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- 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:*\( 32 \times { {20} \choose {5} } \)

**$1$**.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) $2025$

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

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

$28$. 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

(A) $\pi+1$

(B) $2 \pi-2$

(G) $\pi+2$

(D) $2 \pi-1$

$29$. 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

(A) $5$

(B) $9$

(C) $13$

(D) $16$

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 |

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