Download App

Model Paper 6 — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsModel Paper 65 Marks
Question
Find the shortest distance between the given lines. $\vec{r}=(\hat{i}+2 \hat{j}-4 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})$, $\vec{r}=(3 \hat{i}+3 \hat{j}-5 \hat{k})+\mu(-2 \hat{i}+3 \hat{j}+8 \hat{k})$
✓

Answer

Given
$\vec{r}=(\hat{i}+2 \hat{j}-4 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})
$
$\vec{r}=(3 \hat{i}+3 \hat{j}-5 \hat{k})+\mu(-2 \hat{i}+3 \hat{j}+8 \hat{k})$
Here, we have
$\vec{a}_1=\hat{\imath}+2 \hat{j}-4 \hat{k}
$
$\overrightarrow{b_1}=2 \hat{\imath}+3 \hat{j}+6 \hat{k}
$
$\overrightarrow{a_2}=3 \hat{i}+3 \hat{j}-5 \hat{k}
$
$\overrightarrow{b_2}=-2 \hat{\imath}+3 \hat{j}+8 \hat{k}$
Thus,
$\overrightarrow{b_1} \times \overrightarrow{b_2}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\2 & 3 & 6 \\-2 & 3 & 8\end{array}\right|$
$=\hat{i}(24-18)-\hat{j}(16+12)+\hat{k}(6-6)$
$\overrightarrow{b_1} \times \overrightarrow{b_2}=6 \hat{i}-28 \hat{j}+0 \hat{k}$
$\therefore\left|b_1 \times \overrightarrow{b_2}\right|=\sqrt{6^2+(-28)^2+0^2}$
$=\sqrt{36+784+9}$
$=\sqrt{820}$
$\overrightarrow{a_2}-\overrightarrow{a_1}=(3-1) \hat{i}+(3-2) \hat{j}+(-5+4) \hat{k}$
$\therefore \overrightarrow{a_2}-\overrightarrow{a_1}=2 \hat{\imath}+\hat{j}-\hat{k}$
Now, we have
$\left(\overrightarrow{b_1} \times \overrightarrow{b_2}\right) \cdot\left(\overrightarrow{a_2}-\overrightarrow{a_1}\right)=(6 \hat{\imath}-28 \hat{\jmath}+0 \hat{k}) \cdot(2 \hat{\imath}+\hat{\jmath}-\hat{k})$
$=(6 \times 2)+((-28) \times 1)+(0 \times(-1))$
$=12-28+0$
$=-16$
Thus, the shortest distance between the given lines is
$ d =\left|\frac{\left(\overrightarrow{ b _1} \times \overrightarrow{ b _2}\right)\cdot\left(\overrightarrow{a_2}-\overrightarrow{a_1}\right)}{\left|\overrightarrow{b_1} \times \overrightarrow{b_2}\right|}\right|$
$\Rightarrow d =\left|\frac{-16}{\sqrt{820}}\right|$
$\therefore d=\frac{16}{\sqrt{820}} \text { units }$

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 "5 Marks Questions" from Model Paper 6→Model Paper 6 — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

An unbiased die is tossed twice. Find the probability of getting 4, 5, or 6 on the first toss and 1, 2, 3 or 4 on the second toss.
→
Find $\frac{\text{dy}}{\text{dx}}$
$\text{y}=(\sin\text{x})^\text{x}+\sin^{-1}\sqrt{\text{x}}$
→
For each of the differential equation given in find the general solution:
$(1+\text{x}^2)\ \text{dy}+2\text{xy}\ \text{dx}=\cot\text{x}\ \text{dx}\ (\text{x}\neq0)$
→
The pressure p and the volume $v$ of a gas are connected by the relation $pv^{1.4} =$ const. Find the percentage error in p corresponding to a decrease of $1/2\%$ in $v.$
→
Without expanding, show that the values of the following determinant are zero:
$\begin{vmatrix}\sin^223^{\circ}&\sin^267^{\circ}&\cos180^{\circ}\\-\sin^267^{\circ}&-\sin^223^{\circ}&\cos^2180^{\circ}\\\cos180^{\circ}&\sin^223^{\circ}&\sin^267^{\circ}\end{vmatrix}$
→
Find the angle of intersecting of the following curves:
$\text{y}^2=\text{x}\text{ and }\text{x}^2=\text{y}$
→
Find the values of a and b if the slope of the tangent to the curve xy + ax + by = 2 at (1, 1) is 2.
→
Evaluate the following integrals:
$\int\text{x}^2\text{e}^{\text{x}^3}\cos\text{x}^3\text{dx}$
→
Let $f: W \rightarrow W$ be defined as $f(n)=n-1,$ if $n$ is odd and $f(n)=n+1,$ if $n$ is even. Show that $f$ is invertible. Find the inverse of $f$. Here, $W$ is the set of all whole numbers.
→
The cost of 4kg onion, 3kg wheat and 2kg rice is Rs. 60. The cost of 2kg onion, 4kg wheat and 6kg rice is Rs. 90. The cost of 6kg onion 2kg wheat and 3kg rice is Rs. 70. Find the cost of each item per kg by matrix method.
→