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1261
IITB Practice Set: 28
Consider the following algorithm. What does the above algorithm compute?
Consider the following algorithm.What does the above algorithm compute?
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1262
IITB Practice Set: 29
The input is an array $A$ of size $n\geq 5$. As output we require indices $i < j < k$ such that either $A[i] \leq A[ j] \leq A[k]\text{ or }A[i] \geq A[ j] \geq A[k]$. The worst case complexity of the best algorithm for this problem is $\theta$ ______ .
The input is an array $A$ of size $n\geq 5$. As output we require indices $i < j < k$ such that either $A[i] \leq A[ j] \leq A[k]\text{ or }A[i] \geq A[ j] \geq A[k]$. Th...
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1263
IITB Practice Set: 30
How many ordered pairs of positive integers $(a,b)$ are there such that $\text{lcm}(a,b) = n \text{ and gcd}(a,b) = 1?$ Express your answer in terms of the prime factorization of $n$, given as $n = p_1^{d_1} \dots p_t^{d_t}.$
How many ordered pairs of positive integers $(a,b)$ are there such that $\text{lcm}(a,b) = n \text{ and gcd}(a,b) = 1?$Express your answer in terms of the prime factoriza...
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1268
IITB Practice Set: 35
Consider the $C++$ code fragment below. int A = ____________; bool x = (A/100 >= 8) && !(A%10 > 4) && (A%5 == 1); (a) ______ is the $\textbf{maximum three digit (in decimal notation)}$ value of $A$ such that $x$ evaluates to true.
Consider the $C++$ code fragment below.int A = ____________; bool x = (A/100 >= 8) && !(A%10 4) && (A%5 == 1);(a) ______ is the $\textbf{maximum three digit (in decimal ...
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1270
IITB Practice Set: 37
______ is the $\textbf{minimum three digit (in decimal notation)}$ value of $K$ for which the following loop (part of a $C++$ program) iterates at least $1000$ times. unsigned int K=________; for(int x=10,y=K; x!=y; x++,y--) cout << "#";
______ is the $\textbf{minimum three digit (in decimal notation)}$ value of $K$ for which the followingloop (part of a $C++$ program) iterates at least $1000$ times.unsig...
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1274
TIFR2021-Maths-A: 1
For each positive integer $n$, let $s_n=\frac{1}{\sqrt{4n^2-1^2}}+\frac{1}{\sqrt{4n^2-2^2}}+\dots+\frac{1}{\sqrt{4n^2-n^2}}$ Then the $\displaystyle \lim_{n\rightarrow \infty}s_n$ equals $\pi/2$ $\pi/6$ $1/2$ $\infty$
For each positive integer $n$, let$$s_n=\frac{1}{\sqrt{4n^2-1^2}}+\frac{1}{\sqrt{4n^2-2^2}}+\dots+\frac{1}{\sqrt{4n^2-n^2}}$$Then the $\displaystyle \lim_{n\rightarrow \i...
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1275
TIFR2021-Maths-A: 2
The number of bijective maps $g:\mathbb{N}\rightarrow\mathbb{N}$ such that $\sum_{n=1}^\infty\frac{g(n)}{n^2}<\infty$ is $0$ $1$ $2$ $\infty$
The number of bijective maps $g:\mathbb{N}\rightarrow\mathbb{N}$ such that$$\sum_{n=1}^\infty\frac{g(n)}{n^2}<\infty$$is$0$$1$$2$$\infty$
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1276
TIFR2021-Maths-A: 3
The value of $\displaystyle\lim_{n\rightarrow\infty}\prod_{k=2}^{n}\left(1-\frac{1}{k^2}\right)$ is $1/2$ $1$ $1/4$ $0$
The value of $$\displaystyle\lim_{n\rightarrow\infty}\prod_{k=2}^{n}\left(1-\frac{1}{k^2}\right)$$is$1/2$$1$$1/4$$0$
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1277
TIFR2021-Maths-A: 4
The set $S=\{x\in \mathbb{R}|x>0\text{ and } (1+x^2) \tan(2x)=x\}$ is empty nonempty but finite countably infinite uncountable
The set $$S=\{x\in \mathbb{R}|x>0\text{ and } (1+x^2) \tan(2x)=x\}$$isemptynonempty but finitecountably infiniteuncountable
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1278
TIFR2021-Maths-A: 5
The dimension of the real vector space $V=\{f:(-1,1)\rightarrow\mathbb{R}|f$ is infinitely differentiable on $(-1,1)$ and $f^{(n)}(0)=0$ for all $n\geq 0\}$ is $0$ $1$ greater than one, but finite infinite
The dimension of the real vector space$V=\{f:(-1,1)\rightarrow\mathbb{R}|f$ is infinitely differentiable on $(-1,1)$ and $f^{(n)}(0)=0$ for all $n\geq 0\}$is$0$$1$greater...
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1280
TIFR2021-Maths-A: 7
Let $A$ be the set of all real numbers $\lambda \in [0,1]$ such that $\displaystyle\lim_{p\rightarrow 0}\frac{\log(\lambda2^p+(1-\lambda)3^p)}{p}=\lambda \log2+(1-\lambda)\log3$ Then $A=\{0,1\}$ $A=\{0,\frac{1}{2},1\}$ $A=\{0,\frac{1}{3},\frac{1}{2},\frac{2}{3},1\}$ $A=[0,1]$
Let $A$ be the set of all real numbers $\lambda \in [0,1]$ such that$$\displaystyle\lim_{p\rightarrow 0}\frac{\log(\lambda2^p+(1-\lambda)3^p)}{p}=\lambda \log2+(1-\lambda...
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