Assertion (A) & Reason (B) MCQ

MCQ 11 Mark

Assertion (A): The lines $\vec{r}=\vec{a}_1+\lambda \vec{b}_1$ and $\vec{r}=\vec{a}_2+\mu \vec{b}_2$ are perpendicular, when $\vec{b}_1 \cdot \vec{b}_2=0$.
Reason (R): The angle $\theta$ between the lines $\vec{r}=\vec{a}_1+\lambda \vec{b}_1$ and $\vec{r}=\vec{a}_2+\mu \vec{b}_2$ is given by $\cos \theta=\frac{\vec{b}_1 \cdot \vec{b}_2}{\left|\vec{b}_1\right|\left|\vec{b}_2\right|}$.

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

Answer

if lines are perpendicular, then $\theta=\frac{\pi}{2}$
$\begin{array}{l}
\therefore \quad \cos \frac{\pi}{2}=\frac{\vec{b}_1 \cdot \vec{b}_2}{\left|\vec{b}_1\right| \vec{b}_2 \mid} \Rightarrow \cos \frac{\pi}{2}=\frac{\vec{b}_1 \cdot \vec{b}_2}{\left|\vec{b}_1\right| \vec{b}_2 \mid} \\
\Rightarrow \quad \vec{b}_1 \cdot \vec{b}_2=0
\end{array}$
$\therefore \quad$ Both $A$ and $R$ are true and $R$ is the correct explanation of $A$.

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MCQ 21 Mark

Assertion (A) : Quadrilateral formed by vertices $A(0,0,0), B(3,4,5), C(8,8,8)$ and $D(5,4,3)$ is a rhombus. Reason $(R): A B C D$ is a rhombus if $A B=B C=C D=D A$, $A C \neq B D$.

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

Given, $A(0,0,0), B(3,4,5), C(8,8,8)$ and $D(5,4,3)$
$
\begin{array}{l}
A B=\sqrt{3^2+4^2+5^2}=\sqrt{9+16+25}=\sqrt{50}=5 \sqrt{2} \text { units } \\
B C=\sqrt{(8-3)^2+(8-4)^2+(8-5)^2}=\sqrt{25+16+9}=\sqrt{50} \\
=5 \sqrt{2} \text { units } \\
C D=\sqrt{(5-8)^2+(4-8)^2+(3-8)^2}=\sqrt{9+16+25}=\sqrt{50}=5 \sqrt{2} \text { units } \\
D A=\sqrt{5^2+4^2+3^2}=\sqrt{25+16+9}=\sqrt{50}=5 \sqrt{2} \text { units } \\
A C=\sqrt{8^2+8^2+8^2}=\sqrt{3 \times 8^2}=8 \sqrt{3} \text { units } \\
B D=\sqrt{(5-3)^2+(4-4)^2+(3-5)^2}=\sqrt{4+0+4}=\sqrt{8}=2 \sqrt{2} \text { units } \\
\therefore \quad A B=B C=C D=D A, A C \neq B D
\end{array}$
Hence, both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

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MCQ 31 Mark

Assertion (A) : The lines $\vec{r}=\vec{a}_1+\lambda \vec{b}_1$ and $\vec{r}=\vec{a}_2+\mu \vec{b}_2$ are perpendicular, when $\vec{b}_1 \cdot \vec{b}_2=0$.
Reason (R): The angle $\theta$ between the lines $\vec{r}=\vec{a}_1+\lambda \vec{b}_1$ and $\vec{r}=\vec{a}_2+\mu \vec{b}_2$ is given by $\cos \theta=\frac{\vec{b}_1 \cdot \vec{b}_2}{\left|\vec{b}_1\right|\left|\vec{b}_2\right|}$.

✓
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 lines are perpendicular, then $\theta=\frac{\pi}{2}$
$
\therefore \quad \cos \frac{\pi}{2}=\frac{\vec{b}_1 \cdot \vec{b}_2}{\left|\vec{b}_1\right| \vec{b}_2 \mid} \Rightarrow \cos \frac{\pi}{2}=\frac{\vec{b}_1 \cdot \vec{b}_2}{\left|\vec{b}_1\right| \vec{b}_2 \mid}
$
$
\Rightarrow \quad \vec{b}_1 \cdot \vec{b}_2=0
$
$\therefore \quad$ Both $A$ and $R$ are true and $R$ is the correct explanation of $A$.

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MCQ 41 Mark

Assertion (A): The pair of lines given by $\vec{r}=\hat{i}-\hat{j}+\lambda(2 \hat{i}+\hat{k})$ and $\vec{r}=2 \hat{i}-\hat{k}+\mu(\hat{i}+\hat{j}-\hat{k})$ intersect.
Reason (R) : Two lines intersect each other, if they are not parallel and shortest distance $=0$.

✓
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.

Answer

Correct option: A.

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

(a): Here, $\vec{a}_1=\hat{i}-\hat{j}, \vec{b}_1=2 \hat{i}+\hat{k}$
$\vec{a}_2=2 \hat{i}-\hat{k}$ and $\vec{b}_2=\hat{i}+\hat{j}-\hat{k}$
$\because \quad \vec{b}_1 \neq k \vec{b}_2$, for any scalar $k$
$\therefore \quad$ Given lines are not parallel.
Now, $\vec{a}_2-\vec{a}_1=(2 \hat{i}-\hat{k})-(\hat{i}-\hat{j})=\hat{i}+\hat{j}-\hat{k}$
and $\vec{b}_1 \times \vec{b}_2=-\hat{i}+3 \hat{j}+2 \hat{k}$
$
\begin{array}{l}
\Rightarrow\left|\vec{b}_1 \times \vec{b}_2\right|=\sqrt{(-1)^2+(3)^2+(2)^2}=\sqrt{1+9+4}=\sqrt{14} \\
\therefore \text { S.D. }=\left|\frac{\left(\vec{a}_2-\vec{a}_1\right) \cdot\left(\vec{b}_1 \times \vec{b}_2\right)}{\left|\vec{b}_1 \times \vec{b}_2\right|}\right|=\left|\frac{(\hat{i}+\hat{j}-\hat{k}) \cdot(-\hat{i}+3 \hat{j}+2 \hat{k})}{\sqrt{14}}\right|=0
\end{array}
$
Hence, two lines intersect each other.
Two lines intersect each other, if they are not parallel and shortest distance $=0$.

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MCQ 51 Mark

Assertion (A) : The points $(1,2,3),(-2,3,4)$ and $(7,0,1)$ are collinear.
Reason (R): If the points $\left(x_1, y_1, z_1\right),\left(x_2, y_2, z_2\right)$ and $\left(x_3, y_3, z_3\right)$ are collinear, then
$
\frac{x_2-x_1}{x_3-x_2}=\frac{y_2-y_1}{y_3-y_2}=\frac{z_2-z_1}{z_3-z_2} .
$

✓
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): We have, $x_1=1, y_1=2, z_1=3$;
$x_2=-2, y_2=3, z_2=4 \text { and } x_3=7, y_3=0, z_3=1$
Now, $\frac{x_2-x_1}{x_3-x_2}=\frac{y_2-y_1}{y_3-y_2}=\frac{z_2-z_1}{z_3-z_2}$
$
\begin{array}{l}
\Rightarrow \frac{-2-1}{7-(-2)}=\frac{3-2}{0-3}=\frac{4-3}{1-4} \\
\Rightarrow \frac{-3}{9}=\frac{1}{-3}=\frac{1}{-3} \Rightarrow \frac{-1}{3}=\frac{-1}{3}=\frac{-1}{3}
\end{array}
$
$\therefore \quad$ Given points are collinear.
Hence, both assertion and reason are true and reason is the correct explanation of assertion.

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MCQ 61 Mark

Assertion (A) : If the cartesian equation of a line is $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$, then its vector form is $\vec{r}=5 \hat{i}-4 \hat{j}+6 \hat{k}+\lambda(3 \hat{i}+7 \hat{j}+2 \hat{k})$.
Reason (R): The vector equation of line passing through the points $A(\vec{a})$ and parallel to vector $\vec{b}$ is $\vec{r}=\vec{a}-\lambda(\vec{a}-\vec{b})$, where $\lambda \in R$ is a parameter.

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

Answer

Correct option: C.

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

(c) : In assertion the given cartesian equation is
$
\begin{array}{l}
\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2} . \\
\Rightarrow \vec{a}=5 \hat{i}-4 \hat{j}+6 \hat{k} \text { and } \vec{b}=3 \hat{i}+7 \hat{j}+2 \hat{k} .
\end{array}
$
The vector equation of the line is given by $\vec{r}=\vec{a}+\lambda \vec{b}, \lambda \in R$.
$
\Rightarrow \quad \vec{r}=5 \hat{i}-4 \hat{j}+6 \hat{k}+\lambda(3 \hat{i}+7 \hat{j}+2 \hat{k})
$
Thus assertion is true and reason is false.

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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:
Assertion: If the cartesian equation of a line is $\frac{\text{x}-5}{3}=\frac{\text{y}+4}{7}=\frac{\text{z}-6}{2},$ then its vector form is $\vec{\text{r}}=5\hat{\text{i}}-4\hat{\text{j}}+6\hat{\text{k}}+\lambda(3\hat{\text{i}}+7\hat{\text{j}}+2\hat{\text{k}}).$
Reason: The cartesian equation of the line which passes through the point (-2, 4, -5) and parallel to the line given by $\frac{\text{x}+3}{3}=\frac{\text{y}-4}{5}=\frac{\text{z}+8}{6}$ is $\frac{\text{x}+3}{-2}=\frac{\text{y}-4}{4}=\frac{\text{z}+8}{-5}.$

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 correct statement but Reason is wrong statement.

Solution:

In assertion the given cartesian equation is

$\frac{\text{x}-5}{3}=\frac{\text{y}+4}{7}=\frac{\text{z}-6}{2},$

$\Rightarrow\vec{\text{a}}=5\hat{\text{i}}-4\hat{\text{j}}+6\hat{\text{k}}$ and $\vec{\text{b}}=3\hat{\text{i}}+7\hat{\text{j}}+2\hat{\text{k}}$

The vector equation of the line is given by $\vec{\text{r}}=\vec{\text{a}}+\lambda\vec{\text{b}},\lambda\in\text{R}.$

$\Rightarrow\vec{\text{r}}=5\hat{\text{i}}-4\hat{\text{j}}+6\hat{\text{k}}+\lambda(3\hat{\text{i}}+7\hat{\text{j}}+2\text{k})$

Thus Assertion is correct. In reason it is given that the line passes through the point (-2, 4, -5) and is parallel to

Clearly, the direction ratios of line are (3, 5, 6). Now the equation of the line (in cartesian form) is

$\frac{\text{x}-(-2)}{3}=\frac{\text{y}-4}{5}=\frac{\text{z}-(-5)}{6}$

$\Rightarrow\frac{\text{x}+2}{3}=\frac{\text{y}-4}{5}=\frac{\text{z}+5}{6}$

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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: Points A(4, 0, 4), B(1, 2, 3), C(-2, 4, 2) are collinear.
Reason: Three points A, B, C are collinear if AB + BC = AC and AB, BC < AC.

Both Assertion & Reason are individually true & Reason is correct explanation of Assertion.
Both Assertion & Reason are individually true but Reason is not the, correct (proper) explanation of Assertion.
Assertion is true but Reason is false.
Assertion is false but Reason is true.

Answer

Both Assertion & Reason are individually true & Reason is correct explanation of Assertion.

Solution:

Points A(4, 0, 4), B(1, 2, 3), C(-2, 4, 2) are collinear formula to check whether these three points are collinear or not AB + BC = AC to find AB the equation is

$\sqrt{(\text{x}_2-\text{x}_1)^{2}+(\text{y}_2-\text{y}_1)^2+(\text{z}_1-\text{z}_1)^2}.....(1)$

x₁ = 4, y₁ = 0 and z₁ = 4

x₂ = 1, y₂ = 2 and z₂ = 3

by substituting the values in (1) we will get

AB = 3.7 similarly for BC and AC

BC = 3.7

AC = 7.4

hence finally its is known that these points are collinear

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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: The points (1, 2, 3), (-2, 3, 4) and (7, 0, 1) are collinear
Reason: If a line makes angles $\frac{\pi}{2}, \frac{3\pi}{4}$ and $\frac{\pi}{4}$ with X, Y, and Z - axes respectively, then its direction cosines are $0,\frac{-1}{\sqrt{2}}$ and $\frac{1}{\sqrt{2}}$

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

Solution:

We have, $\text{x}_1=1,\text{y}_1=2,\text{z}_1=3;$

$\text{x}_2=-2,\text{y}_2=3,\text{z}_2=4$ and $\text{x}_3=7,\text{y}_3=0,\text{z}_3=1$

Now, $\frac{\text{x}_2-\text{x}_1}{\text{x}_3-\text{x}_2}=\frac{\text{y}_2-\text{y}_1}{\text{y}_3-\text{y}_2}=\frac{\text{z}_2-\text{z}_1}{\text{z}_3-\text{z}_2}$

$\Rightarrow\frac{-2-1}{7-(-2)}=\frac{3-2}{0-3}=\frac{4-3}{1-4}$

$\Rightarrow\frac{-3}{9}=\frac{1}{-3}=\frac{1}{-3}\Rightarrow\frac{-1}{3}=\frac{-1}{3}=\frac{-1}{3}$

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MCQ 101 Mark

Assertion (A) : The point $A(1,0,7)$ is the mirror image of the point $B(1,6,3)$ in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$.
Reason (R) : The line : $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ bisects the line segment joining $A(1,0,7)$ and $B(1,6,3)$.

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) : The direction ratios of the line segment joining $A(1,0,7)$ and $B(1,6,3)$ is $(0,6,-4)$.
The direction ratios of the given line is $(1,2,3)$.
As $1 \cdot 0+6 \cdot 2-4 \cdot 3=0$, we have the lines are perpendicular.
Also the midpoint of $A B$ is $(1,3,5)$. Also, the point $(1,3,5)$ lies on the line.
$\therefore \quad$ Point $A$ is the mirror image of point $B$ in the given line. Also, the line bisects $A B$, so statement $I$ and statement II are true.
Statement 'II' holds even if the line is not perpendicular. This situation is possible.

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Maths Assertion (A) & Reason (B) MCQ

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