Categories

## How to use Vectors and Carpet Theorem in Geometry 1?

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 question…

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.

Done?

Let’s see the solution in the video below:

Categories

## Mahalanobis National Statistics Competition

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.

## Exam Date and Time

The Exam will be live on 15th November 2020 and end on 30th November.

You can give the exam and enjoy the problems at your own time.

## Registration Time and Fee

The Registration will be over by 10th November 2020.

The Registration Fee is Rs 101 for undergraduate students.

The Registration Fee is Rs 241 for masters, Ph.D. students, and data analytics, and ML professionals.

## Syllabus

The syllabus is the undergraduate statistics syllabus of a statistics course at an Indian University/College.

#### Mathematics

• High School Mathematics (10+2 Level)
• One Variable Calculus
• Multiple Variable Calculus
• Linear Algebra

#### Probability

###### (Reference: Mathematical Statistics and Data Analysis by John.A.Rice Chapters 1- 6)
• Probability Space
• Conditional Probability, Independence, Bayes Theorem
• Random Variables, Moments, MGF, Characteristic Function
• Distribution Function, Density
• Discrete, Continuous, and Mixed Random Variables
• Various Probability Distributions and their relationships
• Joint, Marginal and Conditional Distributions
• Functions of Joint Distributions
• Order Statistics and Sampling Distributions
• Limit Theorems

#### Statistics

###### (Reference: Mathematical Statistics and Data Analysis by John.A.Rice Chapters 7 – 14)
• Parametric Estimation Theory*
• Basic Non-Parametric Estimation Theory
• Testing of Hypothesis*
• Simple and Multiple Linear Regression*
• Analysis of Variance*
• Basic Categorical Data Analysis
• Basic Bayesian Inference

* = important

#### Programming Skills

Skill in any mathematical/statistical computational software is a plus. You must take the help of your coding skills to quickly compute stuff.

## Competition Pattern

You have to solve a total of 20 multiple choice and numerical problems in 2 hours.

• Mathematics (3 problems)
• Probability Theory (6 problems)
• Theoretical Statistics (6 problems)
• Applied Statistics (5 problems)

You can take the help of any book or resources or person during the competition.

## Exciting Cash Prizes and Discussion Session

#### 1. Undergraduate Students (current position)

1st Cash Prize: Rs 1000

2nd Cash Prize: Rs 500

3rd Cash Prize: Rs 200

#### 2. Masters, Ph.D. students, and Data Analytics Professionals (current position)

1st Cash Prize: Rs 1500

2nd Cash Prize: Rs 700

3rd Cash Prize: Rs 400

Terms and Conditions Apply* (read at the bottom of the page)

## Demo Competition

Go to Mahalanobis National Statistics Competition.

You will find a Demo Competition over there.

Click on it and enjoy the demo problems.

## Register now

Terms and Conditions Apply*

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.

Categories

## Carpet Strategy in Geometry | Watch and Learn

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 question…

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.

Done?

Let’s see the solution in the video below:

Categories

## Bijection Principle Problem | ISI Entrance TOMATO Obj 22

Here is a video solution for a Problem based on Bijection Principle. This is an Objective question 22 from TOMATO for ISI Entrance. Watch and Learn!

Here goes the question…

Given that: x+y+z=10, where x, y and z are natural numbers. How many such solutions are possible for this equation?

We will recommend you to try the problem yourself.

Done?

Let’s see the solution in the video below:

Categories

## What is the Area of Quadrilateral? | AMC 12 2018 | Problem 13

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 question…

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.

Done?

Let’s see the solution in the video below:

Categories

## Solving Weird Equations using Inequality | TOMATO Problem 78

Here is a video solution for ISI Entrance Number Theory Problems based on solving weird equations using Inequality. Watch and Learn!

Here goes the question…

Solve: 2 \cos ^{2}\left(x^{3}+x\right)=2^{x}+2^{-x}

We will recommend you to try the problem yourself.

Done?

Let’s see the proof in the video below:

Categories

## AM-GM Inequality Problem | ISI Entrance

Here is a video solution for ISI Entrance Number Theory Problems based on AM-GM Inequality Problem. Watch and Learn!

Here goes the question…

a, b, c, d are positive real numbers. Prove that: (1+a)(1+b)(1+c)(1+d) <= 16.

We will recommend you to try the problem yourself.

Done?

Let’s see the proof in the video below:

Categories

## Sum of 8 fourth powers | ISI Entrance Problem

Here is a video solution for ISI Entrance Number Theory Problems based on Sum of 8 fourth powers. Watch and Learn!

1. Can you show that the sum of 8 fourth powers of integers never adds up to 1993?

How can you solve this fourth-degree diophantine equation? Let’s see in the video below:

## ISI MStat Entrance 2020 Problems and Solutions

This post contains Indian Statistical Institute, ISI MStat Entrance 2020 Problems and Solutions. Try to solve them out.

## 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:

\begin{aligned}
T_{1} &= \frac{\left(X_{1}+\ldots+X_{m}\right)}{m} \\
T_{2} &= \frac{\left(X_{2}+\ldots+X_{m+1}\right)}{m} \\
\vdots &=\vdots \\
T_{n-m+1} &= \frac{\left(X_{n-m+1}+\ldots+X_{n}\right)}{m}
\end{aligned}

$\operatorname{Fix} \alpha \in(0,1) .$ 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,

$\widehat{T}{m}=\frac{1}{m} \sum{i=1}^{m} Y_{i}$

(a) Show that $\widehat{T}{m}$ is an unbiased estimator of the population mean.

(b) Which of the following has lower variance: $\widehat{T}{m}$ or $\bar{X}_{n} ?$

Solution

## Objective Paper

 1. C 2. D 3. A 4. B 5. A 6. B 7. C 8. A 9. C 10. A 11. C 12. D 13. C 14. B 15. B 16. C 17. D 18. B 19. B 20. C 21. C 22. D 23. A 24. B 25. D 26. B 27. D 28. D 29. B 30. C

Watch videos related to the ISI MStat Problems here.

Categories

## ISI Entrance 2020 Problems and Solutions – B.Stat & B.Math

This post contains Indian Statistical Institute, ISI Entrance 2020 Problems and Solutions. Try them out.

## 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. $1$ .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

(A) $\frac{1}{1+x^{2}}$ (B) $1+x^{2}$ (C) $\frac{1}{1+x^{4}}$ (D) $1+x^{4}$

$4$ . 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?

(A) $\frac{64 !}{32 ! 2^{32}}$ (B) $64 \choose 2$ $62 \choose 2$ $\ldots$ $4 \choose 2$$2 \choose 2(C) \frac{64 !}{32 ! 32 !} (D) \frac{64 !}{2^{64}} 7 .The integral part of \sum_{n=2}^{9999} \frac{1}{\sqrt{n}} equals (A) 196 (B) 197 (C) 198 (D) 199 8 .Let a_{n} be the number of subsets of {1,2, \ldots, n} that do not contain any two consecutive numbers. Then (A) a_{n}=a_{n-1}+a_{n-2} (B) a_{n}=2 a_{n-1} (C) a_{n}=a_{n-1}-a_{n-2} (D) a_{n}=a_{n-1}+2 a_{n-2} 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: (A) \frac{\sqrt{19}}{\sqrt{7}} (B) \frac{\sqrt{7}}{\sqrt{19}} (C) \frac{\sqrt{12}}{\sqrt{7}} (D) \frac{\sqrt{7}}{\sqrt{12}} 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 (A) \frac{3 r}{2} (B) \frac{4 r}{3} (C) \frac{6 r}{5} (D) \frac{8 r}{5} 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) ? (A) 49 \choose 19 (B) 49 \choose 20 (C) 50 \choose 19 (A) 50 \choose 20 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)$202518$. Which of the following is the sum of an infinite geometric sequence whose terms come from the set$\{1, \frac{1}{2}, \frac{1}{4}, \ldots, \frac{1}{2^{n}}, \ldots\} ?$(A)$\frac{1}{5}$(B)$\frac{1}{7}$(C)$\frac{1}{9}$(D)$\frac{1}{11}19$. 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 (A)$\frac{4}{17}$(B)$ \frac{2}{3}$(C)$\sqrt{\frac{17^{2}-13^{2}}{17^{2}+13^{2}}}$(D)$\sqrt{\frac{17^{2}-13^{2}}{17 \times 13}}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)$2^{n}(n-1)+128$. 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 (A)$\pi+1$(B)$2 \pi-2$(G)$\pi+2$(D)$2 \pi-129$. 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)$\left(n^{2}-2 n+2\right) t30$. 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 (A)$5$(B)$9$(C)$13$(D)$16\$