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Recent questions tagged isi2019-pcb-mathematics
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ISI 2019 | PCB Mathematics | Question: 1
Let $\left\{a_{n}\right\}$ be a decreasing sequence such that $\displaystyle{}\sum_{n=1}^{\infty} a_{n}$ is convergent. Prove that the sequence $\left\{n a_{n}\right\}$ goes to zero as $n \rightarrow \infty$.
Lakshman Patel RJIT
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isi2019-pcb-mathematics
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ISI 2019 | PCB Mathematics | Question: 2
Consider an $n \times n$ matrix $A=I_{n}-\alpha \alpha^{T}$, where $I_{n}$ is the identity matrix of order $n$ and $\alpha$ is an $n \times 1$ column vector such that $\alpha^{T} \alpha=1$. Prove that $A^{2}=A.$
Lakshman Patel RJIT
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Aug 9
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Lakshman Patel RJIT
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isi2019-pcb-mathematics
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ISI 2019 | PCB Mathematics | Question: 3
Let $A$ and $B$ be two invertible real matrices of order $n$. Show that $\det(x A+(1-x) B)=0$ has finitely many solutions for $x.$
Lakshman Patel RJIT
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Aug 9
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Lakshman Patel RJIT
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isi2019-pcb-mathematics
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4
ISI 2019 | PCB Mathematics | Question: 4
Show that for every $\theta \in\left(0, \frac{\pi}{2}\right),$ there exists a unique real number $x_{\theta}$ such that $ (\sin \theta)^{x_{\theta}}+(\cos \theta)^{x_{\theta}}=\frac{3}{2} . $
Lakshman Patel RJIT
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Aug 9
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Lakshman Patel RJIT
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isi2019-pcb-mathematics
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ISI 2019 | PCB Mathematics | Question: 5
Suppose $f$ and $g$ are continuous real valued functions on $[a, b]$ and are differentiable on $(a, b)$. Assume that $g^{\prime}(x) \neq 0$ for any $x \in(a, b)$. Prove that there exists $\xi \in(a, b)$ such that $ \frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}=\frac{f(b)-f(a)}{g(b)-g(a)} $
Lakshman Patel RJIT
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Aug 9
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Lakshman Patel RJIT
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isi2019-pcb-mathematics
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ISI 2019 | PCB Mathematics | Question: 6
Consider the function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ defined by $ f(0,0)=0, \quad f(x, y)=\frac{x y}{x^{2}+y^{2}}, \quad(x, y) \neq(0,0) . $ Prove that the directional derivative of $f$ at $(0,0)$ exists in all directions. Is $f$ continuous at $(0,0)$ ? Justify your answer.
Lakshman Patel RJIT
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Aug 9
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Lakshman Patel RJIT
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ISI 2019 | PCB Mathematics | Question: 7
Solve the differential equation $ x^{2}\left(x^{2}-1\right) \frac{d y}{d x}+x\left(x^{2}+1\right) y=x^{2}-1 . $
Lakshman Patel RJIT
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Aug 9
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Lakshman Patel RJIT
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isi2019-pcb-mathematics
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ISI 2019 | PCB Mathematics | Question: 8
Let $f$ be a real valued function on $\mathbb{R}$. If for all real $x$, $ f(x)+3 f(1-x)=5 $ holds, then show that $f$ is a constant function.
Lakshman Patel RJIT
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Aug 9
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Lakshman Patel RJIT
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ISI 2019 | PCB Mathematics | Question: 9
Let $f:[0,1] \rightarrow[0, \infty)$ be a continuous function. Let $ a=\inf _{0 \leq x \leq 1} f(x) \text { and } b=\sup _{0 \leq x \leq 1} f(x) . $ For every positive integer $m$ ... $c_{m} \in[a, b]$, for all $m \geq 1$, $\displaystyle{}\lim _{m \rightarrow \infty} c_{m}$ exists and find its value.
Lakshman Patel RJIT
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Aug 9
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Lakshman Patel RJIT
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10
ISI 2019 | PCB Mathematics | Question: 10
Let $f_{1}:[0,4] \rightarrow[0,4]$ be defined by $f_{1}(x)=3-(x / 2)$. Define $f_{n}(x)=$ $f_{1}\left(f_{n-1}(x)\right)$ for $n \geq 2$. Prove that $\displaystyle{}\lim _{n \rightarrow \infty} f_{n}(0)$ exists. Find the set of all $x$ such that $\displaystyle{}\lim _{n \rightarrow \infty} f_{n}(x)$ exists and also find the corresponding limits.
Lakshman Patel RJIT
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Aug 9
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Lakshman Patel RJIT
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isi2019-pcb-mathematics
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11
ISI 2019 | PCB Mathematics | Question: 11
Let $m$ be a fixed integer greater than $2.$ Prove that all simple graphs having $n\;(n \geq 3)$ vertices and with $m$ edges are connected if and only if $m>\left(\begin{array}{c}n-1 \\ 2\end{array}\right)$.
Lakshman Patel RJIT
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Aug 9
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Lakshman Patel RJIT
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isi2019-pcb-mathematics
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12
ISI 2019 | PCB Mathematics | Question: 12
Suppose the collection $\left\{A_{1}, \cdots, A_{k}\right\}$ forms a group under matrix multiplication, where each $A_{i}$ is an $n \times n$ real matrix. Let $\displaystyle{}A=\sum_{i=1}^{k} A_{i}$. Show that $A^{2}=k A$. If the trace of $A$ is zero, then show that $A$ is the zero matrix.
Lakshman Patel RJIT
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Aug 9
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Lakshman Patel RJIT
23
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isi2019-pcb-mathematics
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13
ISI 2019 | PCB Mathematics | Question: 13
Let $A$ be an $n \times n$ integer matrix whose entries are all even. Show that the determinant of $A$ is divisible by $2^{n}$. Hence or otherwise, show that if $B$ is an $n \times n$ matrix whose entries are $\pm 1$, then the determinant of $B$ is divisible by $2^{n-1}$.
Lakshman Patel RJIT
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Others
Aug 9
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Lakshman Patel RJIT
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14
ISI 2019 | PCB Mathematics | Question: 14
Let $ A=\left(\begin{array}{rrrr} 1 & 2 & 1 & -1 \\ 2 & 0 & t & 0 \\ 0 & -4 & 5 & 2 \end{array}\right) $ If $\operatorname{rank}(A)=2,$ calculate $t.$
Lakshman Patel RJIT
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Aug 9
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Lakshman Patel RJIT
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isi2019-pcb-mathematics
numerical-answers
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15
ISI 2019 | PCB Mathematics | Question: 15
Let $n, r, s$ be positive integers, each greater than $2.$ Prove that $n^{r}-1$ divides $n^{s}-1$ if and only if $r$ divides $s.$
Lakshman Patel RJIT
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Others
Aug 9
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Lakshman Patel RJIT
18
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16
ISI 2019 | PCB Mathematics | Question: 16
Let $\Omega=\{1,2,3, \ldots, 100\}$. In how many ways $ a_{1}<a_{2}<a_{3}<a_{4}<a_{5}, \quad a_{i} \in \Omega $ can be chosen from $\Omega$ such that $a_{i+1}-a_{i} \geq 2$ for each $i?$
Lakshman Patel RJIT
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Others
Aug 9
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Lakshman Patel RJIT
42
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isi2019-pcb-mathematics
numerical-answers
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17
ISI 2019 | PCB Mathematics | Question: 17
Show that $5|x|+x(x-2) \geq 0$ for every real number $x.$
Lakshman Patel RJIT
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in
Others
Aug 9
by
Lakshman Patel RJIT
41
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isi2019-pcb-mathematics
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18
ISI 2019 | PCB Mathematics | Question: 18
Let $N=1 !+2 !+\cdots+2020!.$ Find the remainder obtained when $N$ is divided by $8 .$
Lakshman Patel RJIT
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Aug 9
by
Lakshman Patel RJIT
29
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isi2019-pcb-mathematics
numerical-answers
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19
ISI 2019 | PCB Mathematics | Question: 19
Let $G$ be a finite group and $H$ the only subgroup of $G$ of order $|H|$. Prove that $H$ is normal in $G.$
Lakshman Patel RJIT
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Others
Aug 9
by
Lakshman Patel RJIT
17
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isi2019-pcb-mathematics
descriptive
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20
ISI 2019 | PCB Mathematics | Question: 20
Let $H$ and $K$ be subgroups of a group $G$ of finite indices $(\text{i.e.}, [G: H]<$ $\infty$ and $[G: K]<\infty).$ Prove that $H \cap K$ is also of finite index $(\text{i.e.}, [G: H \cap K]<\infty).$
Lakshman Patel RJIT
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Others
Aug 9
by
Lakshman Patel RJIT
19
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isi2019-pcb-mathematics
descriptive
1
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21
ISI 2019 | PCB Mathematics | Question: 21
Consider all the permutations of the numbers $1,2, \ldots, 9$. Find the number of permutations which satisfy all of the following: the sum of the numbers lying between $1$ and $2\;($including $1$ and $2)$ is $12,$ the sum of the numbers lying between $2$ ... $4)$ is $34,$ the sum of the numbers lying between $4$ and $5\; ($including $4$ and $5)$ is $45.$
Lakshman Patel RJIT
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Others
Aug 9
by
Lakshman Patel RJIT
72
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isi2019-pcb-mathematics
numerical-answers
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22
ISI 2019 | PCB Mathematics | Question: 22
If $\alpha, \beta, \gamma$ are the roots of the equation $x^{3}+6 x+1=0$, then prove that $ \frac{\alpha}{\beta}+\frac{\beta}{\alpha}+\frac{\beta}{\gamma}+\frac{\gamma}{\beta}+\frac{\gamma}{\alpha}+\frac{\alpha}{\gamma}=-3. $
Lakshman Patel RJIT
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Aug 9
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Lakshman Patel RJIT
24
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isi2019-pcb-mathematics
descriptive
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23
ISI 2019 | PCB Mathematics | Question: 23
Let $X \sim \operatorname{Bin}(n, p)$, and $Y \sim \operatorname{Poisson}\; (\lambda)$. Let $ T=X_{1}+X_{2}+\cdots+X_{Y}, $ with $X_{i} \text{'s i. i. d}.\; \operatorname{Bin}(n, p)\;($and independent to $Y),$ ... $X).$ Compare Expectations of $T$ and $S$ and Variances of $T$ and $S.$
Lakshman Patel RJIT
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Aug 9
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18
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isi2019-pcb-mathematics
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