LFM Stats And Pure

gaokao 2022 Q5 5 marks Linear Programming (Optimize Objective over Linear Constraints) View

If $x , y$ satisfy the constraints $\left\{ \begin{array} { l } x + y \geqslant 2 , \\ x + 2 y \leqslant 4 , \end{array} \right.$ then the maximum value of $z = 2 x - y$ is
A. $- 2$
B. 4
C. 8
D. 12

gaokao 2023 Q14 5 marks Linear Programming (Optimize Objective over Linear Constraints) View

Let $x , y$ satisfy the constraints $\left\{ \begin{array}{l} -2x + 3y \leqslant 3 \\ 3x - 2y \leqslant 3 \\ x + y = 1 \end{array} \right.$ . Let $z = 3x + 2y$ . The maximum value of $z$ is $\_\_\_\_$ .

gaokao 2023 Q14 Linear Programming (Optimize Objective over Linear Constraints) View

If $x , y$ satisfy the constraints $\left\{ \begin{array} { l } x - 3 y \leqslant - 1 \\ x + 2 y \leqslant 9 \\ 3 x + y \geqslant 7 \end{array} \right.$, then the maximum value of $z = 2 x - y$ is \_\_\_\_

gaokao 2023 Q23 10 marks Absolute Value Inequality View

[Elective 4-5: Inequalities]
Given $f(x) = 2|x - a| - a , \ a > 0$ .
(1) Solve the inequality $f(x) < x$ ;
(2) If the area enclosed by $y = f(x)$ and the coordinate axes is 2 , find $a$ .

gaokao 2025 Q4 5 marks Solve Polynomial/Rational Inequality for Solution Set View

The solution set of the inequality $\frac{x-4}{x-1} < 2$ is ( )
A. $\{x \mid -2 \leq x \leq 1\}$
B. $\{x \mid x < -2\}$
C. $\{x \mid -2 \leq x < 1\}$
D. $\{x \mid x > 1\}$

grandes-ecoles 2013 QIII.A.1 Prove or Verify an Algebraic Inequality (AM-GM, Cauchy-Schwarz, etc.) View

Show that if $x$ is a real number different from 1 and from $-1$, then $x ^ { 2 } - 2 x \cos \theta + 1 > 0$ for all $\theta \in \mathbb { R }$.

grandes-ecoles 2017 Q24 Prove or Verify an Algebraic Inequality (AM-GM, Cauchy-Schwarz, etc.) 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 $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ We set $$\mathbb{R}_{2m-1}^0[X] = \{P \in \mathbb{R}_{2m-1}[X] \mid P(-1) = 0 \text{ and } P(1) = 0\}$$
Show that $$\forall P \in \mathbb{R}_{2m-1}^0[X], \quad (P \mid P) \leq 4 (P' \mid P')$$ with strict inequality if $P$ is non-zero.

grandes-ecoles 2018 Q11 Prove or Verify an Algebraic Inequality (AM-GM, Cauchy-Schwarz, etc.) View

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative real $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Let $\delta$ be a real such that $0 \leqslant |\delta| \leqslant \sqrt{\frac{a}{b}}$. Show that, for all real $t$,
$$-b(t - |\delta|)^{2} \leqslant a - \frac{1}{2}bt^{2}$$

grandes-ecoles 2018 Q11 Prove or Verify an Algebraic Inequality (AM-GM, Cauchy-Schwarz, etc.) View

Let $X : \Omega \rightarrow \mathbb{R}$ be a real-valued random variable. We assume that there exist two strictly positive reals $a$ and $b$ such that, for all non-negative reals $t$,
$$\mathbb{P}(|X| \geqslant t) \leqslant a \exp(-bt^{2})$$
Let $\delta$ be a real such that $0 \leqslant |\delta| \leqslant \sqrt{\frac{a}{b}}$. Show that, for all reals $t$,
$$-b(t - |\delta|)^{2} \leqslant a - \frac{1}{2}bt^{2}$$

grandes-ecoles 2018 Q20 Prove or Verify an Algebraic Inequality (AM-GM, Cauchy-Schwarz, etc.) View

Show that $\lambda ^ { \alpha } \leq 1 + \lambda$ for all real $\lambda > 0$ and deduce the inequality:
$$\left| x - \frac { k } { n } \right| ^ { \alpha } \leq n ^ { - \alpha / 2 } \left( 1 + \sqrt { n } \left| x - \frac { k } { n } \right| \right)$$
for all $x \in ]0,1[ , n \in \mathbb { N } ^ { * }$ and $k \in \{ 0 , \ldots , n \}$.

grandes-ecoles 2025 Q24 Prove or Verify an Algebraic Inequality (AM-GM, Cauchy-Schwarz, etc.) View

We consider a convex function $f \in \mathcal{C}(\mathbb{R})$, admitting a minimizer $x_* \in \mathbb{R}$, and $\tau > 0$. The sequence $(x_n)_{n \in \mathbb{N}}$ is defined by $x_{n+1} := p_f(x_n)$. Show that for all $M, N \in \mathbb{N}$ $$|x_N - x_M| \leq \sqrt{2\tau|N-M|}\sqrt{\left|f(x_M) - f(x_N)\right|}$$

isi-entrance 2012 Q28 Set Operations Using Inequality-Defined Sets View

Let $A = \{(x,y) : x^4 + y^2 \leq 1\}$ and $B = \{(x,y) : x^6 + y^4 \leq 1\}$. Which of the following is true?
(A) $A = B$
(B) $A \subset B$ (A is a proper subset of B)
(C) $B \subset A$ (B is a proper subset of A)
(D) Neither $A \subset B$ nor $B \subset A$

isi-entrance 2013 Q8 4 marks Solve Polynomial/Rational Inequality for Solution Set View

The set of all real numbers $x$ satisfying the inequality $x ^ { 3 } ( x + 1 ) ( x - 2 ) \geq 0$ is
(A) the interval $[ 2 , \infty )$
(B) the interval $[ 0 , \infty )$
(C) the interval $[ - 1 , \infty )$
(D) none of the above

isi-entrance 2014 Q1 Prove or Verify an Algebraic Inequality (AM-GM, Cauchy-Schwarz, etc.) View

If $a, b, c, d$ are real numbers such that $a - b^2 \geq 1/4$, $b - c^2 \geq 1/4$, $c - d^2 \geq 1/4$, $d - a^2 \geq 1/4$, find the number of solutions $(a, b, c, d)$.
(A) 0 (B) 1 (C) 2 (D) Infinitely many

isi-entrance 2015 Q7 4 marks Solve Polynomial/Rational Inequality for Solution Set View

The set of all real numbers $x$ such that $x ^ { 3 } ( x + 1 ) ( x - 2 ) \geq 0$ is:
(a) the interval $2 \leq x < \infty$
(b) the interval $0 \leq x < \infty$
(c) the interval $- 1 \leq x < \infty$
(d) none of the above.

isi-entrance 2015 Q7 4 marks Solve Polynomial/Rational Inequality for Solution Set View

The set of all real numbers $x$ such that $x ^ { 3 } ( x + 1 ) ( x - 2 ) \geq 0$ is:
(a) the interval $2 \leq x < \infty$
(b) the interval $0 \leq x < \infty$
(c) the interval $- 1 \leq x < \infty$
(d) none of the above.

isi-entrance 2016 Q8 4 marks Solve Polynomial/Rational Inequality for Solution Set View

The set of all real numbers $x$ satisfying the inequality $x ^ { 3 } ( x + 1 ) ( x - 2 ) \geq 0$ is
(A) the interval $[ 2 , \infty )$
(B) the interval $[ 0 , \infty )$
(C) the interval $[ - 1 , \infty )$
(D) none of the above

isi-entrance 2016 Q8 4 marks Solve Polynomial/Rational Inequality for Solution Set View

The set of all real numbers $x$ satisfying the inequality $x ^ { 3 } ( x + 1 ) ( x - 2 ) \geq 0$ is
(A) the interval $[ 2 , \infty )$
(B) the interval $[ 0 , \infty )$
(C) the interval $[ - 1 , \infty )$
(D) none of the above

isi-entrance 2022 Q27 Optimization Subject to an Algebraic Constraint View

If $x _ { 1 } > x _ { 2 } > \cdots > x _ { 10 }$ are real numbers, what is the least possible value of $$\left( \frac { x _ { 1 } - x _ { 10 } } { x _ { 1 } - x _ { 2 } } \right) \left( \frac { x _ { 1 } - x _ { 10 } } { x _ { 2 } - x _ { 3 } } \right) \cdots \left( \frac { x _ { 1 } - x _ { 10 } } { x _ { 9 } - x _ { 10 } } \right) ?$$ (A) $10 ^ { 10 }$
(B) $10 ^ { 9 }$
(C) $9 ^ { 9 }$
(D) $9 ^ { 10 }$

isi-entrance 2022 Q30 Optimization Subject to an Algebraic Constraint View

The range of values that the function $$f ( x ) = \frac { x ^ { 2 } + 2 x + 4 } { 2 x ^ { 2 } + 4 x + 9 }$$ takes as $x$ varies over all real numbers in the domain of $f$ is:
(A) $\frac { 3 } { 7 } < f ( x ) \leq \frac { 1 } { 2 }$
(B) $\frac { 3 } { 7 } \leq f ( x ) < \frac { 1 } { 2 }$
(C) $\frac { 3 } { 7 } < f ( x ) \leq \frac { 4 } { 9 }$
(D) $\frac { 3 } { 7 } \leq f ( x ) \leq \frac { 1 } { 2 }$

isi-entrance 2023 Q1 Solve Polynomial/Rational Inequality for Solution Set View

For a real number $x$, $$x ^ { 3 } - 7 x + 6 > 0$$ if and only if
(A) $x > 2$.
(B) $- 3 < x < 1$.
(C) $x < - 3$ or $1 < x < 2$.
(D) $- 3 < x < 1$ or $x > 2$.

isi-entrance 2024 Q13 Integer Solutions of an Inequality View

The number of elements in the set $$\left\{x : 0 \leqslant x \leqslant 2,\, \left|x - x^5\right| = \left|x^5 - x^6\right|\right\}$$ is
(A) 2
(B) 3
(C) 4
(D) 5

isi-entrance 2024 Q17 Linear Programming (Optimize Objective over Linear Constraints) View

Let $P = \{(x, y) : x + 1 \geqslant y,\, x \geqslant -1,\, y \geqslant 2x\}$. Then the minimum value of $(x + y)$ where $(x, y)$ varies over the set $P$ is
(A) $-1$
(B) $-3$
(C) $3$
(D) $0$

isi-entrance 2024 Q20 Absolute Value Inequality View

The real number $x$ satisfies $$\frac{|x|^2 - |x| - 2}{2|x| - |x|^2 - 2} > 2$$ if and only if $x$ belongs to
(A) $(-2, -1) \cup (1, 2)$
(B) $(-2/3, 0) \cup (0, 2/3)$
(C) $(-1, -2/3) \cup (2/3, 1)$
(D) $(-1, 0) \cup (0, 1)$

jee-main 2013 Q61 Solve Polynomial/Rational Inequality for Solution Set View

The least integral value $\alpha$ of $x$ such that $\frac { x - 5 } { x ^ { 2 } + 5 x - 14 } > 0$, satisfies :
(1) $\alpha ^ { 2 } + 3 \alpha - 4 = 0$
(2) $\alpha ^ { 2 } - 5 \alpha + 4 = 0$
(3) $\alpha ^ { 2 } - 7 \alpha + 6 = 0$
(4) $\alpha ^ { 2 } + 5 \alpha - 6 = 0$

1
2
3
4
5
6
7
8

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