LFM Stats And Pure

isi-entrance 2023 Q25 Quadratic sandwiching and coefficient determination View

Suppose $a , b , c \in \mathbb { R }$ and $$f ( x ) = a x ^ { 2 } + b x + c , x \in \mathbb { R } .$$ If $0 \leq f ( x ) \leq ( x - 1 ) ^ { 2 }$ for all $x$, and $f ( 3 ) = 2$, then
(A) $a = \frac { 1 } { 2 } , b = - 1 , c = \frac { 1 } { 2 }$.
(B) $a = \frac { 1 } { 3 } , b = - \frac { 1 } { 3 } , c = 0$.
(C) $a = \frac { 2 } { 3 } , b = - \frac { 5 } { 3 } , c = 1$.
(D) $a = \frac { 3 } { 4 } , b = - 2 , c = \frac { 5 } { 4 }$.

isi-entrance 2026 QB4 Quadratic Diophantine Equations and Perfect Squares View

If $n ^ { 2 } + 19 n + 92$ is a perfect square, then the possible values of $n$ may be
(A) - 19
(B) - 8
(C) - 4
(D) - 11

jee-advanced 1999 Q18 Optimization or extremal value of an expression via completing the square View

18. If the roots of the equation $x 2 - 2 a x + a 2 + a - 3 = 0$ are real and less than 3,then:
(A) $a < 2$
(B) $2 < a < 3$
(C) $3 < a < 4$
(D) a $> 4$

jee-advanced 2003 Q13 Parameter range for specific root conditions (location/count) View

If $x ^ { 2 } + ( a - b ) x + ( 1 - a - b ) = 0$ where $a , b$ Î $R$ then find the values of $a$ for which equation has unequal real roots for all values of $b$.

jee-advanced 2003 Q14 Parameter range for specific root conditions (location/count) View

14. If $f ( x ) = x ^ { 2 } + 2 b x + 2 c ^ { 2 }$ and $g ( x ) = - x ^ { 2 } - 2 c x + b ^ { 2 }$ such that min $f ( x ) > \max g ( x )$, then the relation between $b$ and $c$, is :
(a) no real value of $b$ and $c$
(b) $0 <$ c $<$ b $\sqrt { } 2$
(c) $| \mathrm { c } | < | \mathrm { b } | \sqrt { } 2$
(d) $| c | > | b | \sqrt { } 2$

jee-advanced 2008 Q11 Root relationships and Vieta's formulas View

Let $a , b , c , p , q$ be real numbers. Suppose $\alpha , \beta$ are the roots of the equation $x ^ { 2 } + 2 p x + q = 0$ and $\alpha , \frac { 1 } { \beta }$ are the roots of the equation $a x ^ { 2 } + 2 b x + c = 0$, where $\beta ^ { 2 } \notin \{ - 1,0,1 \}$. STATEMENT-1 : $\left( p ^ { 2 } - q \right) \left( b ^ { 2 } - a c \right) \geq 0$ and STATEMENT-2 : $b \neq p a$ or $c \neq q a$
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

jee-advanced 2009 Q37 Parameter range for specific root conditions (location/count) View

The smallest value of $k$, for which both the roots of the equation $$x^{2}-8kx+16\left(k^{2}-k+1\right)=0$$ are real, distinct and have values at least 4, is

jee-advanced 2015 Q49 Parameter range for specific root conditions (location/count) View

Let $S$ be the set of all non-zero real numbers $\alpha$ such that the quadratic equation $\alpha x ^ { 2 } - x + \alpha = 0$ has two distinct real roots $x _ { 1 }$ and $x _ { 2 }$ satisfying the inequality $\left| x _ { 1 } - x _ { 2 } \right| < 1$. Which of the following intervals is(are) a subset(s) of $S$ ?
(A) $\left( - \frac { 1 } { 2 } , - \frac { 1 } { \sqrt { 5 } } \right)$
(B) $\left( - \frac { 1 } { \sqrt { 5 } } , 0 \right)$
(C) $\left( 0 , \frac { 1 } { \sqrt { 5 } } \right)$
(D) $\left( \frac { 1 } { \sqrt { 5 } } , \frac { 1 } { 2 } \right)$

jee-main 2007 Q83 Parameter range for specific root conditions (location/count) View

If the difference between the roots of the equation $x ^ { 2 } + a x + 1 = 0$ is less than $\sqrt { 5 }$, then the set of possible values of $a$ is
(1) $( - 3,3 )$
(2) $( - 3 , \infty )$
(3) $( 3 , \infty )$
(4) $( - \infty , - 3 )$

jee-main 2012 Q61 Parameter range for specific root conditions (location/count) View

The value of k for which the equation $( K - 2 ) x ^ { 2 } + 8 x + K + 4 = 0$ has both roots real, distinct and negative is
(1) 6
(2) 3
(3) 4
(4) 1

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 2013 Q61 Parameter range for specific root conditions (location/count) View

The values of ' $a$ ' for which one root of the equation $x ^ { 2 } - ( a + 1 ) x + a ^ { 2 } + a - 8 = 0$ exceeds 2 and the other is lesser than 2 , are given by :
(1) $3 < a < 10$
(2) $a \geq 10$
(3) $- 2 < a < 3$
(4) $a \leq - 2$

jee-main 2013 Q62 Root relationships and Vieta's formulas View

If the equations $x^2 + 2x + 3 = 0$ and $ax^2 + bx + c = 0, a, b, c \in R$, have a common root, then $a : b : c$ is:
(1) $1 : 3 : 2$
(2) $3 : 1 : 2$
(3) $1 : 2 : 3$
(4) $3 : 2 : 1$

jee-main 2018 Q61 Optimization or extremal value of an expression via completing the square View

If $\lambda \in R$ is such that the sum of the cubes of the roots of the equation $x ^ { 2 } + ( 2 - \lambda ) x + ( 10 - \lambda ) = 0$ is minimum, then the magnitude of the difference of the roots of this equation is :
(1) $4 \sqrt { 2 }$
(2) 20
(3) $2 \sqrt { 5 }$
(4) $2 \sqrt { 7 }$

jee-main 2019 Q61 Vieta's formulas: compute symmetric functions of roots View

If $\lambda$ be the ratio of the roots of the quadratic equation in $x , 3 m ^ { 2 } x ^ { 2 } + m ( m - 4 ) x + 2 = 0$, then the least value of $m$ for which $\lambda + \frac { 1 } { \lambda } = 1$, is :
(1) $2 - \sqrt { 3 }$
(2) $- 2 + \sqrt { } \overline { 2 }$
(3) $4 - 2 \sqrt { 3 }$
(4) $4 - 3 \sqrt { 2 }$

jee-main 2019 Q61 Nature of roots given coefficient constraints View

The number of all possible positive integral value of $\alpha$ for which the roots of the quadratic equation $6x^2 - 11x + \alpha = 0$ are rational numbers is:
(1) 5
(2) 3
(3) 4
(4) 2

jee-main 2019 Q61 Parameter range for specific root conditions (location/count) View

Consider the quadratic equation $( c - 5 ) x ^ { 2 } - 2 c x + ( c - 4 ) = 0 , c \neq 5$. Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $( 0,2 )$ and its other root lies in the interval $( 2,3 )$. Then the number of elements in $S$ is
(1) 11
(2) 12
(3) 18
(4) 10

jee-main 2019 Q61 Parameter range for no real roots (positive definite) View

The number of integral values of $m$ for which the quadratic expression $( 1 + 2 m ) x ^ { 2 } - 2 ( 1 + 3 m ) x + 4 ( 1 + m ) , x \in R$ is always positive, is
(1) 7
(2) 3
(3) 6
(4) 8

jee-main 2019 Q61 Root relationships and Vieta's formulas View

Let $p , q \in Q$. If $2 - \sqrt { 3 }$ is a root of the quadratic equation $x ^ { 2 } + p x + q = 0$, then
(1) $p ^ { 2 } - 4 q + 12 = 0$
(2) $q ^ { 2 } + 4 p + 14 = 0$
(3) $p ^ { 2 } - 4 q - 12 = 0$
(4) $q ^ { 2 } - 4 p - 16 = 0$

jee-main 2019 Q61 Optimization or extremal value of an expression via completing the square View

If $m$ is chosen in the quadratic equation $\left( m ^ { 2 } + 1 \right) x ^ { 2 } - 3 x + \left( m ^ { 2 } + 1 \right) ^ { 2 } = 0$ such that the sum of its roots is greatest, then the absolute difference of the cubes of its roots is:
(1) $4 \sqrt { 3 }$
(2) $10 \sqrt { 5 }$
(3) $8 \sqrt { 3 }$
(4) $8 \sqrt { 5 }$

jee-main 2019 Q62 Parameter range for no real roots (positive definite) View

The number of integral values of $m$ for which the equation, $1 + m ^ { 2 } x ^ { 2 } - 21 + 3 m x + 1 + 8 m = 0$ has no real root, is
(1) 2
(2) 3
(3) Infinitely many
(4) 1

jee-main 2020 Q51 Determining quadratic function from given conditions View

Let $f ( x )$ be a quadratic polynomial such that $f ( - 1 ) + f ( 2 ) = 0$. If one of the roots of $f ( x ) = 0$ is 3 , then its other root lies in
(1) $( - 1,0 )$
(2) $( 1,3 )$
(3) $( - 3 , - 1 )$
(4) $( 0,1 )$

jee-main 2020 Q51 Parameter range for specific root conditions (location/count) View

Consider the two sets: $A = \left\{ m \in R : \right.$ both the roots of $x ^ { 2 } - ( m + 1 ) x + m + 4 = 0$ are real $\}$ and $B = [ - 3,5 )$
Which of the following is not true?
(1) $A - B = ( - \infty , - 3 ) \cup ( 5 , \infty )$
(2) $A \cap B = \{ - 3 \}$
(3) $B - A = ( - 3,5 )$
(4) $A \cup B = R$

jee-main 2020 Q51 Parameter range for specific root conditions (location/count) View

The set of all real values of $\lambda$ for which the quadratic equation $\left( \lambda ^ { 2 } + 1 \right) x ^ { 2 } - 4 \lambda x + 2 = 0$ always have exactly one root in the interval $( 0,1 )$ is :
(1) $( - 3 , - 1 )$
(2) $( 0,2 )$
(3) $( 1,3 ]$
(4) $( 2,4 ]$

jee-main 2020 Q52 Finding roots or coefficients of a quadratic using Vieta's relations View

Let $a , b \in R , a \neq 0$ be such that the equation, $a x ^ { 2 } - 2 b x + 5 = 0$ has a repeated root $\alpha$, which is also a root of the equation, $x ^ { 2 } - 2 b x - 10 = 0$. If $\beta$ is the other root of this equation, then $\alpha ^ { 2 } + \beta ^ { 2 }$ is equal to:
(1) 25
(2) 26
(3) 28
(4) 24

1
2
3
4
5

Load questions...

LFM Stats And Pure

Completing the square and sketching 80

Inequalities 215

Discriminant and conditions for roots 106

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