discrete-mathematics

Set Theory & Algebra

June 21, 2015 by Arjun Suresh 3 Comments

GATE Materials for Set Theory & Algebra

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Answered: GATE CSE 2019 | Question: 48

[youtube https://www.youtube.com/watch?v=B8ElmECvmWw]\ GATE CS 2019,Q48: Let Σ be the set of all bijections from {1, …, 5} to {1, …, 5}, where id denotes - YouTube [...]

Answered: MadeEasy Subject Test 2019: Compiler Design - Parsing

S1: False Refer this: https://stackoverflow.com/questions/34707467/is-every-ll1-grammar-also-an-lr0-grammar S2: False Refer this : https://gateoverflow.in/1251/gate-cse-2007-question-53 S3: TRUE Refer this: https://www.csd.uwo.ca/~mmorenom/CS447/Lectures/IntermediateCode.html/node3.html [...]

Answered: NIELIT 2016 DEC Scientist B (IT) - Section B: 43

D. None of the above As we have only two basic operation which are union and set difference [...]

Answered: GO 2021 Verbal Ability 1: 2

Had I not informed you on time, you would have missed the opportunity. [...]

Answered: GO 2021 Verbal Ability 1: 1

confine – keep or restrict someone or something within certain limits of (space, scope, or time). [...]

Answered: NTA NET NOV 2020

This question is solved: https://gateoverflow.in/349662/ugcnet-oct2020-ii-11 [...]

Answered: NIELIT 2019 Feb Scientist D - Section D: 12

$\frac{1}{2^{2}-1}+\frac{1}{4^{2}-1}+\frac{1}{6^{2}-1}+\cdots+\frac{1}{20^{2}-1}$ $= \frac{1}{(2+1)(2-1)} + \frac{1}{(4+1)(4-1)} +\frac{1}{(6+1)(6-1)} +\cdots +\frac{1}{(20+1)(20-1)}$ $= \frac{2}{2}\left [\frac{1}{(2+1)(2-1)} + \frac{1}{(4+1)(4-1)}+\frac{1}{(6+1)(6-1)} +\cdots +\frac{1}{(20+1)(20-1)}\right]$ $= \frac{1}{2}\left [\frac{2}{(2+1)(2-1)} + \frac{2}{(4+1)(4-1)} +\frac{2}{(6+1)(6-1)} +\cdots +\frac{2}{(20+1)(20-1)}\right]$ $= \frac{1}{2}\left [\frac{(2+1)-(2-1)}{(2+1)(2-1)} + \frac{(4+1)-(4-1)}{(4+1)(4-1)} + \frac{(6+1)-(6-1)}{(6+1)(6-1)}+\cdots +\frac{(20+1)-(20-1)}{(20+1)(20-1)}\right]$ $= \frac{1}{2}\left [\frac{1}{(2-1)} - \frac{1}{(2+1)} + \frac{1}{(4-1)} - \frac{1}{(4+1)} + \frac{1}{(6-1)} - \frac{1}{(6+1)}+\frac{1}{(8-1)} - \frac{1}{(8+1)}+\cdots +\frac{1}{(20-1)} - \frac{1}{(20+1)}\right]$ $= \frac{1}{2}\left [\frac{1}{1} - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7}+\frac{1}{7} - \frac{1}{9}+\cdots +\frac{1}{19} - \frac{1}{21}\right]$ $= \frac{1}{2}\left[\frac{1}{1} - \frac{1}{21}\right]$ $= \frac{1}{2}\left[ \frac{20}{21}\right]$ $= \frac{10}{21}$ Similar question: https://gateoverflow.in/358957/tifr2021-a-11 [...]

Answered: NIELIT 2019 Feb Scientist D - Section D: 13

$1-6+2-7+3-8+\dots\dots$ to $100$ terms $=\underbrace{1-6}_{-5}+\underbrace{2-7}_{-5}+\underbrace{3-8}_{-5}+\dots\dots \underbrace{50-55}_{-5}$ $= (1+2+\dots\dots 50) - (6+7+\dots\dots 55)$ $= (1+2+3+4+5)+\underbrace{(6+7+\dots\dots 50) - (6+7+\dots\dots 50)}_0-(51+52+53+54+55)$ $= (1+2+3+4+5)-(51+52+53+54+55) $ $=\frac{5}{2}(2(1)+(5-1)(1)) - \frac{5}{2}(2(51)+(5-1)(1))$ $(\because$ Sum of $n$ terms of A.P. sequence is $ \frac{n}{2}(2a+(n-1)d)$ $=\frac{5}{2}(6) - \frac{5}{2}(106)$ $=5(3)-5(53)$ $=15-265$ $=-250$ [...]

Answered: NIELIT 2017 OCT Scientific Assistant A - Section A: 52

Let Brian time be Y months and given Adrian invested for whole 12 months. partnership ratio 85000 ×12 : 42500 × y we need to make this as 3:1 in terms of profit so simply dividing by 85000 ×4 on both sides of ratio we'll now have 3 : y/8 that means Brian invested for 8 months [...]

Answered: NIELIT 2019 Feb Scientist C - Section D: 1

Ans is option (D) The minute covers $360^{\circ}$ in $60$ minutes. Hence, it will cover $\frac{360^{\circ}\times 35}{60}=210^{\circ}$ in $35$ minutes. Now, area of a sector with radius $r$ units and $\theta$ angle subtended at the centre ,is : $\frac{1}{2}\times r^{2}\times \theta$ units ($\theta$ in radian) . So, $210^{\circ}=210^{\circ}\times \frac{\pi}{180^{\circ}}$ radians $=\frac{11}{3}$ radians $\therefore$ Area of the sector: $\frac{1}{2}\times 100\times \frac{11}{3}=183.33$ cm$^{2}$ [...]

Answered: NIELIT 2019 Feb Scientist C - Section D: 28

Ans is option (D) Let the original price be $x$ Rs. Price after $p\%$ increase: $x(1+\frac{p}{100})$ Rs. Price after $p\%$ decrease: $x(1+\frac{p}{100})-[\frac{p}{100}\times x(1+\frac{p}{100})]=1$ (given in question) $\therefore$ $x(1+\frac{p}{100})(1-\frac{p}{100})=1$ $\Rightarrow$ $x=\frac{10000}{10000-p^{2}}$ Rs. [...]

Answered: CAT2019-2: 86

$\frac{n^{2}+7n+12}{n^{2}-n-12} = \frac{n^{2}+3n+4n+12}{n^{2}+3n-4n-12}$ $= \frac{n(n+3)+4(n+3)}{n(n+3)-4(n+3)}$ $= \frac{(n+3)(n+4)}{(n+3)(n-4)} = \frac{(n+4)}{(n-4)}$ $\therefore \frac{n^{2}+7n+12}{n^{2}-n-12} = \frac{(n+4)}{(n-4)}$ Now let’s substitute $n$ values given in options $ n = 8 $ $\frac{(n+4)}{(n-4)} = \frac{(8+4)}{(8-4)} =\frac{(12)}{(4)} = 3$ (Positive Integer) $ n = 12 $ $\frac{(n+4)}{(n-4)} = \frac{(12+4)}{(12-4)} =\frac{(16)}{(8)} = 2$ (Positive Integer) $ n = 16 $ $\frac{(n+4)}{(n-4)} = \frac{(16+4)}{(16-4)} =\frac{(20)}{(12)} \sim 1.67$ (Not an integer) $ n = 6 $ $\frac{(n+4)}{(n-4)} = \frac{(6+4)}{(6-4)} =\frac{(10)}{(2)} = 5$ (Positive Integer) Options A, B, D have produced the result as a positive integer. In question, given that we need to find “largest positive integer $n$ such that… [...]

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