
Assertion (A) is false but Reason (R) is true.

Answer

Assertion (A) is true but reason $( R )$ is false. As, $\vec{a} \times \vec{b}=-\vec{b} \times \vec{a}$.

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MCQ 31 Mark

Assertion (A): $(\vec{b} \cdot \vec{c}) \vec{a}$ is a scalar quantity.
Reason $(R)$ : Dot product of two vectors is a scalar quantity.

A
Both Assertion (A) and Reason (R) are true and Reason $(R)$ is the correct explanation of the Assertion (A).
B
Both Assertion (A) and Reason (R) are true but Reason $(R)$ is not the correct explanation of the Assertion (A).
C
Assertion (A) is true but Reason (R) is false.
D
Assertion $( A )$ is false but Reason $( R )$ is true.

Answer

$(\vec{b} \cdot \vec{c}) \vec{a}$ is a vector quantity.
So, Assertion $(A)$ is false but Reason $(R)$ is true.

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Question 41 Mark

Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The magnitude of the resultant of vectors $\overline{\text{a}}=2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $\hat{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}}$
Reason: The magnitude of a vector can never be negative.

Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

Answer

Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Solution:

$\overline{\text{a}}=2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\overline{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}},$

Resultant of $\hat{\text{a}}$ and $\hat{\text{b}}$ is $\hat{\text{a}}+\hat{\text{b}}$

$=(2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})+(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})=3\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}}$

$\therefore|\overline{\text{a}}+\overline{\text{b}}|=\sqrt{3^2+3^2+4^2}=\sqrt{9+9+16}=\sqrt{34}$

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Question 51 Mark

Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The adjacent sides of a parallelogramarealong $\overline{\text{a}}=\hat{\text{i}}+2\hat{\text{j}}$ and $\overline{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}$ The angle between the diagonals is $150^\circ$.
Reason: Two vectors are perpendicular to each other if their dot product is zero.

Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

Answer

Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Solution:

$\vec{\text{a}}=\hat{\text{i}}+2\hat{\text{j}},\vec{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}$

Diagonals of the parallelogram arealong $\vec{\text{a}}+\vec{\text{b}}$ and $\vec{\text{a}}-\vec{\text{b}}$

Now, $\vec{\text{a}}+\vec{\text{b}}=(\hat{\text{i}}+2\hat{\text{j}})+(2\hat{\text{i}}+\hat{\text{j}})=3\hat{\text{i}}+3\hat{\text{j}}$

and $\vec{\text{a}}-\vec{\text{b}}=(\hat{\text{i}}+2\hat{\text{j}})-(2\hat{\text{i}}+\hat{\text{j}})=-\hat{\text{i}}+\hat{\text{j}}$

Let $\theta$ be the angle between these vectors, then

$\cos\theta=\frac{(3\text{i}+3\text{j})(\hat{-\text{i}}+\hat{\text{j}})}{\sqrt{9+9}\sqrt{1+1}}=\frac{-3+3}{\sqrt{18}\sqrt{2}}=0$

$\Rightarrow\theta=90^\circ$

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Question 61 Mark

Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: Three points with position vectors $\vec{\text{a}},\vec{\text{b}}$ and $\vec{\text{c}}$ are collinear if $\vec{\text{a}}\times\vec{\text{b}}+\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{c}}\times\vec{\text{a}}=\vec{0}$
Reason: If $\overrightarrow{\text{AB}}.\overrightarrow{\text{AC}}.=0,$ then $\overrightarrow{\text{AB}}\bot\overrightarrow{\text{AC}}.$

Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

Answer

Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Solution:

If $\text{A},\text{B},\text{C}$ are collinear, then $\overrightarrow{\text{AB}}=\overrightarrow{\text{kAC}}$

$\therefore\overrightarrow{\text{AB}}\ \times\overrightarrow{\text{AC}}=\vec{0}$

$\Rightarrow(\vec{\text{b}}-\vec{\text{a}})\times(\vec{\text{c}}-\vec{\text{a}})=0$

$\Rightarrow\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{a}}\times\vec{\text{b}}+\vec{\text{c}}\times\vec{\text{a}}=\vec{0}$

$\text{i.e}..,\ \vec{\text{a}}\times\vec{\text{b}}+\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{c}}\times\vec{\text{a}}=\vec{0}$

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Question 71 Mark

Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Let $\vec{\text{a}}$ and $\vec{\text{b}}$ be proper vectors and $\theta$ be the angle between them.
Assertion: $(\vec{\text{a}}\times\vec{\text{b}})^2+(\vec{\text{a}}.\vec{\text{b}})^2\neq(\vec{\text{a}})^2(\vec{\text{b}})^2$
Reason: $\sin^2\theta+\cos^2\theta=0$

Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

Answer

Assertion is wrong statement but Reason is correct statement.

Solution:

$(\vec{\text{a}}\times\vec{\text{b}})^2+(\vec{\text{a}}.\vec{\text{b}})^2=|\vec{\text{a}}\times\vec{\text{b}}|^2+(\vec{\text{a}}.\vec{\text{b}})$

$=(\vec{\text{a}}\vec{\text{b}}\sin\theta)^2+(\vec{\text{a}}\vec{\text{b}}\cos\theta)^2=\vec{\text{a}}^2\vec{\text{b}}^2$

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Question 81 Mark

Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: If $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\vec{0},|\vec{\text{a}}|=3,|\vec{\text{b}}|=4,|\vec{\text{c}}|=5,$ then $\vec{\text{a}}.\vec{\text{b}}+\vec{\text{b}}.\vec{\text{c}}+\vec{\text{c}}.\vec{\text{a}}$ is equal to $-25.$
Reason: If $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\vec0,$ then the$\angle\theta$ between $\vec{\text{b}}$ and $\vec{\text{c}}$ is given by $\cos\theta=\frac{\vec{\text{a}}^2-\vec{\text{b}}^2-\vec{\text{c}}^2}{2\vec{\text{b}}{\vec{\text{c}}}}$

Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

Answer

Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.

Solution:

We have, $|\vec{\text{a}}|=3,|\vec{\text{b}}|=4,|\vec{\text{c}}|=5, $ and

$\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=\vec{0}\Rightarrow(\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}})^2=0$

$\Rightarrow|\vec{\text{a}}|^2+|\vec{\text{b}}|^2+|\vec{\text{c}}|^2+2(\vec{\text{a}}.\vec{\text{b}}+\vec{\text{b}}.\vec{\text{c}}+\vec{\text{c}}.\vec{\text{a}})=0$

$\Rightarrow(3)^2+(4)^2+(5)^2+2(\vec{\text{a}}.\vec{\text{b}}+\vec{\text{b}}.\vec{\text{c}}+\vec{\text{c}}.\vec{\text{a}})=0$

$\Rightarrow\vec{\text{a}}.\vec{\text{b}}+\vec{\text{b}}.\vec{\text{c}}+\vec{\text{c}}.\vec{\text{a}}=\frac{1}{2}[9+16+25]=-\frac{1}{2}(50)=-25$

Now, $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=0\Rightarrow\vec{\text{b}}+\vec{\text{c}}=-\vec{\text{a}}$

$\Rightarrow|\vec{\text{b}}+\vec{\text{c}}|^2=|-\vec{\text{a}}|^2\Rightarrow\vec{\text{b}}^2+\vec{\text{c}}^2+2\vec{\text{b}}.\vec{\text{c}}=\vec{\text{a}}^2$

$\Rightarrow\vec{\text{b}}^2+\vec{\text{c}}^2+2\vec{\text{b}}\vec{\text{c}}\cos\theta=\vec{\text{a}}^2$

$\Rightarrow\cos\theta\frac{\vec{\text{a}}^2-\vec{\text{b}}^2-\vec{\text{c}}^2}{2\vec{\text{b}}\vec{\text{c}}}$

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Question 91 Mark

Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: If $\vec{\text{a}}=2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}},\vec{\text{a}}=-\hat{\text{i}}+3\hat{\text{j}}-4\hat{\text{k}}$ then projection of on .
Reason: Projection of $\vec{\text{a}}$ on $\vec{\text{b}}=\frac{3}{\sqrt{26}}$

Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

Answer

Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Solution:

Projection of $\vec{\text{a}}$ on $\vec{\text{b}}$ $=\frac{\vec{\text{a}}\vec{\text{b}}}{\sqrt{|\vec{\text{b}}|}}$

$\frac{(2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}})(-\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}})}{\sqrt{(-1)^2+(3)^2+(4)^2}}=\frac{-2+9-4}{\sqrt{26}}=\frac{3}{\sqrt{26}}$

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Question 101 Mark

Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Let $\overline{\text{a}}=\hat{\text{i}}+\hat{\text{j}}=3\hat{\text{k}}$ and $\overline{\text{b}}=\hat{2\text{i}}+\hat{\text{j}}=\hat{\text{k}}$
Assertion: Vectors $\overline{\text{a}}$ and $\overline{\text{b}}$ are perpendicular to each other.
Reason: $\overline{\text{a}}.\overline{\text{b}}=0$

Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

Answer

Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.

Solution:

$\overline{\text{a}}=\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}},\overline{\text{b}}=\hat{\text{2i}}+\hat{\text{j}}-\hat{\text{k}}$

$\overline{\text{a}}.\overline{\text{b}}=(\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}).(2\hat{\text{i}}+\text{j}+\hat{\text{k}})$

$=1.2+1.1+(-3).1=2+1-3=0$

$\Rightarrow\cos\theta=0\Rightarrow\theta=\frac{\pi}{2}$

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Question 111 Mark

Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: The unit vector in the direction of sum of the vectors $\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}$ and $2\hat{\text{j}}+6\hat{\text{k}}$ is $-\frac{1}{7}(3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}}).$
Reason: Let $\overline{\text{a}}$ be a non - zero vector, then $\frac{\overline{\text{a}}}{|\overline{\text{a}}|}$ is a unit vector parallel to $\overline{\text{a}}$.

Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

Answer

Assertion is wrong statement but Reason is correct statement.

Solution:

Sum of the given vectors

$=(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})+(2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}})+(2\hat{\text{j}}+6\hat{\text{k}})=3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}}$

$\therefore$ The unit vector in the direction of the sum of the given vectors

$=\frac{3\hat{\text{i}}+2\hat{\text{j}}+6\hat{{\text{k}}}}{\sqrt{3^2+\text{2}^2+6^2}}=\frac{3\hat{\text{i}}+2\hat{\text{j}}+6\hat{{\text{k}}}}{\sqrt{9+4+36}}=\frac{1}{7}(3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}})$

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Question 121 Mark

Directions: In these questions, a statement of Assertion is followed by a statement of Reason is given.Choose the correct answer out of the following choices:
Assertion: If $ (\vec{\text{a}}\times\vec{\text{b}})+(\vec{\text{a}}.\vec{\text{b}})=400$ and $|\vec{\text{a}}|=4,$ then $|\vec{\text{b}}|=9.$
Reason: If $\vec{\text{a}}$ and $\vec{\text{b}}$ are any two vectors, then $(\vec{\text{a}}\times\vec{\text{b}})^2$ is equal to $(\vec{\text{a}})^2(\vec{\text{b}})^2-(\vec{\text{a}}.\vec{\text{b}})^2.$

Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
Assertion is correct statement but Reason is wrong statement.
Assertion is wrong statement but Reason is correct statement.

Answer

Assertion is wrong statement but Reason is correct statement.

Solution:

$(\vec{\text{a}}\times\vec{\text{b}})^2+(\vec{\text{a}}.\vec{\text{b}})^2=400,|\vec{\text{a}}|=4$

We know that,

$(\vec{\text{a}}\times\vec{\text{b}})^2+(\vec{\text{a}}.\vec{\text{b}})^2=|\vec{\text{a}}|^2|\vec{\text{b}}|^2$

$=400=(4)^2|\vec{\text{b}}|^2\Rightarrow16|\vec{\text{b}}|^2=400$

$\Rightarrow|\vec{\text{b}}|^2=25\Rightarrow|\vec{\text{b}}|=5$

Hence, Assertion is wrong.

$(\vec{\text{a}}\times\vec{\text{b}})^2+(\vec{\text{a}}.\vec{\text{b}})^2=|\vec{\text{a}}\times\vec{\text{b}}|^2+(\vec{\text{a}}.\vec{\text{b}})^2$

$=(\vec{\text{a}}\vec{\text{b}}\sin\theta)^2+(\vec{\text{a}}\vec{\text{b}}\cos\theta)^2=\vec{\text{a}}^2\vec{\text{b}}^2$

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Question 131 Mark

Directions: In the following questions, the Assertions (A) and Reason(s) (R) have been put forward. Read both the statements carefully and choose the correct alternative from the following:
Assertion: In $\triangle\text{ABC},\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{CA}}=0.$
Reason: If $\overrightarrow{\text{OA}}=\overrightarrow{\text{a}},\overrightarrow{\text{OB}},\overrightarrow{\text{b}},$ then $\overrightarrow{\text{AB}}=\overrightarrow{\text{a}}+\overrightarrow{\text{b}}.$

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
Assertion is correct but Reason is incorrect.
Both Assertion and Reason are incorrect.

Answer

Assertion is correct but Reason is incorrect.

Solution:

In $\triangle\text{ABC},\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{AC}}=-\overrightarrow{\text{CA}}$

$\Rightarrow\overrightarrow{\text{AB}}+\overrightarrow{\text{BC}}+\overrightarrow{\text{CA}}=\overrightarrow{0}$

$\overrightarrow{\text{OA}}+\overrightarrow{\text{AB}}=\overrightarrow{\text{OB}}$ is the triangle law of addition.

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MCQ 141 Mark

Let $\vec{a}=\hat{i}+\hat{j}-3 \hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}+\hat{k}$.
Assertion (A): Vectors $\vec{a}$ and $\vec{b}$ are perpendicular to each other.
Reason (R) : $\vec{a} \cdot \vec{b}=0$

A
Both (A) and (R) are true and (R) is the correct explanation of (A).
B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
D
(A) is false but (R) is true.

Answer

$
\begin{array}{l}
\text { (a): } \vec{a}=\hat{i}+\hat{j}-3 \hat{k}, \vec{b}=2 \hat{i}+\hat{j}+\hat{k} \\
\begin{aligned}
\vec{a} \cdot \vec{b} & =(\hat{i}+\hat{j}-3 \hat{k}) \cdot(2 \hat{i}+\hat{j}+\hat{k}) \\
& =1 \cdot 2+1 \cdot 1+(-3) \cdot 1=2+1-3=0 \\
\Rightarrow & \cos \theta=0 \Rightarrow \theta=\frac{\pi}{2}
\end{aligned}
\end{array}
$
Hence, $\vec{a}$ and $\vec{b}$ are perpendicular to each other.
Hence, both assertion and reason are true and reason is the correct explanation of assertion.

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MCQ 151 Mark

Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors and $\theta$ be the angle between then.
Assertion (A) : $(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2 \neq|\vec{a}|^2|\vec{b}|^2$
Reason (R) : $\sin ^2 \theta+\cos ^2 \theta=1$

A
Both (A) and (R) are true and (R) is the correct explanation of (A).
B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
✓
(A) is false but (R) is true.

Answer

Correct option: D.

(A) is false but (R) is true.

(d) :
$
\begin{aligned}
(\vec{a} \times \vec{b})^2 & +(\vec{a} \cdot \vec{b})^2=|\vec{a} \times \vec{b}|^2+(\vec{a} \cdot \vec{b})^2 \\
& =(|\vec{a}|| \vec{b} \mid \sin \theta)^2+(|\vec{a}||\vec{b}| \cos \theta)^2=|\vec{a}|^2|\vec{b}|^2
\end{aligned}
$
Hence, Assertion is false.
But $\sin ^2 \theta+\cos ^2 \theta=1$
Hence, reason is true.

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MCQ 161 Mark

Assertion (A) : If $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0},|\vec{a}|=3$, $|\vec{b}|=4,|\vec{c}|=5$, then $\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$ is equal to -25 .
Reason (R) : If $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$, then the angle $\theta$ between $\vec{b}$ and $\vec{c}$ is given by
$
\cos \theta=\frac{|\vec{a}|^2-|\vec{b}|^2-|\vec{c}|^2}{2|\vec{b}||\vec{c}|^2} .
$

A
Both (A) and (R) are true and (R) is the correct explanation of (A).
✓
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
D
(A) is false but (R) is true.

Answer

Correct option: B.

Both (A) and (R) are true but (R) is not the correct explanation of (A).

(b) : We have, $|\vec{a}|=3,|\vec{b}|=4,|\vec{c}|=5$ and
$
\begin{array}{l}
\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0} \Rightarrow(\vec{a}+\vec{b}+\vec{c})^2=0 \\
\Rightarrow|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0 \\
\Rightarrow(3)^2+(4)^2+(5)^2+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0
\end{array}
$
$
\begin{array}{l}
\Rightarrow \vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}=-\frac{1}{2}[9+16+25]=-\frac{1}{2}(50)=-25 \\
\text { Now, } \vec{a}+\vec{b}+\vec{c}=\overrightarrow{0} \Rightarrow \vec{b}+\vec{c}=-\vec{a} \\
\Rightarrow(\vec{b}+\vec{c})^2=(-\vec{a})^2 \Rightarrow|\vec{b}|^2+|\vec{c}|^2+2 \vec{b} \cdot \vec{c}=|\vec{a}|^2 \\
\Rightarrow|\vec{b}|^2+|\vec{c}|^2+2|\vec{b}||\vec{c}| \cos \theta=|\vec{a}|^2 \\
\Rightarrow \cos \theta=\frac{|\vec{a}|^2-|\vec{b}|^2-|\vec{c}|^2}{2|\vec{b}||\vec{c}|}
\end{array}
$
Hence, both assertion and reason are true but reason is not the correct explanation of assertion.

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MCQ 171 Mark

Assertion (A) : The adjacent sides of a parallelogram are along $\vec{a}=\hat{i}+2 \hat{j}$ and $\vec{b}=2 \hat{i}+\hat{j}$. The angle between the diagonals is $150^{\circ}$.
Reason (R) : Two vectors are perpendicular to each other if their dot product is zero.

A
Both (A) and (R) are true and (R) is the correct explanation of (A).
B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
✓
(A) is false but (R) is true.

Answer

Correct option: D.

(A) is false but (R) is true.

(d) : $\vec{a}=\hat{i}+2 \hat{j}, \vec{b}=2 \hat{i}+\hat{j}$
Diagonals of the parallelogram are along $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$.
Now, $\vec{a}+\vec{b}=(\hat{i}+2 \hat{j})+(2 \hat{i}+\hat{j})=3 \hat{i}+3 \hat{j}$
and $\vec{a}-\vec{b}=(\hat{i}+2 \hat{j})-(2 \hat{i}+\hat{j})=-\hat{i}+\hat{j}$
Let $\theta$ be the angle between these vectors, then
$
\begin{array}{l}
\cos \theta=\frac{(3 \hat{i}+3 \hat{j}) \cdot(-\hat{i}+\hat{j})}{\sqrt{9+9} \sqrt{1+1}}=\frac{-3+3}{\sqrt{18} \sqrt{2}}=0 \\
\Rightarrow \theta=90^{\circ}
\end{array}
$
Hence, assertion is false and reason is true.

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MCQ 181 Mark

Assertion (A) : The projection of the vector $3 \hat{i}-\hat{j}-2 \hat{k}$ on the vector $\hat{i}+2 \hat{j}-3 \hat{k}$ is $\frac{7}{\sqrt{14}}$.
Reason (R) : The projection of a vector $\vec{a}$ on another vector $\vec{b}$ is $\frac{(\vec{a} \cdot \vec{b})}{|\vec{b}|}$.

✓
Both (A) and (R) are true and (R) is the correct explanation of (A).
B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
D
(A) is false but (R) is true.

Answer

Correct option: A.

Both (A) and (R) are true and (R) is the correct explanation of (A).

(a) : Required projection $=\frac{(3 \hat{i}-\hat{j}-2 \hat{k}) \cdot(\hat{i}+2 \hat{j}-3 \hat{k})}{\sqrt{1^2+2^2+(-3)^2}}$
$
\frac{3-2+6}{\sqrt{1+4+9}}=\frac{7}{\sqrt{14}}
$
Also, projection of vector $\vec{a}$ on $\vec{b}=(\vec{a} \cdot \hat{b})=\left(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}\right)$
Hence, both assertion and reason are true and reason is the correct explanation of assertion.

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MCQ 191 Mark

Assertion (A) : The unit vector in the direction of sum of the vectors $\hat{i}+\hat{j}+\hat{k}, 2 \hat{i}-\hat{j}-\hat{k}$ and $2 \hat{j}+6 \hat{k}$ is $\frac{1}{7}(3 \hat{i}-2 \hat{j}+6 \hat{k})$.
Reason (R) : Let $\vec{a}$ be a non-zero vector, then $\frac{\vec{a}}{|\vec{a}|}$ is a unit vector parallel to $\vec{a}$.

A
Both (A) and (R) are true and (R) is the correct explanation of (A).
B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
✓
(A) is false but (R) is true.

Answer

Correct option: D.

(A) is false but (R) is true.

(d) : Sum of the given vectors
$
=(\hat{i}+\hat{j}+\hat{k})+(2 \hat{i}-\hat{j}-\hat{k})+(2 \hat{j}+6 \hat{k})=3 \hat{i}+2 \hat{j}+6 \hat{k}
$
$\therefore \quad$ The unit vector in the direction of the sum of the given vectors is given by
$
=\frac{3 \hat{i}+2 \hat{j}+6 \hat{k}}{\sqrt{3^2+2^2+6^2}}=\frac{3 \hat{i}+2 \hat{j}+6 \hat{k}}{\sqrt{9+4+36}}=\frac{1}{7}(3 \hat{i}+2 \hat{j}+6 \hat{k})
$
Hence, Assertion is false.
Also, $\frac{\vec{a}}{|\vec{a}|}$ is a unit vector which is parallel to $\vec{a}$.
Hence, Reason is true.

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MCQ 201 Mark

Assertion (A) : Three points with position vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are collinear if $\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=\overrightarrow{0}$
Reason (R): If $\overrightarrow{A B} \cdot \overrightarrow{A C}=0$, then $\overrightarrow{A B} \perp \overrightarrow{A C}$.

A
Both (A) and (R) are true and (R) is the correct explanation of (A).
✓
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
D
(A) is false but (R) is true.

Answer

Correct option: B.

Both (A) and (R) are true but (R) is not the correct explanation of (A).

(b) : If $A, B, C$ are collinear, then $\overrightarrow{A B}=k \overrightarrow{A C}$
$
\begin{array}{l}
\therefore \overrightarrow{A B} \times \overrightarrow{A C}=\overrightarrow{0} \Rightarrow(\vec{b}-\vec{a}) \times(\vec{c}-\vec{a})=\overrightarrow{0} \\
\Rightarrow \vec{b} \times \vec{c}+\vec{a} \times \vec{b}+\vec{c} \times \vec{a}=\overrightarrow{0} \text { i.e., } \vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=\overrightarrow{0}
\end{array}
$
Hence, both assertion and reason are true but reason is not the correct explanation of assertion.

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MCQ 211 Mark

Assertion (A) : If the points $\vec{P}=(\vec{a}+\vec{b}-\vec{c})$, $\vec{Q}=(2 \vec{a}+\vec{b})$ and $\vec{R}=(\vec{b}+t \vec{c})$ are collinear, where $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar vectors, then the value of $t$ is -2 .
Reason (R) : If $P, Q, R$ are collinear, then
$
\overrightarrow{P Q} \| \overrightarrow{P R} \text { or } \overrightarrow{P Q}=\lambda \overrightarrow{P R}, \lambda \in R
$

✓
Both (A) and (R) are true and (R) is the correct explanation of (A).
B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
D
(A) is false but (R) is true.

Answer

Correct option: A.

Both (A) and (R) are true and (R) is the correct explanation of (A).

(a) : If $P, Q, R$ are collinear, then $\overrightarrow{P Q} \| \overrightarrow{P R}$ or $\overrightarrow{P Q}=\lambda \overrightarrow{P R}, \lambda \in R$
$
\begin{array}{l}
\Rightarrow \quad(2 \vec{a}+\vec{b})-(\vec{a}+\vec{b}-\vec{c})=\lambda[(\vec{b}+t \vec{c})-(\vec{a}+\vec{b}-\vec{c})] \\
\Rightarrow \quad(\vec{a}+\vec{c})=\lambda[-\vec{a}+(t+1) \vec{c}] \\
\Rightarrow \quad \vec{a}+\vec{c}=-\lambda \vec{a}+\lambda(t+1) \vec{c}
\end{array}
$
On comparing, we get
$
\begin{array}{l}
-\lambda=1 \Rightarrow \lambda=-1 \\
\text { and } \lambda(t+1)=1 \Rightarrow-(t+1)=1 \\
\Rightarrow-t-1=1 \Rightarrow t=-2
\end{array}
$
Hence, both assertion are true and reason is the correct explanation of assertion.

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MCQ 221 Mark

Assertion (A) : The magnitude of resultant of vectors $\vec{a}=2 \hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}+3 \hat{k}$ is $\sqrt{34}$.
Reason (R) : The magnitude of a vector can never be negative.

A
Both (A) and (R) are true and (R) is the correct explanation of (A).
✓
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
D
(A) is false but (R) is true.

Answer

Correct option: B.

Both (A) and (R) are true but (R) is not the correct explanation of (A).

(b) : $\vec{a}=2 \hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}+2 \hat{j}+3 \hat{k}$
Addition of $\vec{a}$ and $\vec{b}$ is $\vec{a}+\vec{b}$
$
\begin{array}{l}
\vec{a}+\vec{b}=(2 \hat{i}+\hat{j}+\hat{k})+(\hat{i}+2 \hat{j}+3 \hat{k})=3 \hat{i}+3 \hat{j}+4 \hat{k} \\
\therefore|\vec{a}+\vec{b}|=\sqrt{3^2+3^2+4^2}=\sqrt{9+9+16}=\sqrt{34}
\end{array}
$
Also, the magnitude of a vector can never be negative. Hence, both Assertion and Reason are correct but Reason is not the correct explanation of Assertion.

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MCQ 231 Mark

Assertion (A) : If $(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=400$ and $|\vec{a}|=4$, then $|\vec{b}|=9$.
Reason (R) : If $\vec{a}$ and $\vec{b}$ are any two vectors, then $(\vec{a} \times \vec{b})^2$ is equal to $|\vec{a}|^2|\vec{b}|^2-(\vec{a} \cdot \vec{b})^2$.

A
Both (A) and (R) are true and (R) is the correct explanation of (A).
B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
✓
(A) is false but (R) is true.

Answer

Correct option: D.

(A) is false but (R) is true.

(d): $(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=400,|\vec{a}|=4$
Now, $(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=|\vec{a}|^2|\vec{b}|^2$
$\Rightarrow 400=(4)^2|\vec{b}|^2 \Rightarrow 16|\vec{b}|^2=400$
$\Rightarrow|\vec{b}|^2=25 \Rightarrow|\vec{b}|=5$
Hence, assertion is false.
$
\begin{array}{l}
(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=|\vec{a} \times \vec{b}|^2+(\vec{a} \cdot \vec{b})^2 \\
=(|\vec{a}||\vec{b}| \sin \theta)^2+(|\vec{a}||\vec{b}| \cos \theta)^2=|\vec{a}|^2|\vec{b}|^2 \\
\Rightarrow(\vec{a} \times \vec{b})^2=|\vec{a}|^2|\vec{b}|^2-(\vec{a} \cdot \vec{b})^2
\end{array}
$
Hence, reason is true.

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