Here is a video solution for a Problem based on using Vectors and Carpet Theorem in Geometry 1? This problem is helpful for Math Olympiad, ISI & CMI Entrance, and other math contests. Watch and Learn!

Here goes the questionâ€¦

Given ABCD is a quadrilateral and P and Q are 2 points on AB and CD respectively, such that AP/AB = CQ/CD. Show that: in the figure below, the area of the green part = the white part.

We will recommend you to try the problem yourself.

Mahalanobis National Statistics Competition = MNStatC

organized by Cheenta Statistics Department

with exciting cash prizes.

What is MNStatC?

Mahalanobis National Statistics Competition (MNStatC) is a national level statistics competition, aimed at undergraduate students as well as masters, Ph.D. students, and data analytics, and ML professionals. MNStatC plans to test your core mathematics, probability, and statistics (theoretical + applied) skills.

MNStatC aims

to engage the data lovers to explore the beauty of data through effective problem-solving.

to enhance the mathematics, probability, and statistics skills for data analytics and machine learning professionals and equip them for the interviews.

to let the undergraduate, masters, and Ph.D. students to taste a flavor of out of the box statistics using the same knowledge they learned in their colleges.

to build a community and team of authentic and serious data lovers, who plan to change the world through their data-driven minds.

to raise awareness of the importance of core statistics, probability, and mathematics in the rising field of Data Science, ML, and AI.

During prize collection, if you cannot share the proper proof of your college id card or office id card with your name in the application, we will be moving on to the next deserving candidate for the prize.

This is to stop the insurgence of various intelligent scam applicants.

Example: Suppose, you are not an undergraduate and you decide to enroll in the undergraduate exam and you become first in the undergraduate category. We will ask for proof that you are an undergraduate. If you are an undergraduate passout, you belong to the masters’ group. If you cannot provide the proof, we will give the cash prize to the next candidate.

The cash prize may vary (increase or decrease) depending on the number of candidates applying.

Here is a video solution for a Problem based on Carpet Strategy in Geometry. This problem is helpful for Math Olympiad, ISI & CMI Entrance, and other math contests. Watch and Learn!

Here goes the questionâ€¦

Suppose ABCD is a square and X is a point on BC such that AX and DX are joined to form a triangle AXD. Similarly, there is a point Y on AB such that DY and CY are joined to form the triangle DYC. Compare the area of the triangles to the area of the square.

We will recommend you to try the problem yourself.

Here is a video solution for a Problem based on finding the area of a quadrilateral. This question is from American Mathematics Competition, AMC 12, 2018. Watch and Learn!

Here goes the questionâ€¦

Connect the centroids of the four triangles in a square. Can you find the area of the quadrilateral?

We will recommend you to try the problem yourself.

Subjective Paper – ISI MStat Entrance 2020 Problems and Solutions

Let \(f(x)=x^{2}-2 x+2\). Let \(L_{1}\) and \(L_{2}\) be the tangents to its graph at \(x=0\) and \(x=2\) respectively. Find the area of the region enclosed by the graph of \(f\) and the two lines \(L_{1}\) and \(L_{2}\).

Solution

Find the number of \(3 \times 3\) matrices \(A\) such that the entries of \(A\) belong to the set \(\mathbb{Z}\) of all integers, and such that the trace of \(A^{t} A\) is 6 . \(\left(A^{t}\right.\) denotes the transpose of the matrix \(\left.A\right)\).

Solution

Consider \(n\) independent and identically distributed positive random variables \(X_{1}, X_{2}, \ldots, X_{n},\) Suppose \(S\) is a fixed subset of \({1,2, \ldots, n}\) consisting of \(k\) distinct elements where \(1 \leq k<n\) (a) Compute \(\mathbb{E}\left[\frac{\sum_{i \in S} X_{i}}{\sum_{i=1}^{n} X_{i}}\right]\)

(b) Assume that \(X_{i}\) ‘s have mean \(\mu\) and variance \(\sigma^{2}, 0<\sigma^{2}<\infty\). If \(j \notin S,\) show that the correlation between \(\left(\sum_{i \in S} X_{i}\right) X_{j}\) and \(\sum_{i \in S} X_{i}\) lies between -\(\frac{1}{\sqrt{k+1}} \text { and } \frac{1}{\sqrt{k+1}}\).

Solution

Let \(X_{1,} X_{2}, \ldots, X_{n}\) be independent and identically distributed random variables. Let \(S_{n}=X_{1}+\cdots+X_{n}\). For each of the following statements, determine whether they are true or false. Give reasons in each case.

(a) If \(S_{n} \sim E_{x p}\) with mean \(n,\) then each \(X_{i} \sim E x p\) with mean 1 .

(b) If \(S_{n} \sim B i n(n k, p),\) then each \(X_{i} \sim B i n(k, p)\)

Solution

Let \(U_{1}, U_{2}, \ldots, U_{n}\) be independent and identically distributed random variables each having a uniform distribution on (0,1) . Let \( X=\min \{U_{1}, U_{2}, \ldots, U_{n}\} \), \( Y=\max \{U_{1}, U_{2}, \ldots, U_{n}\} \)

Evaluate \(\mathbb{E}[X \mid Y=y]\) and \( \mathbb{E}[Y \mid X=x] \).

Solution

Suppose individuals are classified into three categories \(C_{1}, C_{2}\) and \(C_{3}\) Let \(p^{2},(1-p)^{2}\) and \(2 p(1-p)\) be the respective population proportions, where \(p \in(0,1)\). A random sample of \(N\) individuals is selected from the population and the category of each selected individual recorded.

For \(i=1,2,3,\) let \(X_{i}\) denote the number of individuals in the sample belonging to category \(C_{i} .\) Define \(U=X_{1}+\frac{X_{3}}{2}\)

(a) Is \(U\) sufficient for \(p ?\) Justify your answer.

(b) Show that the mean squared error of \(\frac{U}{N}\) is \(\frac{p(1-p)}{2 N}\)

Solution

Consider the following model: \( y_{i}=\beta x_{i}+\varepsilon_{i} x_{i}, \quad i=1,2, \ldots, n \), where \(y_{i}, i=1,2, \ldots, n\) are observed; \(x_{i}, i=1,2, \ldots, n\) are known positive constants and \(\beta\) is an unknown parameter. The errors \(\varepsilon_{1}, \varepsilon_{2}, \ldots, \varepsilon_{n}\) are independent and identically distributed random variables having the probability density function \[ f(u)=\frac{1}{2 \lambda} \exp \left(-\frac{|u|}{\lambda}\right), \quad-\infty<u<\infty \] and \(\lambda\) is an unknown parameter.

(a) Find the least squares estimator of \(\beta\).

(b) Find the maximum likelihood estimator of \(\beta\).

Solution

Assume that \(X_{1}, \ldots, X_{n}\) is a random sample from \(N(\mu, 1),\) with \(\mu \in \mathbb{R}\). We want to test \(H_{0}: \mu=0\) against \(H_{1}: \mu=1\). For a fixed integer \(m \in{1, \ldots, n},\) the following statistics are defined:

\(\operatorname{Fix} \alpha \in(0,1) .\) Consider the test

Reject \(H_{0}\) if \( \max \{T_{i}: 1 \leq i \leq n-m+1\}>c_{m, \alpha}\)

Find a choice of \(c_{m, \alpha} \in \mathbb{R}\) in terms of the standard normal distribution function \(\Phi\) that ensures that the size of the test is at most \(\alpha\).

Solution

A finite population has \(N\) units, with \(x_{i}\) being the value associated with the \(i\) th unit, \(i=1,2, \ldots, N\). Let \(\bar{x}{N}\) be the population mean. A statistician carries out the following experiment.

Step 1: Draw an SRSWOR of size \(n({1}\) and denote the sample mean by \( \bar{X}{n}\)

Step 2: Draw a SRSWR of size \(m\) from \(S_{1}\). The \(x\) -values of the sampled units are denoted by \(\{Y_{1}, \ldots, Y_{m}\}\)

An estimator of the population mean is defined as,

Subjective Paper – ISI Entrance 2020 Problems and Solutions

Let \( \iota \) be a root of the equation \( x^2 + 1 = 0 \) and let \( \omega \) be a root of the equation \( x^2 + x + 1 = 0 \). Construct a polynomial $$ f(x) = a_0 + a_1 x + \cdots + a_n x^n $$ where \( a_0, a_1, \cdots , a_n \) are all integers such that \( f (\iota + \omega) = 0 \).

Answer: \( f(x) = x^4 + 2x^3 + 5x^2 + 4x + 1 \)

Let \( a \) be a fixed real number. Consider the equation $$(x+2)^2 (x+7)^2 + a = 0, x \in \mathbb{R} $$ where \( \mathbb{R} \) is the set of real numbers. For what values of \(a \), will the equation have exactly one root?

Answer: \( – (2.5)^4 \)

Let \( A \) and \( B \) be variable points on the \(x\)-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 (a) \( P \) and the origin are on the opposite sides of \(AB\) and, (b) \(PC\) is a line of length \(d\) which is perpendicular to \(AB\). Find the locus of \(P\).

Answer: Line segment connecting \( (d, d) \) to \( \sqrt{2} d, \sqrt{2} d \)

Let a real-valued sequence \( \{x_n\}_{n \geq 1} \) be such that $$ \displaystyle{\lim_{n \to \infty} n x_n = 0 }. $$ Find all possible real values of \( t \) such that \( \displaystyle{\lim_{n \to \infty} x_n \cdot (\log n)^t = 0 }. \)

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).

Prove that the family of curves $$ \displaystyle{ \frac{x^2}{a^2 + \lambda} + \frac{y^2}{b^2 + \lambda} = 1} $$ satisfies $$ \displaystyle { \frac{dy}{dx} (a^2 – b^2) = (x + y \frac{dy}{dx})(x \frac{dy}{dx} – y ) } $$

Consider a right-angled triangle with integer-valued sides \( a < b < c \) where \(a, b, c\) are pairwise co-prime. Let \( d = c – b \). Suppose \( d \) divides \(a \). Then (a) Prove that \( d \leq 2 \) (b) Find all such triangles (i.e. all possible triplets \(a, b, c\) ) with permeter less than \(100 \).

A finite sequence of numbers \( (a_1, \cdots , a_n ) \) is said to be alternating if $$ \displaystyle{ a_1 > a_2, a_2 < a_3, a_2 > a_4, a_4 < a_5, \cdots \\ \\ \textrm{or} \quad a_1 < a_2, a_2 > a_3, a_3< a_4, a_4 > a_5} $$ How many alternatig sequences of length \(5 \), with distinct number \( a_1, \cdots , a_5 \) can be formed such that \( a_i \in \{ 1, 2, \cdots , 20 \} \) for \( i = 1, \cdots , 5 \)?

Answer: \( 32 \times { {20} \choose {5} } \)

Objective Paper – ISI Entrance 2020 Problems and Solutions

$1$ .The number of subsets of ${1,2,3, \ldots, 10}$ having an odd number of elements is (A) $1024$ (B) $512$ (C) $256 $ (D)$ 50$

$2$ .For the function on the real line $\mathbb{R}$ given by $f(x)=|x|+|x+1|+e^{x}$, which of the following is true?

(A) It is differentiable everywhere. (B) It is differentiable everywhere except at $x=0$ and $x=-1$ (C) It is differentiable everywhere except at $x=1 / 2$ (D) It is differentiable everywhere except at $x=-1 / 2$

$3$ .If $f, g$ are real-valued differentiable functions on the real line $\mathbb{R}$ such that $f(g(x))=x$ and $f^{\prime}(x)=1+(f(x))^{2},$ then $g^{\prime}(x)$ equals

$4$ . The number of real solutions of $e^{x}=\sin (x)$ is (A) $0$ (B) $1$ (C) $2$ (D) infinite.

$5$ . What is the limit of $\sum_{k=1}^{n} \frac{e^{-k / n}}{n}$ as $n$ tends to $\infty ?$ (A) The limit does not exist. (B) $\infty$ (C) $1-e^{-1}$ (D) $e^{-0.5}$

$6$ . A group of 64 players in a chess tournament needs to be divided into 32 groups of 2 players each. In how many ways can this be done?

$9$ . There are $128$ numbers $1,2, \ldots, 128$ which are arranged in a circular pattern in clockwise order. We start deleting numbers from this set in a clockwise fashion as follows. First delete the number $2,$ then skip the next available number (which is 3) and delete 4. Continue in this manner, that is, after deleting a number, skip the next available number clockwise and delete the number available after that, till only one number What is the last number left?

(A) $1$ (B) $63$ (C) $127$ (D) None of the above.

$10$ . Let $z$ and $w$ be complex numbers lying on the circles of radii 2 and 3 respectively, with centre $(0,0) .$ If the angle between the corresponding vectors is 60 degrees, then the value of $|z+w| /|z-w|$ is:

$11$ . Two vertices of a square lie on a circle of radius $r$ and the other two vertices lie on a tangent to this circle. Then the length of the side of the square is

$12$ . For a real number $x,$ let $[x]$ denote the greatest integer less than or equal to $x .$ Then the number of real solutions of $|2 x-[x]|=4$ is

(A) $4$ (B) $3$ (C) $2$ (D) $1$

$13$ . Let $f, g$ be differentiable functions on the real line $\mathbb{R}$ with $f(0)>g(0)$ Assume that the set $M={t \in \mathbb{R} \mid f(t)=g(t)}$ is non-empty and that $f^{\prime}(t) \geq g^{\prime}(t)$ for all $t \in M .$ Then which of the following is necessarily true?

(A) If $t \in M,$ then $t<0$. (B) For any $t \in M, f^{\prime}(t)>g^{\prime}(t)$ (C) For any $t \notin M, f(t)>g(t)$ (D) None of the above.

$14$ . Consider the sequence $1,2,2,3,3,3,4,4,4,4,5,5,5,5,5, \ldots$ obtained by writing one $1,$ two $2$ ‘s, three $3$ ‘s and so on. What is the $2020^{\text {th }}$ term in the sequence?

(A) $62$ (B)$ 63$ (C) $64$ (D) $65$

$15$.Let $A=\{x_{1}, x_{2}, \ldots, x_{50}\}$ and $B=\{y_{1}, y_{2}, \ldots, y_{20}\}$ be two sets of real numbers. What is the total number of functions $f: A \rightarrow B$ such that $f$ is onto and $f\left(x_{1}\right) \leq f\left(x_{2}\right) \leq \cdots \leq f\left(x_{50}\right) ?$

$16$. The number of complex roots of the polynomial $z^{5}-z^{4}-1$ which have modulus $1$ is

(A) $0$ (B) $1$ (C) $2$ (D) more than $2$

$17$. The number of real roots of the polynomial $$ p(x)=\left(x^{2020}+2020 x^{2}+2020\right)\left(x^{3}-2020\right)\left(x^{2}-2020\right) $$

(A) $2$ (B)$3$ (C) $2023$ (D) $2025$

$18$. Which of the following is the sum of an infinite geometric sequence whose terms come from the set $\{1, \frac{1}{2}, \frac{1}{4}, \ldots, \frac{1}{2^{n}}, \ldots\} ?$

$19$. If $a, b, c$ are distinct odd natural numbers, then the number of rational roots of the polynomial $a x^{2}+b x+c$

(A) must be $0 $. (B) must be $1$ . (C) must be $2$ . (D) cannot be determined from the given data.

$20$. Let $A, B, C$ be finite subsets of the plane such that $A \cap B, B \cap C$ and $C \cap A$ are all empty. Let $S=A \cup B \cup C$. Assume that no three points of $S$ are collinear and also assume that each of $A, B$ and $C$ has at least 3 points. Which of the following statements is always true?

(A) There exists a triangle having a vertex from each of $A, B, C$ that does not contain any point of $S$ in its interior. (B) Any triangle having a vertex from each of $A, B, C$ must contain a point of $S$ in its interior. (C) There exists a triangle having a vertex from each of $A, B, C$ that contains all the remaining points of $S$ in its interior. (D) There exist 2 triangles, both having a vertex from each of $A, B, C$ such that the two triangles do not intersect.

$21$. Shubhaangi thinks she may be allergic to Bengal gram and takes a test that is known to give the following results:

(i)For people who really do have the allergy, the test says “Yes” $90 \%$ of the time.

(ii)For people who do not have the allergy, the test says “Yes” $15 \%$ of the time.

If $2 \%$ of the population has the allergy and Shubhangi’s test says “Yes” then the chances that Shubhaangi does really have the allergy are

(A) $1 / 9$ (B) $6 / 55$ (C) $1 / 11$ (D) cannot be determined from the given data.

$22$. If $\sin \left(\tan ^{-1}(x)\right)=\cot \left(\sin ^{-1}\left(\sqrt{\frac{13}{17}}\right)\right)$ then $x$ is

$23$. If the word PERMUTE is permuted in all possible ways and the different resulting words are written down in alphabetical order (also known as dictionary order $)$, irrespective of whether the word has meaning or not, then the $720^{\text {th }}$ word would be:

(A) EEMPRTU (B) EUTRPME (C) UTRPMEE (D) MEET-PUR.

$24$. The points (4,7,-1),(1,2,-1),(-1,-2,-1) and (2,3,-1) in $\mathbb{R}^{3}$ are the vertices of a

(A) rectangle which is not a square. (B) rhombus. (C) parallelogram which is not a rectangle. (D) trapezium which is not a parallelogram.

$25$. Let $f(x), g(x)$ be functions on the real line $\mathbb{R}$ such that both $f(x)+g(x)$ and $f(x) g(x)$ are differentiable. Which of the following is FALSE?

(A) $f(x)^{2}+g(x)^{2}$ is necessarily differentiable. (B) $f(x)$ is differentiable if and only if $g(x)$ is differentiable. (C) $f(x)$ and $g(x)$ are necessarily continuous. (D) If $f(x)>g(x)$ for all $x \in \mathbb{R},$ then $f(x)$ is differentiable.

$26$. Let $S$ be the set consisting of all those real numbers that can be written as $p-2 a$ where $p$ and $a$ are the perimeter and area of a right-angled triangle having base length 1 . Then $S$ is

(A)$(2, \infty)$ (B) $(1, \infty)$ (C) $(0, \infty)$ (D) the real line $\mathbb{R}$.

$27$. Let $S={1,2, \ldots, n} .$ For any non-empty subset $A$ of $S$, let l(a) denote the largest number in $A .$ If $f(n)=\sum_{A \subseteq S} l(A),$ that is, $f(n)$ is the sum of the numbers $l(A)$ while $A$ ranges over all the nonempty subsets of $S$, then $f(n)$ is

( A )$ 2^{n}(n+1)$ (B) $2^{n}(n+1)-1$ ( C) $2^{n}(n-1)$ (D) $2^{n}(n-1)+1$

$28$. The area of the region in the plane $\mathbb{R}^{2}$ given by points $(x, y)$ satisfying $|y| \leq 1$ and $x^{2}+y^{2} \leq 2$ is

$29$. Let $n$ be a positive integer and $t \in(0,1) .$ Then $\sum_{r=0} r\left(\begin{array}{l}n \ r\end{array}\right) t^{r}(1-t)^{n-r}$equals

(A) $n t$

(B)$(n-1)(1-t)$

(C) $n t+(n-1)(1-t)$

(D) $\left(n^{2}-2 n+2\right) t$

$30$. For any real number $x,$ let $[x]$ be the greatest integer $m$ such that $m \leq x$ Then the number of points of discontinuity of the function $g(x)=\left[x^{2}-2\right]$ on the interval$ (-3,3)$ is