LFM Stats And Pure

gaokao 2022 Q7 5 marks Identifying the Correct Graph of a Function View

The graph of the function $f ( x ) = \left( 3 ^ { x } - 3 ^ { - x } \right) \cos x$ on the interval $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$ is approximately [see figures A, B, C, D in the original paper].

gaokao 2022 Q8 5 marks Identifying the Correct Graph of a Function View

The figure on the right is the approximate graph of one of the following four functions on the interval $[ - 3,3 ]$ . The function is
A. $y = \frac { - x ^ { 3 } + 3 x } { x ^ { 2 } + 1 }$
B. $y = \frac { x ^ { 3 } - x } { x ^ { 2 } + 1 }$
C. $y = \frac { 2 x \cos x } { x ^ { 2 } + 1 }$
D. $y = \frac { 2 \sin x } { x ^ { 2 } + 1 }$

gaokao 2023 Q10 5 marks Number of Solutions / Roots via Curve Analysis View

Let $f(x)$ be the function obtained by shifting $y = \cos\left(2x + \frac{\pi}{4}\right)$ to the left by $\frac{\pi}{6}$ units. The number of intersection points of $y = f(x)$ and $y = \frac{1}{2}x - \frac{1}{2}$ is
A. $1$
B. $2$
C. $3$
D. $4$

gaokao 2024 Q6 5 marks Number of Solutions / Roots via Curve Analysis View

Let $f ( x ) = a ( x + 1 ) ^ { 2 } - 1 , g ( x ) = \cos x + 2 a x$. When $x \in ( - 1,1 )$, the curves $y = f ( x )$ and $y = g ( x )$ have exactly one intersection point. Then $a =$
A. $- 1$
B. $\frac { 1 } { 2 }$
C. 1
D. 2

gaokao 2025 Q10 6 marks Function Properties from Symmetry or Parity View

Given that $f(x)$ is an odd function defined on $\mathbb{R}$, and when $x > 0$, $f(x) = (x^2 - 3)e^x + 2$, then
A. $f(0) = 0$
B. When $x < 0$, $f(x) = -(x^2 - 3)e^{-x} - 2$
C. $f(x) < 0$ if and only if $x > \sqrt{3}$
D. $x = -1$ is a local maximum point of $f(x)$

grandes-ecoles 2011 QIV.A Sketching a Curve from Analytical Properties View

The function $h$ is defined on $\mathbb{R}$ by $$h : \mathbb{R} \longrightarrow \mathbb{R}, \quad u \longmapsto u - [u] - 1/2$$ where $[u]$ denotes the integer part of $u$.
Carefully draw the graph of the application $h$ on the interval $[-1, 1]$.

grandes-ecoles 2013 Q5 Variation Table and Monotonicity from Sign of Derivative View

We are given four real numbers $a \leqslant b \leqslant c \leqslant d$ such that $a + d = b + c$. Study the variations of the function $x \mapsto |x - a| - |x - b| - |x - c| + |x - d|$; show that it takes positive values. A reasoned argument supported by a graphical representation would be welcome.

grandes-ecoles 2015 QI.A.1 Sketching a Curve from Analytical Properties View

We denote $\mathcal{D}$ the set of functions from $\mathbb{R}$ to $\mathbb{R}$ of class $\mathcal{C}^{\infty}$ and with compact support. We denote $\varphi$ the function defined by: $$\begin{cases} \varphi(x) = 0 & \text{if } |x| \geqslant 1 \\ \varphi(x) = \exp\left(-\frac{x^2}{1-x^2}\right) & \text{if } |x| < 1 \end{cases}$$
a) Study the variations of $\varphi$. b) Sketch the graph of $\varphi$. c) Show that $\varphi$ is $\mathcal{C}^{\infty}$. d) Show that $\mathcal{D}$ is a vector space over $\mathbb{R}$ not reduced to $\{0\}$.

grandes-ecoles 2015 Q3a Sketching a Curve from Analytical Properties View

We recall that the function $\phi$ is defined on $] - 1 , + \infty [$ by $\phi ( s ) = s - \ln ( 1 + s )$.
Sketch the graph of $\phi$. Show that $\phi$ defines by restriction to the intervals $] - 1,0 [$ and $] 0 , + \infty [$ respectively

a bijection $\left. \phi _ { - } : \right] - 1,0 [ \rightarrow ] 0 , + \infty [$,
a bijection $\left. \phi _ { + } : \right] 0 , + \infty [ \rightarrow ] 0 , + \infty [$.

We denote $\left. \phi _ { - } ^ { - 1 } : \right] 0 , + \infty [ \rightarrow ] - 1,0 \left[ \right.$ and $\left. \phi _ { + } ^ { - 1 } : \right] 0 , + \infty [ \rightarrow ] 0 , + \infty [$ the inverse bijections.

grandes-ecoles 2017 Q18 Function Properties from Symmetry or Parity View

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The two endomorphisms $T$ and $M$ of $E$ are defined by $$\forall P \in \mathbb{R}_{2m}[X], \quad T(P) = P' \text{ and } M(P) = P^*$$ where $P^*(X) = P(-X)$. We set $F^+ = \operatorname{ker}(M - \operatorname{Id}_E)$, $F^- = \operatorname{ker}(M + \operatorname{Id}_E)$.
What are the spaces $F^+$ and $F^-$ in this case?

grandes-ecoles 2017 Q20 Function Properties from Symmetry or Parity View

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The two endomorphisms $T$ and $M$ of $E$ are defined by $T(P) = P'$ and $M(P) = P^*$ where $P^*(X) = P(-X)$. We set $$\mathbb{R}_k^0[X] = \{P \in \mathbb{R}_k[X] \mid P(-1) = 0 \text{ and } P(1) = 0\}$$ The subspace $G$ consists of elements $u \in E$ satisfying (a) $u \in \operatorname{Im}(T)$ and (b) $\forall v \in E, S(u,v) = 0$, where $S(P,Q) = P(1)Q(1) - P(-1)Q(-1)$.
Determine the subspace $G$. Is hypothesis (H5) satisfied?

grandes-ecoles 2018 Q3 Sketching a Curve from Analytical Properties View

Study the variations of $g_{\sigma}$. Show that the second derivative of $g_{\sigma}$ vanishes and changes sign at exactly two points. Give the shape of the graph of $g_{\sigma}$ and mark the two points mentioned.

grandes-ecoles 2018 Q9 Sketching a Curve from Analytical Properties View

Give the shape of the representative curve of $\zeta(x) = \sum_{n=1}^{+\infty} \frac{1}{n^x}$.

grandes-ecoles 2018 Q30 Variation Table and Monotonicity from Sign of Derivative View

Show that for all $x \in [0, 1[$
$$\frac{x^{2}}{2} + (x - 1)\ln(1 - x) \leqslant \ln(2 + x) - \ln(2 - x)$$
One may perform a function study.

grandes-ecoles 2018 Q30 Variation Table and Monotonicity from Sign of Derivative View

Show that for all $x \in [0, 1[$
$$\frac{x^{2}}{2} + (x - 1) \ln(1 - x) \leqslant \ln(2 + x) - \ln(2 - x)$$
One may perform a function study.

grandes-ecoles 2018 Q31 Number of Solutions / Roots via Curve Analysis View

Deduce that for all $x \in [0, 1[$
$$1 + \exp\left(\frac{x^{2}}{2}\right)(1 - x)^{x - 1} \leqslant \frac{4}{2 - x}$$

grandes-ecoles 2018 Q31 Number of Solutions / Roots via Curve Analysis View

Deduce that for all $x \in [0, 1[$
$$1 + \exp\left(\frac{x^{2}}{2}\right)(1 - x)^{x - 1} \leqslant \frac{4}{2 - x}$$

grandes-ecoles 2020 Q5 Sketching a Curve from Analytical Properties View

Let $f(x) = xe^x$ and let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$. Sketch, on the same diagram, the curves $\mathcal { C } _ { f }$ and $\mathcal { C } _ { W }$ representing the functions $f$ and $W$. Specify the tangent lines to the two curves at the point with abscissa 0 as well as the tangent line to $\mathcal { C } _ { W }$ at the point with abscissa $- \mathrm { e } ^ { - 1 }$.

grandes-ecoles 2020 Q6 Sketching a Curve from Analytical Properties View

For all $s \in [0,1]$, the function $k_s$ is defined by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ Let $s \in ]0,1[$. Sketch the graph of $k_s$ on $[0,1]$.

grandes-ecoles 2020 Q6 Sketching a Curve from Analytical Properties View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ Let $s \in ]0,1[$. Sketch the graph of $k_s$ on $[0,1]$.

grandes-ecoles 2020 Q7 Continuity and Discontinuity Analysis of Piecewise Functions View

For all $(s,t) \in [0,1]^2$, $K(s,t) = k_s(t)$ where $$k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ Show that $K$ is continuous on $[0,1] \times [0,1]$.

grandes-ecoles 2020 Q7 Continuity and Discontinuity Analysis of Piecewise Functions View

In this part, $E$ denotes the vector space of functions $f : [0,1] \rightarrow \mathbb{R}$ continuous, equipped with the inner product defined by, $$\forall (f,g) \in E^2, \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, \mathrm{d}t$$ For all $s \in [0,1]$, we define the function $k_s$ by, $$\forall t \in [0,1], \quad k_s(t) = \begin{cases} t(1-s) & \text{if } t < s \\ s(1-t) & \text{if } t \geqslant s. \end{cases}$$ We also note, for all $(s,t) \in [0,1]^2$, $K(s,t) = k_s(t)$. Show that $K$ is continuous on $[0,1] \times [0,1]$.

grandes-ecoles 2020 Q10 Number of Solutions / Roots via Curve Analysis View

Let $f(x) = xe^x$, and let $V$ and $W$ denote the inverses of $f|_{]-\infty,-1]}$ and $f|_{[-1,+\infty[}$ respectively. For a real parameter $m$, we consider the inequality with unknown $x \in \mathbb { R }$
$$x \mathrm { e } ^ { x } \leqslant m \tag{I.2}$$
Using the functions $V$ and $W$, determine, according to the values of $m$, the solutions of (I.2). Illustrate graphically the different cases.

grandes-ecoles 2020 Q11 Number of Solutions / Roots via Curve Analysis View

Let $V$ and $W$ denote the inverses of $f|_{]-\infty,-1]}$ and $f|_{[-1,+\infty[}$ respectively, where $f(x) = xe^x$. For non-zero real parameters $a$ and $b$, we consider the equation with unknown $x \in \mathbb { R }$
$$\mathrm { e } ^ { a x } + b x = 0 \tag{I.3}$$
Determine, according to the values of $a$ and $b$, the number of solutions of (I.3). Explicitly express the possible solutions using the functions $V$ and $W$.

grandes-ecoles 2023 Q8 Sketching a Curve from Analytical Properties View

Throughout this problem, $I = ]-1, +\infty[$, and $f(x) = \int_0^{\pi/2} (\sin(t))^x \mathrm{~d}t$.
Sketch the graph of $f$ by making best use of the previous results.

1
2
3
4
5
6
7
8
9
10

Load questions...

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