MCQ 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 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.
- 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.
- C
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
AnswerCorrect option: B. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
$\overline{\text{a}}=2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\overline{\text{b}}=\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}},$
Resultant of $\hat{\text{a}}$ and $\hat{\text{b}}$ is $\hat{\text{a}}+\hat{\text{b}}$
$=(2\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})+(\hat{\text{i}}+2\hat{\text{j}}+3\hat{\text{k}})=3\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}}$
$\therefore|\overline{\text{a}}+\overline{\text{b}}|=\sqrt{3^2+3^2+4^2}=\sqrt{9+9+16}=\sqrt{34}$
View full question & answer→MCQ 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 :
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.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
AnswerCorrect option: A. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
$\vec{\text{a}}=\hat{\text{i}}+2\hat{\text{j}},\vec{\text{b}}=2\hat{\text{i}}+\hat{\text{j}}$
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
$\cos\theta=\frac{(3\text{i}+3\text{j})(\hat{-\text{i}}+\hat{\text{j}})}{\sqrt{9+9}\sqrt{1+1}}=\frac{-3+3}{\sqrt{18}\sqrt{2}}=0$
$\Rightarrow\theta=90^\circ$
View full question & answer→MCQ 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 :
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.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
AnswerCorrect option: A. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
$\overline{\text{a}}=\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}},\overline{\text{b}}=\hat{\text{2i}}+\hat{\text{j}}-\hat{\text{k}}$
$\overline{\text{a}}.\overline{\text{b}}=(\hat{\text{i}}+\hat{\text{j}}-3\hat{\text{k}}).(2\hat{\text{i}}+\text{j}+\hat{\text{k}})$
$=1.2+1.1+(-3).1=2+1-3=0$
$\Rightarrow\cos\theta=0$
$\Rightarrow\theta=\frac{\pi}{2}$
View full question & answer→MCQ 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 : 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 $(\vec{\text{a}})^2(\vec{\text{b}})^2-(\vec{\text{a}}.\vec{\text{b}})^2.$
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- ✓
Assertion is wrong statement but Reason is correct statement.
AnswerCorrect option: D. Assertion is wrong statement but Reason is correct statement.
$(\vec{\text{a}}\times\vec{\text{b}})^2+(\vec{\text{a}}.\vec{\text{b}})^2=400,|\vec{\text{a}}|=4$
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.
$(\vec{\text{a}}\times\vec{\text{b}})^2+(\vec{\text{a}}.\vec{\text{b}})^2=|\vec{\text{a}}\times\vec{\text{b}}|^2+(\vec{\text{a}}.\vec{\text{b}})^2$
$=(\vec{\text{a}}\vec{\text{b}}\sin\theta)^2+(\vec{\text{a}}\vec{\text{b}}\cos\theta)^2=\vec{\text{a}}^2\vec{\text{b}}^2$
View full question & answer→MCQ 51 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 $\overrightarrow{\text{AB}}\bot\overrightarrow{\text{AC}}.$
- 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.
- C
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
AnswerCorrect option: B. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
If $\text{A},\text{B},\text{C}$ are collinear, then $\overrightarrow{\text{AB}}=\overrightarrow{\text{k AC}}$
$\therefore\overrightarrow{\text{AB}}\ \times\overrightarrow{\text{AC}}=\vec{0}$
$\Rightarrow(\vec{\text{b}}-\vec{\text{a}})\times(\vec{\text{c}}-\vec{\text{a}})=0$
$\Rightarrow\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{a}}\times\vec{\text{b}}+\vec{\text{c}}\times\vec{\text{a}}=\vec{0}$
$\text{i.e}..,\ \vec{\text{a}}\times\vec{\text{b}}+\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{c}}\times\vec{\text{a}}=\vec{0}$
View full question & answer→MCQ 61 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 : $\sin^2\theta+\cos^2\theta=0$
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- ✓
Assertion is wrong statement but Reason is correct statement.
AnswerCorrect option: D. Assertion is wrong statement but Reason is correct statement.
$(\vec{\text{a}}\times\vec{\text{b}})^2+(\vec{\text{a}}.\vec{\text{b}})^2=|\vec{\text{a}}\times\vec{\text{b}}|^2+(\vec{\text{a}}.\vec{\text{b}})$
$=(\vec{\text{a}}\vec{\text{b}}\sin\theta)^2+(\vec{\text{a}}\vec{\text{b}}\cos\theta)^2=\vec{\text{a}}^2\vec{\text{b}}^2$
View full question & answer→MCQ 71 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 $\overrightarrow{\text{AB}}=\overrightarrow{\text{a}}+\overrightarrow{\text{b}}.$
- A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion.
- B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.
- ✓
Assertion is correct but Reason is incorrect.
- D
Both Assertion and Reason are incorrect.
AnswerCorrect option: C. Assertion is correct but Reason is incorrect.
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.
View full question & answer→MCQ 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 $\cos\theta=\frac{\vec{\text{a}}^2-\vec{\text{b}}^2-\vec{\text{c}}^2}{2\vec{\text{b}}{\vec{\text{c}}}}$
- 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.
- C
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
AnswerCorrect option: B. Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
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, $\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}=0$
$\Rightarrow\vec{\text{b}}+\vec{\text{c}}=-\vec{\text{a}}$
$\Rightarrow|\vec{\text{b}}+\vec{\text{c}}|^2=|-\vec{\text{a}}|^2$
$\Rightarrow\vec{\text{b}}^2+\vec{\text{c}}^2+2\vec{\text{b}}.\vec{\text{c}}=\vec{\text{a}}^2$
$\Rightarrow\vec{\text{b}}^2+\vec{\text{c}}^2+2\vec{\text{b}}\vec{\text{c}}\cos\theta=\vec{\text{a}}^2$
$\Rightarrow\cos\theta\frac{\vec{\text{a}}^2-\vec{\text{b}}^2-\vec{\text{c}}^2}{2\vec{\text{b}}\vec{\text{c}}}$
View full question & answer→MCQ 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.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- D
Assertion is wrong statement but Reason is correct statement.
AnswerCorrect option: A. Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
Projection of $\vec{\text{a}}$ on $\vec{\text{b}}$
$=\frac{\vec{\text{a}}\vec{\text{b}}}{\sqrt{|\vec{\text{b}}|}}$
$\frac{(2\hat{\text{i}}+3\hat{\text{j}}-\hat{\text{k}})(-\hat{\text{i}}+3\hat{\text{j}}+4\hat{\text{k}})}{\sqrt{(-1)^2+(3)^2+(4)^2}}$
$=\frac{-2+9-4}{\sqrt{26}}=\frac{3}{\sqrt{26}}$
View full question & answer→MCQ 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 $\overline{\text{a}}$.
- A
Assertion and Reason both are correct statements and Reason is the correct explanation of Assertion.
- B
Assertion and Reason both are correct statements but Reason is not the correct explanation of Assertion.
- C
Assertion is correct statement but Reason is wrong statement.
- ✓
Assertion is wrong statement but Reason is correct statement.
AnswerCorrect option: D. Assertion is wrong statement but Reason is correct statement.
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
$=\frac{3\hat{\text{i}}+2\hat{\text{j}}+6\hat{{\text{k}}}}{\sqrt{3^2+\text{2}^2+6^2}}=\frac{3\hat{\text{i}}+2\hat{\text{j}}+6\hat{{\text{k}}}}{\sqrt{9+4+36}}=\frac{1}{7}(3\hat{\text{i}}+2\hat{\text{j}}+6\hat{\text{k}})$
View full question & answer→