ISI Entrance 2020 Problems and Solutions - B.Stat & B.Math

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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 $𝑓 (𝑥) = 𝑎 0 + 𝑎 1 𝑥 +\dots+ 𝑎 𝑛 𝑥 𝑛$ 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 $(𝑥 +2) 2 (𝑥 +7) 2 + 𝑎 =0, 𝑥 \in ℝ$ where 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 2‾√𝑑,2‾√𝑑
Let a real-valued sequence {𝑥𝑛}𝑛≥1 be such that $lim 𝑛 \to\infty 𝑛 𝑥 𝑛 =0 .$ 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 $𝑥 2 𝑎 2 + 𝜆 + 𝑦 2 𝑏 2 + 𝜆 =1$ satisfies $𝑑𝑦 𝑑𝑥 (𝑎 2 - 𝑏 2) =(𝑥 + 𝑦 𝑑𝑦 𝑑𝑥)(𝑥 𝑑𝑦 𝑑𝑥 - 𝑦)$
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≤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 $𝑎 1 > 𝑎 2, 𝑎 2 < 𝑎 3, 𝑎 2 > 𝑎 4, 𝑎 4 < 𝑎 5, \dots or 𝑎 1 < 𝑎 2, 𝑎 2 > 𝑎 3, 𝑎 3 < 𝑎 4, 𝑎 4 > 𝑎 5$ How many alternatig sequences of length 5, with distinct number 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

ISI-BStat-BMath-2020-UGA Download

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) = (𝑥 2020 +2020 𝑥 2 +2020) (𝑥 3 -2020)(𝑥 2 -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

(Created by students). Please suggest changes in the comment section.

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

Some useful links

ISI Entrance Solutions and Problems

ISI BStat 2010 Solution and Strategy - Video

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 $𝑓 (𝑥) = 𝑎 0 + 𝑎 1 𝑥 +\dots+ 𝑎 𝑛 𝑥 𝑛$ 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 $(𝑥 +2) 2 (𝑥 +7) 2 + 𝑎 =0, 𝑥 \in ℝ$ where 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 2‾√𝑑,2‾√𝑑
Let a real-valued sequence {𝑥𝑛}𝑛≥1 be such that $lim 𝑛 \to\infty 𝑛 𝑥 𝑛 =0 .$ 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 $𝑥 2 𝑎 2 + 𝜆 + 𝑦 2 𝑏 2 + 𝜆 =1$ satisfies $𝑑𝑦 𝑑𝑥 (𝑎 2 - 𝑏 2) =(𝑥 + 𝑦 𝑑𝑦 𝑑𝑥)(𝑥 𝑑𝑦 𝑑𝑥 - 𝑦)$
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≤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 $𝑎 1 > 𝑎 2, 𝑎 2 < 𝑎 3, 𝑎 2 > 𝑎 4, 𝑎 4 < 𝑎 5, \dots or 𝑎 1 < 𝑎 2, 𝑎 2 > 𝑎 3, 𝑎 3 < 𝑎 4, 𝑎 4 > 𝑎 5$ How many alternatig sequences of length 5, with distinct number 𝑎1,⋯,𝑎5 can be formed such that 𝑎𝑖∈{1,2,⋯,20} for 𝑖=1,⋯,5?

Answer: 32×(205)