Binomial Theorem - Class 11

Let $f: [1, \infty) \to (-\infty, \frac{1}{6}]$ be defined as $f(x) = \sum_{k=0}^{5} \binom{5}{k} \frac{(-1)^k}{k+1} x^{k+1}$ Let $g(x)$ be the inverse function of $f(x)$. Find the value of $14 \int_{-21/2}^{0} g(x) dx$

Let the function $f(x,y)$ be defined by the expansion: $f(x,y) = \sum_{r=0}^{251} \binom{251}{r} x^{251-r} y^r \left( \frac{x^2+y^2}{xy} \right)^r \ln\left( 1 + \frac{2y}{x} \right)$ A curve $C$ in the first quadrant is defined implicitly by the differential equation: $x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = 251 \cdot 3^{251} \ln(3)$ If $(1, y_0)$ is the unique point on the curve $C$ for $x=1$, find the value of $\frac{dy}{dx}$ at this point.

Let $A = [a_{ij}]$ be a $5 \times 5$ matrix where $a_{ij}$ is the coefficient of $x^2$ in the binomial expansion of $(1+x)^{i+j+2}$ for $1 \le i, j \le 5$. If $B = [b_{ij}]$ is the adjoint matrix of $A$, then the value of $\sum_{i=1}^5 \sum_{j=1}^5 (b_{ij}^2 + 2^{b_{ij}})$ is equal to

Consider the binomial expansion of $(|x-2| + |x-5|)^{15}$. If the $6^{\text{th}}$ term is the strictly numerically greatest term in the expansion, then the sum of all possible integral values of $x$ is:

A discrete random variable $X$ has a probability mass function given by $P(X = k) = \lambda \binom{6}{k} \left(\frac{1}{2}\right)^6 I(k \ge 2)$ for $k \in \{0, 1, 2, 3, 4, 5, 6\}$, where $\lambda$ is a constant and $I(A)$ is the indicator function which equals $1$ if $A$ is true and $0$ otherwise. Find the value of $361 \text{ Var}(X)$.

Let $x = \sum_{k=1}^\infty \frac{9}{6^k}$ and $y = \sum_{k=1}^\infty \frac{1}{5^k}$. Define $S_m = \sum_{n=1}^m \sum_{r=1}^n \binom{n}{r} x^r y^n$. Find the value of $L_1 + L_2$, where $L_1 = \lim_{m \to \infty} [S_m]$ and $L_2 = \left[ \lim_{m \to \infty} S_m \right]$, and $[ \cdot ]$ denotes the greatest integer function.

Let $i = \sqrt{-1}$. Define the complex numbers $A$ and $B$ as: $A = \frac{1}{16} \sum_{k=0}^{10} \binom{10}{k} i^k, \quad B = -\frac{5}{16} \sum_{k=0}^{8} \binom{8}{k} i^k$ Let $z_1$ and $z_2$ be the roots of the quadratic equation $z^2 - Az + B = 0$. Find the value of $|z_1 - \bar{z_2}|^2 + |z_2 - \bar{z_1}|^2$.

Let $n = \lim_{k \to \infty} \sum_{r=1}^{2k} \frac{5}{k} \left[ \frac{5r}{k} \right]$ where $[x]$ denotes the greatest integer less than or equal to $x$. Find the term independent of $x$ in the binomial expansion of $\left( x \sqrt{x} - \frac{1}{x} \right)^n$.

Let $k$ be the term independent of $x$ in the binomial expansion of $\left( \frac{x}{\sqrt{2}} + \frac{\alpha}{x^2} \right)^9$, where $\alpha \in \mathbb{R}$. Let $A = \begin{pmatrix} k-2 & 2 & 2 \\ 2 & k-2 & 2 \\ 2 & 2 & k-2 \end{pmatrix}$ Let $S$ be the set of all real values of $\alpha$ for which $\text{adj}(A)$ is the null matrix. Then the value of $\sum_{\alpha \in S} 21\alpha^3$ is equal to:

Let $S$ be the set of all prime numbers $p \le 367$. For each $p \in S$, let $n_p$ be the exponent of $p$ in the prime factorization of $367!$. Find the number of primes $p \in S$ such that the binomial expansion of $\left( x^2 + \frac{1}{x^3} \right)^{n_p}$ does \textbf{not} contain a term independent of $x$.

Let $n = \int_{0}^{\pi/2} \frac{18 \sin 2x}{(1 + \sin 2x)^2} dx$. If the term independent of $x$ in the binomial expansion of $\left( x^2 \arcsin(\alpha) + \frac{\arccos(\alpha)}{x} \right)^n$ is equal to $\frac{15 \pi^6}{2}$, then the number of real values of $\alpha$ is

Let $M$ be a $7 \times 7$ matrix whose $(i,j)$-th entry is given by the coefficient of $t^5$ in the polynomial expansion of $(1+t)^i (1+xt)^j$, for $1 \le i, j \le 7$. Let $f(x) = \det(M^3)$. Find the coefficient of $x^4$ in the polynomial $f(x)$.

Let $c_r$ be the coefficient of $x^r$ in the binomial expansion of $(x^2 - 1)^5$. Let $P(x) = \frac{c_2 x^2 + c_1 x + c_0}{x^3 - x}$. Let $f(x) = e^{\int P(x) dx}$ (where $C$ is the constant of integration) be defined on $(-1, 1) \setminus \{0\}$. If $C$ is chosen such that $f\left(\frac{1}{2}\right) = \frac{9}{32}$, find the value of $192 \int_{-1/2}^{1/2} f(x) dx$

Let $f(x) = |x - 2|^{17} + |x + 2|^{17}$. For $x \in (-2, 2)$, $f(x)$ can be expressed as a polynomial $P(x) = \sum_{r=0}^{17} C_r x^r$. Find the value of $r$ for which $C_r$ is maximum.

Let $S_n$ be the set of all polynomials $Q(x)$ of degree $n$ with real coefficients such that the coefficient of $x^n$ is $1$. Let $m_n = \min_{Q \in S_n} \int_{0}^{1} (Q(x))^2 dx$. Find the value of $\lim_{n \to \infty} n \cdot m_n \cdot \binom{2n}{n}^2$.

Let $f: \mathbb{Z} \to \mathbb{Z}$ be a function where $f(r)$ is defined as the coefficient of $x^r$ in the binomial expansion of $(1+x)^{15}$. Let $S$ be the set of all real numbers $x$ satisfying the equation: $f([x]^2 - [x] - 4) = f([x]^2 - [x] - 5)$ where $[t]$ denotes the greatest integer less than or equal to $t$. Which of the following represents the set $S$?

Let $M = \begin{pmatrix} \binom{2}{0} & \binom{4}{2} \\ \binom{3}{3}-1 & \binom{5}{5} \end{pmatrix}$ and $G_n$ be the greatest binomial coefficient in the expansion of $(1+x)^n$ for a positive integer $n$. If the matrix $A = G_n M^2 - 5 \text{adj}(2M)$ is singular, then the value of $\sum_{r=0}^n \binom{n}{r}^2$ is

Let $N$ be a random variable representing the number of heads obtained when 20 fair coins are tossed, given that at least one head appears. An integer $R$ is then chosen uniformly at random from the set $\{0, 1, 2, \dots, N\}$. What is the probability that the binomial coefficient $\binom{N}{R}$ is the strictly maximum value in the set $\left\{ \binom{N}{k} : 0 \le k \le N \right\}$?

Let $M_n$ be the greatest binomial coefficient in the expansion of $(1+x)^{2n}$. Which of the following inequalities holds strictly true for all integers $n \ge 1$?

Let $M_n$ be the greatest binomial coefficient in the expansion of $(1+x)^{2n}$, and let $A_n = \sum_{k=0}^{2n} \binom{2n}{k} e^{k/n}$. Given the asymptotic expansion $n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n \exp\left(\frac{1}{12n}\right)$ for large $n$, evaluate the limit: $\lim_{n \to \infty} \left( \frac{A_n}{M_n \sqrt{\pi n} \, e} \right)^n$

Let $n$ be the unique positive integer such that the numerically greatest term in the expansion of $(1 + 5x)^n$ at $x=2$ is the $15^{\text{th}}$ term. Let $X$ be a random variable following a binomial distribution $B(n, p)$. It is given that the variance of $X$ is $\frac{10}{3}$ and the probability mass function $P(X=k)$ attains its maximum value uniquely at $k=5$. Find the value of $10 \frac{P(X=5)}{P(X=4)}$.

Let $M$ be the integer $k \in [1, 270]$ that maximizes the sequence $a_k = k^2 (271-k)^3$. If the $M^{\text{th}}$ term is the unique numerically greatest term in the binomial expansion of $(1+x)^{271}$, then the range of positive values of $x$ is:

Let $M(x) = \begin{pmatrix} 0 & 1 & 2x & 0 \\ -1 & 0 & 0 & 2x \\ -2x & 0 & 0 & 1 \\ 0 & -2x & -1 & 0 \end{pmatrix}$. For $x > \frac{1}{2}$, a polynomial $P(x)$ is defined as $P(x) = \sqrt{|M(x)|}$, where $|M(x)|$ denotes the determinant of $M(x)$. Find the algebraic value of the numerically greatest term in the binomial expansion of $(P(x))^{11}$ evaluated at $x = \frac{3}{4}$.

Let $n$ be the number of ways to distribute 20 identical items among 3 distinct boxes such that exactly one box is empty. Consider the binomial expansion of $(5 - 2x)^n$ in increasing powers of $x$. If $x = 2$, let $k$ be the index of the algebraically greatest term in the expansion. Find the value of $k$.

Let $S$ be the sum of the numerically greatest terms in the binomial expansion of $\left( \sqrt{1+\sin 2\theta} + \sqrt{1-\sin 2\theta} \right)^{14}$, where $\theta \in \left( \frac{3\pi}{4}, \pi \right)$ and $\tan \theta = -\frac{1}{5}$. Find the value of $\frac{13^7 \cdot S}{1001 \cdot 2^{12} \cdot 3^8}$.

Let $\alpha = \lim_{n \to \infty} \sum_{k=1}^n \frac{12n}{(n+k)^2}$, $\beta = \lim_{n \to \infty} \sum_{k=1}^n \frac{3\pi}{10n} \sin\left(\frac{k\pi}{2n}\right)$, and $\gamma = \lim_{n \to \infty} \sum_{k=1}^n \frac{k^2}{9n^3}$. If $C$ is the coefficient of $x^3$ in the expansion of $(1 + \beta x + \gamma x^2)^\alpha$, find the value of $[C]$, where $[x]$ denotes the greatest integer less than or equal to $x$.

A university has 25 new students who need to be assigned to 5 distinct dormitories, denoted as $D_1, D_2, D_3, D_4, D_5$, with exactly 5 students in each dormitory. Among the 25 students, there are 5 specific students $S_1, S_2, S_3, S_4, S_5$. Let $N$ be the total number of ways to assign the students such that for all $i \in \{1, 2, 3, 4, 5\}$, student $S_i$ is NOT assigned to dormitory $D_i$. Find the exponent of the highest power of 5 that divides $N$.

Let $A$ and $B$ be two $11 \times 11$ matrices with row and column indices ranging from $0$ to $10$. The entries of $A$ and $B$ are given by: $A_{i,j} = \frac{10!}{i! j! (10 - i - j)!}$ $B_{i,j} = (-1)^{i-j} \binom{i}{j} 2^j$ (Note: $\frac{1}{k!} = 0$ and $\binom{n}{k} = 0$ for $k < 0$ or $k > n$). Let $C = AB$ and $P = \prod_{k=0}^{10} \binom{10}{k}$. What is the value of $\det(C)$?

Let $a_n(x)$ be the coefficient of $t^n$ in the expansion of $\left(1 + t + \frac{x}{|x|} t^2\right)^n$ for $x \neq 0$. Let $D$ be the set of all $x \in \mathbb{R} \setminus \{0\}$ for which the series $\sum_{n=1}^{\infty} a_n(x) x^n$ converges absolutely. If $D = (-a, 0) \cup (0, b)$, find the value of $\frac{1}{a^2} + \frac{1}{b^2}$.

A biased 3-sided die with faces labeled $A, B,$ and $C$ is rolled 6 times. The probabilities of landing on $A, B,$ and $C$ on any given roll are $\frac{1}{2}, \frac{1}{4},$ and $\frac{1}{4}$ respectively. Let $E$ be the event that every face appears at least once during the 6 rolls. If $X$ is the random variable representing the number of times face $A$ appears, find the conditional expectation $E[X \mid E]$.

Let $F(x) = \int_{0}^{x^2} \frac{1}{\sqrt{279 - 25t}} dt$. The function $F'(x)$ is expanded as an infinite Maclaurin series $\sum_{n=0}^{\infty} a_n x^{2n+1}$. Let $R$ be the radius of convergence of this series, and $k = [R]$ be the greatest integer less than or equal to $R$. Find the value of $\int_{0}^{k} \frac{1}{x^2 - 6x + 25} dx$.

Let $L = \lim_{x \to \infty} \left( \frac{\sqrt[3]{x^3 + 162 x^2}}{\sqrt[4]{x^4 + 152 x^3}} - \frac{16}{x} \right)^{x^2}$. Find the value of $\ln L$.

Let $A$ be a $3 \times 3$ matrix where its entries are given by $A_{ij} = \text{coefficient of } x^{j-1} \text{ in the binomial expansion of } (1-3x)^{-(n+i-1)}$ for $i, j \in \{1, 2, 3\}$. Given that the coefficient of $x^2$ in the expansion of $(1-x)^{-n}$ is $231$, find the value of $\det(\text{adj}(2A^T))$.

Let $x, y, z$ be the roots of the equation $24t^3 - 26t^2 + 9t - 1 = 0$. For a real number $t$ such that $|t| < 1$, define the vector: $\vec{v}_t = \sum_{r=0}^{\infty} (-t)^r \left[ \binom{-1}{r} \hat{i} + \binom{-2}{r} \hat{j} + \binom{-3}{r} \hat{k} \right]$ Find the volume of the parallelepiped whose coterminous edges are the vectors $\vec{v}_x, \vec{v}_y$, and $\vec{v}_z$.

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \int_0^x (x-t) \left( \sum_{k=0}^{n} \binom{n}{k} \sin^k |t| \right) dt$ where $n$ is a positive integer. It is given that $\lim_{x \to 0^+} \frac{f''(x) - 1 - n x}{x^2} = 253$ Evaluate the integral $I = \int_{-\pi/2}^{\pi/2} \frac{f''(x) \cos x}{1+e^x} dx$

Let $f(x) = \begin{vmatrix} \binom{x}{1} & \binom{x}{2} & \binom{x}{3} & 1 \\ \binom{2}{1} & \binom{2}{2} & \binom{2}{3} & 1 \\ \binom{4}{1} & \binom{4}{2} & \binom{4}{3} & 1 \\ \binom{6}{1} & \binom{6}{2} & \binom{6}{3} & 1 \end{vmatrix}$. Find the value of $\sum_{x=7}^{10} f(x)$.

Let $S_m(x) = \sum_{k=1}^n k^m \binom{n}{k} x^k$ for $m \in \mathbb{N}$ and $x > 0$. A function $f(x)$ is defined as $f(x) = \frac{S_3(x)}{S_1(x)} - \left(\frac{S_2(x)}{S_1(x)}\right)^2$. If the maximum value of $f(x)$ for $x \in (0, \infty)$ is $20$, then the value of $n$ is:

Consider the function $f(x) = \left(1 + \frac{x}{2}\right)^{-3}$ and its quadratic Maclaurin polynomial $P(x) = \sum_{k=0}^{2} \binom{-3}{k} \left(\frac{x}{2}\right)^k$. Let $S_1$ be the set of all integer solutions to the inequality $\frac{P(x) - 1 + \frac{3}{2}x}{x^2 - 2x - 15} \le 0$ Let $D$ be the interval of convergence of the infinite binomial series expansion of $f(x)$. If $S_2 = S_1 \cap D$, find the number of elements in the set $S_1 \setminus S_2$.

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined as: $f(x) = \begin{cases} \prod_{r=1}^6 \left(1 + \binom{6}{r} x\right) - 1, & x \ge 0 \\ \lambda |x| + \frac{\mu}{2} x^2, & x < 0 \end{cases}$ If $f(x)$ is twice differentiable at $x=0$, then the value of $\mu + \lambda$ is:

Let $A$ be a $3 \times 3$ real invertible matrix and let $B = (AA^T)^2 - 2AA^T$. If $\text{Tr}(B) = 2$, $\text{Tr}(B^2) = 10$, and $\text{Tr}(B^3) = 26$, which of the following is the value of $\det(AA^T + I)$?

Let $S$ be the set of all integers $m \in [-50, 10]$ such that the definite integral $I(m) = \int_{m}^{-41-m} \frac{39}{x^2+41x+40} \ln\left( -\frac{x+1}{x+40} \right) dx$ evaluates to a real number equal to zero. Find the sum of all elements in $S$.

Let $C_r$ be the coefficient of $x^r$ in the expansion of $(1+x)^4(1+x)^6$. Let $M$ be the maximum value among $C_r$ for $0 \le r \le 10$. If $a = C_4$ and $b = C_6$, consider the set $S$ of all positive integers $x \le 260$ that satisfy the inequality: $\frac{x^3 + a^3 + b^3 - 3xab}{x - M} \ge 0$ Find the sum of all elements in $S$.

Let $Z$ be a random variable taking values in the set $\{0, 1, \dots, n\}$. The probability generating function of $Z$, defined as $G(t) = \sum_{k=0}^n P(Z=k) t^k$, is given by the relation: $G(t) = \frac{\text{Coefficient of } x^{2n} \text{ in } (1+tx)^n (1+x)^{2n}}{\binom{3n}{2n}}$ If $W = 3Z - 2n$, which of the following expressions represents the variance of $W$?

Urn $A$ contains $367$ balls, of which $187$ are Red and $180$ are Blue. Urn $B$ contains $367$ balls, of which $180$ are Red and $187$ are Blue. A random variable $K$ follows a hypergeometric distribution, representing the number of Red balls obtained when drawing $187$ balls without replacement from a third Urn $C$ containing $367$ Red and $367$ Blue balls.

Given $K=k$, two independent experiments are performed: Experiment 1: $k$ balls are drawn without replacement from Urn $A$. Let $X$ be the number of Red balls drawn. Experiment 2: $187-k$ balls are drawn without replacement from Urn $B$. Let $Y$ be the number of Red balls drawn.

Find the unconditional probability $P(X + Y = 187)$.

Let $p_1, p_2, p_3, p_4, p_5$ be the first five prime numbers. Define a $5 \times 5$ matrix $M = [m_{ij}]$ such that $m_{ij} = \sum_{k=0}^5 (-1)^k \binom{p_i}{k} \binom{p_j}{5-k}$ for $1 \le i, j \le 5$, where $\binom{n}{k} = 0$ if $n < k$. Find the value of $\det(M)$.

Let $a_r = \sum_{k=0}^r \binom{2024}{k} \binom{2025}{r-k}$ for $0 \le r \le 4049$. Define the polynomial $P(x) = \sum_{r=0}^{4049} a_r x^r$ and the function $f(x) = \ln(P(x))$ for $|x| < 1$. If $g(x) = \int_0^x f(t) dt$ and $d_k$ denotes the coefficient of $x^k$ in the Maclaurin series expansion of $g(x)$, find the value of $\left\lfloor \sum_{k=2}^{4050} |d_k| \right\rfloor$, where $\lfloor \cdot \rfloor$ denotes the greatest integer function.

Let $\vec{p} = \left( \sum_{k=0}^2 \binom{2}{k}^2 \right) \hat{i} + \left( \sum_{k=0}^2 (-1)^k \binom{2}{k}^2 \right) \hat{j} + \binom{4}{2} \hat{k}$ and $\vec{q} = \left( \sum_{k=0}^2 \binom{2}{k} \binom{3}{k} \right) \hat{i} + \left( \sum_{k=0}^2 (-1)^k \binom{2}{k} \binom{3}{k} \right) \hat{j} + \binom{5}{2} \hat{k}$. Let $\vec{w} = (a - 2) \hat{i} + 4 \hat{j} + 3 \hat{k}$. It is given that the vectors $\vec{p}, \vec{q},$ and $\vec{w}$ are coplanar. A point $C$ with position vector $\vec{c}$ lies on the line passing through the points with position vectors $\vec{p}$ and $\vec{q}$, such that $\vec{c} = (1-\lambda)\vec{p} + \lambda\vec{q}$. If the scalar projection of $\vec{c}$ on the vector $\vec{v} = -\hat{i} + (a-1)\hat{j} - \hat{k}$ is $-12\sqrt{2}$, then the value of $2\lambda$ is:

Let $f(\alpha) = \int_0^{\sqrt{3}} \lfloor x^2 \rfloor (1+\alpha x^2)^3 (1+\alpha - \alpha x^2)^3 x \, dx$, where $\lfloor \cdot \rfloor$ denotes the greatest integer function. The value of $\left. \frac{d^3 f}{d\alpha^3} \right|_{\alpha=0}$ is:

Let $[t]$ denote the greatest integer less than or equal to $t$. Evaluate the integral: $I = \int_{1 \le |x| \le 5/3} \left[ \left| x - \sqrt{x^2-1} \right| \right] \frac{(x + \sqrt{x^2-1})^3 - (x - \sqrt{x^2-1})^3}{\sqrt{x^2-1}} \, dx$

Let $\{y\}$ denote the fractional part of $y$. Evaluate the following limit: $\lim_{x \to 3} \frac{\sum_{k=1}^{4} \left( \{ (x + \sqrt{x^2-1})^k \} + (x - \sqrt{x^2-1})^k \right) - 4}{x - 3}$