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# 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\$

ISI Entrance Solutions and Problems

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