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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 𝜄 be a root of the equation 𝑥2+1=0 and let 𝜔 be a root of the equation 𝑥2+𝑥+1=0. Construct a polynomial
𝑓(𝑥)=𝑎0+𝑎1𝑥+⋯+𝑎𝑛𝑥𝑛
where 𝑎0,𝑎1,⋯,𝑎𝑛 are all integers such that 𝑓(𝜄+𝜔)=0.

• Let 𝑎 be a fixed real number. Consider the equation
(𝑥+2)2(𝑥+7)2+𝑎=0,𝑥
where is the set of real numbers. For what values of 𝑎, will the equation have exactly one root?

• 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 2𝑑,2𝑑
• Let a real-valued sequence {𝑥𝑛}𝑛≥1 be such that
lim𝑛→∞𝑛𝑥𝑛=0.
Find all possible real values of 𝑡 such that lim𝑛→∞𝑥𝑛⋅(log𝑛)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
𝑥2𝑎2+𝜆+𝑦2𝑏2+𝜆=1
satisfies
𝑑𝑦𝑑𝑥(𝑎2𝑏2)=(𝑥+𝑦𝑑𝑦𝑑𝑥)(𝑥𝑑𝑦𝑑𝑥𝑦)
• Consider a right-angled triangle with integer-valued sides 𝑎<𝑏<𝑐 where 𝑎,𝑏,𝑐 are pairwise co-prime. Let 𝑑=𝑐𝑏. Suppose 𝑑 divides 𝑎. Then
(a) Prove that 𝑑≤2
(b) Find all such triangles (i.e. all possible triplets 𝑎,𝑏,𝑐 ) with permeter less than 100.
• A finite sequence of numbers (𝑎1,⋯,𝑎𝑛) is said to be alternating if
𝑎1>𝑎2,𝑎2<𝑎3,𝑎2>𝑎4,𝑎4<𝑎5,or𝑎1<𝑎2,𝑎2>𝑎3,𝑎3<𝑎4,𝑎4>𝑎5
How many alternatig sequences of length 5, with distinct number 𝑎1,⋯,𝑎5 can be formed such that 𝑎𝑖∈{1,2,⋯,20} for 𝑖=1,⋯,5?

Objective Paper - ISI Entrance 2020 Problems and Solutions

Problem 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)$50$

Problem 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 $x=-1 / 2$

Problem 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) $1+x^{4}$

Problem 4

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

Problem 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) $e^{-0.5}$

Problem 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}} Problem 7 The integral part of \sum_{n=2}^{9999} \frac{1}{\sqrt{n}} equals (A) 196 (B) 197 (C) 198 (D) 199 Problem 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} Discussion and Solution Problem 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. Problem 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}} Problem 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} Problem 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 Problem 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. Problem 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 Problem 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 Problem 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 Problem 17 The number of real roots of the polynomial 𝑝(𝑥)=(𝑥2020+2020𝑥2+2020)(𝑥3−2020)(𝑥2−2020) (A) 2 (B)3 (C) 2023 (D) 2025 Problem 18 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\} ? (A) \frac{1}{5} (B) \frac{1}{7} (C) \frac{1}{9} (D) \frac{1}{11} Problem 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. Problem 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. Problem 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. Problem 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}} Problem 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. Problem 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. Problem 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. Problem 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}. Problem 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 Problem 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 (A) \pi+1 (B) 2 \pi-2 (G) \pi+2 (D) 2 \pi-1 Problem 29 Let n be a positive integer and t \in(0,1) . Then \sum_{r=0} r\left( 𝑛 𝑟 \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 Problem 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 (A) 5 (B) 9 (C) 13 (D) 16 Objective Paper - Answer Key Some useful links ISI Entrance Solutions and Problems This post contains Indian Statistical Institute, ISI Entrance 2020 Problems and Solutions. Try them out. Subjective Paper - ISI Entrance 2020 Problems and Solutions • Let 𝜄 be a root of the equation 𝑥2+1=0 and let 𝜔 be a root of the equation 𝑥2+𝑥+1=0. Construct a polynomial 𝑓(𝑥)=𝑎0+𝑎1𝑥+⋯+𝑎𝑛𝑥𝑛 where 𝑎0,𝑎1,⋯,𝑎𝑛 are all integers such that 𝑓(𝜄+𝜔)=0. Answer: 𝑓(𝑥)=𝑥4+2𝑥3+5𝑥2+4𝑥+1 • Let 𝑎 be a fixed real number. Consider the equation (𝑥+2)2(𝑥+7)2+𝑎=0,𝑥 where 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 2𝑑,2𝑑 • Let a real-valued sequence {𝑥𝑛}𝑛≥1 be such that lim𝑛→∞𝑛𝑥𝑛=0. Find all possible real values of 𝑡 such that lim𝑛→∞𝑥𝑛⋅(log𝑛)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 𝑥2𝑎2+𝜆+𝑦2𝑏2+𝜆=1 satisfies 𝑑𝑦𝑑𝑥(𝑎2𝑏2)=(𝑥+𝑦𝑑𝑦𝑑𝑥)(𝑥𝑑𝑦𝑑𝑥𝑦) • Consider a right-angled triangle with integer-valued sides 𝑎<𝑏<𝑐 where 𝑎,𝑏,𝑐 are pairwise co-prime. Let 𝑑=𝑐𝑏. Suppose 𝑑 divides 𝑎. Then (a) Prove that 𝑑≤2 (b) Find all such triangles (i.e. all possible triplets 𝑎,𝑏,𝑐 ) with permeter less than 100. • A finite sequence of numbers (𝑎1,⋯,𝑎𝑛) is said to be alternating if 𝑎1>𝑎2,𝑎2<𝑎3,𝑎2>𝑎4,𝑎4<𝑎5,or𝑎1<𝑎2,𝑎2>𝑎3,𝑎3<𝑎4,𝑎4>𝑎5 How many alternatig sequences of length 5, with distinct number 𝑎1,⋯,𝑎5 can be formed such that 𝑎𝑖∈{1,2,⋯,20} for 𝑖=1,⋯,5? Answer: 32×(205) Objective Paper - ISI Entrance 2020 Problems and Solutions Problem 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 Problem 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 Problem 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} Problem 4 The number of real solutions of e^{x}=\sin (x) is (A) 0 (B) 1 (C) 2 (D) infinite. Problem 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} Problem 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) $\frac{64 !}{2^{64}}$

Problem 7

The integral part of $\sum_{n=2}^{9999} \frac{1}{\sqrt{n}}$ equals

(A) $196$ (B) $197$ (C) $198$ (D) $199$

Problem 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) $a_{n}=a_{n-1}+2 a_{n-2}$

Discussion and Solution

Problem 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.

Problem 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) $\frac{\sqrt{7}}{\sqrt{12}}$

Problem 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) $\frac{8 r}{5}$

Problem 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) $1$

Problem 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.

Problem 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) $65$

Problem 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) $50 \choose 20$

Problem 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 $2$

Problem 17

The number of real roots of the polynomial

𝑝(𝑥)=(𝑥2020+2020𝑥2+2020)(𝑥3−2020)(𝑥2−2020)

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

Problem 18

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

(A) $\frac{1}{5}$ (B) $\frac{1}{7}$ (C) $\frac{1}{9}$ (D) $\frac{1}{11}$

Problem 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.

Problem 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.

Problem 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.

Problem 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) $\sqrt{\frac{17^{2}-13^{2}}{17 \times 13}}$

Problem 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.

Problem 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.

Problem 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.

Problem 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 $\mathbb{R}$.

Problem 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) $2^{n}(n-1)+1$

Problem 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

(A) $\pi+1$
(B) $2 \pi-2$
(G) $\pi+2$
(D) $2 \pi-1$

Problem 29

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

(A) $n t$

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

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

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

Problem 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

(A) $5$
(B) $9$
(C) $13$
(D) $16$

ISI Entrance Solutions and Problems

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### 5 comments on “ISI Entrance 2020 Problems and Solutions - B.Stat & B.Math”

1. Samprit Chakraborty says:

HELLO SIR I THINK 13 SHOULD BE D.NONE.
Take f(x)=x+1 and g(x)= x^2

1. Shruti bansal says:

Pls check that for ur assumed function the condition of f'(t)>=g'(t) is not satisfied for all t belonging to m

2. Dipak Kumar Das says:

What will be the cut off for B.math 2020

3. SUDIP MONDAL says:

25 -C.........take a case f(X)={1 for x>=0, and -1 for x<0 g(x)=-f(x)

I think you are right. I wrongly chose option D.