Found problems: 42
2018 ISI Entrance Examination, 8
Let $n\geqslant 3$. Let $A=((a_{ij}))_{1\leqslant i,j\leqslant n}$ be an $n\times n$ matrix such that $a_{ij}\in\{-1,1\}$ for all $1\leqslant i,j\leqslant n$. Suppose that $$a_{k1}=1~~\text{for all}~1\leqslant k\leqslant n$$ and $~~\sum_{k=1}^n a_{ki}a_{kj}=0~~\text{for all}~i\neq j$.
Show that $n$ is a multiple of $4$.
2022 ISI Entrance Examination, 2
Consider the function $$f(x)=\sum_{k=1}^{m}(x-k)^{4}~, \qquad~ x \in \mathbb{R}$$ where $m>1$ is an integer. Show that $f$ has a unique minimum and find the point where the minimum is attained.
2019 ISI Entrance Examination, 4
Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that $$\frac{1}{2y}\int_{x-y}^{x+y}f(t)\, dt=f(x)\qquad\forall~x\in\mathbb{R}~\&~y>0$$ Show that there exist $a,b\in\mathbb{R}$ such that $f(x)=ax+b$ for all $x\in\mathbb{R}$.
2020 ISI Entrance Examination, 6
Prove that the family of curves $$\frac{x^2}{a^2+\lambda}+\frac{y^2}{b^2+\lambda}=1$$ satisfies $$\frac{dy}{dx}(a^2-b^2)=\left(x+y\frac{dy}{dx}\right)\left(x\frac{dy}{dx}-y\right)$$
2015 ISI Entrance Examination, 3
Consider the set $S = {1,2,3,\ldots , j}$. Let $m(A)$ denote the maximum element of $A$. Prove that
$$\sum_ {A\subseteq S} m(A) = (j-1)2^j +1$$
2018 ISI Entrance Examination, 2
Suppose that $PQ$ and $RS$ are two chords of a circle intersecting at a point $O$. It is given that $PO=3 \text{cm}$ and $SO=4 \text{cm}$. Moreover, the area of the triangle $POR$ is $7 \text{cm}^2$. Find the area of the triangle $QOS$.
2020 ISI Entrance Examination, 8
A finite sequence of numbers $(a_1,\cdots,a_n)$ is said to be alternating if $$a_1>a_2~,~a_2<a_3~,~a_3>a_4~,~a_4<a_5~,~\cdots$$ $$\text{or ~}~~a_1<a_2~,~a_2>a_3~,~a_3<a_4~,~a_4>a_5~,~\cdots$$ How many alternating sequences of length $5$ , with distinct numbers $a_1,\cdots,a_5$ can be formed such that $a_i\in\{1,2,\cdots,20\}$ for $i=1,\cdots,5$ ?
2019 ISI Entrance Examination, 3
Let $\Omega=\{z=x+iy~\in\mathbb{C}~:~|y|\leqslant 1\}$. If $f(z)=z^2+2$, then draw a sketch of $$f\Big(\Omega\Big)=\{f(z):z\in\Omega\}$$ Justify your answer.
2019 ISI Entrance Examination, 6
For all natural numbers $n$, let $$A_n=\sqrt{2-\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}\quad\text{(n many radicals)}$$ [b](a)[/b] Show that for $n\geqslant 2$, $$A_n=2\sin\frac{\pi}{2^{n+1}}$$ [b](b)[/b] Hence or otherwise, evaluate the limit $$\lim_{n\to\infty} 2^nA_n$$
2020 ISI Entrance Examination, 3
Let $A$ and $B$ be variable points on $x-$axis and $y-$axis respectively such that the line segment $AB$ is in the first quadrant and of a fixed length $2d$ . Let $C$ be the mid-point of $AB$ and $P$ be a point such that
[b](a)[/b] $P$ and the origin are on the opposite sides of $AB$ and,
[b](b)[/b] $PC$ is a line segment of length $d$ which is perpendicular to $AB$ .
Find the locus of $P$ .
1994 Moldova Team Selection Test, 2
Prove that every positive rational number can be expressed uniquely as a finite sum of the form $$a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},$$ where $a_n$ are integers such that $0 \leq a_n \leq n-1$ for all $n > 1$.
2015 ISI Entrance Examination, 6
Find all $n\in \mathbb{N} $ so that 7 divides $5^n + 1$
2012 ISI Entrance Examination, 2
Consider the following function
\[g(x)=(\alpha+|x|)^{2}e^{(5-|x|)^{2}}\]
[b]i)[/b] Find all the values of $\alpha$ for which $g(x)$ is continuous for all $x\in\mathbb{R}$
[b]ii)[/b]Find all the values of $\alpha$ for which $g(x)$ is differentiable for all $x\in\mathbb{R}$.
2019 ISI Entrance Examination, 1
Prove that the positive integers $n$ that cannot be written as a sum of $r$ consecutive positive integers, with $r>1$, are of the form $n=2^l~$ for some $l\geqslant 0$.
2021 ISI Entrance Examination, 8
A pond has been dug at the Indian Statistical Institute as an inverted truncated pyramid with a square base (see figure below). The depth of the pond is 6m. The square at the bottom has side length 2m and the top square has side length 8m. Water is filled in at a rate of $\tfrac{19}{3}$ cubic meters per hour. At what rate is the water level rising exactly $1$ hour after the water started to fill the pond?
[img]https://cdn.artofproblemsolving.com/attachments/0/9/ff8cac4bb4596ec6c030813da7e827e9a09dfd.png[/img]
2022 ISI Entrance Examination, 8
Find the minimum value of $$\big|\sin x+\cos x+\tan x+\cot x+\sec x+\operatorname{cosec}x\big|$$ for real numbers $x$ not multiple of $\frac{\pi}{2}$.
2020 ISI Entrance Examination, 5
Prove that the largest pentagon (in terms of area) that can be inscribed in a circle of radius $1$ is regular (i.e., has equal sides).