LFM Stats And Pure

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 2024 Q2 Polynomial Degree and Structural Properties View

Let $P \in \mathbb{K}[X]$. Determine the degree of $\Delta(P)$ as a function of that of $P$, where $\Delta(P) = P(X+1) - P(X)$.

grandes-ecoles 2024 Q3 Polynomial Degree and Structural Properties View

Show that, for all $d \in \mathbb{N}^{*}$, $\Delta$ induces an endomorphism on $\mathbb{K}_{d}[X]$, where $\Delta(P) = P(X+1) - P(X)$.

grandes-ecoles 2024 Q7 Proof of Polynomial Divisibility or Identity View

We denote by $\Delta_{d}$ the endomorphism of $\mathbb{K}_{d}[X]$ induced by $\Delta$, where $\Delta(P) = P(X+1) - P(X)$. Let $d \in \mathbb{N}^{*}$. Determine an annihilating polynomial of $\Delta_{d}$. Is the endomorphism $\Delta_{d}$ diagonalisable?

grandes-ecoles 2025 Q31 Polynomial Degree and Structural Properties View

We define on $[ 0,1 ]$ the function $P _ { n }$ by:
$$\forall x \in [ 0,1 ] , \quad P _ { n } ( x ) = \frac { 1 } { n ! } \frac { \mathrm { d } ^ { n } \left( x ^ { n } ( 1 - x ) ^ { n } \right) } { \mathrm { d } x ^ { n } } .$$
Let $n \in \mathbb { N } ^ { * }$. Justify that $P _ { n }$ is a polynomial function on $[ 0,1 ]$ of degree $n$ with coefficients in $\mathbb { Z }$.

isi-entrance 2019 Q14 Polynomial Construction from Root/Value Conditions View

Let $P ( X ) = X ^ { 4 } + a _ { 3 } X ^ { 3 } + a _ { 2 } X ^ { 2 } + a _ { 1 } X + a _ { 0 }$ be a polynomial in $X$ with real coefficients. Assume that
$$P ( 0 ) = 1 , P ( 1 ) = 2 , P ( 2 ) = 3 , \text { and } P ( 3 ) = 4 .$$
Then, the value of $P ( 4 )$ is
(A) 5
(B) 24
(C) 29
(D) not determinable from the given data.

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 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) $p ( 2 ) = 11$

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 $P = \mathbf{ST}$.

kyotsu-test 2012 QCourse2-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
$$P = (x - \mathbf{N})(x + \mathbf{O}a + \mathbf{P}).$$
Hence $x$ must be $\mathbf{Q}$.
Furthermore, the smallest possible $a$ is $\mathbf{R}$, and in this case the value of $P$ is $P = \mathbf{ST}$.

taiwan-gsat 2021 Q5 5 marks Remainder by Quadratic or Higher Divisor View

Let $f ( x )$ be a real polynomial function of degree 3 satisfying the condition that the remainder when $( x + 1 ) f ( x )$ is divided by $x ^ { 3 } + 2$ is $x + 2$. If $f ( 0 ) = 4$, what is the value of $f ( 2 )$?
(1) 8
(2) 10
(3) 15
(4) 18
(5) 20

taiwan-gsat 2023 Q6 5 marks Sum of Coefficients and Coefficient Relationships View

A person calculates the remainder when the polynomial $f(x) = x^{3} + ax^{2} + bx + c$ is divided by $g(x) = ax^{3} + bx^{2} + cx + d$, where $a, b, c, d$ are real numbers and $a \neq 0$. He mistakenly read it as $g(x)$ divided by $f(x)$, and after calculation obtained the remainder as $-3x - 17$. Assuming the correct remainder when $f(x)$ is divided by $g(x)$ equals $px^{2} + qx + r$, what is the value of $p$?
(1) $-3$ (2) $-1$ (3) $0$ (4) $2$ (5) $3$

taiwan-gsat 2023 Q14 5 marks Remainder by Quadratic or Higher Divisor View

Let $a$ and $b$ be real numbers (where $a > 0$). If the polynomial $ax^{2} + (2a+b)x - 12$ divided by $x^{2} + (2-a)x - 2a$ gives a remainder of 6, then the ordered pair $(a, b) = $ (14--1), 14--2).

taiwan-gsat 2024 Q9 5 marks Euclidean Division: Quotient and Remainder Determination View

Given that when polynomial $f ( x )$ is divided by $x ^ { 2 } + 5 x + 1$, the quotient is $x ^ { 3 } + 7 x ^ { 2 } + x + 3$, select the options that could be $f ( x )$.
(1) $2 \left( x ^ { 3 } + 7 x ^ { 2 } + x + 3 \right) \left( x ^ { 2 } + 5 x + 1 \right)$
(2) $\left( x ^ { 3 } + 7 x ^ { 2 } + x + 3 \right) \left( x ^ { 2 } + 5 x + 1 \right) - x$
(3) $\left( x ^ { 3 } + 7 x ^ { 2 } + x + 3 \right) \left( x ^ { 2 } + 5 x + 1 \right) + x ^ { 2 }$
(4) $\left( x ^ { 3 } + 7 x ^ { 2 } + x + 4 \right) \left( x ^ { 2 } + 5 x + 1 \right) - x$
(5) $\left( x ^ { 3 } + 7 x ^ { 2 } + x + 4 \right) \left( x ^ { 2 } + 5 x + 1 \right) - x ^ { 2 }$

taiwan-gsat 2024 Q14 5 marks Remainder by Quadratic or Higher Divisor View

It is known that $f(x), g(x), h(x)$ are all real-coefficient cubic polynomials, and their remainders when divided by $x^{2} - 2x + 3$ are $x + 1$, $x - 3$, and $-2$ respectively. If $xf(x) + ag(x) + bh(x)$ is divisible by $x^{2} - 2x + 3$, where $a, b$ are real numbers, then $a =$ (14－1)(14－2), $b =$ (14－3).

taiwan-gsat 2025 Q13 5 marks Euclidean Division: Quotient and Remainder Determination View

A real-coefficient cubic polynomial $f ( x )$ divided by $x + 6$ gives quotient $q ( x )$ and remainder 3. If $q ( x )$ has a maximum value of 8 at $x = - 6$, then the coordinates of the center of symmetry of the graph $y = f ( x )$ are ((13-1) (13-2), (13-3)).

turkey-yks 2010 Q9 Divisibility and Factor Determination View

$$P(x) = 2x^{3} - (m+1)x^{2} - nx + 3m - 1$$
Given that the polynomial is completely divisible by $x^{2} - x$, what is $m - n$?
A) $\frac{-1}{3}$
B) $\frac{-1}{2}$
C) $\frac{3}{2}$
D) $2$
E) $3$

turkey-yks 2010 Q15 Polynomial Construction from Root/Value Conditions View

Let $P(x)$ be a third-degree polynomial function such that $$P(-4) = P(-3) = P(5) = 0, \quad P(0) = 2$$ Given this, what is $P(1)$?
A) $\frac{7}{3}$
B) $\frac{8}{3}$
C) $\frac{7}{4}$
D) $\frac{9}{4}$
E) $\frac{8}{5}$

turkey-yks 2011 Q3 Sum of Coefficients and Coefficient Relationships View

Given that $t ^ { 3 } - 2 = 0$, which of the following is the equivalent of $\frac { 1 } { t ^ { 2 } + t + 1 }$ in terms of $t$?
A) $t + 1$
B) $\mathrm { t } - 2$
C) $t - 1$
D) $t ^ { 2 } + 1$
E) $t ^ { 2 } + 3$

turkey-yks 2011 Q16 Sum of Coefficients and Coefficient Relationships View

Real coefficient polynomials $P ( x ) , Q ( x )$ and $R ( x )$ are given. For the polynomial $\mathrm { P } ( \mathrm { x } )$ whose constant term is nonzero,
$$P ( x ) = Q ( x ) \cdot R ( x + 1 )$$
the equality is satisfied. If the constant term of P is twice the constant term of Q, what is the sum of the coefficients of R?
A) $\frac { 2 } { 3 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 3 } { 4 }$
D) 1
E) 2

turkey-yks 2013 Q18 Remainder by Quadratic or Higher Divisor View

$$P ( x ) = x ^ { 11 } - 2 x ^ { 10 } + x - 2$$
What is the remainder when this polynomial is divided by $x ^ { 2 } - 5 x + 6$?
A) $3 ^ { 10 } + 1$
B) $3 ^ { 10 } - 1$
C) $3 ^ { 11 } + 1$
D) $3 ^ { 11 } - 1$

turkey-yks 2015 Q21 Remainder Theorem with Composed or Shifted Arguments View

The third-degree polynomial $\mathrm { P } ( \mathrm { x } )$ with leading coefficient 1 is divisible without remainder by $x ^ { 2 } + 4$. The remainder obtained from dividing the polynomial $P ( 2 x )$ by $2 x - 3$ is 52.
Accordingly, what is the value of $\mathbf { P } ( 2 )$?
A) 20
B) 22
C) 24
D) 26
E) 28

turkey-yks 2016 Q21 Remainder Theorem with Composed or Shifted Arguments View

$$P ( x ) = x ^ { 3 } - m x + 1$$
The remainder when $P ( x - 1 )$ is divided by $x + 1$ equals the remainder when $P ( x + 1 )$ is divided by $x - 1$.
Accordingly, what is m?
A) 2
B) 4
C) 6
D) - 1
E) - 8

turkey-yks 2016 Q22 Polynomial Construction from Root/Value Conditions View

A third-degree polynomial $\mathrm { P } ( \mathrm { x } )$ with real coefficients and leading coefficient 1 satisfies the equalities
$$P ( 1 ) = P ( 3 ) = P ( 5 ) = 7$$
Accordingly, what is the value of $\mathbf { P } ( \mathbf { 0 } )$?
A) - 1
B) - 4
C) - 8
D) 4
E) 8

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LFM Stats And Pure

Completing the square and sketching 61

Inequalities 181

Discriminant and conditions for roots 65

Solving quadratics and applications 131

Curve Sketching 228

Factor & Remainder Theorem 61

Polynomial Division & Manipulation 35

Binomial Theorem (positive integer n) 197

Function Transformations 28

Composite & Inverse Functions 211

Partial Fractions 12

Generalised Binomial Theorem 10

Complex Numbers Arithmetic 207

Complex Numbers Argand & Loci 159

Permutations & Arrangements 255

Combinations & Selection 237

Measures of Location and Spread 181

Probability Definitions 212

Tree Diagrams 18

Principle of Inclusion/Exclusion 33

Independent Events 35

Conditional Probability 167

Discrete Probability Distributions 176

Uniform Distribution 3

Binomial Distribution 101

Geometric Distribution 8

Hypergeometric Distribution 9

Negative Binomial Distribution 1

Modelling and Hypothesis Testing 22

Hypothesis test of binomial distributions 2

Data representation 21

Continuous Probability Distributions and Random Variables 322

Continuous Uniform Random Variables 3

Geometric Probability 16

Normal Distribution 87

Approximating Binomial to Normal Distribution 4

Sign Change & Interval Methods 26

Fixed Point Iteration 9

Newton-Raphson method 6