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 $\overrightarrow{\text{AB}}=\overrightarrow{\text{kAC}}$
$\therefore \overrightarrow{\mathrm{AB}} \times \overrightarrow{\mathrm{AC}}=\overrightarrow{0}$
$\Rightarrow(\vec{\text{b}}-\vec{\text{a}})\times(\vec{\text{c}}-\vec{\text{a}})=0$
$\Rightarrow\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{a}}\times\vec{\text{b}}+\vec{\text{c}}\times\vec{\text{a}}=\vec{0}$
$\text{i.e}..,\ \vec{\text{a}}\times\vec{\text{b}}+\vec{\text{b}}\times\vec{\text{c}}+\vec{\text{c}}\times\vec{\text{a}}=\vec{0}$
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→MCQ 111 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.
- ✓
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.
AnswerCorrect option: A. Both Assertion $(A)$ and Reason $(R)$ are true and the Reason $(R)$ is the correct explanation of the Assertion $(A).$
Given, $\vec{a}=6 \hat{i}+2 \hat{j}-8 \hat{k}, \vec{b}=10 \hat{i}-2 \hat{j}-6 \hat{k}$ and $\vec{c}=4 \hat{i}-4 \hat{j}+2 \hat{k}$
$\Rightarrow \vec{a}+\vec{c} =(6 \hat{i}+2 \hat{j}-8 \hat{k})+(4 \hat{i}-4 \hat{j}+2 \hat{k})$
$ =10 \hat{i}-2 \hat{j}-6 \hat{k}=\vec{b}$
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
$\cos \theta=\frac{\vec{a} \cdot \vec{c}}{|\vec{a}| \cdot|\vec{c}|}$
$=\frac{6 \cdot 4+2(-4)+(-8)(2)}{\sqrt{6^2+2^2+(-8)^2} \sqrt{4^2+(-4)^2+2^2}}$
$=\frac{24-8-16}{\sqrt{104} \sqrt{36}}=0$
$\Rightarrow \cos \theta=\cos 90^{\circ}$
$\Rightarrow \theta=90^{\circ}$
$\therefore$ Assertion $(A)$ is true.
$\therefore$ Assertion $( A )$ is true.
Hence, both Assertion $(A)$ and Reason $(R)$ are true and Reason $(R)$ is the correct explanation of Assertion $(A).$
View full question & answer→MCQ 121 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.
AnswerAssertion (A) is true but reason $( R )$ is false. As, $\vec{a} \times \vec{b}=-\vec{b} \times \vec{a}$.
View full question & answer→MCQ 131 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.
View full question & answer→MCQ 141 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.
AnswerCorrect option: B. Both $(A)$ and $(R)$ are true but $(R)$ is not the correct explanation of $(A)$.
If $A, B, C$ are collinear, then $\overrightarrow{A B}=k \overrightarrow{A C}$
$\therefore \overrightarrow{A B} \times \overrightarrow{A C}=\overrightarrow{0} \Rightarrow(\vec{b}-\vec{a}) \times(\vec{c}-\vec{a})=\overrightarrow{0}$
$\Rightarrow \vec{b} \times \vec{c}+\vec{a} \times \vec{b}+\vec{c} \times \vec{a}=\overrightarrow{0} \text { i.e., } \vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}=\overrightarrow{0}$
Hence, both assertion and reason are true but reason is not the correct explanation of assertion.
View full question & answer→MCQ 151 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.
AnswerCorrect option: A. Both $(A)$ and$ (R)$ are true and $(R)$ is the correct explanation of $(A).$
If $P, Q, R$ are collinear, then $\overrightarrow{P Q} \| \overrightarrow{P R}$ or $\overrightarrow{P Q}=\lambda \overrightarrow{P R}, \lambda \in R$
$\Rightarrow (2 \vec{a}+\vec{b})-(\vec{a}+\vec{b}-\vec{c})=\lambda[(\vec{b}+t \vec{c})-(\vec{a}+\vec{b}-\vec{c})]$
$\Rightarrow (\vec{a}+\vec{c})=\lambda[-\vec{a}+(t+1) \vec{c}]$
$\Rightarrow \vec{a}+\vec{c}=-\lambda \vec{a}+\lambda(t+1) \vec{c}$
On comparing, we get $-\lambda=1$
$\Rightarrow \lambda=-1$
and $\lambda(t+1)=1$
$\Rightarrow-(t+1)=1$
$\Rightarrow-t-1=1$
$\Rightarrow t=-2$
Hence, both assertion are true and reason is the correct explanation of assertion.
View full question & answer→MCQ 161 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.
AnswerCorrect option: D. $(A)$ is false but $(R)$ is true.
$(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=400,|\vec{a}|=4$
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.
$(\vec{a} \times \vec{b})^2+(\vec{a} \cdot \vec{b})^2=|\vec{a} \times \vec{b}|^2+(\vec{a} \cdot \vec{b})^2$
$=(|\vec{a}||\vec{b}| \sin \theta)^2+(|\vec{a}||\vec{b}| \cos \theta)^2=|\vec{a}|^2|\vec{b}|^2$
$\Rightarrow(\vec{a} \times \vec{b})^2=|\vec{a}|^2|\vec{b}|^2-(\vec{a} \cdot \vec{b})^2$
Hence, reason is true.
View full question & answer→MCQ 171 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.
AnswerCorrect 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.
View full question & answer→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.
AnswerCorrect 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.
View full question & answer→MCQ 191 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.
AnswerCorrect option: B. Both $(A)$ and $(R)$ are true but $(R)$ is not the correct explanation of $(A).$
We have, $|\vec{a}|=3,|\vec{b}|=4,|\vec{c}|=5$ and
$\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$
$\Rightarrow(\vec{a}+\vec{b}+\vec{c})^2=0$
$\Rightarrow|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0$
$\Rightarrow(3)^2+(4)^2+(5)^2+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0$
$\Rightarrow \vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}=-\frac{1}{2}[9+16+25]=-\frac{1}{2}(50)=-25$
Now, $\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$
$\Rightarrow \vec{b}+\vec{c}=-\vec{a}$
$\Rightarrow(\vec{b}+\vec{c})^2=(-\vec{a})^2$
$\Rightarrow|\vec{b}|^2+|\vec{c}|^2+2 \vec{b} \cdot \vec{c}=|\vec{a}|^2$
$\Rightarrow|\vec{b}|^2+|\vec{c}|^2+2|\vec{b}||\vec{c}| \cos \theta=|\vec{a}|^2$
$\Rightarrow \cos \theta=\frac{|\vec{a}|^2-|\vec{b}|^2-|\vec{c}|^2}{2|\vec{b}||\vec{c}|}$
Hence, both assertion and reason are true but reason is not the correct explanation of assertion.
View full question & answer→MCQ 201 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.
AnswerCorrect option: D. $(A)$ is false but $(R)$ is true.
$\vec{a}=\hat{i}+2 \hat{j}, \vec{b}=2 \hat{i}+\hat{j}$
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
$\cos \theta=\frac{(3 \hat{i}+3 \hat{j}) \cdot(-\hat{i}+\hat{j})}{\sqrt{9+9} \sqrt{1+1}}=\frac{-3+3}{\sqrt{18} \sqrt{2}}=0$
$\Rightarrow \theta=90^{\circ}$
Hence, assertion is false and reason is true.
View full question & answer→MCQ 211 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 $(R) : \vec{a} \cdot \vec{b}=0$
- ✓
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.
AnswerCorrect option: A. Both $(A)$ and $(R)$ are true and $(R)$ is the correct explanation of $(A).$
$\vec{a}=\hat{i}+\hat{j}-3 \hat{k}, \vec{b}=2 \hat{i}+\hat{j}+\hat{k}$
$\vec{a} \cdot \vec{b} =(\hat{i}+\hat{j}-3 \hat{k}) \cdot(2 \hat{i}+\hat{j}+\hat{k})$
$ =1 \cdot 2+1 \cdot 1+(-3) \cdot 1$
$=2+1-3$
$=0$
$\Rightarrow \cos \theta=0$
$\Rightarrow \theta=\frac{\pi}{2}$
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.
View full question & answer→MCQ 221 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.
AnswerCorrect 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.
View full question & answer→MCQ 231 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.
AnswerCorrect option: B. Both $(A)$ and $(R)$ are true but $(R)$ is not the correct explanation of $(A)$.
$\vec{a}=2 \hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}+2 \hat{j}+3 \hat{k}$
Addition of $\vec{a}$ and $\vec{b}$ is $\vec{a}+\vec{b}$
$\vec{a}+\vec{b}=(2 \hat{i}+\hat{j}+\hat{k})+(\hat{i}+2 \hat{j}+3 \hat{k})=3 \hat{i}+3 \hat{j}+4 \hat{k}$
$\therefore|\vec{a}+\vec{b}|=\sqrt{3^2+3^2+4^2}=\sqrt{9+9+16}=\sqrt{34}$
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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