Found problems: 19
2017 CMI B.Sc. Entrance Exam, 4
The domain of a function $f$ is $\mathbb{N}$ (The set of natural numbers). The function is defined as follows : $$f(n)=n+\lfloor\sqrt{n}\rfloor$$ where $\lfloor k\rfloor$ denotes the nearest integer smaller than or equal to $k$. Prove that, for every natural number $m$, the following sequence contains at least one perfect square $$m,~f(m),~f^2(m),~f^3(m),\cdots$$ The notation $f^k$ denotes the function obtained by composing $f$ with itself $k$ times.
2023 CMI B.Sc. Entrance Exam, 3
Consider the polynomial $p(x) = x^4 + ax^3 + bx^2 + cx + d$.
It is given that $p(x)$ has its only root at $x = r$ i.e $p(r) = 0$.
$\textbf{(a)}$ Show that if $a, b, c, d$ are rational then $r$ is rational.
$\textbf{(b)}$ Show that if $a, b, c, d$ are integers then $r$ is an integer.
[hide=Hint](Hint: Consider the roots of $p'(x)$ )[/hide]
2023 CMI B.Sc. Entrance Exam, 6
Consider a positive integer $a > 1$. If $a$ is not a perfect square then at the next move we add $3$ to it and if it is a perfect square we take the square root of it. Define the trajectory of a number $a$ as the set obtained by performing this operation on $a$. For example the cardinality of $3$ is $\{3, 6, 9\}$.
Find all $n$ such that the cardinality of $n$ is finite.
The following part problems may attract partial credit.
$\textbf{(a)}$Show that the cardinality of the trajectory of a number cannot be $1$ or $2$.
$\textbf{(b)}$Show that $\{3, 6, 9\}$ is the only trajectory with cardinality $3$.
$\textbf{(c)}$ Show that there for all $k \geq 3$, there exists a number such that the cardinality
of its trajectory is $k$.
$\textbf{(d)}$ Give an example of a number with cardinality of trajectory as infinity.
2017 CMI B.Sc. Entrance Exam, 2
Let $L$ be the line of intersection of the planes $~x+y=0~$ and $~y+z=0$.
[b](a)[/b] Write the vector equation of $L$, i.e. find $(a,b,c)$ and $(p,q,r)$ such that $$L=\{(a,b,c)+\lambda(p,q,r)~~\vert~\lambda\in\mathbb{R}\}$$
[b](b)[/b] Find the equation of a plane obtained by $x+y=0$ about $L$ by $45^{\circ}$.
2023 CMI B.Sc. Entrance Exam, 5
In whatever follows $f$ denotes a differentiable function from $\mathbb{R}$ to $\mathbb{R}$. $f \circ f$
denotes the composition of $f(x)$.
$\textbf{(a)}$ If $f\circ f(x) = f(x) \forall x \in \mathbb{R}$ then for all $x$, $f'(x) =$ or $f'(f(x)) =$. Fill in the
blank and justify.
$\textbf{(b)}$Assume that the range of $f$ is of the form $ \left(-\infty , +\infty \right), [a, \infty ),(- \infty , b], [a, b] $.
Show that if $f \circ f = f$, then the range of $f$ is $\mathbb{R}$. [hide=Hint](Hint: Consider a maximal
element in the range of f)[/hide]
$\textbf{(c)}$ If $g$ satisfies $g \circ g \circ g = g$, then $g$ is onto. Prove that $g$ is either strictly increasing or strictly decreasing. Furthermore show that if $g$ is strictly increasing, then $g$ is unique.
2017 CMI B.Sc. Entrance Exam, 1
Answer the following questions :
[b](a)[/b] Evaluate $~~\lim_{x\to 0^{+}} \Big(x^{x^x}-x^x\Big)$
[b](b)[/b] Let $A=\frac{2\pi}{9}$, i.e. $40$ degrees. Calculate the following $$1+\cos A+\cos 2A+\cos 4A+\cos 5A+\cos 7A+\cos 8A$$
[b](c)[/b] Find the number of solutions to $$e^x=\frac{x}{2017}+1$$
2019 CMI B.Sc. Entrance Exam, 4
Let $ABCD$ be a parallelogram $.$ Let $O$ be a point in its interior such that $\angle AOB + \angle DOC = 180^{\circ} . $
Show that $,\angle ODC = \angle OBC . $
2019 CMI B.Sc. Entrance Exam, 3
Evaluate $\int_{ 0 }^{ \infty } ( 1 + x^2 )^{-( m + 1 )} \mathrm{d}x$ where $m \in \mathbb{N} $
2017 CMI B.Sc. Entrance Exam, 5
Each integer is colored with exactly one of $3$ possible colors -- black, red or white -- satisfying the following two rules : the negative of a black number must be colored white, and the sum of two white numbers (not necessarily distinct) must be colored black.
[b](a)[/b] Show that, the negative of a white number must be colored black and the sum of two black numbers must be colored white.
[b](b)[/b] Determine all possible colorings of the integers that satisfy these rules.
2017 CMI B.Sc. Entrance Exam, 6
You are given a regular hexagon. We say that a square is inscribed in the hexagon if it can be drawn in the interior such that all the four vertices lie on the perimeter of the hexagon.
[b](a)[/b] A line segment has its endpoints on opposite edges of the hexagon. Show that, it passes through the centre of the hexagon if and only if it divides the two edges in the same ratio.
[b](b)[/b] Suppose, a square $ABCD$ is inscribed in the hexagon such that $A$ and $C$ are on the opposite sides of the hexagon. Prove that, centre of the square is same as that of the hexagon.
[b](c)[/b] Suppose, the side of the hexagon is of length $1$. Then find the length of the side of the inscribed square whose one pair of opposite sides is parallel to a pair of opposite sides of the hexagon.
[b](d)[/b] Show that, up to rotation, there is a unique way of inscribing a square in a regular hexagon.
2019 CMI B.Sc. Entrance Exam, 2
$(a)$ Count the number of roots of $\omega$ of the equation $z^{2019} - 1 = 0 $ over complex numbers that satisfy
\begin{align*}
\vert \omega + 1 \vert \geq \sqrt{2 + \sqrt{2}}
\end{align*}
$(b)$ Find all real numbers $x$ that satisfy following equation $:$
\begin{align*}
\frac{ 8^x + 27^x }{ 12^x + 18^x } = \frac{7}{6}
\end{align*}
2023 CMI B.Sc. Entrance Exam, 1
We will consider odd natural numbers $n$ such that$$n|2023^n-1$$
$\textbf{a.}$ Find the smallest two such numbers.
$\textbf{b.}$ Prove that there exists infinitely many such $n$
2018 CMI B.Sc. Entrance Exam, 1
Answer the following questions :
$\textbf{(a)}~$ A natural number $k$ is called stable if there exist $k$ distinct natural numbers $a_1, a_2,\cdots, a_k$, each $a_i>1$, such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_k}=1$$ Show that if $k$ is stable, then $(k+1)$ is also stable. Using this or otherwise, find all stable numbers.
$\textbf{(b)}$ Let $f$ be a differentiable function defined on a subset $A$ of the real numbers. Define $$f^*(y):=\max_{x\in A} \left\{yx-f(x)\right\}$$ whenever the above maximum is finite.
For the function $f(x)=\ln x$, determine the set of points for which $f^*$ is defined and find an expression for $f^*(y)$ involving only $y$ and constants.
2018 CMI B.Sc. Entrance Exam, 2
Answer the following questions :
$\textbf{(a)}$ Find all real solutions of the equation $$\Big(x^2-2x\Big)^{x^2+x-6}=1$$ Explain why your solutions are the only solutions.
$\textbf{(b)}$ The following expression is a rational number. Find its value. $$\sqrt[3]{6\sqrt{3}+10} -\sqrt[3]{6\sqrt{3}-10}$$
2018 CMI B.Sc. Entrance Exam, 6
Imagine the unit square in the plane to be a [i]carrom board[/i]. Assume the [i]striker[/i] is just a point, moving with no friction (so it goes forever), and that when it hits an edge, the angle of reflection is equal to the angle of incidence, as in real life. If the striker ever hits a corner it falls into the pocket and disappears. The trajectory of the striker is completely determined by its starting point $(x,y)$ and its initial velocity $\overrightarrow{(p,q)}$.
If the striker eventually returns to its initial state (i.e. initial position and initial velocity), we define its [i]bounce number[/i] to be the number of edges it hits before returning to its initial state for the $1^{\text{st}}$ time.
For example, the trajectory with initial state $[(.5,.5);\overrightarrow{(1,0)}]$ has bounce number $2$ and it returns to its initial state for the $1^{\text{st}}$ time in $2$ time units. And the trajectory with initial state $[(.25,.75);\overrightarrow{(1,1)}]$ has bounce number $4$.
$\textbf{(a)}$ Suppose the striker has initial state $[(.5,.5);\overrightarrow{(p,q)}]$. If $p>q\geqslant 0$ then what is its velocity after it hits an edge for the $1^{\text{st}}$ time ? What if $q>p\geqslant 0$ ?
$\textbf{(b)}$ Draw a trajectory with bounce number $5$ or justify why it is impossible.
$\textbf{(c)}$ Consider the trajectory with initial state $[(x,y);\overrightarrow{(p,0)}]$ where $p$ is a positive integer. In how much time will the striker $1^{\text{st}}$ return to its initial state ?
$\textbf{(d)}$ What is the bounce number for the initial state $[(x,y);\overrightarrow{(p,q)}]$ where $p,q$ are relatively prime positive integers, assuming the striker never hits a corner ?
2017 CMI B.Sc. Entrance Exam, 3
Let $p(x)$ be a polynomial of degree strictly less than $100$ and such that it does not have $(x^3-x)$ as a factor. If $$\frac{d^{100}}{dx^{100}}\bigg(\frac{p(x)}{x^3-x}\bigg)=\frac{f(x)}{g(x)}$$ for some polynomials $f(x)$ and $g(x)$ then find the smallest possible degree of $f(x)$.
2023 CMI B.Sc. Entrance Exam, 4
In a class there are n students with unequal heights.
$\textbf{(a)}$ Find the number of orderings of the students such that the shortest person
is not at the front and the tallest person is not at the end.
$\textbf{(b)}$ Define the [i]badness[/i] of an ordering as the maximum number $k$ such that there
are $k$ many people with height greater than in front of a person. For example:
the sequence $66, 61, 65, 64, 62, 70$ has [i]badness [/i] $3$ since there are $3$ numbers greater
than $62$ in front of it. Let $f_k(n)$ denote the number of orderings of $n$ with [i]badness[/i] $k$. Find $f_k(n)$.
[hide=hint](Hint: Consider $g_k(n)$ as the number of orderings of n with [i]badness [/i]less than
or equal to $k$)[/hide]
2019 CMI B.Sc. Entrance Exam, 1
For a natural number $n$ denote by Map $( n )$ the set of all functions $f : \{ 1 , 2 , 3 , \cdots , n \} \rightarrow \{ 1 , 2 , 3 , \cdots , n \} . $ For $ f , g \in $ Map$( n ) , f \circ g $ denotes the function in Map $( n )$ that sends $x \rightarrow f ( g ( x ) ) . $ \\
\\
$(a)$ Let $ f \in$ Map $( n ) . $ If for all $x \in \{ 1 , 2 , 3 , \cdots , n \} f ( x ) \neq x , $ show that $ f \circ f \neq f $
\\$(b)$ Count the number of functions $ f \in$ Map $( n )$ such that $ f \circ f = f $
2019 CMI B.Sc. Entrance Exam, 5
Three positive reals $x , y , z $ satisfy \\
$x^2 + y^2 = 3^2 \\
y^2 + yz + z^2 = 4^2 \\
x^2 + \sqrt{3}xz + z^2 = 5^2 .$ \\
Find the value of $2xy + xz + \sqrt{3}yz$