Sequences and Series

Consider $M$ with $r = \dfrac{1025}{513}$. Let $k$ be the number of all those circles $C_n$ that are inside $M$. Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then

Consider $M$ with $r = \dfrac{(2^{199} - 1)\sqrt2}{2^{198}}$. The number of all those circles $D_n$ that are inside $M$ is

If $e^{\left(\cos^2x+\cos^4x+\cos^6x+\dots\infty\right)\log_e2}$ satisfies the equation $t^2-9t+8=0$, then the value of $\dfrac{2\sin x}{\sin x+\sqrt3\cos x}\left(0<x<\dfrac\pi2\right)$ is :

Let A = {n ∈ ℕ : n is a 3-digit number}

B = {9k + 2 : k ∈ ℕ}

and C = {9k + l : k ∈ ℕ} for some l (0 < l < 9)

If the sum of all the elements of the set A ∩ (B ∪ C) is 274 × 400, then l is equal to _______.

The minimum value of α for which the equation $\dfrac4{\sin x}+\dfrac1{1-\sin x}=\alpha$ has at least one solution in $\left(0,\dfrac\pi2\right)$ is _______.

$\displaystyle\lim_{n\to\infty}\tan\left\{\sum_{r=1}^n\tan^{-1}\left(\frac1{1+r+r^2}\right)\right\}$ is equal to _______.

Let $m$ be the minimum possible value of $\log_3\left(3^{y_1} + 3^{y_2} + 3^{y_3}\right)$, where $y_1, y_2, y_3$ are real numbers for which $y_1 + y_2 + y_3 = 9$. Let $M$ be the maximum possible value of $(\log_3 x_1 + \log_3 x_2 + \log_3 x_3)$, where $x_1, x_2, x_3$ are positive real numbers for which $x_1 + x_2 + x_3 = 9$. Then the value of $\log_2(m^3) + \log_3(M^2)$ is ______

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive integers in arithmetic progression with common difference 2. Also, let $b_1, b_2, b_3, \ldots$ be a sequence of positive integers in geometric progression with common ratio 2. If $a_1 = b_1 = c$, then the number of all possible values of $c$, for which the equality

$$2(a_1 + a_2 + \cdots + a_n) = b_1 + b_2 + \cdots + b_n$$

holds for some positive integer $n$, is ______

Let the function $f:[0,1]\to\mathbb R$ be defined by

$$f(x) = \frac{4^x}{4^x + 2}.$$

Then the value of

$$f\left(\frac1{40}\right) + f\left(\frac2{40}\right) + f\left(\frac3{40}\right) + \cdots + f\left(\frac{39}{40}\right) - f\left(\frac12\right)$$

is ______

Let $\alpha$ and $\beta$ be the roots of $x^2 - x - 1 = 0$, with $\alpha > \beta$. For all positive integers $n$, define

$$a_n = \frac{\alpha^n - \beta^n}{\alpha - \beta},\quad n \ge 1,$$

$$b_1 = 1\quad\text{and}\quad b_n = a_{n-1} + a_{n+1},\quad n \ge 2.$$

Then which of the following options is/are correct?

Define the collections $\{E_1, E_2, E_3, \ldots\}$ of ellipses and $\{R_1, R_2, R_3, \ldots\}$ of rectangles as follows:

$E_1: \dfrac{x^2}9 + \dfrac{y^2}4 = 1$;

$R_1$: rectangle of largest area, with sides parallel to the axes, inscribed in $E_1$;

$E_n$: ellipse $\dfrac{x^2}{a_n^2} + \dfrac{y^2}{b_n^2} = 1$ of largest area inscribed in $R_{n-1}$, $n > 1$;

$R_n$: rectangle of largest area, with sides parallel to the axes, inscribed in $E_n$, $n > 1$.

Then which of the following options is/are correct?

Let $AP(a; d)$ denote the set of all the terms of an infinite arithmetic progression with first term $a$ and common difference $d > 0$. If

$$AP(1;3)\cap AP(2;5)\cap AP(3;7) = AP(a;d)$$

then $a + d$ equals ______

For non-negative integers $n$, let

$$f(n) = \frac{\displaystyle\sum_{k=0}^n\sin\left(\frac{k+1}{n+2}\pi\right)\sin\left(\frac{k+2}{n+2}\pi\right)}{\displaystyle\sum_{k=0}^n\sin^2\left(\frac{k+1}{n+2}\pi\right)}$$

Assuming $\cos^{-1}x$ takes values in $[0,\pi]$, which of the following options is/are correct?

The value of

$$\sec^{-1}\left(\frac14\sum_{k=0}^{10}\sec\left(\frac{7\pi}{12} + \frac{k\pi}2\right)\sec\left(\frac{7\pi}{12} + \frac{(k+1)\pi}2\right)\right)$$

in the interval $\left[-\frac\pi4, \frac{3\pi}4\right]$ equals ______

Which of the following is the only CORRECT combination?

Which of the following is the only CORRECT combination?

Let $X$ be the set consisting of the first 2018 terms of the arithmetic progression $1, 6, 11, \ldots$, and $Y$ be the set consisting of the first 2018 terms of the arithmetic progression $9, 16, 23, \ldots$. Then, the number of elements in the set $X \cup Y$ is _____.

The number of real solutions of the equation

$$\sin^{-1}\left(\sum_{i=1}^{\infty} x^{i+1} - x \sum_{i=1}^{\infty} \left(\frac{x}{2}\right)^i\right) = \frac{\pi}{2} - \cos^{-1}\left(\sum_{i=1}^{\infty} \left(-\frac{x}{2}\right)^i - \sum_{i=1}^{\infty} (-x)^i\right)$$

lying in the interval $\left(-\frac{1}{2}, \frac{1}{2}\right)$ is _____ .

(Here, the inverse trigonometric functions $\sin^{-1} x$ and $\cos^{-1} x$ assume values in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $[0, \pi]$, respectively.)

The sides of a right angled triangle are in arithmetic progression. If the triangle has area 24, then what is the length of its smallest side?

If the triangle $PQR$ varies, then the minimum value of

$$\cos(P + Q) + \cos(Q + R) + \cos(R + P)$$

is