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Syllabus: Sets, Relations, Functions, Partial orders, Lattices, Monoids, Groups.

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}&\textbf{2024-1} &\textbf{2024-2} &\textbf{2023} & \textbf{2022} & \textbf{2021-1}&\textbf{2021-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count} &1&1&0& 1&0&1&0&0.83&1
\\\hline\textbf{2 Marks Count} &1&1&2& 0 &2&1&0&1.16&2
\\\hline\textbf{Total Marks} & 3&3&4&1&4&3&\bf{1}&\bf{3}&\bf{4}\\\hline
\end{array}}}$$

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Recent questions in Set Theory & Algebra

#281
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Kenneth Rosen Edition 7 Exercise 2.4 Question 36 (Page No. 169)
Version$1:$ Use the identity $\dfrac{1}{k(k+1)} = \dfrac{\left(\frac{1}{k-1}\right)}{(k+1)}$ and question $35$ ... $35$ to compute $\displaystyle{}\sum_{k=1}^{n} 1/(k(k+1)).$
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#282
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Kenneth Rosen Edition 7 Exercise 2.4 Question 35 (Page No. 169)
Show that $\displaystyle{}\sum_{j=1}^{n}(a_{j} - a_{j-1}) = a_{n} -a_{0,}$ where $a_{0}, a_{1},\dots,a_{n}$ is a sequence of real numbers. This type of sum is called telescoping.
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#283
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Kenneth Rosen Edition 7 Exercise 2.4 Question 34 (Page No. 169)
Compute each of these double sums.$\displaystyle{}\sum_{i=1}^{3}\sum_{j=1}^{2}(i-j)$\displaystyle{}\sum_{i=0}^{3}\sum_{j=0}^{2}(3i+2j)$\displaystyle{}\sum_{i=1}^{3}\sum_{j=0}^{2}j$\displaystyle{}\sum_{i=0}^{2}\sum_{j=0}^{3}i^{2}j^{3}$
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#284
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Kenneth Rosen Edition 7 Exercise 2.4 Question 33 (Page No. 169)
Compute each of these double sums.$\displaystyle{}\sum_{i=1}^{2}\sum_{j=1}^{3}(i+j)$\displaystyle{}\sum_{i=0}^{2}\sum_{j=0}^{3}(2i+3j)$\displaystyle{}\sum_{i=1}^{3}\sum_{j=0}^{2}i$\displaystyle{}\sum_{i=0}^{2}\sum_{j=1}^{3}ij$
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#285
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Kenneth Rosen Edition 7 Exercise 2.4 Question 32 (Page No. 169)
Find the value of each of these sums.$\displaystyle{}\sum_{j=0}^{8}(1+(-1)^{j})$\displaystyle{}\sum_{j=0}^{8}(3^{j}-2^{j})$\displaystyle{}\sum_{j=0}^{8}(2\cdot 3^{j} + 3\cdot 2^{j})$\displaystyle{}\sum_{j=0}^{8}(2^{j+1}-2^{j})$
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#286
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Kenneth Rosen Edition 7 Exercise 2.4 Question 31 (Page No. 169)
What is the value of each of these sums of terms of a geometric progression?$\displaystyle{}\sum_{j=0}^{8}3\cdot 2^{j}$\displaystyle{}\sum_{j=1}^{8}2^{j}$\displaystyle{}\sum_{j=2}^{8}(-3)^{j}$\displaystyle{}\sum_{j=0}^{8}2\cdot (-3)^{j}$
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#287
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Kenneth Rosen Edition 7 Exercise 2.4 Question 30 (Page No. 169)
What are the values of these sums, where $S = \{1, 3, 5, 7\}?$\displaystyle{}\sum_{j\in S}j$\displaystyle{}\sum_{j\in S}j^{2}$\displaystyle{}\sum_{j\in S}(1/j)$\displaystyle{}\sum_{j\in S}1$
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#288
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Kenneth Rosen Edition 7 Exercise 2.4 Question 29 (Page No. 169)
What are the values of these sums?$\displaystyle{}\sum_{k=1}^{5}(k+1)$\displaystyle{}\sum_{j=0}^{4}(-2)^{j}$\displaystyle{}\sum_{i=1}^{10}3$\displaystyle{}\sum_{j=0}^{8}(2^{j+1}-2^{j})$
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#289
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Kenneth Rosen Edition 7 Exercise 2.4 Question 28 (Page No. 169)
Let $a_{n}$ be the $nth$ term of the sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6,\dots,$ constructed by including the integer $k$ exactly $k$ times. Show that $a_{n} =\lfloor \sqrt{2n}+\frac{1}{2} \rfloor.$
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#290
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Kenneth Rosen Edition 7 Exercise 2.4 Question 27 (Page No. 169)
Show that if $a_{n}$ denotes the $nth$ positive integer that is not a perfect square, then $a_{n} = n + \{\sqrt{n}\},$ where $\{x\}$ denotes the integer closest to the real number $x.$
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#291
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Kenneth Rosen Edition 7 Exercise 2.4 Question 26 (Page No. 169)
For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list.Assuming that your formula or ... , 1, 1, 1,\dots$2, 4, 16, 256, 65536, 4294967296,\dots$
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#292
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Kenneth Rosen Edition 7 Exercise 2.4 Question 25 (Page No. 169)
For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list.Assuming that your formula or ... , 686,\dots$2, 3, 7, 25, 121, 721, 5041, 40321,\dots$
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#293
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Kenneth Rosen Edition 7 Exercise 2.4 Question 24 (Page No. 169)
Find a recurrence relation for the balance $B(k)$ owed at the end of $k$ months on a loan at a rate of $r$ if a payment $P$ is made on the loan each ... what the monthly payment $P$ should be so that the loan is paid off after $T$ months.
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#294
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Kenneth Rosen Edition 7 Exercise 2.4 Question 23 (Page No. 169)
Find a recurrence relation for the balance $B(k)$ owed at the end of $k$ months on a loan of $\$5000$ at a rate of $7\%$ if a payment of $\$100$ ... in terms of $B(k − 1);$ the monthly interest is $(0.07/12)B(k − 1).]$
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#295
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Kenneth Rosen Edition 7 Exercise 2.4 Question 22 (Page No. 168 - 169)
An employee joined a company in $2009$ with a starting salary of $\$50,000.$ Every year this employee receives a raise of $\$1000$ plus $5\%$ of the ... $n$ years after $2009.$
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#296
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Kenneth Rosen Edition 7 Exercise 2.4 Question 21 (Page No. 168)
A factory makes custom sports cars at an increasing rate. In the first month, only one car is made, in the second month, two cars are made, and so on, ... formula for the number of cars produced in the first $n$ months by this factory
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#297
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Kenneth Rosen Edition 7 Exercise 2.4 Question 20 (Page No. 168)
Assume that the population of the world in $2010$ was $6.9$ billion and is growing at the rate of $1.1\%$ a year.Set up a recurrence relation for the population ... $n$ years after $2010.$What will the population of the world be in $2030?$
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#298
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Kenneth Rosen Edition 7 Exercise 2.4 Question 19 (Page No. 168)
Suppose that the number of bacteria in a colony triples every hour.Set up a recurrence relation for the number of bacteria after n hours have elapsed.If $100$ bacteria ... a new colony, how many bacteria will be in the colony in $10$ hours?
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#299
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Kenneth Rosen Edition 7 Exercise 2.4 Question 18 (Page No. 168)
A person deposits $\$1000$ in an account that yields $ ... end of $n$ years.How much money will the account contain after $100$ years?
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#300
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Kenneth Rosen Edition 7 Exercise 2.4 Question 17 (Page No. 168)
Find the solution to each of these recurrence relations and initial conditions. Use an iterative approach such as that used in Example $10.$ ... } = na_{n−1}, a_{0} = 5$a_{n} = 2na_{n−1}, a_{0} = 1$
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