Maths Assertion (A) & Reason (B) MCQ

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

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

$\Rightarrow\theta=90^\circ$

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

Solution:

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

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

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

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

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

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

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

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

2. 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 $\hat{\text{a}}+\hat{\text{b}}$

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

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

4. 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.

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

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

2.  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 $\overrightarrow{\text{AB}}=\overrightarrow{\text{kAC}}$

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

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

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

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

4.  Assertion is wrong statement but Reason is correct statement.

Solution:

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

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

3. 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.

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

Solution:

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

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

4. 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

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