Download App

Sets — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSSets1 Mark
Question
If $\text{X}=\{8^\text{n}-7\text{x}-1:\text{n}\in\text{N}\}$ and $\text{Y}=\{49(\text{n}-1):\text{n}\in\text{N}\},$ then prove thet $\text{X}\subseteq\text{Y}.$
✓

Answer

$\text{X}=\{8^\text{n}-7\text{x}-1:\text{n}\in\text{N}\}$ $\text{Y}=\{49(\text{n}-1):\text{n}\in\text{N}\}$ In order to show that $\text{X}\subseteq\text{Y}$ we shoe the every element of X is an element of Y. So let $\text{x}\in\text{X}\Rightarrow\text{x}=8^\text{n}-7\text{m}-1$ for same $\text{m}\in\text{N}$ $\Rightarrow\text{x = (1 + 7)}^\text{m}- 7\text{m} - 1$ $=(^\text{m}\text{C}_01^\text{m}+^\text{m}\text{C}_11^\text{m-1}7+...+^\text{m}\text{C}_\text{m-1}1^17^\text{m-1}+^\text{m}\text{C}_\text{m}7^\text{m})-7\text{m}-1$ [using binomail expansion] $=1+7\text{m}+^\text{m}\text{C}_27^2+^\text{m}\text{C}_37^3+...+^\text{m}\text{C}_\text{m}7^\text{m}-7\text{m}-1$ $=\ ^\text{m}\text{C}_27^2+\ ^\text{m}\text{C}_37^3+...+\ ^\text{m}\text{C}_\text{m}7^\text{m}$ $=49(^\text{m}\text{C}_2+^\text{m}\text{C}_3+...+^\text{m}\text{C}_\text{m}7^\text{m}),\ \text{m}\geq2 $ $=49\text{t}_\text{m},\ \text{m}\geq2,$ where $\text{t}_\text{m}=^\text{m}\text{C}_2+^\text{m}\text{C}_37+...+^\text{m}\text{C}_\text{m}7^\text{m-2}$ Is some positive integer depending on $\text{m}\geq2$ For $\text{m} = 1$$\text{x} = 1^8 - 7 × 1 - 1$
$=8 - 8$
$= 0$
Hence, X contains all positive integral multiples of 49. Also, Y consistes all positive integral multiples of 49, including 0, for n = 1. Thuse, we coclude that $\text{X}\subseteq\text{Y}.$

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 "(Each question 1 marks)" from Sets→Sets — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Let R be the relation on Z defined by R = {(a, b): $a,b \in {Z}$, a - b is an even integer. Find the domain and range of R.
→
Compute derivative of $g(x)=\cot x$
→
The given set is the example of an empty set or not? Set of all even natural numbers divisible by 5.
→
Write the sum of first n odd natural numbers.
→
A group consist of 4 girls and 7 boys. In how many ways, a team of 5 members be selected, if the team has at least one boy and one girl ?
→
Find the locus of the mid-points of the portion of the line $\text{x}\sin\theta+ \text{y}\cos\theta =\text{p}$ intercepted between the axes.
→
Find the sum of the sequence 7, 77, 777, ........ to n.
→
Find the degree measure to the radian measure $\left( {use\;\pi \frac{{22}}{7}} \right)\frac{{5\pi }}{3}$
→
Is x : x is a natural number, x < 5 and, x > 7 null set?
→
Let A = {1, 2, 3, 4, 5, 6}. Define a relation R from A to A by R = {(x, y) : y = x + 1 }. Write down the domain, codomain and range of R
→