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MATHS Assertion (A) & Reason (B) MCQ

VECTOR ALGEBRA · Rajasthan Board (RBSE) STD 12 Science (Rajasthan - English Medium). 10 questions.

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Assertion (A) & Reason (B) MCQ

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Question 11 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.
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
Answer
  1. 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 21 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: $\overline{\text{a}}.\overline{\text{b}}=0$
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
Answer
  1. 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 31 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 $\cos\theta=\frac{\vec{\text{a}}^2-\vec{\text{b}}^2-\vec{\text{c}}^2}{2\vec{\text{b}}{\vec{\text{c}}}}$
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
Answer
  1. 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 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.
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
Answer
  1. 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: 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 $(\vec{\text{a}})^2(\vec{\text{b}})^2-(\vec{\text{a}}.\vec{\text{b}})^2.$
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion. 
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion. 
  3. Assertion is correct statement but Reason is wrong statement. 
  4. Assertion is wrong statement but Reason is correct statement.
Answer
  1. 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 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 $\overrightarrow{\text{AB}}\bot\overrightarrow{\text{AC}}.$
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
Answer
  1.  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: $\sin^2\theta+\cos^2\theta=0$
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion. 
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion. 
  3. Assertion is correct statement but Reason is wrong statement. 
  4. Assertion is wrong statement but Reason is correct statement.
Answer
  1.  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 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 $\overrightarrow{\text{AB}}=\overrightarrow{\text{a}}+\overrightarrow{\text{b}}.$
  1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
  2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
  3. Assertion is correct but Reason is incorrect.
  4. Both Assertion and Reason are incorrect.
Answer
  1. 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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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 $\vec{\text{b}}=\frac{3}{\sqrt{26}}$
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
Answer
  1.  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:
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 $\overline{\text{a}}$.
  1. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
  2. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
  3. Assertion is correct statement but Reason is wrong statement.
  4. Assertion is wrong statement but Reason is correct statement.
Answer
  1. 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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