Sequences and Series - Class 11

Let $z_1, z_2 \in \mathbb{C}$ be non-zero complex numbers such that $|z_1|^2 + |z_2|^2 = 32$ and $\arg(z_1) = \arg(z_2)$. Let $w_1 = z_1 e^{i\alpha}$ and $w_2 = z_2 e^{i\beta}$, where $\alpha \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right]$ and $\beta \in \left[\frac{5\pi}{6}, \frac{7\pi}{6}\right]$. What is the maximum possible value of $|w_1 + w_2|$?

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors in $\mathbb{R}^3$ such that $|\vec{a}| + \sqrt{2}|\vec{b}| + \sqrt{3}|\vec{c}| = 9$. If $V$ represents the volume of the parallelepiped formed by the vectors $\vec{a}, \vec{b}, \vec{c}$ as coterminous edges, find the maximum possible value of $[V]$, where $[x]$ denotes the greatest integer less than or equal to $x$.

Let $f(x) = 2x + \frac{18}{x}$ for $x > 0$. Let $x_0$ be the point where $f(x)$ attains its minimum value, and let $m$ be this minimum value. Let $g: (0, \infty) \to \mathbb{R}$ be a continuous function such that $g(1) = -1$ and its derivative is given by: $g'(x) = \begin{cases} \cos(\pi x) - \pi x \sin(\pi x) & \text{for } 0 < x < x_0 \\ (x+1)e^{x-x_0} & \text{for } x > x_0 \end{cases}$ Find the value of $g(m)$.

Let $x_1, x_2, \dots, x_6$ be positive real numbers satisfying $\sum_{i=1}^6 x_i = 12$ and $\sum_{i=1}^6 x_i^2 = 24$. Let $S = \{1, 2, 3, 4, 5, 6\}$. For each $i \in S$, define the set $A_i = \{ i, (i + \lfloor x_i \rfloor) \pmod 6 \}$, where $0 \pmod 6$ is taken as $6$. Let $N$ be the number of permutations $\pi$ of $S$ such that $\pi(i) \notin A_i$ for all $i \in S$. Find the value of $N$.

Let $f(x) = \int_{\pi/6}^{x} \left( \frac{25}{2}\sin^2 t + 2\csc^2 t \right) dt$ for $x \in (0, \pi/2)$. If $m_0$ is the minimum possible slope of the tangent to the curve $y = f(x)$, find the value of $\sin^{-1}(\sin m_0) + \cos^{-1}(\cos m_0)$.

Let $S_n = \sum_{k=1}^n \frac{k^2+k-1}{(k^2+1)(k^2+2k+2)}$. Find the value of $\sum_{n=1}^{50} \left[ 10 S_n \right]$ where $[x]$ denotes the greatest integer less than or equal to $x$.

Let $\alpha = e^{i \frac{2\pi}{11}}$. Evaluate the sum $S = \sum_{k=1}^{10} \sqrt{ (\alpha^k - \alpha^{k+1})^2 }$ where $\sqrt{z}$ denotes the principal square root of $z$ (i.e., for $z = re^{i\theta}$ with $\theta \in (-\pi, \pi]$, $\sqrt{z} = \sqrt{r}e^{i\theta/2}$).

For each integer $n \ge 1$, let $F_n(x) = \int_0^x e^{x-t} (t^n - t^{n+1}) dt$. Define the function $H(x) = \sum_{n=1}^{\infty} F_n(x)$ for $x \in [0, 1]$. Let $A$ be the left-hand derivative of $H(x)$ at $x=1$, and $B = \sum_{n=1}^{\infty} F_n'(1)$. Find the value of $A - B$.

Let $P = \begin{pmatrix} 2 & 3 \\ 0 & 5 \end{pmatrix}$ and $I$ be the $2 \times 2$ identity matrix. Let $t_k$ denote the element in the first row and second column of the matrix $P^k - P^{k-1}$ for $k \ge 1$, where $P^0 = I$. A $3 \times 3$ matrix $M$ is constructed such that its elements are given by $M_{i,j} = \sum_{k=1}^{i+j} t_k$ for $1 \le i, j \le 3$. The value of $\det(M)$ is:

Let $I_n = \int_{\frac{1}{n+1}}^{\frac{1}{n}} \left( 2x \cos^{-1}\left(\cos \frac{\pi}{x}\right) - \pi \frac{\sin\left(\frac{\pi}{x}\right)}{\left| \sin\left(\frac{\pi}{x}\right) \right|} \right) dx$. Find the value of $\frac{100}{\pi} \sum_{n=1}^{99} I_n$.

Let $f(x) = \sum_{n=1}^{\infty} x^3 (1 + x|x|)^{n-1}$. Evaluate the limit: $L = \lim_{x \to 0} \left( 1 + f(x) \right)^{\frac{1}{|x|}}$

Evaluate the integral: $I = \int_{0}^{\pi/2} \left( x \sin x + \sum_{n=1}^{\infty} \frac{(\sin x - \cos x)^n}{n} \right) dx$

Let a differential equation be given by $\alpha \left( \frac{d^4y}{dx^4} \right)^5 + \beta \left( \frac{d^3y}{dx^3} \right)^7 + \gamma \left( \frac{d^2y}{dx^2} \right)^4 + \left( \frac{dy}{dx} \right)^6 = e^x$ where the coefficients are defined as: $\alpha = \sum_{n=1}^{\infty} \frac{1}{n(n+1)(n+2)} - \frac{1}{4}$ $\beta = \sum_{n=1}^{\infty} \frac{2^{n-1}}{3^n} - 1$ $\gamma = \sum_{n=1}^{\infty} \frac{1}{n(n+2)}$ If $O$ and $D$ represent the order and degree of the differential equation respectively, find the value of $O + D$.

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{6}$. A sequence of position vectors is defined for $k \in \mathbb{N}$ as: $\vec{r}_k = \vec{\Omega} + \sum_{n=1}^{\infty} \left( \frac{k}{k+1} \right)^n \vec{a} + \sum_{n=1}^{\infty} \left( \frac{k^2}{k^2+1} \right)^n \vec{b}$ where $\vec{\Omega} = \left( \lim_{x \to 0} \frac{\sin x - x}{x^3} \right) \vec{a} + \left( \int_{0}^{\pi/2} \ln(\tan x) dx \right) \vec{b} + \left( \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n} \right) (\vec{a} \times \vec{b})$. Find the area of the triangle formed by the points with position vectors $\vec{r}_1, \vec{r}_2,$ and $\vec{r}_3$.

Let $S$ be the set of all real numbers $x$ for which the function $P(X=k) = (24x^4 - 62x^3 + 27x^2 + 2x - 1) x^{k-1}, \quad k = 1, 2, 3, \dots$ represents a valid probability mass function of a discrete random variable $X$. Find the sum of all elements in $S$.

Let $z_1, z_2, z_3$ be the roots of the cubic equation $z^3 - 3(1+i)z^2 + pz + q = 0$, where $p, q \in \mathbb{C}$. If $z_1, z_2, z_3$ form an arithmetic progression and satisfy $|z_1| + |z_3| = 2\sqrt{2}$, then the difference between the maximum and minimum possible values of $|p|$ is:

Let $f(x) = x^2 - 4$ and $g(x) = |x-3| - 4$ for all $x \in \mathbb{R}$. Define $H(x) = f(g(x))$. Let $S$ be the set of all real numbers $x$ such that $H'(x) = 0$, and $D$ be the set of all real numbers $x$ such that $H(x)$ is not differentiable. If $\alpha = \sum_{x \in S} x$, $\beta = n(S)$, and $\gamma = n(D)$, where $n(A)$ denotes the number of elements in set $A$, find the value of $\alpha^2 + \beta^2 + \gamma^2$.

Let $x, y, z$ be three consecutive terms of an arithmetic progression with common difference $d$. Let $\vec{v} = x\hat{i} + y\hat{j} + z\hat{k}$ be a vector such that its $y$-component is $2$. Consider a variable vector $\vec{n} = (d+1)\hat{i} + \hat{j} + \hat{k}$. If $M$ is the maximum possible value of the scalar projection of $\vec{v}$ on $\vec{n}$ for $d \in \mathbb{R}$, then $M$ equals:

An urn contains 15 balls numbered $1, 2, 3, \dots, 15$. Four balls are drawn simultaneously from the urn. Given that the sum of the numbers on the drawn balls is 34, let $p$ be the probability that no three of the drawn numbers form an arithmetic progression. Then $p$ is equal to

Evaluate the following infinite series: $S = \sum_{n=1}^{\infty} \tan^{-1}\left( \frac{(n-1)2^n}{(n^2 - 5n + 6)2^{2n+1} + 3(3n - 7)2^n + 10} \right)$

Let $r \in (-1, 1)$ be the common ratio of an infinite geometric progression. Define the function $f(r) = \sum_{k=1}^{\infty} k^2 r^{k-1}$. A differentiable function $y = y(x)$ satisfies the differential equation $(x^2+1)^2 \frac{dy}{dx} + 2x(x^2+1)y = f(r)$ with the initial condition $y(0) = 0$. Let $m_T$ be the slope of the tangent to the curve $y = y(x)$ at $x = 0$. It is given that the function $g(x) = \left( m_T - \frac{4-2r}{(1-r)^2} + 1 \right) x + \cos x$ is strictly increasing on $\mathbb{R}$. If the maximum possible interval for $r$ is $[a, b)$, find the value of $a + b$.

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2 - 2kx + (k^2 + k - 2) = 0$, where $k \in \mathbb{R}$. An Arithmetic-Geometric Progression is defined by its partial sum $S_n = \sum_{m=1}^{n} (m|\alpha| + |\beta|) r^{m-1}$, where the common ratio is $r = \frac{|\alpha + \beta|}{|\alpha| + |\beta|}$. Let $D$ be the set of all values of $k$ for which $\lim_{n \to \infty} S_n$ exists and is finite. Which of the following represents the set $D$?

Let $S(x) = \sum_{n=1}^{\infty} n^2 x^n$ for $x \in (-1, 1)$. Define the function $F(x) = \int_{S(x)}^{-S(x)} \frac{t^2}{1+e^t} dt$. If $M$ is the maximum value of $F(x)$ on the interval $(-1, 1)$, then the value of $1944 \sqrt{3} M$ is

Let $R$ be the term independent of $x$ in the binomial expansion of $\left( x - \frac{1}{x^2} \right)^9$. Let $S_n$ denote the sum of the first $n$ terms of an Arithmetic-Geometric Progression whose $k$-th term is given by $t_k = (2k+1) \left( \frac{R}{84} \right)^{k-1}$. Find the value of $S_{57} + S_{84}$.

For $k \ge 2$, let $C_k$ be the coefficient of $x y z^{k-2}$ in the multinomial expansion of $(x+y+z)^k$, and let $C_1 = 0$. For $k \ge 1$, let $D_k$ be the coefficient of $x y^{k-1}$ in the binomial expansion of $(x+y)^k$. A sequence is defined by $T_k = (C_k + D_k + 1) k!$. If $S_n = \sum_{k=1}^n T_k$, find the maximum integer $m$ such that $5^m$ divides $S_{125}$.

Let the sequence of points $P_n(x_n, y_n)$ be defined by $x_n = \left( \sum_{k=1}^n \frac{1}{k(k+1)} \right)^2 \quad \text{and} \quad y_n = 2 \sum_{k=1}^n \frac{1}{k(k+1)}$ for $n \in \mathbb{N}$. Let $L_n$ be the straight line passing through $P_n$ and $P_{n+1}$. Let $S$ be the set of all $n \in \mathbb{N}$ such that the origin $(0,0)$ and the point $A(15, 17)$ lie on the same side of the line $L_n$. Find the sum of all elements in $S$.

Let $S = \sum_{n=1}^{99} (-1)^n \tan^{-1}\left( \frac{2n+1}{1 - n - n^2} \right)$. Find the value of $S$.

Let $T_n$ be a sequence of distinct real numbers in a Geometric Progression. Let $T_1 = 1$ and $r$ be its common ratio. Define the determinant $\Delta(r) = \begin{vmatrix} 1 & 1 & 1 \\ T_1+T_2 & T_2+T_3 & T_3+T_4 \\ T_1^2+T_2^2 & T_2^2+T_3^2 & T_3^2+T_4^2 \end{vmatrix}$ If $\Delta(r) \ge 0$, which of the following represents the complete set of valid values for $r$?

Let $y = f(x)$ be a real-valued function satisfying the implicit relation: $y^3 + 3y = -28 \sum_{n=1}^{\infty} \frac{x^{3^{n-1}} + x^{2 \cdot 3^{n-1}}}{1 - x^{3^n}}$ Find the value of $\frac{dy}{dx}$ at $x = -1$.

Let $t_1, t_2, t_3$ be three distinct positive real numbers in a geometric progression such that $t_1 t_2 t_3 = 8$. A moving point $P(x, y)$ is defined by its coordinates $x = t_1 + t_2 + t_3$ and $y = t_1 t_2 + t_2 t_3 + t_3 t_1$. If the locus of $P$ intersects the region $(x - 3)^2 + (y - 4)^2 \le R^2$, then the infimum of $R^2$ is

A variable plane $P$ has non-zero intercepts on the $x, y,$ and $z$ axes, which respectively form the first, second, and third terms of a geometric progression with common ratio $r$ ($r \neq 0, 1$). The plane $P$ passes through the point $(1, 1, 1)$. Let $L$ be the line passing through the origin with direction ratios $(1, -r, 0)$. The orthogonal projection of the line $L$ onto the plane $P$ is the line $L'$. If $L'$ lies in the plane $z = c$, then the maximum value of $c$ as $r$ varies is:

Let $\{I_n\}_{n=1}^{\infty}$ be a sequence of independent and identically distributed indicator random variables such that $P(I_n = 1) = p$ and $P(I_n = 0) = 1 - p$, where $p \in (0, 1)$. A random variable $S$ is defined by the infinite series $S = \sum_{n=1}^{\infty} I_n p^n$. If the expected value of $S$ is given by $E[S] = \frac{1}{6}$, then the probability that $S \ge \frac{10}{27}$ is equal to:

Let $a, b, c$ be distinct positive real numbers in Harmonic Progression. Consider the system of linear equations in variables $x, y, z$: $a x + b y + c z = 0$ $(a^2+ac) x + k y + (c^2+ac) z = 0$ $f(a) x + f(b) y + f(c) z = 0$ If this system has a non-trivial solution for every strictly increasing function $f: \mathbb{R}^+ \to \mathbb{R}$, then the value of $k$ is:

Let $f(x) = 1+\sin x$ and $g(x) = 1-\sin x$. Let $H(x)$ be the harmonic mean of $f(x)$ and $g(x)$. Let $y(x)$ be a twice differentiable function on $[0, \pi]$ satisfying the differential equation $H(x) y'' - H'(x) y' = x \cos x$ If $y(\pi) = \pi$, then the value of $y''\left(\frac{\pi}{2}\right) + y(0)$ is:

Let $H_1, H_2, H_3, \dots$ be a sequence of positive real numbers such that $H_{n+1}$ is the harmonic mean of $H_n$ and $H_{n+2}$ for all $n \ge 1$. If $H_1 = \frac{1}{5}$ and $H_2 = \frac{1}{9}$, evaluate the infinite series: $S = \sum_{n=1}^{\infty} \tan^{-1}\left( \frac{8 H_n H_{n+2}}{1 + H_n H_{n+2}} \right)$

Let $X$ be a random variable taking values in $\{1, 2, 3\}$ with probabilities $p_1, p_2, p_3$ respectively. It is given that $p_1, p_2, p_3$ form a strictly decreasing Harmonic Progression and their product is $\frac{3}{128}$. An experiment consists of 4 independent observations of $X$. Let $E$ be the event that the value 2 is observed exactly once, and $F$ be the event that the value 2 is observed at least once. The conditional probability $P(E|F)$ is:

Let $H$ be the harmonic mean of the roots of the polynomial equation $x^3 - 11x^2 + 36x - 36 = 0$. Consider the curve $C: y = 2x^2 - (3H+2)x + H^2 + 5H$ and the line $L: y = (H-2)x + c$. If the $x$-coordinates of the intersection points of $C$ and $L$ are roots of an equation with even multiplicity, find the value of $x_0 + y_0 + c$, where $(x_0, y_0)$ is the point of contact.

Let $S_k(n) = \sum_{i=1}^n i^k$ for $k \in \{1, 2, 3\}$. It is known that $S_1(n) = a_1 n^2 + b_1 n + c_1$, $S_2(n) = a_2 n^3 + b_2 n^2 + c_2 n$, and $S_3(n) = a_3 n^4 + b_3 n^3 + c_3 n^2$. Let $M$ be the $3 \times 3$ matrix defined by $M = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}$. Consider the system of linear equations $M \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} \alpha \\ \beta \\ \gamma \end{bmatrix}$. If this system has infinitely many solutions, which of the following relations must hold true?

Let $S_p(n) = \sum_{k=1}^n k^p$ for $p \in \mathbb{N}$. Evaluate the following limit: $\lim_{n \to \infty} n^2 \left( 17 \left| \frac{S_5(n)}{n^6} - \frac{1}{6} - \frac{1}{2n} - \frac{1}{2n^2} \right| + 19 \left| \frac{S_6(n)}{n^7} - \frac{1}{7} - \frac{1}{2n} - \frac{3}{4n^2} \right| \right)$

Let $S_n = \sum_{r=1}^n r^3$. Consider the function $f(n) = \frac{S_n}{n^2} + \frac{32}{n+1}$ defined for all $n \in \mathbb{N}$. Find the minimum value of $f(n)$ subject to the condition $|n^2 - 3n - 2| \ge 26$.

Let $\omega = e^{i2\pi/3}$. Evaluate the sum $S = \sum_{n=1}^{\infty} \text{Arg}\left( \left( \frac{n^2-\omega}{n^2-\omega^2} \right)^2 \right)$ where $\text{Arg}(z) \in (-\pi, \pi]$ denotes the principal argument.

Let $A_k = \begin{pmatrix} \frac{1}{k(k+1)} & \frac{2}{k(k+2)} \\ \frac{2}{(k+1)(k+3)} & \frac{1}{(k+2)(k+3)} \end{pmatrix}$ for $k \in \mathbb{N}$. Evaluate $\det \left( \sum_{k=1}^{\infty} A_k \right)$.

Let $S(x) = \sum_{n=1}^{\infty} \frac{1}{(n - \log_2 x)(n + 1 - \log_2 x)}$. Let $y=y(x)$ be the solution to the initial value problem $x \ln 2 \frac{dy}{dx} - S(x) y = -S(x) y^2, \quad y(1) = \frac{1}{2}$ defined on its maximal interval of existence. Find the number of solutions to the equation $y(x) = x$.

Let $S = \sum_{n=1}^{\infty} \frac{\lfloor \frac{n+1}{2} \rfloor}{n(n+1)(n+2)}$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Then the value of $S$ is:

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors such that no two are orthogonal. A sequence of vectors is defined as $\vec{v}_k = x_k (\vec{a} \times (\vec{b} \times \vec{c})) + y_k (\vec{b} \times (\vec{c} \times \vec{a})) + z_k (\vec{c} \times (\vec{a} \times \vec{b}))$ where $x_k = \frac{4(k^2+k-1)}{(k+2)!}$, $y_k = \frac{2^k(k-1)}{(k+1)!}$, and $z_k = \frac{\lambda}{k(k+1)(k+2)}$. If $\sum_{k=1}^\infty \vec{v}_k = \vec{0}$, then the value of $\lambda$ is

Let $\alpha_n$ and $\beta_n$ be the roots of the quadratic equation $x^2 - n x + 1 = 0$. For each integer $n \ge 0$, define the matrix $A_n = \begin{pmatrix} \alpha_n + \beta_n & \alpha_n & \beta_n \\ \alpha_n & \alpha_n + \beta_n & \beta_n \\ \beta_n & \alpha_n & \alpha_n + \beta_n \end{pmatrix}$ Find the value of $\sum_{n=0}^{10} \det(\text{adj}(A_n) + I)$ where $I$ is the $3 \times 3$ identity matrix.

Let $\alpha$ and $\beta$ be the roots of the equation $z^2 - \sqrt{2}z + 1 = 0$. Let $z_0 = \frac{\sqrt{3} + i}{2}$. A sequence is defined as $T_n = (z_0 \alpha)^n + (\overline{z_0} \beta)^n$ for $n \in \mathbb{N}$. Find the value of $\sum_{n=1}^{95} (T_n^2 + T_n)$.

Let $Q(x) = x^2 - px + q$ be a quadratic polynomial with real roots $\alpha$ and $\beta$ ($\alpha < \beta$). It is given that $p = \frac{1}{\pi} \int_{\alpha}^{\beta} \frac{x^2 - q}{\sqrt{-Q(x)}} \, dx$ Furthermore, the improper integral $I = \int_{0}^{\infty} \frac{dx}{x^2 + px + q}$ converges to a finite value, and satisfies the relation $q = e^{I \sqrt{p^2 - 4q}}$. Find the value of $p^2 + q^2$.

For each $k \in \{1, 2, 3\}$, let $E_k$ be the quadratic equation $x^2 - \frac{k}{2}x + \frac{k^2-2}{4} = 0$. Let $S$ be the set of all valid real vectors $\vec{v}_k = l_k \hat{i} + m_k \hat{j} + n_k \hat{k}$ such that $l_k, m_k, n_k$ are the direction cosines of a line in 3D space, where $l_k$ and $m_k$ are the two roots of $E_k$ (with $l_k > m_k$) and $n_k > 0$. Let $W$ be the vector space spanned by the elements of $S$ over the field of real numbers. What is the dimension of $W$?

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2 - px + q = 0$, where $p = \lim_{x \to 0} \frac{\ln(1+4x)}{x}$ and $q = \lim_{x \to 0} \frac{1 - \cos(2\sqrt{2}x)}{x^2}$. A sequence $\{t_r\}$ is defined for $r \in \mathbb{N}$ as $t_r = \sum_{j=0}^{r-1} \alpha^{r-1-j} \beta^j$. If $n = t_3$, find the sum of the coefficients of $x^{t_r}$ (for all $r \in \mathbb{N}$) in the binomial expansion of $(1+x)^n$.