Download App

Three Dimensional Geometry — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSThree Dimensional Geometry1 Mark
Question
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 $=0$.
✓

Answer

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

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "Assertion (A) & Reason (B) MCQ" from Three Dimensional Geometry→Three Dimensional Geometry — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Assertion $(A)$ : If $A=\frac{1}{3}\left[\begin{array}{ccc}1 & -2 & 2 \\ -2 & 1 & 2 \\ -2 & -2 & -1\end{array}\right]$, then $A\left(A^T\right)=I$
Reason $(R)$ : For any square matrix $A,\left(A^T\right)^T=A$
→
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: Domain and Range of a relation R = {{x, y) : x -2y = 0} defined on the set A = {1, 2, 3, 4} are respectively {1, 2, 3, 4} and {2, 4, 6, 8}.
Reason: Domain and Range of a relation R are respectively the sets {$\text{a}:\text{a}\in\text{A}$ and $(\text{a},\text{b})\in\text{R}.$} and {$\text{b}:\text{b}\in\text{A}$ and $(\text{a},\text{b})\in\text{R}.$}
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are fals.
→
Assertion (A) : The domain for
$f(x)=\sin ^{-1}\left(\frac{1+x^2}{2 x}\right)$ is $\{0,1\}$.
Reason (R) : $\sin ^{-1} x$ is defined only if $x \in[-1,1]$.
→
Assertion (A) : Principal value of $\sin ^{-1}\left(\sin \left(\frac{2 \pi}{3}\right)\right)$ is $\frac{\pi}{3}$.
Reason (R) : Principal value branch of $\sin ^{-1}$ function is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.
→
Assertion $(A)$: If $e^{x y}+\log (x y)+\cos (x y)+5=0$, then $\frac{d y}{d x}=-\frac{y}{x}$.
Reason $(R) : \frac{d}{d x}(x y)=0 \Rightarrow \frac{d y}{d x}=\frac{-y}{x}$
→
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 $B(1,6,3)$.
→
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 (A) The value of the constant ‘k’ so that $\text{f(x)}=\begin{cases}\text{kx}^2,\text{if x}\leq2\\3,\text{if x}>2\end{cases}$ is continuous at x = 2 is $\text{k}=\frac{4}{3}$
Reason(R) A function f(x) is continuous at a point x= a of its domain if $\lim\limits_{\text{x}\rightarrow 0}\text{f(x)}=\text{f(x)}$
  1. Both A and R are true and R is the correct explanation of A
  2. Both A and R are true but R is NOT the correct explanation of A.
  3. A is true but R is false
  4. A is false but R is true 
→
Assertion (A) : If $f^{\prime}(x)=(x-1)^3(x-2)^8$, then $f(x)$ has neither maximum nor minimum at $x=2$.
Reason $( R ): f^{\prime}(x)$ changes sign from negative to positive at $x=2$.
→
Directions: In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion: The equation of the line joining (1, 2) and (3, 6) using determinants is y = 3x.
Reason: The area of $\triangle\text{PAB}$ is zero if P(x, y) is a point on the line joining a A and B.
  1. Both A and R are true and R is the correct explanation of A.
  2. Both A and R are true but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
  5. Both A and R are false.
→
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
$\overrightarrow{P Q} \| \overrightarrow{P R} \text { or } \overrightarrow{P Q}=\lambda \overrightarrow{P R}, \lambda \in R$
→