LFM Stats And Pure

bac-s-maths 2019 Q3 5 marks True/False or Multiple-Statement Verification View

Specify whether each of the following statements is true or false by justifying your answer.

Let $m$ be a real number and let the equation $( E )$ : $2 z ^ { 2 } + ( m - 5 ) z + m = 0$. a. Statement 1 : ``For $m = 4$, the equation ( $E$ ) admits two real solutions.'' b. Statement 2 : ``There exists only one value of $m$ such that ( $E$ ) admits two complex solutions that are pure imaginary numbers.''
In the complex plane, we consider the set $S$ of points $M$ with affixe $z$ satisfying: $$| z - 6 | = | z + 5 i |$$ Statement 3 : ``The set $S$ is a circle.''
We equip space with an orthonormal coordinate system ( $\mathrm { O } ; \vec { \imath } , \vec { \jmath } , \vec { k }$ ). We denote $d$ the line with parametric representation: $$d : \left\{ \begin{aligned} x & = - 1 + t \\ y & = 2 - t \quad t \in \mathbb { R } . \\ z & = 3 + t \end{aligned} \right.$$ We denote $d ^ { \prime }$ the line passing through the point $\mathrm { B } ( 4 ; 4 ; - 6 )$ and with direction vector $\vec { v } ( 5 ; 2 ; - 9 )$. Statement 4 : ``The lines $d$ and $d ^ { \prime }$ are coplanar.''
We consider the cube ABCDEFGH. Statement 5 : ``The vector $\overrightarrow { \mathrm { DE } }$ is a normal vector to the plane (ABG).''

bac-s-maths 2025 Q4A Probability involving discriminant conditions View

Consider the set of non-zero relative integers between $-30$ and $30$; this set can be written as follows: $\{-30; -29; -28; \ldots -1; 1; \ldots; 28; 29; 30\}$. It contains 60 elements. We choose from this set successively and without replacement a relative integer $a$ then a relative integer $c$.

How many different pairs $(a; c)$ can we obtain?

Consider the event $M$: ``the equation $ax^2 + 2x + c = 0$ has two distinct real solutions'', where $a$ and $c$ are the relative integers previously chosen.

Show that event $M$ occurs if and only if $ac < 1$.
Explain why the opposite event $\bar{M}$ contains 1740 outcomes.
What is the probability of event $M$? Round the result to $10^{-2}$.

cmi-entrance 2011 QA6 3 marks Nature of roots given coefficient constraints View

The equation $x ^ { 2 } + b x + c = 0$ has nonzero real coefficients satisfying $b ^ { 2 } > 4 c$. Moreover, exactly one of $b$ and $c$ is irrational. Consider the solutions $p$ and $q$ of this equation.
(A) Both $p$ and $q$ must be rational.
(B) Both $p$ and $q$ must be irrational.
(C) One of $p$ and $q$ is rational and the other irrational.
(D) We cannot conclude anything about rationality of $p$ and $q$ unless we know $b$ and $c$.

csat-suneung 2010 Q8 3 marks Counting solutions or configurations satisfying a quadratic system View

For a real number $a$, let $f ( a )$ be the number of elements in the set $$\left\{ x \mid a x ^ { 2 } + 2 ( a - 2 ) x - ( a - 2 ) = 0 , x \text { is a real number } \right\}$$ Which of the following statements in are correct? [3 points]
Remarks ᄀ. $\lim _ { a \rightarrow 0 } f ( a ) = f ( 0 )$ ㄴ. There are 2 real numbers $c$ such that $\lim _ { a \rightarrow c + 0 } f ( a ) \neq \lim _ { a \rightarrow c - 0 } f ( a )$. ㄷ. The function $f ( a )$ is discontinuous at 3 points.
(1) ᄂ
(2) ᄃ
(3) ᄀ, ᄂ
(4) ㄴ,ㄷ
(5) ᄀ, ᄂ, ᄃ

csat-suneung 2016 Q10 3 marks Parameter range for specific root conditions (location/count) View

For a sequence $\left\{ a _ { n } \right\}$, the curve $y = x ^ { 2 } - ( n + 1 ) x + a _ { n }$ intersects the $x$-axis, and the curve $y = x ^ { 2 } - n x + a _ { n }$ does not intersect the $x$-axis. What is the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n ^ { 2 } }$? [3 points]
(1) $\frac { 1 } { 20 }$
(2) $\frac { 1 } { 10 }$
(3) $\frac { 3 } { 20 }$
(4) $\frac { 1 } { 5 }$
(5) $\frac { 1 } { 4 }$

csat-suneung 2017 Q18 4 marks Determining quadratic function from given conditions View

A quadratic function $f ( x )$ with leading coefficient 1 satisfies $$\lim _ { x \rightarrow a } \frac { f ( x ) - ( x - a ) } { f ( x ) + ( x - a ) } = \frac { 3 } { 5 }$$ When the two roots of the equation $f ( x ) = 0$ are $\alpha$ and $\beta$, what is the value of $| \alpha - \beta |$? (Here, $a$ is a constant.) [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

csat-suneung 2019 Q11 3 marks Parameter range for no real roots (positive definite) View

When $0 \leq \theta < 2 \pi$, the range of all values of $\theta$ such that the quadratic equation in $x$ $$6 x ^ { 2 } + ( 4 \cos \theta ) x + \sin \theta = 0$$ has no real roots is $\alpha < \theta < \beta$. What is the value of $3 \alpha + \beta$? [3 points]
(1) $\frac { 5 } { 6 } \pi$
(2) $\pi$
(3) $\frac { 7 } { 6 } \pi$
(4) $\frac { 4 } { 3 } \pi$
(5) $\frac { 3 } { 2 } \pi$

csat-suneung 2020 Q6 3 marks Sufficient/necessary condition or logical relationship involving a quadratic View

For two conditions on real number $x$: $$\begin{aligned} & p : x = a , \\ & q : 3 x ^ { 2 } - a x - 32 = 0 \end{aligned}$$ What is the value of positive $a$ such that $p$ is a sufficient condition for $q$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

gaokao 2015 Q8 Finding roots or coefficients of a quadratic using Vieta's relations View

8. If $a$ and $b$ are two distinct zeros of the function $f ( x ) = x ^ { 2 } - p x + q$ ($p > 0$, $q > 0$), and the three numbers $a$, $b$, and $-2$ can be arranged to form an arithmetic sequence, and can also be arranged to form a geometric sequence, then the value of $p + q$ equals
A. 6
B. 7
C. 8
D. 9

gaokao 2015 Q12 Parameter range for specific root conditions (location/count) View

12. For the quadratic function $f ( x ) = a x ^ { 2 } + b x + c$ (where $a$ is a non-zero constant), four students each give a conclusion. Exactly one conclusion is wrong. The wrong conclusion is
A. $-1$ is a zero of $f ( x )$
B. $1$ is an extremum point of $f ( x )$
C. $3$ is an extremum value of $f ( x )$
D. The point $( 2,8 )$ lies on the curve $y = f ( x )$

II. Fill in the Blanks

gaokao 2015 Q15 5 marks Probability involving discriminant conditions View

A number $p$ is randomly chosen from the interval $[ 0,5 ]$. The probability that the equation $x ^ { 2 } + 2 p x + 3 p - 2 = 0$ has two negative roots is $\_\_\_\_$ .

grandes-ecoles 2013 QIII.A.3 Compute eigenvalues of a given matrix View

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, with $\varphi_A(x,y)$ the determinant of $A(x,y) = \left(\begin{array}{cc} a-x & b-y \\ c+y & d-x \end{array}\right)$, what do the solutions of the equation $\varphi_A(x,0) = 0$ represent for $A$? Specify the number of real eigenvalues of $A$ according to the value of $\Delta_A = (a-d)^2 + 4bc$.

grandes-ecoles 2014 QIII.C.6 Location and bounds on roots View

We assume $\alpha = 1$. We denote $T_n$ the unique polynomial eigenvector of $\varphi_1$ of degree $n$, of norm 1 (with respect to $S_1$) and with positive leading coefficient. For $n \in \mathbb{N}^*$, determine the roots of $T_n$.

iran-konkur 2014 Q148 Probability involving discriminant conditions View

148- In the equation $ax^2 + bx = 5$, coefficient $a$ is chosen randomly from the interval $[1,3]$ and coefficient $b$ is chosen randomly from the interval $[-3, 0]$. With which probability is the set of solutions of this equation more than $\dfrac{2}{3}$?
$$\frac{4}{9} \ (1) \hspace{2cm} \frac{5}{9} \ (2) \hspace{2cm} \frac{7}{12} \ (3) \hspace{2cm} \frac{5}{6} \ (4)$$

iran-konkur 2019 Q103 Parameter range for specific root conditions (location/count) View

103. For which values of $m$, the equation $0 = (2m-1)x^2 + 6x + m - 2 = 0$ has two real roots?
(1) $-2 < m < 2.5$ (2) $-2 < m < 3.5$
(3) $-1 < m < 3.5$ (4) $-1 < m < 2.5$

iran-konkur 2022 Q106 Root relationships and Vieta's formulas View

106-- $\alpha$ and $\beta$ are roots of the equation $x^2 + 6x + a = 0$. If $0 < \alpha < \beta < 0$ and $12\sqrt{2} + 85 = 12\sqrt{2} + 85$, and $3\alpha^2 + 2\beta^2 = 12\sqrt{2} + 85$, what is the value of $a$?
(1) $1$ (2) $\dfrac{13}{4}$ (3) $\dfrac{21}{5}$ (4) $2$

isi-entrance 2012 Q20 Number of Solutions / Roots via Curve Analysis View

Let $f(x) = x^4 + x^2 + x - 1$. Which of the following is true?
(A) $f$ has exactly two real roots
(B) $f$ has no real roots
(C) $f$ has four real roots
(D) $f$ has exactly two real roots, one of which is $-1$

isi-entrance 2015 Q18 4 marks Parameter range for no real roots (positive definite) View

The set of values of $m$ for which $m x ^ { 2 } - 6 m x + 5 m + 1 > 0$ for all real $x$ is
(a) $m < \frac { 1 } { 4 }$
(b) $m \geq 0$
(c) $0 \leq m \leq \frac { 1 } { 4 }$
(d) $0 \leq m < \frac { 1 } { 4 }$.

isi-entrance 2016 Q66 4 marks Parameter range for no real roots (positive definite) View

The set of values of $m$ for which $mx^2 - 6mx + 5m + 1 > 0$ for all real $x$ is
(A) $m < \frac{1}{4}$
(B) $m \geq 0$
(C) $0 \leq m \leq \frac{1}{4}$
(D) $0 \leq m < \frac{1}{4}$

isi-entrance 2017 Q1 Nature of roots given coefficient constraints View

Consider a quadratic equation $ax^2 + 2bx + c = 0$, where $a, b$ and $c$ are positive real numbers. If the equation has no real roots, then which of the following is true?
(A) $a, b, c$ cannot be in AP or HP, but can be in GP.
(B) $a, b, c$ cannot be in GP or HP, but can be in AP.
(C) $a, b, c$ cannot be in AP or GP, but can be in HP.
(D) $a, b, c$ cannot be in AP, GP or HP.

isi-entrance 2018 Q28 Condition for repeated (equal/double) roots View

For which values of $\theta$, with $0 < \theta < \pi / 2$, does the quadratic polynomial in $t$ given by $t ^ { 2 } + 4 t \cos \theta + \cot \theta$ have repeated roots?
(A) $\frac { \pi } { 6 }$ or $\frac { 5 \pi } { 18 }$
(B) $\frac { \pi } { 6 }$ or $\frac { 5 \pi } { 12 }$
(C) $\frac { \pi } { 12 }$ or $\frac { 5 \pi } { 18 }$
(D) $\frac { \pi } { 12 }$ or $\frac { 5 \pi } { 12 }$

isi-entrance 2020 Q2 Condition for repeated (equal/double) roots View

Let $a$ be a fixed real number. Consider the equation
$$(x + 2)^{2}(x + 7)^{2} + a = 0, \quad x \in \mathbb{R},$$
where $\mathbb{R}$ is the set of real numbers. For what values of $a$, will the equation have exactly one double-root?

isi-entrance 2020 Q19 Nature of roots given coefficient constraints View

If $a , b , c$ are distinct odd natural numbers, then the number of rational roots of the polynomial $a x ^ { 2 } + b x + c$
(A) must be 0 .
(B) must be 1 .
(C) must be 2 .
(D) cannot be determined from the given data.

isi-entrance 2021 Q15 Count or characterize roots using extremum values View

The polynomial $x ^ { 4 } + 4 x + c = 0$ has at least one real root if and only if
(A) $c < 2$.
(B) $c \leq 2$.
(C) $c < 3$.
(D) $c \leq 3$.

isi-entrance 2021 Q21 Multiplicity and derivative analysis of roots View

The number of different values of $a$ for which the equation $x ^ { 3 } - x + a = 0$ has two identical real roots is
(A) 0 .
(B) 1 .
(C) 2 .
(D) 3 .

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LFM Stats And Pure

Completing the square and sketching 80

Inequalities 215

Discriminant and conditions for roots 106

II. Fill in the Blanks

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