LFM Stats And Pure

grandes-ecoles 2021 Q14 Proof of Inequalities Involving Series or Sequence Terms View

Let $n$ be a non-zero natural number. Let $f \in \mathcal{S}_n$. Using the result of question 13, deduce that $$\forall \theta \in \mathbb{R}, \quad |f'(\theta)| \leqslant n \|f\|_{L^\infty(\mathbb{R})} \tag{I.4}$$

grandes-ecoles 2022 Q11 Location and bounds on roots View

Let $\alpha \in \mathbf{R}$. Prove that if $\alpha$ is a root of a polynomial $P$ in $\mathbf{R}[X]$ with strictly positive coefficients, then $\alpha < 0$.

grandes-ecoles 2022 Q12 Algebraic Number Theory and Minimal Polynomials View

Prove that every divisor of a Hurwitz polynomial is a Hurwitz polynomial.

grandes-ecoles 2022 Q12 Polynomial Degree and Structural Properties View

Let $n \in \mathbb{N}^*$, $W$ be a monic polynomial of degree $n$, and set $Q = \frac { 1 } { 2 ^ { n - 1 } } T _ { n } - W$. Show that $Q$ is a polynomial of degree at most $n - 1$.

grandes-ecoles 2022 Q13 Coefficient and structural properties of special polynomial families View

Let $P$ be an irreducible Hurwitz polynomial in $\mathbf{R}[X]$ with positive leading coefficient. Prove that all coefficients of $P$ are strictly positive.

grandes-ecoles 2022 Q14 Determine coefficients or parameters from root conditions View

Let $n \in \mathbf{N}^{*}$. Let $(z_{1}, z_{2}, \ldots, z_{n}) \in \mathbf{C}^{n}$. We define the two polynomials $P(X)$ and $Q(X)$ in $\mathbf{C}[X]$ by: $$P(X) = \prod_{k=1}^{n}(X - z_{k}) \quad \text{and} \quad Q(X) = \prod_{(k,l) \in \llbracket 1;n \rrbracket^{2}}(X - z_{k} - z_{l})$$
We assume $n = 2$ and $P \in \mathbf{R}_{2}[X]$. If the coefficients of $Q$ are strictly positive, is $P$ then a Hurwitz polynomial?

grandes-ecoles 2022 Q15 Direct Proof of a Stated Identity or Equality View

Let $A$ and $B$ be two polynomials in $\mathbf{R}[X]$ whose coefficients are all strictly positive. Prove that the coefficients of the product $AB$ are also strictly positive.

grandes-ecoles 2022 Q16 Location and bounds on roots View

Let $n \in \mathbf{N}^{*}$. Let $(z_{1}, z_{2}, \ldots, z_{n}) \in \mathbf{C}^{n}$. We define the two polynomials $P(X)$ and $Q(X)$ in $\mathbf{C}[X]$ by: $$P(X) = \prod_{k=1}^{n}(X - z_{k}) \quad \text{and} \quad Q(X) = \prod_{(k,l) \in \llbracket 1;n \rrbracket^{2}}(X - z_{k} - z_{l})$$
Prove that if $P$ and $Q$ are in $\mathbf{R}[X]$, then we have the equivalence: $P$ is a Hurwitz polynomial if and only if the coefficients of $P$ and $Q$ are strictly positive.

grandes-ecoles 2025 Q1 Reciprocal and antireciprocal polynomial properties View

Let $P \in \mathbf{C}[X]$ of degree $p$. We write $P = \sum_{k=0}^{p} a_k X^k$, where $a_0, \ldots, a_p$ are complex numbers, and $a_p \neq 0$.
Show that $P$ is reciprocal if and only if for every integer $k$, $0 \leq k \leq p$, we have the equality $a_k = a_{p-k}$.

grandes-ecoles 2025 Q1 Factored form and root structure from polynomial identities View

Show that $p_0$, the reciprocal polynomial of $p$, satisfies $$\forall x \in \mathbf{R}^* \quad p_0(x) = x^n p(1/x)$$ and deduce that $$p_0 = a_n \prod_{j=1}^{n} \left(1 - \alpha_j X\right)$$

grandes-ecoles 2025 Q2 Reciprocal and antireciprocal polynomial properties View

Let $P$ be a polynomial of degree $p$ written in factored form $P = a_p \prod_{i=1}^{d} (X - \lambda_i)^{m_i}$, where $\lambda_1, \ldots, \lambda_d$ are the distinct complex roots of $P$ and $m_1, \ldots, m_d$ their multiplicities.
Write in factored form the polynomial $X^p P\left(\frac{1}{X}\right)$ and prove that if $P$ is reciprocal then for every integer $i$, $1 \leq i \leq d$, $\lambda_i$ is nonzero and $\frac{1}{\lambda_i}$ is a root of $P$ with multiplicity $m_i$.

grandes-ecoles 2025 Q2 Factored form and root structure from polynomial identities View

Let $P$ be a polynomial of degree $p$ written in factored form $P = a_p \prod_{i=1}^{d} (X - \lambda_i)^{m_i}$, where $\lambda_1, \ldots, \lambda_d$ are the distinct complex roots of $P$ and $m_1, \ldots, m_d$ their multiplicities. Write in factored form the polynomial $X^p P\left(\frac{1}{X}\right)$ and prove that if $P$ is reciprocal then for every integer $i$, $1 \leq i \leq d$, $\lambda_i$ is nonzero and $\frac{1}{\lambda_i}$ is a root of $P$ with multiplicity $m_i$.

grandes-ecoles 2025 Q3 Reciprocal and antireciprocal polynomial properties View

Let $Q$ be a polynomial of degree $p$. We say that $Q$ is antireciprocal if $$Q(X) = -X^p Q\left(\frac{1}{X}\right)$$ Show that if $Q$ is antireciprocal, 1 is a root of $Q$ and there exists a polynomial $P$ that is constant or reciprocal such that $Q = (X-1)P$.

grandes-ecoles 2025 Q5 Reciprocal and antireciprocal polynomial properties View

Let $R$ be a non-constant polynomial of $\mathbf{C}[X]$ having the following property: Every root $a$ of $R$ is nonzero and $\frac{1}{a}$ is a root of $R$ with the same multiplicity as $a$.
Deduce that $R$ is reciprocal or antireciprocal.

iran-konkur 2022 Q117 Remainder by Quadratic or Higher Divisor View

117- The polynomial $p(x) = x^{2n+1} + 2x^{2n} + x^6 + 3x^5 + 16x + 16a$ is divisible by $x + 2$ for every natural number $n$.
For $n=1$, what is the remainder when $p(x)$ is divided by $x^2 + 2x - 3$?
(1) $-15x + 24$ (2) $-15x + 14$ (3) $-5x + 34$ (4) $-5x + 44$

isi-entrance 2022 Q18 Existence or counting of roots with specified properties View

Let $p$ and $q$ be two non-zero polynomials such that the degree of $p$ is less than or equal to the degree of $q$, and $p ( a ) q ( a ) = 0$ for $a = 0,1,2 , \ldots , 10$. Which of the following must be true?
(A) degree of $q \neq 10$
(B) degree of $p \neq 10$
(C) degree of $q \neq 5$
(D) degree of $p \neq 5$

isi-entrance 2023 Q7 Vieta's formulas: compute symmetric functions of roots View

(a) Let $n \geq 1$ be an integer. Prove that $X ^ { n } + Y ^ { n } + Z ^ { n }$ can be written as a polynomial with integer coefficients in the variables $\alpha = X + Y + Z$, $\beta = X Y + Y Z + Z X$ and $\gamma = X Y Z$.
(b) Let $G _ { n } = x ^ { n } \sin ( n A ) + y ^ { n } \sin ( n B ) + z ^ { n } \sin ( n C )$, where $x , y , z , A , B , C$ are real numbers such that $A + B + C$ is an integral multiple of $\pi$. Using (a) or otherwise, show that if $G _ { 1 } = G _ { 2 } = 0$, then $G _ { n } = 0$ for all positive integers $n$.

isi-entrance 2023 Q24 Divisibility and Factor Determination View

The polynomial $x ^ { 10 } + x ^ { 5 } + 1$ is divisible by
(A) $x ^ { 2 } + x + 1$.
(B) $x ^ { 2 } - x + 1$.
(C) $x ^ { 2 } + 1$.
(D) $x ^ { 5 } - 1$.

jee-advanced 1999 Q14 Polynomial Construction from Root/Value Conditions View

14. If $\mathrm { f } ( \mathrm { x } ) \left| \begin{array} { c c c } 1 & x & x + 1 \\ 2 x & x ( x - 1 ) & ( x + 1 ) x \\ 3 x ( x - 1 ) & x ( x - 1 ) ( x - 2 ) & ( x + 1 ) x ( x - 1 ) \end{array} \right|$ then $\mathrm { f } ( 100 )$ is equal to :
(A) 0
(B) 1
(C) 100
(D) $\quad - 100$

jee-advanced 2020 Q17 Root Interlacing and Sign Conditions View

For a polynomial $g ( x )$ with real coefficients, let $m _ { g }$ denote the number of distinct real roots of $g ( x )$. Suppose $S$ is the set of polynomials with real coefficients defined by
$$S = \left\{ \left( x ^ { 2 } - 1 \right) ^ { 2 } \left( a _ { 0 } + a _ { 1 } x + a _ { 2 } x ^ { 2 } + a _ { 3 } x ^ { 3 } \right) : a _ { 0 } , a _ { 1 } , a _ { 2 } , a _ { 3 } \in \mathbb { R } \right\}$$
For a polynomial $f$, let $f ^ { \prime }$ and $f ^ { \prime \prime }$ denote its first and second order derivatives, respectively. Then the minimum possible value of ( $m _ { f ^ { \prime } } + m _ { f ^ { \prime \prime } }$ ), where $f \in S$, is $\_\_\_\_$

jee-main 2017 Q61 Polynomial Construction from Root/Value Conditions View

Let $p ( x )$ be a quadratic polynomial such that $p ( 0 ) = 1$. If $p ( x )$ leaves remainder 4 when divided by $x - 1$ and it leaves remainder 6 when divided by $x + 1$ then:
(1) $p ( - 2 ) = 19$
(2) $p ( 2 ) = 19$
(3) $p ( - 2 ) = 11$
(4) $p ( 2 ) = 11$

jee-main 2022 Q66 Limit Involving Derivative Definition of Composed Functions View

Let $f(x)$ be a polynomial function such that $f(x) + f ^ { \prime } (x) + f ^ { \prime \prime } (x) = x ^ { 5 } + 64$. Then, the value of $\lim _ { x \rightarrow 1 } \frac { f(x) } { x - 1 }$ is equal to
(1) $- 15$
(2) $15$
(3) $- 60$
(4) $60$

jee-main 2022 Q86 Factorization and Root Analysis View

The number of distinct real roots of the equation $x ^ { 5 } \left( x ^ { 3 } - x ^ { 2 } - x + 1 \right) + x \left( 3 x ^ { 3 } - 4 x ^ { 2 } - 2 x + 4 \right) - 1 = 0$ is $\_\_\_\_$.

jee-main 2023 Q65 Sketching a Curve from Analytical Properties View

The combined equation of the two lines $ax + by + c = 0$ and $a'x + b'y + c' = 0$ can be written as $(ax + by + c)(a'x + b'y + c') = 0$. The equation of the angle bisectors of the lines represented by the equation $2x^2 + xy - 3y^2 = 0$ is
(1) $3x^2 + 5xy + 2y^2 = 0$
(2) $x^2 - y^2 + 10xy = 0$
(3) $3x^2 + xy - 2y^2 = 0$
(4) $x^2 - y^2 - 10xy = 0$

kyotsu-test 2012 QCourse1-I-Q2 Factorization and Root Analysis View

Consider the polynomial
$$P = x^2 + 2(a-1)x - 8a - 8.$$
(1) Let $a$ be a rational number. If the value of $P$ is a rational number when $x = 1 - \sqrt{2}$, then $a =$ $\mathbf{K}$ and in this case the value of $P$ is $P =$ $\mathbf{LM}$.
(2) Let $x$ and $a$ be positive integers. We are to investigate $x$ and $a$ which are such that the value of $P$ is a prime number.
When we factorize $P$, we have
$$P = (x - \mathbf{N}\mathbf{N})(x + \mathbf{O}a + \mathbf{P}).$$
Hence $x$ must be $\mathbf{Q}$.
Furthermore, the smallest possible $a$ is $\mathbf{R}$, and in this case the value of $P$ is $P = \mathbf{ST}$.

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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