LFM Stats And Pure

grandes-ecoles 2020 Q5 View

For $\alpha \in \mathbb{R}$, recall, without giving a proof, the power series expansion of $( 1 + x ) ^ { \alpha }$ on $]-1,1[$.
Justify the formula: $$\forall x \in ]-1,1[ , \quad \frac { 1 } { \sqrt { 1 - x } } = \sum _ { n = 0 } ^ { + \infty } \frac { \binom { 2 n } { n } } { 4 ^ { n } } x ^ { n }$$

grandes-ecoles 2020 Q25 Functional Equations and Identities via Series View

We consider a natural integer $n$ and a complex number $a$. We define a family of polynomials $(A_0, A_1, \ldots, A_n)$ by setting $$A_0 = 1 \quad \text{and, for all } k \in \llbracket 1, n \rrbracket, \quad A_k = \frac{1}{k!} X(X - ka)^{k-1}.$$ Using the result of Question 24, deduce Abel's binomial identity: $$\forall (a, x, y) \in \mathbb{C}^3, \quad (x+y)^n = y^n + \sum_{k=1}^{n} \binom{n}{k} x(x - ka)^{k-1}(y + ka)^{n-k}.$$

grandes-ecoles 2020 Q26 Functional Equations and Identities via Series View

We consider a natural integer $n$ and a complex number $a$. Using Abel's binomial identity $$\forall (a, x, y) \in \mathbb{C}^3, \quad (x+y)^n = y^n + \sum_{k=1}^{n} \binom{n}{k} x(x - ka)^{k-1}(y + ka)^{n-k},$$ establish the relation $$\forall (a, y) \in \mathbb{C}^2, \quad ny^{n-1} = \sum_{k=1}^{n} \binom{n}{k} (-ka)^{k-1}(y + ka)^{n-k}.$$

grandes-ecoles 2021 Q14 Convergence proof and limit determination View

We set, for all $t \in I$, $$f ( t ) = \sum _ { n = 0 } ^ { + \infty } C _ { n } t ^ { n } \quad \text { and } \quad g ( t ) = 2 t f ( t ) .$$ Show that $\varepsilon$ is continuous on $I \backslash \left\{ \frac { 1 } { 4 } \right\}$. Deduce $$\forall t \in I , \quad g ( t ) = 1 - \sqrt { 1 - 4 t } .$$

grandes-ecoles 2021 Q17 View

Justify the existence of a sequence of real numbers $\left( a _ { n } \right) _ { n \in \mathbb { N } }$ such that $$\forall x \in ] - 1,1 \left[ , \quad \sqrt { 1 + x } = 1 + \sum _ { n = 0 } ^ { + \infty } a _ { n } x ^ { n + 1 } , \right.$$ and, for all $n \in \mathbb { N }$, express $a _ { n }$ using a binomial coefficient.

jee-advanced 2016 Q51 Determine Parameters from Conditions on Coefficients or Terms View

Let $m$ be the smallest positive integer such that the coefficient of $x^2$ in the expansion of $(1+x)^2 + (1+x)^3 + \cdots + (1+x)^{49} + (1+mx)^{50}$ is $(3n+1)\,{}^{51}C_3$ for some positive integer $n$. Then the value of $n$ is

jee-main 2012 Q66 Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View

If $f ( y ) = 1 - ( y - 1 ) + ( y - 1 ) ^ { 2 } - ( y - 1 ) ^ { 3 } + \ldots - ( y - 1 ) ^ { 17 }$ then the coefficient of $y ^ { 2 }$ in it is
(1) ${ } ^ { 17 } \mathrm { C } _ { 2 }$
(2) ${ } ^ { 17 } \mathrm { C } _ { 3 }$
(3) ${ } ^ { 18 } \mathrm { C } _ { 2 }$
(4) ${ } ^ { 18 } \mathrm { C } _ { 3 }$

jee-main 2015 Q78 Count Integral or Rational Terms in a Binomial Expansion View

The sum of coefficients of integral powers of $x$ in the binomial expansion of $(1 - 2\sqrt{x})^{50}$ is:
(1) $\frac{1}{2}(3^{50} + 1)$
(2) $\frac{1}{2}(3^{50})$
(3) $\frac{1}{2}(3^{50} - 1)$
(4) $\frac{1}{2}(2^{50} + 1)$

jee-main 2019 Q64 Determine Parameters from Conditions on Coefficients or Terms View

The positive value of $\lambda$ for which the co-efficient of $x ^ { 2 }$ in the expansion $x ^ { 2 } \left( \sqrt { x } + \frac { \lambda } { x ^ { 2 } } \right) ^ { 10 }$ is 720, is
(1) $\sqrt { 5 }$
(2) 3
(3) 4
(4) $2 \sqrt { 2 }$

jee-main 2021 Q64 Find a Specific Coefficient in a Product of Binomial/Polynomial Expressions View

The coefficient of $x ^ { 256 }$ in the expansion of $( 1 - x ) ^ { 101 } \left( x ^ { 2 } + x + 1 \right) ^ { 100 }$ is:
(1) ${ } ^ { 100 } C _ { 16 }$
(2) ${ } ^ { 100 } C _ { 15 }$
(3) ${ } ^ { - 100 } C _ { 16 }$
(4) ${ } ^ { - 100 } C _ { 15 }$

jee-main 2021 Q64 Integer Part or Limit Involving Conjugate Surd Binomial Expansions View

The lowest integer which is greater than $\left( 1 + \frac { 1 } { 10 ^ { 100 } } \right) ^ { 10 ^ { 100 } }$ is
(1) 3
(2) 4
(3) 2
(4) 1

jee-main 2024 Q64 View

The sum of the coefficient of $x ^ { 2 / 3 }$ and $x ^ { - 2 / 5 }$ in the binomial expansion of $\left( x ^ { 2 / 3 } + \frac { 1 } { 2 } x ^ { - 2 / 5 } \right) ^ { 9 }$ is
(1) $21/4$
(2) $63/16$
(3) $19/4$
(4) $69/16$

jee-main 2025 Q19 Determine Parameters from Conditions on Coefficients or Terms View

If in the expansion of $( 1 + x ) ^ { \mathrm { p } } ( 1 - x ) ^ { \mathrm { q } }$, the coefficients of $x$ and $x ^ { 2 }$ are 1 and $-2$, respectively, then $\mathrm { p } ^ { 2 } + \mathrm { q } ^ { 2 }$ is equal to :
(1) 18
(2) 13
(3) 8
(4) 20

jee-main 2025 Q64 View

Q64. The sum of the coefficient of $x ^ { 2 / 3 }$ and $x ^ { - 2 / 5 }$ in the binomial expansion of $\left( x ^ { 2 / 3 } + \frac { 1 } { 2 } x ^ { - 2 / 5 } \right) ^ { 9 }$ is
(1) $21 / 4$
(2) $63 / 16$
(3) $19 / 4$
(4) $69 / 16$

jee-main 2025 Q83 View

Q83. If the constant term in the expansion of $\left( 1 + 2 x - 3 x ^ { 3 } \right) \left( \frac { 3 } { 2 } x ^ { 2 } - \frac { 1 } { 3 x } \right) ^ { 9 }$ is p , then 108 p is equal to

turkey-yks 2024 Q16 Determine Parameters from Conditions on Coefficients or Terms View

Let $m$ and $n$ be natural numbers. If the constant term in the expansion of
$$\left(x + \frac{5}{x^{m}}\right)^{n}$$
is 60, what is $m + n$?
A) 36 B) 35 C) 31 D) 27 E) 23

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LFM Stats And Pure

Completing the square and sketching 80

Inequalities 215

Discriminant and conditions for roots 106

Solving quadratics and applications 106

Curve Sketching 295

Factor & Remainder Theorem 90

Polynomial Division & Manipulation 44

Binomial Theorem (positive integer n) 244

Function Transformations 65

Composite & Inverse Functions 249

Partial Fractions 22

Generalised Binomial Theorem 16

Generalised Binomial Theorem and Partial Fractions 3

Complex Numbers Arithmetic 283

Complex Numbers Argand & Loci 135

Permutations & Arrangements 276

Combinations & Selection 268

Measures of Location and Spread 233

Probability Definitions 413

Tree Diagrams 5

Principle of Inclusion/Exclusion 23

Independent Events 51

Conditional Probability 127

Discrete Probability Distributions 210

Binomial Distribution 107

Geometric Distribution 11

Hypergeometric Distribution 10

Negative Binomial Distribution 2

Modelling and Hypothesis Testing 38

Hypothesis test of binomial distributions 12

Data representation 29

Continuous Probability Distributions and Random Variables 30

Continuous Uniform Random Variables 15

Geometric Probability 28

Normal Distribution 96

Approximating Binomial to Normal Distribution 7

Sign Change & Interval Methods 74

Fixed Point Iteration 12