LFM Pure and Mechanics

csat-suneung 2024 Q5 3 marks Recovering Function Values from Derivative Information View

A polynomial function $f(x)$ satisfies $$f'(x) = 3x(x-2), \quad f(1) = 6$$ Find the value of $f(2)$. [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

csat-suneung 2024 Q8 3 marks Definite Integral Evaluation (Computational) View

A cubic function $f(x)$ satisfies $$xf(x) - f(x) = 3x^4 - 3x$$ for all real numbers $x$. Find the value of $\int_{-2}^{2} f(x)\,dx$. [3 points]
(1) 12
(2) 16
(3) 20
(4) 24
(5) 28

csat-suneung 2025 Q7 3 marks Finding a Function from an Integral Equation View

A polynomial function $f(x)$ satisfies $$\int_{0}^{x} f(t)\, dt = 3x^{3} + 2x$$ for all real numbers $x$. What is the value of $f(1)$? [3 points]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15

csat-suneung 2025 Q9 4 marks Definite Integral Evaluation (Computational) View

For the function $f(x) = 3x^{2} - 16x - 20$, $$\int_{-2}^{a} f(x)\, dx = \int_{-2}^{0} f(x)\, dx$$ When this condition is satisfied, what is the value of the positive number $a$? [4 points]
(1) 16
(2) 14
(3) 12
(4) 10
(5) 8

csat-suneung 2025 Q17 3 marks Recovering Function Values from Derivative Information View

For a polynomial function $f(x)$, $f'(x) = 9x^{2} + 4x$ and $f(1) = 6$. What is the value of $f(2)$? [3 points]

csat-suneung 2025 Q26C 3 marks Definite Integral Evaluation (Computational) View

As shown in the figure, a solid figure has as its base the region enclosed by the curve $y = \sqrt{\frac{x+1}{x(x + \ln x)}}$, the $x$-axis, and the two lines $x = 1$ and $x = e$. When the cross-section of this solid figure cut by a plane perpendicular to the $x$-axis is a square, what is the volume of this solid figure? [3 points]
(1) $\ln(e+1)$
(2) $\ln(e+2)$
(3) $\ln(e+3)$
(4) $\ln(2e+1)$
(5) $\ln(2e+2)$

csat-suneung 2026 Q15 4 marks Accumulation Function Analysis View

The function $f ( x )$ is $$f ( x ) = \begin{cases} - x ^ { 2 } & ( x < 0 ) \\ x ^ { 2 } - x & ( x \geq 0 ) \end{cases}$$ and for a positive number $a$, the function $g ( x )$ is $$g ( x ) = \left\{ \begin{array} { c l } a x + a & ( x < - 1 ) \\ 0 & ( - 1 \leq x < 1 ) \\ a x - a & ( x \geq 1 ) \end{array} \right.$$ Let $k$ be the maximum value of $a$ such that the function $h ( x ) = \int _ { 0 } ^ { x } ( g ( t ) - f ( t ) ) d t$ has exactly one extremum. When $a = k$, what is the value of $k + h ( 3 )$? [4 points]
(1) $\frac { 9 } { 2 }$
(2) $\frac { 11 } { 2 }$
(3) $\frac { 13 } { 2 }$
(4) $\frac { 15 } { 2 }$
(5) $\frac { 17 } { 2 }$

csat-suneung 2026 Q17 3 marks Antiderivative Verification and Construction View

For the function $f ( x ) = 4 x ^ { 3 } - 2 x$, let $F ( x )$ be an antiderivative with $F ( 0 ) = 4$. Find the value of $F ( 2 )$. [3 points]

gaokao 2015 Q11 Definite Integral Evaluation (Computational) View

11. $\int _ { 0 } ^ { 2 } ( x - 1 ) d x = $ $\_\_\_\_$.

grandes-ecoles 2010 QI.C.1 Convergence and Evaluation of Improper Integrals View

We denote by $E$ the vector space of polynomial functions on $\mathbb{R}$ and, for all $n \in \mathbb{N}$, we denote by $E_n$ the vector subspace of $E$ formed by polynomial functions of degree at most $n$.
Show that, for every pair $(f, g)$ of $E \times E$, the function: $$x \mapsto f(x) g(x) \frac{1}{\sqrt{1 - x^2}}$$ is integrable on the interval $]-1,1[$.

grandes-ecoles 2011 QI.A.1 Integral Inequalities and Limit of Integral Sequences View

Let $f$ be a real function, defined, continuous and decreasing on $[ a , + \infty [$, where $a \in \mathbb { R }$. Show that for every integer $k \in \left[ a + 1 , + \infty \left[ \right. \right.$, we have $\int _ { k } ^ { k + 1 } f ( x ) d x \leqslant f ( k ) \leqslant \int _ { k - 1 } ^ { k } f ( x ) d x$.

grandes-ecoles 2011 QVI.C Integral Inequalities and Limit of Integral Sequences View

For all $i \in \{1,2,3,4\}$, we set $$\theta(N_{i}) = \frac{1}{2N_{i}} + \int_{0}^{+\infty} \frac{h(u)}{(u+N_{i})^{2}} du$$
VI.C.1) Show that for all $i \in \{1,2,3,4\}$ $$0 < \theta(N_{i}) < \frac{1}{N_{i}}$$
VI.C.2) Show the existence of a strictly positive real $K$ such that for all $i \in \{1,2,3,4\}$ $$N_{i} = K e^{\mu \varepsilon_{i}} e^{-\theta(N_{i})}$$

grandes-ecoles 2011 QII.A Definite Integral Evaluation (Computational) View

We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ Moreover, we set: $$\forall (P, Q) \in (\mathbb{R}[X])^2, \quad \langle P, Q \rangle = \int_0^1 P(t) Q(t) \, dt$$
Show that the map $(P, Q) \mapsto \langle P, Q \rangle$ is an inner product on $\mathbb{R}[X]$.

grandes-ecoles 2011 QII.D Definite Integral Evaluation (Computational) View

We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ We define the sequence of polynomials $\left(L_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} L_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad L_n = \frac{1}{P_n^{(n)}(1)} P_n^{(n)} \end{array}\right.$$
II.D.1) For all $n \in \mathbb{N}$, we set $I_n = \int_0^1 P_n(u) \, du$.
Calculate, for all $n \in \mathbb{N}$, the value of $I_n$.
II.D.2) Deduce for all $n \in \mathbb{N}$ the relation: $\langle L_n, L_n \rangle = \frac{1}{2n+1}$.

grandes-ecoles 2011 QII.F Definite Integral Evaluation (Computational) View

We define the sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ by: $$\left\{\begin{array}{l} P_0 = 1 \\ \forall n \in \mathbb{N}^*, \quad P_n = [X(X-1)]^n \end{array}\right.$$ Moreover, we set: $$\forall (P, Q) \in (\mathbb{R}[X])^2, \quad \langle P, Q \rangle = \int_0^1 P(t) Q(t) \, dt$$ The family $\left(K_n\right)_{n \in \mathbb{N}}$ is the unique family of polynomials such that for all $n \in \mathbb{N}$, the degree of $K_n$ equals $n$ with strictly positive leading coefficient, and for all $N \in \mathbb{N}$, $\left(K_n\right)_{0 \leqslant n \leqslant N}$ is an orthonormal basis of $\mathbb{R}_N[X]$ for $\langle \cdot, \cdot \rangle$.
Calculate $K_0$, $K_1$ and $K_2$.

grandes-ecoles 2011 QIII.B.1 Definite Integral Evaluation (Computational) View

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ and we denote by $\|\cdot\|$ the associated norm.
Let $n \in \mathbb{N}$. Show that there exists a unique polynomial $\Pi_n \in \mathbb{R}_n[X]$ such that $$\left\|\Pi_n - f\right\| = \min_{Q \in \mathbb{R}_n[X]} \|Q - f\|$$

grandes-ecoles 2011 QIII.B.5 Definite Integral Evaluation (Computational) View

For all $n \in \mathbb{N}^*$, we define the matrix $H_n$ by: $$\forall (i,j) \in \llbracket 1; n \rrbracket^2, \quad (H_n)_{i,j} = \frac{1}{i+j-1}$$ We extend to $C^0([0;1], \mathbb{R})$ the inner product $\langle \cdot, \cdot \rangle$ by setting $$\forall f, g \in C^0([0;1], \mathbb{R}), \quad \langle f, g \rangle = \int_0^1 f(t) g(t) \, dt$$ For each $n \in \mathbb{N}$, $\Pi_n$ denotes the unique polynomial in $\mathbb{R}_n[X]$ minimizing $\|Q - f\|$ over $\mathbb{R}_n[X]$.
Determine explicitly $\Pi_2$ when $f$ is the function defined for all $t \in [0,1]$ by $f(t) = \frac{1}{1+t^2}$.

grandes-ecoles 2011 QIV.A.3 Definite Integral Evaluation (Computational) View

For $n \in \mathbb{N}^*$ and $(i,j) \in \llbracket 1, n \rrbracket^2$, we denote by $h_{i,j}^{(-1,n)}$ the coefficient at position $(i,j)$ of the matrix $H_n^{-1}$ and we denote by $s_n$ the sum of the coefficients of the matrix $H_n^{-1}$, that is: $$s_n = \sum_{1 \leqslant i,j \leqslant n} h_{i,j}^{(-1,n)}$$ We define, for all $n \in \mathbb{N}^*$, the polynomial $S_n$ by: $S_n = a_0^{(n)} + a_1^{(n)} X + \cdots + a_{n-1}^{(n)} X^{n-1}$, where $\left(a_p^{(n)}\right)_{0 \leqslant p \leqslant n-1}$ is the unique solution of the system in IV.A.2.
Show that $$\forall Q = \alpha_0 + \alpha_1 X + \cdots + \alpha_{n-1} X^{n-1} \in \mathbb{R}_{n-1}[X], \quad \langle S_n, Q \rangle = \sum_{p=0}^{n-1} \alpha_p$$

grandes-ecoles 2011 Q15 Definite Integral Evaluation (Computational) View

We consider a rectangle $]a,b[ \times ]c,d[$ of the plane $\mathbb{R}^{2}$, with $a < b$ and $c < d$. Calculate the real number $V(]a,b[ \times ]c,d[)$. What does it represent? (One may use functions of the type $$(x,y) \mapsto f(x,y) = \phi(x)\varphi(y)$$ where $\phi$ and $\varphi$ are well-chosen continuous and piecewise affine functions).

grandes-ecoles 2011 Q16 Integral Inequalities and Limit of Integral Sequences View

Let $\mathcal{A}$ and $\mathcal{B}$ be two open bounded non-empty subsets of $\mathbb{R}^{2}$ and $\lambda \in ]0,1[$. Verify that $\lambda\mathcal{A} + (1-\lambda)\mathcal{B}$ is an open bounded subset of $\mathbb{R}^{2}$. Then show that $$V(\lambda\mathcal{A} + (1-\lambda)\mathcal{B}) \geq V(\mathcal{A})^{\lambda} V(\mathcal{B})^{1-\lambda}$$ To prove this inequality, you will use the following admitted result. For all $f \in C(\mathcal{A})$ and $g \in C(\mathcal{B})$, the function $h$ determined by: $$\forall Z \in \mathbb{R}^{2},\quad h(Z) = \sup\left\{f(X)^{\lambda} g(Y)^{1-\lambda} \,/\, X, Y \in \mathbb{R}^{2},\, Z = \lambda X + (1-\lambda) Y\right\}$$ defines a continuous function on $\mathbb{R}^{2}$.

grandes-ecoles 2011 Q17 Integral Inequalities and Limit of Integral Sequences View

Let $u : \mathbb{R}^{2} \rightarrow ]0,+\infty[$ be a continuous and log-concave function in the sense of Part II. Prove that the preceding inequality remains true if we replace the application $V$ by the application $\gamma$ defined for all open bounded (non-empty) subsets $\mathcal{A}$ of $\mathbb{R}^{2}$ by $$\gamma(\mathcal{A}) = \sup_{f \in C(\mathcal{A})} \iint_{\mathbb{R}^{2}} f(x,y)\,u(x,y)\,dx\,dy$$

grandes-ecoles 2012 QII.E.1 Piecewise/Periodic Function Integration View

Let $g : [ 0,1 ] \rightarrow \mathbb { R }$ be the function such that $g ( x ) = 1 / x$ if $x \geqslant \mathrm { e } ^ { - 1 }$ and $g ( x ) = 0$ otherwise. We fix a real $\varepsilon \in ] 0 , \mathrm { e } ^ { - 1 } [$. We define two continuous applications $g ^ { + } , g ^ { - } : [ 0,1 ] \rightarrow \mathbb { R }$ as follows:

$g ^ { + }$ is affine on $\left[ \mathrm { e } ^ { - 1 } - \varepsilon , \mathrm { e } ^ { - 1 } \right]$ and coincides with $g$ on $\left[ 0 , \mathrm { e } ^ { - 1 } - \varepsilon \right] \cup \left[ \mathrm { e } ^ { - 1 } , 1 \right]$;
$g ^ { - }$ is affine on $\left[ \mathrm { e } ^ { - 1 } , \mathrm { e } ^ { - 1 } + \varepsilon \right]$ and coincides with $g$ on $\left[ 0 , \mathrm { e } ^ { - 1 } \left[ \cup \left[ \mathrm { e } ^ { - 1 } + \varepsilon , 1 \right] \right. \right.$.

Calculate $\int _ { 0 } ^ { 1 } g ^ { + } ( t ) \mathrm { d } t$ and $\int _ { 0 } ^ { 1 } g ^ { - } ( t ) \mathrm { d } t$.

grandes-ecoles 2013 QIII.A.2 Convergence and Evaluation of Improper Integrals View

Study the convergence of the improper integrals $$\int _ { 0 } ^ { \pi } \ln ( \sin \theta ) d \theta \quad \int _ { 0 } ^ { \pi } \ln ( 1 - \cos \theta ) d \theta \quad \int _ { 0 } ^ { \pi } \ln ( 1 + \cos \theta ) d \theta$$
Deduce that, for all $x \in \mathbb { R }$, the integral $\int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$ converges.

grandes-ecoles 2013 QIII.A.3 Integral Inequalities and Limit of Integral Sequences View

Show that, as $x$ tends to $+ \infty$, $$2 \pi \ln ( x ) - \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$$ has a limit, which one will determine.

grandes-ecoles 2013 QIII.A.4 Integral Equation with Symmetry or Substitution View

Show that $x \mapsto \int _ { 0 } ^ { \pi } \ln \left( x ^ { 2 } - 2 x \cos \theta + 1 \right) \mathrm { d } \theta$ is an even function of the variable $x \in \mathbb { R }$.

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LFM Pure and Mechanics

Indices and Surds 107

Exponential Functions 100

Laws of Logarithms 200

Exponential Equations & Modelling 58

Arithmetic Sequences and Series 271

Geometric Sequences and Series 203

Differentiation from First Principles 37

Tangents, normals and gradients 93

Stationary points and optimisation 384

Applied differentiation 164

Indefinite & Definite Integrals 533

Areas Between Curves 83

Areas by integration 109

Volumes of Revolution 61

Numerical integration 32

Vectors Introduction & 2D 204

Vectors 3D & Lines 233

Travel graphs 11

SUVAT in 2D & Gravity 6

Constant acceleration (SUVAT) 32

Variable acceleration (1D) 40

Variable acceleration (vectors) 26

Forces, equilibrium and resultants 9

Newton's laws and connected particles 18

Pulley systems 5

Motion on a slope 4

Friction 10

Momentum and Collisions 15

Moments 13

Parametric curves and Cartesian conversion 11

Parametric differentiation 15

Parametric integration 4

Projectiles 40