Core Syllabus Module

Quadratic Equations

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Chapter Elite-20 Challenge Zone

Target Advanced

JEE Advanced 2022

LEVELJEE Advanced

Let $∣ M ∣$ denote the determinant of a square matrix $M$ . Let $g : [0, \frac{π}{2}] \to R$ be the function defined by $g (θ) = f (θ) - 1 + f (\frac{π}{2} - θ) - 1$ where $f (θ) = \frac{1}{2} 1 - sin θ - 1 sin θ 1 - sin θ 1 sin θ 1 + sin π sin (θ - \frac{π}{4}) cot (θ + \frac{π}{4}) cos (θ + \frac{π}{4}) - cos \frac{π}{2} lo g_{e} (\frac{π}{4}) tan (θ - \frac{π}{4}) lo g_{e} (\frac{4}{π}) tan π$ . Let $p (x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g (θ)$ , and $p (2) = 2 - 2$ . Then, which of the following is/are TRUE ?

* Multiple Correct Options

(A)

p (\frac{3 + 2}{4}) < 0

(B)

p (\frac{1 + 3 2}{4}) > 0

(C)

p (\frac{5 2 - 1}{4}) > 0

(D)

p (\frac{5 - 2}{4}) < 0

JEE Main 2019 (12 April)

LEVELJEE Advanced

If $α, β$ and $γ$ are three consecutive terms of a non-constant G.P. such that the equations $x^{2} + 2 α x + β = 0$ and $x^{2} + β x - 1 = 0$ have a common root, then $α (β + γ)$ is equal to :

(A)

β γ

(B)

(C)

α γ

(D)

α β

JEE Advanced 1997

LEVELJEE Advanced

Let $p$ and $q$ be roots of the equation $x^{2} - A x + A = 0$ and let $r$ and $s$ be the roots of the equation $x^{2} - 18 x + B = 0$ . If $p < q < r < s$ are in arithmetic progression, then $A = \dots$ and $B = \dots$ .

JEE Main 2024 (04 Apr Shift 2)

LEVELJEE Advanced

Let $S = {sin^{2} 2 θ : (sin^{4} θ + cos^{4} θ) x^{2} + (sin 2 θ) x + (sin^{6} θ + cos^{6} θ) = 0 has real roots}$ . If $α$ and $β$ be the smallest and largest elements of the set $S$ , respectively, then $3 ((α - 2)^{2} + (β - 1)^{2})$ equals _________

JEE Main 2022 (25 July Shift 2)

LEVELJEE Advanced

Let $f (x)$ be a quadratic polynomial with leading coefficient 1 such that $f (0) = p, p \neq = 0$ and $f (1) = \frac{1}{3}$ . If the equation $f (x) = 0$ and $f (f (f (f (x)))) = 0$ have a common real root, then $f (- 3)$ is equal to

JEE Advanced 2005

LEVELJEE Advanced

Find the range of values of $t$ for which $2 sin t = \frac{1 - 2 x + 5 x ^{2}}{3 x ^{2} - 2 x - 1}$ , $t \in [- π /2, π /2]$ .

JEE Main 2020 (8 January Shift 2)

LEVELJEE Advanced

Let $S$ be the set of all real roots of the equation, $3^{x} (3^{x} - 1) + 2 = ∣ 3^{x} - 1∣ + ∣ 3^{x} - 2∣$ . Then $S$ :

contains at least four elements

(D)

contains exactly two elements.

JEE Main 2021 (25 July Shift 1)

LEVELJEE Main

The number of real roots of the equation $e^{6 x} - e^{4 x} - 2 e^{3 x} - 12 e^{2 x} + e^{x} + 1 = 0$ is :

JEE Main 2024 (31 Jan Shift 1)

LEVELJEE Main

For $0 < c < b < a$ , let $(a + b - 2 c) x^{2} + (b + c - 2 a) x + (c + a - 2 b) = 0$ and $α \neq = 1$ be one of its root. Then, among the two statements \\ (I) If $α \in (- 1, 0)$ , then $b$ cannot be the geometric mean of $a$ and $c$ \\ (II) If $α \in (0, 1)$ , then $b$ may be the geometric mean of $a$ and $c$

(A)

Both (I) and (II) are true

(B)

Neither (I) nor (II) is true

JEE Main 2020 (9 January Shift 1)

LEVELJEE Main

The number of real roots of the equation, $e^{4 x} + e^{3 x} - 4 e^{2 x} + e^{x} + 1 = 0$ is:

(A)

3

(B)

4

(C)

1

(D)

2

JEE Main 2019 (10 January Shift 1)

LEVELJEE Main

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 :

JEE Main 2020 - 9 Jan (Morning)

LEVELJEE Main

Find number of real roots of equation $e^{4 x} + e^{3 x} - 4 e^{2 x} + e^{x} + 1 = 0$ is

JEE Main 2020 (8 January Shift 2)

LEVELJEE Main

Let $S$ be the set of all real roots of the equation, $3^{x} (3^{x} - 1) + 2∣ 3^{x} - 1∣ + ∣ 3^{x} - 2∣ = 0$ . Then $S$ :

at least four elements

(D)

infinite elements

JEE Main 2025 (January)

LEVELJEE Main

If the set of all $a \in R$ for which the equation $2 x^{2} + (a - 5) x + 15 = 3 a$ has no real root, is the interval $(α, β)$ , and $X = {x \in Z : α < x < β}$ then $\sum_{x \in X} x^{2}$ is equal to :

Comprehension Passage

Let

p, q

be integers and let

α, β

be the roots of the equation,

x^{2} - x - 1 = 0

, where

α \neq = β

. For

n = 0, 1, 2, ...

, let

a_{n} = p α^{n} + q β^{n}

. FACT : If

a

and

b

are rational numbers and

a + b 5 = 0

, then

a = 0 = b

Question 1:

$a_{12} =$

(A)

a_{11} - a_{10}

(B)

a_{11} + a_{10}

(C)

2 a_{11} + a_{10}

(D)

a_{11} + 2 a_{10}

Question 2:

If $a_{4} = 28$ , then $p + 2 q =$

The sum of the squares of the roots of $∣ x + 2 ∣^{2} + ∣ x - 2∣ - 2 = 0$ and the squares of the roots of $x^{2} - 2∣ x - 3∣ - 5 = 0$ , is

JEE Main 2025 (January)

LEVELJEE Main

Let $α, β$ be the roots of the equation $x^{2} - a x - b = 0$ with $I m (α) < I m (β)$ . Let $P_{n} = α^{n} - β^{n}$ . If $P_{3} = - 57 i, P_{4} = - 37 i, P_{5} = 117 i$ and $P_{6} = 457 i,$ then $∣ α^{4} + β^{4} ∣$ is equal to

JEE Main 2021 (01 Sep Shift 2)

LEVELJEE Main

The numbers of pairs $(a, b)$ of real numbers, such that whenever $α$ is a root of the equation $x^{2} + a x + b = 0$ , $α^{2} - 2$ is also a root of this equation, is :

JEE Main 2024 (31 Jan Shift 2)

LEVELJEE Main

Let $a, b, c$ be the length of three sides of a triangle satisfying the condition $(a^{2} + b^{2}) x^{2} - 2 b (a + c) x + (b^{2} + c^{2}) = 0$ . If the set of all possible values of $x$ is the interval $(α, β)$ , then $12 (α^{2} + β^{2})$ is equal to

JEE Main 2005

LEVELJEE Main

If both the roots of the quadratic equation $x^{2} - 2 k x + k^{2} + k - 5 = 0$ are less than 5, then $k$ lies in the interval

(A)

(5, 6]

(B)

(6, \infty)

(C)

(- \infty, 4)

(D)

[4, 5]

Micro-Topic Strategy Matrix

Solution of Quadratic Equations

Relation Between Roots and Coefficients

Nature of Roots

Transformation of Equations

Common Roots

Maximum and Minimum Values of Quadratic Expressions

Location of Roots

Chapter Elite-20 Challenge Zone

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α,β and γ are three consecutive terms of a non-constant G.P. such that the equations x2+2αx+β=0 and x2+βx−1=0 have a common root, then α(β+γ) is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let p and q be roots of the equation x2−Ax+A=0 and let r and s be the roots of the equation x2−18x+B=0. If p<q<r<s are in arithmetic progression, then A=… and B=….

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the range of values of t for which 2sint=3x2−2x−11−2x+5x2​, t∈[−π/2,π/2].

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S be the set of all real roots of the equation, 3x(3x−1)+2=∣3x−1∣+∣3x−2∣. Then S:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of real roots of the equation e6x−e4x−2e3x−12e2x+ex+1=0 is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of real roots of the equation, e4x+e3x−4e2x+ex+1=0 is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find number of real roots of equation e4x+e3x−4e2x+ex+1=0 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let S be the set of all real roots of the equation, 3x(3x−1)+2∣3x−1∣+∣3x−2∣=0. Then S:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the set of all a∈R for which the equation 2x2+(a−5)x+15=3a has no real root, is the interval (α,β), and X={x∈Z:α<x<β} then ∑x∈X​x2 is equal to :

Comprehension Passage

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a12​=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a4​=28, then p+2q=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The sum of the squares of the roots of ∣x+2∣2+∣x−2∣−2=0 and the squares of the roots of x2−2∣x−3∣−5=0, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The numbers of pairs (a,b) of real numbers, such that whenever α is a root of the equation x2+ax+b=0, α2−2 is also a root of this equation, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If both the roots of the quadratic equation x2−2kx+k2+k−5=0 are less than 5, then k lies in the interval

If $α, β$ and $γ$ are three consecutive terms of a non-constant G.P. such that the equations $x^{2} + 2 α x + β = 0$ and $x^{2} + β x - 1 = 0$ have a common root, then $α (β + γ)$ is equal to :

Let $p$ and $q$ be roots of the equation $x^{2} - A x + A = 0$ and let $r$ and $s$ be the roots of the equation $x^{2} - 18 x + B = 0$ . If $p < q < r < s$ are in arithmetic progression, then $A = \dots$ and $B = \dots$ .

Find the range of values of $t$ for which $2 sin t = \frac{1 - 2 x + 5 x ^{2}}{3 x ^{2} - 2 x - 1}$ , $t \in [- π /2, π /2]$ .

Let $S$ be the set of all real roots of the equation, $3^{x} (3^{x} - 1) + 2 = ∣ 3^{x} - 1∣ + ∣ 3^{x} - 2∣$ . Then $S$ :

The number of real roots of the equation $e^{6 x} - e^{4 x} - 2 e^{3 x} - 12 e^{2 x} + e^{x} + 1 = 0$ is :

The number of real roots of the equation, $e^{4 x} + e^{3 x} - 4 e^{2 x} + e^{x} + 1 = 0$ is:

Find number of real roots of equation $e^{4 x} + e^{3 x} - 4 e^{2 x} + e^{x} + 1 = 0$ is

Let $S$ be the set of all real roots of the equation, $3^{x} (3^{x} - 1) + 2∣ 3^{x} - 1∣ + ∣ 3^{x} - 2∣ = 0$ . Then $S$ :

If the set of all $a \in R$ for which the equation $2 x^{2} + (a - 5) x + 15 = 3 a$ has no real root, is the interval $(α, β)$ , and $X = {x \in Z : α < x < β}$ then $\sum_{x \in X} x^{2}$ is equal to :

$a_{12} =$

If $a_{4} = 28$ , then $p + 2 q =$

The sum of the squares of the roots of $∣ x + 2 ∣^{2} + ∣ x - 2∣ - 2 = 0$ and the squares of the roots of $x^{2} - 2∣ x - 3∣ - 5 = 0$ , is

The numbers of pairs $(a, b)$ of real numbers, such that whenever $α$ is a root of the equation $x^{2} + a x + b = 0$ , $α^{2} - 2$ is also a root of this equation, is :

If both the roots of the quadratic equation $x^{2} - 2 k x + k^{2} + k - 5 = 0$ are less than 5, then $k$ lies in the interval