LFM Stats And Pure

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 2010 Q19 3 marks Solving an equation via substitution to reduce to quadratic form View

Find the product of all real roots of the irrational equation $\sqrt { x ^ { 2 } - 7 x + 15 } = x ^ { 2 } - 7 x + 9$. [3 points]

csat-suneung 2011 Q4 3 marks Solving an equation via substitution to reduce to quadratic form View

For the irrational equation $$\sqrt { 4 x ^ { 2 } - 5 x + 7 } - 4 x ^ { 2 } + 5 x = 1$$ What is the product of all real roots? [3 points]
(1) $- \frac { 1 } { 2 }$
(2) $- \frac { 3 } { 2 }$
(3) $- \frac { 5 } { 2 }$
(4) $- \frac { 7 } { 2 }$
(5) $- \frac { 9 } { 2 }$

csat-suneung 2014 Q24 3 marks Solving an equation via substitution to reduce to quadratic form View

Find the product of all real roots of the irrational equation $\sqrt { 2 x ^ { 2 } - 6 x } = x ^ { 2 } - 3 x - 4$, and call it $k$. Find the value of $k ^ { 2 }$. [3 points]

csat-suneung 2015 Q13 3 marks Evaluating an algebraic expression given a constraint View

For the function $f ( x ) = x ( x + 1 ) ( x - 4 )$, answer the following. For the matrix $A = \left( \begin{array} { l l } 2 & 1 \\ 0 & 3 \end{array} \right)$, what is the sum of all constant values $a$ that satisfy $A \binom { 0 } { f ( a ) } = \binom { 0 } { 0 }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

csat-suneung 2015 Q24 3 marks Solving an equation via substitution to reduce to quadratic form View

For the irrational equation $x ^ { 2 } - 6 x - \sqrt { x ^ { 2 } - 6 x - 1 } = 3$, let $k$ be the product of all real roots. Find the value of $k ^ { 2 }$. [3 points]

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 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 Q14 5 marks Optimization or extremal value of an expression via completing the square View

Let $a , b > 0 , a + b = 5$. The maximum value of $\sqrt { a + 1 } + \sqrt { b + 3 }$ is $\_\_\_\_$ .

gaokao 2015 Q20 15 marks Determining quadratic function from given conditions View

20. (15 points) Let the function $f ( x ) = x ^ { 2 } + a x + b , ( a , b \in R )$ .
(1) When $b = \frac { a ^ { 2 } } { 4 } + 1$ , find the expression for the minimum value $g ( a )$ of the function $f ( x )$ on $[ - 1,1 ]$ ;
(2) Given that the function $f ( x )$ has a zero on $[ - 1,1 ]$ , $0 \leq b - 2 a \leq 1$ , find the range of values for $b$ .

gaokao 2020 Q3 5 marks Finding a ratio or relationship between variables from an equation View

The Great Pyramid of Egypt is one of the ancient wonders of the world. Its shape can be viewed as a regular square pyramid. The area of a square with side length equal to the height of the pyramid equals the area of one lateral triangular face of the pyramid. The ratio of the height of a lateral triangular face to the side length of the base square is
A. $\frac { \sqrt { 5 } - 1 } { 4 }$
B. $\frac { \sqrt { 5 } - 1 } { 2 }$
C. $\frac { \sqrt { 5 } + 1 } { 4 }$
D. $\frac { \sqrt { 5 } + 1 } { 2 }$

gaokao 2020 Q3 5 marks Finding a ratio or relationship between variables from an equation View

The Great Pyramid of Khufu in Egypt is one of the ancient wonders of the world. Its shape can be viewed as a regular square pyramid. The area of a square with side length equal to the height of the pyramid equals the area of one lateral triangular face of the pyramid. Then the ratio of the height of the lateral triangle to the base of the square is
A. $\frac { \sqrt { 5 } - 1 } { 4 }$
B. $\frac { \sqrt { 5 } - 1 } { 2 }$
C. $\frac { \sqrt { 5 } + 1 } { 4 }$
D. $\frac { \sqrt { 5 } + 1 } { 2 }$

isi-entrance 2026 QB5 Counting solutions or configurations satisfying a quadratic system View

Let $a , b$ and $c$ be three real numbers. Then the equation $\frac { 1 } { x - a } + \frac { 1 } { x - b } + \frac { 1 } { x - c } = 0$
(A) always have real roots.
(B) can have real or complex roots depending on the values of $a , b$ and $c$.
(C) always have real and equal roots.
(D) always have real roots, which are not necessarily equal.

jee-advanced 2014 Q51 Evaluating an algebraic expression given a constraint View

Let $a, b, c$ be positive integers such that $\frac{b}{a}$ is an integer. If $a, b, c$ are in geometric progression and the arithmetic mean of $a, b, c$ is $b + 2$, then the value of $$\frac{a^2 + a - 14}{a + 1}$$ is

jee-advanced 2018 Q8 Counting solutions or configurations satisfying a quadratic system View

The number of 5 digit numbers which are divisible by 4, with digits from the set $\{ 1,2,3,4,5 \}$ and the repetition of digits is allowed, is $\_\_\_\_$.

jee-advanced 2020 Q1 Finding roots or coefficients of a quadratic using Vieta's relations View

Suppose $a , b$ denote the distinct real roots of the quadratic polynomial $x ^ { 2 } + 20 x - 2020$ and suppose $c , d$ denote the distinct complex roots of the quadratic polynomial $x ^ { 2 } - 20 x + 2020$. Then the value of
$$a c ( a - c ) + a d ( a - d ) + b c ( b - c ) + b d ( b - d )$$
is
(A) 0
(B) 8000
(C) 8080
(D) 16000

jee-main 2012 Q61 Determining quadratic function from given conditions View

If $a , b , c \in \mathrm { R }$ and 1 is a root of equation $a x ^ { 2 } + b x + c = 0$, then the curve $y = 4 a x ^ { 2 } + 3 b x + 2 c , a \neq 0$ intersect $x$-axis at
(1) two distinct points whose coordinates are always rational numbers
(2) no point
(3) exactly two distinct points
(4) exactly one point

jee-main 2012 Q62 Optimization or extremal value of an expression via completing the square View

If the sum of the square of the roots of the equation $x^{2} - (\sin\alpha - 2)x - (1+\sin\alpha) = 0$ is least, then $\alpha$ is equal to
(1) $\frac{\pi}{6}$
(2) $\frac{\pi}{4}$
(3) $\frac{\pi}{3}$
(4) $\frac{\pi}{2}$

jee-main 2014 Q61 Optimization or extremal value of an expression via completing the square View

If $a \in R$ and the equation $- 3 ( x - [ x ] ) ^ { 2 } + 2 ( x - [ x ] ) + a ^ { 2 } = 0$ (where $[ x ]$ denotes the greatest integer $\leq x )$ has no integral solution, then all possible values of $a$ lie in the interval
(1) $( - 2 , - 1 )$
(2) $( - \infty , - 2 ) \cup ( 2 , \infty )$
(3) $( - 1,0 ) \cup ( 0,1 )$
(4) $( 1,2 )$

jee-main 2014 Q61 Finding roots or coefficients of a quadratic using Vieta's relations View

If $\frac { 1 } { \sqrt { \alpha } } , \frac { 1 } { \sqrt { \beta } }$ are the roots of the equation $a x ^ { 2 } + b x + 1 = 0 , ( a \neq 0 , a , b \in R )$, then the equation $x \left( x + b ^ { 3 } \right) + \left( a ^ { 3 } - 3 a b x \right) = 0$ has roots:
(1) $\sqrt { \alpha \beta }$ and $\alpha \beta$
(2) $\alpha ^ { - \frac { 3 } { 2 } }$ and $\beta ^ { - \frac { 3 } { 2 } }$
(3) $\alpha \beta ^ { \frac { 1 } { 2 } }$ and $\alpha ^ { \frac { 1 } { 2 } } \beta$
(4) $\alpha ^ { \frac { 3 } { 2 } }$ and $\beta ^ { \frac { 3 } { 2 } }$

jee-main 2014 Q61 Solving an equation via substitution to reduce to quadratic form View

The equation $\sqrt { 3 x ^ { 2 } + x + 5 } = x - 3$, where $x$ is real, has
(1) no solution
(2) exactly four solutions
(3) exactly one solution
(4) exactly two solutions

jee-main 2014 Q62 Finding roots or coefficients of a quadratic using Vieta's relations View

If equations $a x ^ { 2 } + b x + c = 0 , ( a , b , c \in R , a \neq 0 )$ and $2 x ^ { 2 } + 3 x + 4 = 0$ have a common root, then $a : b : c$ equals :
(1) $2 : 3 : 4$
(2) $4 : 3 : 2$
(3) $1 : 2 : 3$
(4) $3 : 2 : 1$

jee-main 2016 Q88 Counting solutions or configurations satisfying a quadratic system View

The sum of all real values of $x$ satisfying the equation $(x^2 - 5x + 5)^{x^2 + 4x - 60} = 1$ is: (1) 3 (2) $-4$ (3) 6 (4) 5

jee-main 2017 Q61 Optimization or extremal value of an expression via completing the square View

If, for a positive integer $n$, the quadratic equation,
$$x(x + 1) + (x + 1)(x + 2) + \ldots + (x + \overline{n-1})(x + n) = 10n$$
has two consecutive integral solutions, then $n$ is equal to:
(1) 12
(2) 9
(3) 10
(4) 11

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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