How many questions from Quadratic Equations appear in JEE?

Quadratic Equations has appeared in 148 of the last 186 JEE Main and JEE Advanced papers — about 80% of them — contributing 170 questions in total across those papers.

Is Quadratic Equations an important chapter for JEE?

Judged by how often it is actually tested, it appears in roughly 80% of papers. Chapters above about 50% are effectively guaranteed to show up every session, so they repay thorough preparation; lower-frequency chapters are better treated as targeted revision.

Where do these Quadratic Equations questions come from?

Every question is from an official JEE Main or JEE Advanced paper, transcribed from the original paper and tagged to this chapter. Answers follow the official answer key.

Quadratic Equations JEE Previous Year Questions with Answers

Q1·MathematicsSingle correctJEE Advanced 2026

Match each entry in List-I to the correct entry in List-II and choose the correct option.

List-I	List-II
P.If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2 + x + 1 = 0$ , then the quadratic equation with roots $\frac{1}{(\alpha + 1)^{2026}}$ and $\frac{1}{(\beta + 1)^{2026}}$ is	1. $x^2 + x + 1 = 0$
Q.If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2 + x + 1 = 0$ , then the quadratic equation with roots $\frac{1}{(\alpha + 1)^{2027}}$ and $\frac{1}{(\beta + 1)^{2027}}$ is	2. $x^2 - x + 1 = 0$
R.If $\gamma$ and $\delta$ are the distinct roots of the equation $x^2 - x + 1 = 0$ , then the value of $\frac{1}{(\gamma - 1)^{2026}} + \frac{1}{(\delta - 1)^{2026}}$ is	3. $x^2 + x - 1 = 0$
S.If $p$ and $r$ are the distinct roots of the equation $x^2 + x - 1 = 0$ , then the value of $\frac{1}{(p + 1)^3} + \frac{1}{(r + 1)^3}$ is	4. $-1$
	5. $-4$

(A)(P) → (1), (Q) → (2), (R) → (5), (S) → (4)
(B)(P) → (3), (Q) → (1), (R) → (4), (S) → (5)
(C)(P) → (1), (Q) → (2), (R) → (4), (S) → (5)
(D)(P) → (2), (Q) → (3), (R) → (5), (S) → (4)

Q2·MathematicsMultiple correctJEE Advanced 2026

Let

a, b, c

be positive integers in arithmetic progression such that the equation

ax^{2} + bx + c = 0

has only integer solutions. Then which of the following statements is (are) TRUE ?

(A) $c - b$ is an integer multiple of $a$
(B)Both the roots of the equation $ax^{2} + bx + c = 0$ are odd integers
(C)If $c = 15$ , then $ab = 8$
(D)If $b = 8$ , then $x = 3$ is a root of the equation $ax^{2} + bx + c = 0$

Q3·MathematicsNumericalJEE Main 2026

The sum of squares of all the real solutions of the equation

\log_{(x+1)}(2x^2 + 5x + 3) = 4 - \log_{(2x+3)}(x^2 + 2x + 1)

is equal to ________.

Q4·MathematicsSingle correctJEE Main 2026

Let e

_1

and e

_2

be two distinct roots of the equation x

^2

− ax + 2 = 0. Let the sets {a ∈ ℝ : e

_1

and e

_2

are the eccentricities of hyperbolas} = (α, β), and {a ∈ ℝ : e

_1

and e

_2

are the eccentricities of an ellipse and a hyperbola, respectively} = (γ, ∞). Then α

^2

+ β

^2

+ γ

^2

is equal to:

(A)18
(B)22
(C)26
(D)34

Q5·MathematicsSingle correctJEE Main 2026

Let one root of the quadratic equation in x: (k

^2

− 15k + 27)x

^2

+ 9(k − 1)x + 18 = 0 be twice the other. Then the length of the latus rectum of the parabola y

^2

= 6kx is equal to:

(A)4
(B)6
(C)8
(D)12

Q6·MathematicsSingle correctJEE Main 2026

Let

\lim_{x \to 2} \frac{(\tan(x-2))(rx^{2} + (p-2)x - 2p)}{(x-2)^{2}} = 5

for some

r, p \in \mathbb{R}

. If the set of all possible values of

q

, such that the roots of the equation

rx^{2} - px + q = 0

lie in

(0, 2)

, be the interval

(\alpha, \beta]

, then

4(\alpha + \beta)

equals :

(A)11
(B)13
(C)17
(D)21

Q7·MathematicsSingle correctJEE Main 2026

Let α, β be the roots of the equation

x^2 - x + p = 0

and γ, δ be the roots the equation

x^2 - 4x + q = 0

; p, q ∈ Z. If α, β, γ, δ are in G.P., then

|p + q|

equals :

(A)16
(B)32
(C)34
(D)38

Q8·MathematicsSingle correctJEE Main 2026

Let

a, b \in \mathbb{C}

. Let

\alpha, \beta

be the roots of the equation

x^2 + ax + b = 0

. If

\beta - \alpha = \sqrt{11}

and

\beta^2 - \alpha^2 = 3i\sqrt{11}

, then

(\beta^3 - \alpha^3)^2

is equal to:

(A)160
(B)176
(C)194
(D)187

Q9·MathematicsSingle correctJEE Main 2026

If the quadratic equation

(\lambda + 2)x^{2} - 3\lambda x + 4\lambda = 0

,

\lambda \neq -2

, has two positive roots, then the number of possible integral values of

\lambda

is:

(A)1
(B)2
(C)3
(D)4

Q10·MathematicsSingle correctJEE Main 2026

If the set of all solutions of

|x^2 + x - 9| = |x| + |x^2 - 9|

is

[\alpha, \beta] \cup [\gamma, \infty)

, then

(\alpha^2 + \beta^2 + \gamma^2)

is equal to:

(A) $9$
(B) $18$
(C) $36$
(D) $72$

Q11·MathematicsSingle correctJEE Main 2026

Let

\alpha, \beta

be the roots of the equation

x^2 - 3x + r = 0

, and

\frac{\alpha}{2}, 2\beta

be the roots of the equation

x^2 + 3x + r = 0

. If the roots of the equation

x^2 + 6x = m

are

2\alpha + \beta + 2r

and

\alpha - 2\beta - \frac{r}{2}

, then

m

is equal to:

(A) $-135$
(B) $-567$
(C) $135$
(D) $567$

Q12·MathematicsSingle correctJEE Main 2026

Let α, α + 2, α ∈ ℤ, be the roots of the quadratic equation

x(x+2)+(x+1)(x+3)+(x+2)(x+4)+\ldots+(x+n-1)(x+n+1)=4n

for some n ∈ ℕ. Then n + α is equal to :

(A)0
(B)1
(C)2
(D)3

Q13·MathematicsSingle correctJEE Main 2026

Let

S = \{x^3 + ax^2 + bx + c : a, b, c \in N

and

a, b, c \le 20\}

be a set of polynomials. Then the number of polynomials in S, which are divisible by

x^2 + 2

, is

(A)20
(B)6
(C)120
(D)10

Q14·MathematicsSingle correctJEE Main 2026

If

\alpha

,

\beta

, where

\alpha < \beta

, are the roots of the equation

\lambda x^2 - (\lambda + 3)x + 3 = 0

such that

\dfrac{1}{\alpha} - \dfrac{1}{\beta} = \dfrac{1}{3}

, then the sum of all possible values of

\lambda

is :

(A)6
(B)2
(C)4
(D)8

Q15·MathematicsSingle correctJEE Main 2026

Let the arithmetic mean of

\dfrac{1}{a}

and

\dfrac{1}{b}

be

\dfrac{5}{16}

, a > 2. If α is such that a, 4, α, b are in A.P., then the equation

\alpha x^{2} - ax + 2(\alpha - 2b) = 0

has :

(A)One root in (1,4) and another in (–2,0)
(B)One root in (0,2) and another in (– 4, –2)
(C)Complex roots of magnitude less than 2
(D)Both roots in the interval (–2, 0)

Q16·MathematicsSingle correctJEE Main 2026

The number of the real solutions of the equation :

\mathrm{x}\left|\mathrm{x} + 3\right| + \left|\mathrm{x} - 1\right| - 2 = 0

is

(A)3
(B)2
(C)5
(D)4

Q17·MathematicsSingle correctJEE Main 2026

The smallest positive integral value of a, for which all the roots of

x^{4} - ax^{2} + 9 = 0

are real and distinct, is equal to

(A)9
(B)3
(C)4
(D)7

Q18·MathematicsSingle correctJEE Main 2026

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in:

(A)24 days
(B)42 days
(C)30 days
(D)36 days

Q19·MathematicsSingle correctJEE Main 2026

The sum of all the real solutions of the equation

\log_{(x+3)}(6x^2 + 28x + 30) = 5 - 2\log_{(6x+10)}(x^2 + 6x + 9)

is equal to :

(A)2
(B)1
(C)0
(D)4

Q20·MathematicsSingle correctJEE Main 2026

If

\alpha

and

\beta

(

\alpha < \beta

) are the roots of the equation

\left(-2 + \sqrt{3}\right)\left(\left|\sqrt{x} - 3\right|\right) + \left(x - 6\sqrt{x}\right) + \left(9 - 2\sqrt{3}\right) = 0

,

x \ge 0

, then

\sqrt{\frac{\beta}{\alpha}} + \sqrt{\alpha\beta}

is equal to :

(A)8
(B)9
(C)10
(D)11

Q21·MathematicsSingle correctJEE Main 2026

The number of distinct real solutions of the equation

x|x + 4| + 3|x + 2| + 10 = 0

is

(A)3
(B)1
(C)0
(D)2

Q22·MathematicsSingle correctJEE Main 2026

Let

\alpha

,

\beta

be the roots of the quadratic equation

12x^{2} - 20x + 3\lambda = 0

,

\lambda \in \mathbb{Z}

. If

\frac{1}{2} \le |\beta - \alpha| \le \frac{3}{2}

, then the sum of all possible values of

\lambda

is :

(A)6
(B)1
(C)3
(D)4

Q23·MathematicsSingle correctJEE Main 2026

The sum of all the roots of the equation

(x-1)^{2}-5|x-1|+6=0

, is:

(A)4
(B)3
(C)1
(D)5

Q24·MathematicsSingle correctJEE Main 2026

Let α and β be the roots of equation

x^2 + 2ax + (3a + 10) = 0

such that α < 1 < β. Then the set of all possible values of a is :

(A) $\left(-\infty, \frac{-11}{5}\right) \cup (5, \infty)$
(B) $(-\infty, -2) \cup (5, \infty)$
(C) $(-\infty, -3)$
(D) $\left(-\infty, \frac{-11}{5}\right)$

Q25·MathematicsSingle correctJEE Main 2026

The positive integer n, for which the solutions of the equation

x(x+2) + (x + 2)(x + 4) + \ldots + (x + 2n - 2)(x+2n) = \frac{8n}{3}

are two consecutive even integers, is :-

(A)3
(B)6
(C)12
(D)9

Q26·MathematicsSingle correctJEE Advanced 2025

Let R denote the set of all real numbers. Let

a_i, b_i \in R

for

i \in \{1, 2, 3\}

. Define the function f: R → R, g : R → R, and h : R → R by

f(x) = a_1 + 10x + a_2x^2 + a_3x^3 + x^4

,

g(x) = b_1 + 3x + b_2x^2 + b_3x^3 + x^4

,

h(x) = f(x + 1) - g(x + 2)

. If

f(x) \neq g(x)

for every

x \in R

, then the coefficient of

x^3

in h(x) is

(A)8
(B)2
(C)− 4
(D)− 6

Q27·MathematicsSingle correctJEE Main 2025

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:

(A) $26$
(B) $36$
(C) $30$
(D) $24$

Q28·MathematicsSingle correctJEE Main 2025

Let the set of all values of

p\in\mathbb{R}

, for which both the roots of the equation

x^2-(p+2)x+(2p+9)=0

are negative real numbers, be the interval

(\alpha,\beta]

. Then

\beta-2\alpha

is equal to:

(A)0
(B)9
(C)5
(D)20

Q29·MathematicsSingle correctJEE Main 2025

The number of real roots of the equation

x|x-2|+3|x-3|+1=0

is:

(A)4
(B)2
(C)1
(D)3

Q30·MathematicsSingle correctJEE Main 2025

Consider the equation

x^2+4x-n=0

, where

n\in[20,100]

is a natural number. Then the number of all distinct values of

n

, for which the given equation has integral roots, is equal to

(A)7
(B)8
(C)6
(D)5

Q31·MathematicsSingle correctJEE Main 2025

Let the product of

\omega_1=(8+i)\sin\theta+(7+4i)\cos\theta

and

\omega_2=(1+8i)\sin\theta+(4+7i)\cos\theta

be

\alpha+i\beta

,

i=\sqrt{-1}

. Let

p

and

q

be the maximum and minimum values of

\alpha+\beta

respectively. Then

p+q

is equal to:

(A)140
(B)130
(C)160
(D)150

Q32·MathematicsSingle correctJEE Main 2025

Let the equation

x(x+2)(12-k)=2

have equal roots. Then the distance of the point

\left(k,\dfrac{k}{2}\right)

from the line

3x+4y+5=0

is:

(A)15
(B) $5\sqrt3$
(C) $15\sqrt5$
(D)12

Q33·MathematicsSingle correctJEE Main 2025

Let

\alpha

and

\beta

be the roots of

x^2+\sqrt3\,x-16=0

, and

\gamma

and

\delta

be the roots of

x^2+3x-1=0

. If

P_n=\alpha^n+\beta^n

and

Q_n=\gamma^n+\delta^n

, then

\dfrac{P_{25}+\sqrt3\,P_{24}}{2P_{23}}+\dfrac{Q_{25}-Q_{23}}{Q_{24}}

is equal to:

(A)3
(B)4
(C)5
(D)7

Q34·MathematicsSingle correctJEE Main 2025

Let

P_n=\alpha^n+\beta^n

,

n\in N

. If

P_{10}=123

,

P_9=76

,

P_8=47

and

P_1=1

, then the quadratic equation having roots

\dfrac{1}{\alpha}

and

\dfrac{1}{\beta}

is:

(A) $x^2-x+1=0$
(B) $x^2+x-1=0$
(C) $x^2-x-1=0$
(D) $x^2+x+1=0$

Q35·MathematicsIntegerJEE Main 2025

If the set of all

a\in\mathbb{R}-\{1\}

, for which the roots of the equation

(1-a)x^2+2(a-3)x+9=0

are positive is

(-\infty,-\alpha]\cup[\beta,\gamma)

, then

2\alpha+\beta+\gamma

is equal to ______.

Q36·MathematicsSingle correctJEE Main 2025

If the set of all

a\in R

, for which the equation

2x^2+(a-5)x+15=3a

has no real root, is the interval

(\alpha,\beta)

, and

X=\{x\in Z:\alpha<x<\beta\}

, then

\displaystyle\sum_{x\in X}x^2

is equal to

(A)2109
(B)2129
(C)2139
(D)2119

Q37·MathematicsSingle correctJEE Main 2025

The number of solutions of the equation

\left(\dfrac{9}{x}-\dfrac{9}{\sqrt{x}}+2\right)\left(\dfrac{2}{x}-\dfrac{7}{\sqrt{x}}+3\right)=0

is:

(A)2
(B)4
(C)1
(D)3

Q38·MathematicsSingle correctJEE Main 2025

The sum, of the squares of all the roots of the equation

x^2+|2x-3|-4=0

, is:

(A) $3(3-\sqrt{2})$
(B) $6(3-\sqrt{2})$
(C) $6(2-\sqrt{2})$
(D) $3(2-\sqrt{2})$

Q39·MathematicsSingle correctJEE Main 2025

If

\alpha+i\beta

and

\gamma+i\delta

are the roots of

x^2-(3-2i)x-(2i-2)=0

,

i=\sqrt{-1}

, then

\alpha\gamma+\beta\delta

is equal to:

(A)6
(B)2
(C)-2
(D)-6

Q40·MathematicsSingle correctJEE Main 2025

The product of all the rational roots of the equation

(x^2-9x+11)^2-(x-4)(x-5)=3

, is equal to :

(A) $14$
(B) $7$
(C) $28$
(D) $21$

Q41·MathematicsSingle correctJEE Main 2025

The number of real solution(s) of the equation

x^2+3x+2=\min\{|x-3|,|x+2|\}

is :

(A)2
(B)0
(C)3
(D)1

Q42·MathematicsIntegerJEE Main 2025

If the equation

a(b-c)x^2+b(c-a)x+c(a-b)=0

has equal roots, where

a+c=15

and

b=\dfrac{36}{5}

, then

a^2+c^2

is equal to _______

Q43·MathematicsIntegerJEE Main 2025

Let

\alpha,\beta

be the roots of the equation

x^2-ax-b=0

with

\operatorname{Im}(\alpha)<\operatorname{Im}(\beta)

. Let

P_n=\alpha^n-\beta^n

. If

P_3=-5\sqrt7\,i,\ P_4=-3\sqrt7\,i,\ P_5=11\sqrt7\,i

and

P_6=45\sqrt7\,i

, then

|\alpha^4+\beta^4|

is equal to __________.

Q44·MathematicsSingle correctJEE Main 2025

The product of all solutions of

e^{5(\log_e x)^2+3}=x^{8}\,(x>0)

is:

(A) $e^{8/5}$
(B) $e^{6/5}$
(C)e
(D) $e^{2}$

Q45·MathematicsSingle correctJEE Main 2025

Let

\alpha_0

and

\beta_0

be the distinct roots of

2x^2+(\cos\theta)x-1=0

,

\theta\in(0,2\pi)

. If

m

and

M

are the minimum and the maximum values of

\alpha_0^4+\beta_0^4

, then

16(M+m)

equals:

(A)24
(B)25
(C)27
(D)17

Q46·MathematicsMultiple correctJEE Advanced 2024

Let

R^{2}

denote

R \times R

. Let

S = \left\{(a, b, c) : a, b, c \in R \text{ and } ax^{2} + 2bxy + cy^{2} > 0 \text{ for all } (x, y) \in R^{2} - \{(0,0)\}\right\}

Then which of the following statements is (are) TRUE?

(A) $\left(2, \frac{7}{2}, 6\right) \in S$
(B)If $\left(3, b, \frac{1}{12}\right) \in S$ , then $|2b| < 1$
(C)For any given $(a, b, c) \in S$ , then the system of linear equations $ax + by = 1$ $bx + cy = -1$ has a unique solution.
(D)For any given $(a, b, c) \in S$ , then the system of linear equations $(a + 1)x + by = 0$ $bx + (c + 1)y = 0$ has a unique solution.

Q47·MathematicsIntegerJEE Advanced 2024

Let

f(x) = x^{4} + ax^{3} + bx^{2} + c

be a polynomial with real coefficients such that

f(1) = -9

. Suppose that

i\sqrt{3}

is a root of the equation

4x^{3} + 3ax^{2} + 2bx = 0

, where

i = \sqrt{-1}

. If

\alpha_{1}, \alpha_{2}, \alpha_{3},

and

\alpha_{4}

are all the roots of the equation

f(x) = 0

, then

\left|\alpha_{1}\right|^{2} + \left|\alpha_{2}\right|^{2} + \left|\alpha_{3}\right|^{2} + \left|\alpha_{4}\right|^{2}

is equal to ______ .

Q48·MathematicsSingle correctJEE Main 2024

Let

\alpha, \beta

be the roots of the equation

x^2 + 2\sqrt{2}\,x - 1 = 0

. The quadratic equation, whose roots are

\alpha^4 + \beta^4

and

\frac{1}{10}(\alpha^6 + \beta^6)

, is :

(A) $x^2 - 190x + 9466 = 0$
(B) $x^2 - 195x + 9466 = 0$
(C) $x^2 - 195x + 9506 = 0$
(D) $x^2 - 180x + 9506 = 0$

Q49·MathematicsSingle correctJEE Main 2024

Let

\alpha, \beta

(

\alpha>\beta

) be the roots of the equation

x^{2}-\sqrt{2}x-\sqrt{3}=0

. Let

P_{n}=\alpha^{n}-\beta^{n}

,

n\in\mathbb{N}

. Then

\left(11\sqrt{3}-10\sqrt{2}\right)P_{10}+\left(11\sqrt{2}+10\right)P_{11}-11P_{12}

is equal to:

(A) $10\sqrt{2}\,P_{9}$
(B) $10\sqrt{3}\,P_{9}$
(C) $11\sqrt{2}\,P_{9}$
(D) $11\sqrt{3}\,P_{9}$

Q50·MathematicsNumericalJEE Main 2024

The number of distinct real roots of the equation

|x+1|\,|x+3|-4\,|x+2|+5=0

, is _______ .

Q51·MathematicsSingle correctJEE Main 2024

The sum of all the solutions of the equation

(8)^{2x}-16\cdot(8)^x+48=0

is:

(A) $1+\log_6(8)$
(B) $\log_8(6)$
(C) $1+\log_8(6)$
(D) $\log_8(4)$

Q52·MathematicsSingle correctJEE Main 2024

Let

\alpha,\beta

be the distinct roots of the equation

x^{2}-(t^{2}-5t+6)x+1=0

,

t\in\mathbb{R}

and

a_{n}=\alpha^{n}+\beta^{n}

. Then the minimum value of

\dfrac{a_{2023}+a_{2025}}{a_{2024}}

is

(A) $\tfrac{1}{4}$
(B) $-\tfrac{1}{2}$
(C) $-\tfrac{1}{4}$
(D) $\tfrac{1}{2}$

Q53·MathematicsNumericalJEE Main 2024

Let

x_{1},x_{2},x_{3},x_{4}

be the solution of the equation

4x^{4}+8x^{3}-17x^{2}-12x+9=0

and

(4+x_{1}^{2})(4+x_{2}^{2})(4+x_{3}^{2})(4+x_{4}^{2})=\dfrac{125}{16}m

. Then the value of

m

is _______.

Q54·MathematicsNumericalJEE Main 2024

Let

\alpha,\beta

be roots of

x^{2}+\sqrt{2}x-8=0

. If

U_{n}=\alpha^{n}+\beta^{n}

, then

\frac{U_{10}+\sqrt{12}\,U_{9}}{2U_{8}}

is equal to ______.

Q55·MathematicsNumericalJEE Main 2024

The number of distinct real roots of the equation

|x|\,|x + 2| - 5|x + 1| - 1 = 0

is ___.

Q56·MathematicsNumericalJEE Main 2024

The number of real solutions of the equation

x\,|x+5|+2|x+7|-2=0

is __________.

Q57·MathematicsSingle correctJEE Main 2024

If

2

and

6

are the roots of the equation

ax^2+bx+1=0

, then the quadratic equation, whose roots are

\dfrac{1}{2a+b}

and

\dfrac{1}{6a+b}

, is:

(A) $2x^2+11x+12=0$
(B) $4x^2+14x+12=0$
(C) $x^2+10x+16=0$
(D) $x^2+8x+12=0$

Q58·MathematicsNumericalJEE Main 2024

Let

S=\{\sin^2 2\theta:(\sin^4\theta+\cos^4\theta)x^2+(\sin 2\theta)x+(\sin^6\theta+\cos^6\theta)=0 \text{ has real roots}\}

. If

\alpha

and

\beta

be the smallest and largest elements of the set S, respectively, then

3\left((\alpha-2)^2+(\beta-1)^2\right)

equals

Q59·MathematicsSingle correctJEE Main 2024

Let

S=\{x\in\mathbb{R}:(\sqrt3+\sqrt2)^x+(\sqrt3-\sqrt2)^x=10\}

. Then the number of elements in

S

is:

(A) $4$
(B) $0$
(C) $2$
(D) $1$

Q60·MathematicsNumericalJEE Main 2024

Let

P=\{z\in\mathbb{C}:|z+2-3i|\le\sqrt{11}\}

and

Q=\{z\in\mathbb{C}:z(1+i)+\bar z(1-i)\le-8\}

. Let in

P\cap Q

,

|z-3+2i|

be the maximum and minimum at

z_1

and

z_2

respectively. If

|z_1|^2+2|z_2|^2=\alpha+\beta\sqrt2

, where

\alpha,\beta

are integers, then

\alpha+\beta

equals ___

Q61·MathematicsSingle correctJEE Main 2024

Let

\alpha

and

\beta

be the roots of the equation

px^2+qx-r=0

, where

p\neq 0

. If

p

,

q

and

r

be the consecutive terms of a non-constant GP and

\dfrac{1}{\alpha}+\dfrac{1}{\beta}=\dfrac{3}{4}

, then the value of

(\alpha-\beta)^2

is:

(A) $\dfrac{80}{9}$
(B)9
(C) $\dfrac{20}{3}$
(D)8

Q62·MathematicsSingle correctJEE Main 2024

For

0<c<b<a

, let

(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)=0

and

\alpha\ne 1

be one of its root. Then, among the two statements (I) If

\alpha\in(-1,0)

, then b cannot be the geometric mean of a and c (II) If

\alpha\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
(C)Only (II) is true
(D)Only (I) is true

Q63·MathematicsNumericalJEE Main 2024

Let

a

,

b

,

c

be the length of three sides of a triangle satisfying the condition

(a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0

. If the set of all possible values of

x

is the interval

(\alpha,\beta)

, then

12(\alpha^2+\beta^2)

is equal to ______.

Q64·MathematicsSingle correctJEE Main 2024

Let a be the sum of all coefficients in the expansion of

\left(1-2x+2x^2\right)^{2023}\left(3-4x^2+2x^3\right)^{2024}

and

b=\displaystyle\lim_{x\to 0}\left(\dfrac{\int_0^x \frac{\log(1+t)}{t^{2024}+1}dt}{x^2}\right)

. If the equations

cx^2+dx+e=0

and

2bx^2+ax+4=0

have a common root, where

c,d,e\in R

, then

d:c:e

equals

(A) $2:1:4$
(B) $4:1:4$
(C) $1:2:4$
(D) $1:1:4$

Q65·MathematicsSingle correctJEE Main 2024

Let S be the set of positive integral values of a for which

\dfrac{ax^2+2(a+1)x+9a+4}{x^2-8x+32}<0

,

\forall x\in R

. Then, the number of elements in S is

(A) $1$
(B) $0$
(C) $\infty$
(D) $3$

Q66·MathematicsNumericalJEE Main 2024

The number of real solutions of the equation

x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0

is ______.

Q67·MathematicsNumericalJEE Main 2024

Let

\alpha,\beta\in\mathbb{N}

be roots of the equation

x^2-70x+\lambda=0

, where

\dfrac\lambda2,\dfrac\lambda3\notin\mathbb{N}

. If

\lambda

assumes the minimum possible value, then

\dfrac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}

is equal to ___

Q68·MathematicsNumericalJEE Main 2024

Let

\alpha,\beta

be the roots of the equation

x^2-\sqrt6\,x+3=0

such that

\text{Im}(\alpha)>\text{Im}(\beta)

. Let

a,b

be integers not divisible by 3 and

n

be a natural number such that

\dfrac{\alpha^{99}}{\beta}+\alpha^{98}=3^n(a+ib),\ i=\sqrt{-1}

. Then

n+a+b

is equal to ___.

Q69·MathematicsNumericalJEE Main 2024

Let

\alpha,\beta

be the roots of the equation

x^2-x+2=0

with

\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)

. Then

\alpha^6+\alpha^4+\beta^4-5\alpha^2

is equal to ______.

Q70·MathematicsSingle correctJEE Main 2023

The number of real roots of the equation

x|x|-5|x+2|+6=0

, is

(A)5
(B)3
(C)6
(D)4

Q71·MathematicsSingle correctJEE Main 2023

The set of all

a\in\mathbb{R}

for which the equation

x|x-1|+|x+2|+a=0

has exactly one real root is:

(A) $(-6,-3)$
(B) $(-\infty,\infty)$
(C) $(-6,\infty)$
(D) $(-\infty,-3)$

Q72·MathematicsSingle correctJEE Main 2023

Let

\alpha,\beta

be the roots of the quadratic equation

x^2+\sqrt6\,x+3=0

. Then

\dfrac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}

is equal to

(A)729
(B)72
(C)81
(D)9

Q73·MathematicsSingle correctJEE Main 2023

The number of integral solutions

x

of

\log_{\left( x + \frac{7}{2} \right)} \left( \frac{x - 7}{2x - 3} \right)^2 \ge 0

is

(A)6
(B)8
(C)5
(D)7

Q74·MathematicsNumericalJEE Main 2023

If

a

and

b

are the roots of equation

x^2 - 7x - 1 = 0

, then the value of

\frac{a^{21} + b^{21} + a^{17} + b^{17}}{a^{19} + b^{19}}

is equal to _______ .

Q75·MathematicsNumericalJEE Main 2023

Let a, b, c be three distinct positive real numbers such that

(2a)^{\log_e a}=(bc)^{\log_e b}

and

b^{\log_e 2}=a^{\log_e c}

. Then

6a+5bc

is equal to _________.

Q76·MathematicsNumericalJEE Main 2023

The number of elements in the set

\{n\in\mathbb{Z}:|n^2-10n+19|<6\}

is _________.

Q77·MathematicsSingle correctJEE Main 2023

Let

\alpha,\beta,\gamma

be the three roots of the equation

x^{3}+bx+c=0

. If

\beta\gamma=1=-\alpha

, then

b^{3}+2c^{3}-3\alpha^{3}-6\beta^{3}-8\gamma^{3}

is equal to

(A)21
(B) $\frac{169}{8}$
(C)19
(D) $\frac{155}{8}$

Q78·MathematicsNumericalJEE Main 2023

Let m and n be the numbers of real roots of the quadratic equations

x^{2}-12x+[x]+31=0

and

x^{2}-5|x+2|-4=0

respectively, where

[x]

denotes the greatest integer

\leq x

. Then

m^{2}+mn+n^{2}

is equal to

Q79·MathematicsSingle correctJEE Main 2023

The sum of all the roots of the equation

|x^{2}-8x+15|-2x+7=0

is:

(A) $9+\sqrt3$
(B) $11+\sqrt3$
(C) $9-\sqrt3$
(D) $11-\sqrt3$

Q80·MathematicsSingle correctJEE Main 2023

The number of integral values of k, for which one root of the equation

2x^2-8x+k=0

lies in the interval

(1,2)

and its other root lies in the interval

(2,3)

, is:

(A) $3$
(B) $0$
(C) $2$
(D) $1$

Q81·MathematicsSingle correctJEE Main 2023

Let

S=\{x:x\in\mathbb{R}\text{ and }(\sqrt{3}+\sqrt{2})^{x^2-4}+(\sqrt{3}-\sqrt{2})^{x^2-4}=10\}

. Then

n(S)

is equal to

(A) $4$
(B) $0$
(C) $6$
(D) $2$

Q82·MathematicsSingle correctJEE Main 2023

The number of real roots of the equation

\sqrt{x^2 - 4x + 3} + \sqrt{x^2 - 9} = \sqrt{4x^2 - 14x + 6}

, is :

(A) $3$
(B) $1$
(C) $2$
(D) $0$

Q83·MathematicsSingle correctJEE Main 2023

The equation

e^{4x}+8e^{3x}+13e^{2x}-8e^{x}+1=0,\ x\in\mathbb{R}

has:

(A)four solutions two of which are negative
(B)two solutions and only one of them is negative
(C)two solutions and both are negative
(D)no solution

Q84·MathematicsNumericalJEE Main 2023

If the value of the real number

a > 0

for which

x^2 - 5ax + 1=0

and

x^2 - ax - 5=0

have a common real root is

\dfrac{3}{\sqrt{2\beta}}

, then

\beta

is equal to _______.

Q85·MathematicsSingle correctJEE Main 2023

Let

\lambda \neq 0

be a real number. Let

\alpha, \beta

be the roots of the equation

14x^2 - 31x + 3\lambda = 0

and

\alpha, \gamma

be the roots of the equation

35x^2 - 53x + 4\lambda = 0

. Then

\dfrac{3\alpha}{\beta}

and

\dfrac{4\alpha}{\gamma}

are the roots of the equation

(A) $49x^2 - 245x + 250 = 0$
(B) $7x^2 + 245x - 250 = 0$
(C) $7x^2 - 245x + 250 = 0$
(D) $49x^2 + 245x + 250 = 0$

Q86·MathematicsNumericalJEE Main 2023

Let

a\in\mathbb{R}

and let

\alpha,\beta

be the roots of the equation

x^{2}+60^{\frac{1}{4}}x+a=0

. If

\alpha^{4}+\beta^{4}=-30

, then the product of all possible values of

a

is

Q87·MathematicsNumericalJEE Main 2023

Let

S=\Big\{\alpha:\log_2(9^{2\alpha-4}+13)-\log_2\big(\tfrac{5}{2}\cdot 3^{2\alpha-4}+1\big)=2\Big\}

. Then the maximum value of

\beta

for which the equation

x^2-2\left(\displaystyle\sum_{\alpha\in S}\alpha\right)^2 x+\sum_{\alpha\in S}(\alpha+1)^2\,\beta=0

has real roots, is _______.

Q88·MathematicsSingle correctJEE Main 2023

The number of real solutions of the equation

3\left(x^2+\dfrac{1}{x^2}\right)-2\left(x+\dfrac{1}{x}\right)+5=0

, is

(A) $0$
(B) $3$
(C) $4$
(D) $2$

Q89·MathematicsNumericalJEE Main 2023

Let

\lambda\in\mathbb{R}

and let the equation

E

be

|x|^2-2|x|+|\lambda-3|=0

. Then the largest element in the set

S=\{x+\lambda:x \text{ is an integer solution of } E\}

is _______ .

Q90·MathematicsIntegerJEE Advanced 2022

The product of all positive real values of x satisfying the equation

x^{\left(16\left(\log_{5}x\right)^{3} - 68\log_{5}x\right)} = 5^{-16}

is ________.

Q91·MathematicsNumericalJEE Main 2022

Let

\alpha, \beta

(\alpha > \beta)

be the roots of the quadratic equation

x^2 - x - 4 = 0

. If

P_n = \alpha^n - \beta^n

,

n \in \mathbb{N}

, then

\frac{P_{15}P_{16} - P_{14}P_{16} - P_{15}^2 + P_{14}P_{15}}{P_{13}P_{14}}

is equal to ______.

Q92·MathematicsNumericalJEE Main 2022

The sum of all real values of x for which

\frac{3x^{2} - 9x + 17}{x^{2} + 3x + 10} = \frac{5x^{2} - 7x + 19}{3x^{2} + 5x + 12}

is equal to ____.

Q93·MathematicsNumericalJEE Main 2022

For

p, q \in R

, consider the real valued function

f(x) = (x - p)^{2} - q

,

x \in R

and q > 0. Let

a_{1}

,

a_{2}

,

a_{3}

and

a_{4}

be in an arithmetic progression with mean p and positive common difference. If

\left| f\left(a_{i}\right) \right| = 500

for all i = 1, 2, 3, 4, then the absolute difference between the roots of f(x) = 0 is ______.

Q94·MathematicsSingle correctJEE Main 2022

Let

f(x)=ax^{2}+bx+c

be such that

f(1)=3

,

f(-2)=\lambda

and

f(3)=4

. If

f(0)+f(1)+f(-2)+f(3)=14

, then

\lambda

is equal to

(A) $-4$
(B) $\frac{13}{2}$
(C) $\frac{23}{2}$
(D)4

Q95·MathematicsSingle correctJEE Main 2022

Let

\alpha

,

\beta

be the roots of the equation

x^{2}-\sqrt{2}x+\sqrt{6}=0

and

\frac{1}{\alpha^{2}}+1, \frac{1}{\beta^{2}}+1

be the roots of the equation

x^{2}+ax+b=0

. Then the roots of the equation

x^{2}-(a+b-2)x+(a+b+2)=0

are :

(A)non-real complex numbers
(B)real and both negative
(C)real and both positive
(D)real and exactly one of them is positive

Q96·MathematicsNumericalJEE Main 2022

Let

f\left(x\right)=2x^{2}-x-1

and

S=\left\{n\in\mathbb{Z}:\left|f\left(n\right)\right|\le 800\right\}

. Then, the value of

\sum_{n\in S}f\left(n\right)

is equal to ________.

Q97·MathematicsSingle correctJEE Main 2022

If

\alpha, \beta

are the roots of the equation

x^{2}-\left(5+3^{\sqrt{\log_{3}5}}-5^{\sqrt{\log_{5}3}}\right)+3\left(3^{(\log_{3}5)^{\frac{1}{3}}}-5^{(\log_{5}3)^{\frac{2}{3}}}-1\right)=0

then the equation, whose roots are

\alpha+\frac{1}{\beta}

and

\beta+\frac{1}{\alpha}

,

(A) $3x^{2}-20x-12=0$
(B) $3x^{2}-10x-4=0$
(C) $3x^{2}-10x+2=0$
(D) $3x^{2}-20x+16=0$

Q98·MathematicsNumericalJEE Main 2022

The number of distinct real roots of the equation x

^{5}

(x

^{3}

− x

^{2}

− x +1) +x (3x

^{3}

− 4x

^{2}

− 2x + 4) − 1 = 0 is

Q99·MathematicsNumericalJEE Main 2022

If for some p, q, r ∈ R, not all have same sign, one of the roots of the equation (p

^{2}

+ q

^{2}

)x

^{2}

− 2q(p + r)x + q

^{2}

+ r

^{2}

= 0 is also a root of the equation x

^{2}

+ 2x − 8 = 0, then

\frac{q^{2}+r^{2}}{p^{2}}

is equal to-

Q100·MathematicsSingle correctJEE Main 2022

The minimum value of the sum of the squares of the roots of

x^{2}+(3-a)x+1=2a

is:

(A)4
(B)5
(C)6
(D)8

Q101·MathematicsNumericalJEE Main 2022

Let a, b be two non-zero real numbers. If p and r are the roots of the equation

x^{2} - 8ax + 2a = 0

and q and s are the roots of the equation

x^{2} + 12bx + 6b = 0

, such that

\frac{1}{p}, \frac{1}{q}, \frac{1}{r}, \frac{1}{s}

are in A.P., then

a^{-1} - b^{-1}

is equal to ________.

Q102·MathematicsNumericalJEE Main 2022

The number of real solutions of the equation

e^{4x} + 4e^{3x} - 58e^{2x} + 4e^{x} + 1 = 0

is ______.

Q103·MathematicsSingle correctJEE Main 2022

Let

f(x)

be a quadratic polynomial such that

f(-2) + f(3) = 0

. If one of the roots of

f(x) = 0

is

-1

, then the sum of the roots of

f(x) = 0

is equal to :

(A) $\frac{11}{3}$
(B) $\frac{7}{3}$
(C) $\frac{13}{3}$
(D) $\frac{14}{3}$

Q104·MathematicsNumericalJEE Main 2022

Let

\alpha

,

\beta

be the roots of the equation

x^{2}-4\lambda x+5=0

and

\alpha

,

\gamma

be the roots of the equation

x^{2}-\left(3\sqrt{2}+2\sqrt{3}\right)x+7+3\lambda\sqrt{3}=0

. If

\beta+\gamma=3\sqrt{2}

, then

(\alpha+2\beta+\gamma)^{2}

is equal to :

Q105·MathematicsNumericalJEE Main 2022

If the sum of all the roots of the equation

e^{2x}-11e^{x}-45e^{-x}+\frac{81}{2}=0

is

\log_e

P, then p is equal to ________.

Q106·MathematicsSingle correctJEE Main 2022

The number of distinct real roots of

x^{4} - 4x + 1 = 0

is :

(A)4
(B)2
(C)1
(D)0

Q107·MathematicsNumericalJEE Main 2022

Let p and q be two real numbers such that

p+q=3

and

p^{4}+q^{4}=369

. Then

\left(\frac{1}{p}+\frac{1}{q}\right)^{-2}

is equal to ________.

Q108·MathematicsNumericalJEE Main 2022

The sum of the cubes of all the roots of the equation

x^{4} - 3x^{3} - 2x^{2} + 3x + 1 = 10

is ______.

Q109·MathematicsSingle correctJEE Main 2022

Let

a, b \in R

be such that the equation

ax^2 - 2bx + 15 = 0

has a repeated root

\alpha

. If

\alpha

and

\beta

are the roots of the equation

x^2 - 2bx + 21 = 0

, then

\alpha^2 + \beta^2

is equal to:

(A)37
(B)58
(C)68
(D)92

Q110·MathematicsSingle correctJEE Main 2022

If the sum of the squares of the reciprocals of the roots

\alpha

and

\beta

of the equation

3x^{2}+\lambda x-1=0

is 15, then

6(\alpha^{3}+\beta^{3})^{2}

is equal to :

(A)18
(B)24
(C)36
(D)96

Q111·MathematicsSingle correctJEE Main 2022

The sum of all the real roots of the equation

(e^{2x} - 4)(6e^{2x} - 5e^{x} + 1) = 0

is

(A) $\log_e 3$
(B) $-\log_e 3$
(C) $\log_e 6$
(D) $-\log_e 6$

Q112·MathematicsIntegerJEE Advanced 2021

For

x \in \mathbb{R}

, then number of real roots of the equation

3x^2 - 4\left|x^2 - 1\right| + x - 1 = 0

is ____.

Q113·MathematicsNumericalJEE Main 2021

Let f(x) be a polynomial of degree 3 such that

f(k) = -\frac{2}{k}

for k = 2, 3, 4, 5. Then the value of 52 − 10 f(10) is equal to :

Q114·MathematicsSingle correctJEE Main 2021

The numbers of pairs (a, b) of real numbers, such that whenever α is a root of the equation

x^{2} + ax + b = 0

,

\alpha^{2} - 2

is also a root of this equation, is :

(A)6
(B)2
(C)4
(D)8

Q115·MathematicsSingle correctJEE Main 2021

The number of real roots of the equation

e^{4x} + 2e^{3x} - e^{x} - 6 = 0

is :

(A)2
(B)4
(C)1
(D)0

Q116·MathematicsSingle correctJEE Main 2021

The sum of the roots of the equation

x + 1 - 2\log_2(3 + 2^x) + 2\log_4(10 - 2^{-x}) = 0

, is :

(A) $\log_2 14$
(B) $\log_2 11$
(C) $\log_2 12$
(D) $\log_2 13$

Q117·MathematicsSingle correctJEE Main 2021

The set of all values of k > −1, for which the equation

(3x^{2} + 4x + 3)^{2} - (k+1)(3x^{2} + 4x + 3)(3x^{2} + 4x + 2) + k(3x^{2} + 4x + 2)^{2} = 0

has real roots, is :

(A) $\left(1, \frac{5}{2}\right]$
(B)[2, 3)
(C) $\left[-\frac{1}{2}, 1\right)$
(D) $\left(\frac{1}{2}, \frac{3}{2}\right] - \{1\}$

Q118·MathematicsNumericalJEE Main 2021

The number of distinct real roots of the equation

3x^{4} + 4x^{3} - 12x^{2} + 4 = 0

is _________.

Q119·MathematicsSingle correctJEE Main 2021

If

x^2 + 9y^2 - 4x + 3 = 0

,

x, y \in \mathbb{R}

, then x and y respectively lie in the intervals:

(A) $\left[-\frac{1}{3}, \frac{1}{3}\right]$ and $\left[-\frac{1}{3}, \frac{1}{3}\right]$
(B) $\left[-\frac{1}{3}, \frac{1}{3}\right]$ and $[1, 3]$
(C) $[1, 3]$ and $[1, 3]$
(D) $[1, 3]$ and $\left[-\frac{1}{3}, \frac{1}{3}\right]$

Q120·MathematicsNumericalJEE Main 2021

Let

\lambda \ne 0

be in

\mathbf{R}

. If

\alpha

and

\beta

are the roots of the equation

x^{2} - x + 2\lambda = 0

, and

\alpha

and

\gamma

are the roots of equation

3x^{2} - 10x + 27\lambda = 0

, then

\frac{\beta\gamma}{\lambda}

is equal to ________.

Q121·MathematicsNumericalJEE Main 2021

The sum of all integral values of k

(k \neq 0)

for which the equation

\frac{2}{x-1} - \frac{1}{x-2} = \frac{2}{k}

in x has no real roots, is ________.

Q122·MathematicsSingle correctJEE Main 2021

Let

\alpha = \max_{x \in R} \left\{ 8^{2\sin 3x} . 4^{4\cos 3x} \right\}

and

\beta = \min_{x \in R} \left\{ 8^{2\sin 3x} . 4^{4\cos 3x} \right\}

. If

8x^2 + bx + c = 0

is a quadratic equation whose roots are

\alpha^{1/5}

and

\beta^{1/5}

, then the value of c − b is equal to :

(A)42
(B)43
(C)47
(D)50

Q123·MathematicsSingle correctJEE Main 2021

Let

\alpha,\beta

be two roots of the equation

x^2+\left(20\right)^{\frac{1}{4}}x+\left(5\right)^{\frac{1}{2}}=0

Then

\alpha^8+\beta^8

is equal to :

(A)50
(B)100
(C)10
(D)160

Q124·MathematicsNumericalJEE Main 2021

The number of real roots of the equation

e^{4x}

−

e^{3x}

−

4e^{2x}

−

e^{x}

+ 1 = 0 is equal to…………

Q125·MathematicsNumericalJEE Main 2021

If

\alpha, \beta

are roots of the equation

x^2 + 5\left( \sqrt{2} \right)x + 10 = 0

,

\alpha > \beta

and

P_n = \alpha^n - \beta^n

for each positive integer n, then the value of

\left( \frac{P_{17}P_{20} + 5\sqrt{2}P_{17}P_{19}}{P_{18}P_{19} + 5\sqrt{2}P_{18}^2} \right)

is equal to..........

Q126·MathematicsSingle correctJEE Main 2021

The number of real solutions of the equation,

x^2 - \left|x\right| - 12 = 0

is :

(A)1
(B)4
(C)3
(D)2

Q127·MathematicsNumericalJEE Main 2021

If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of

a^{4} + b^{4} + c^{4}

is equal to.....

Q128·MathematicsSingle correctJEE Main 2021

The number of real roots of the equation

e^{6x} - e^{4x} - 2e^{3x} - 12e^{2x} + e^{x} + 1 = 0

is :

(A)1
(B)4
(C)2
(D)6

Q129·MathematicsSingle correctJEE Main 2021

If

\alpha

and

\beta

are the distinct roots of the equation

x^2+\left(3\right)^{\frac{1}{4}}x+3^{\frac{1}{2}} = 0

, then the value of

\alpha^{96}\left(\alpha^{12}-1\right)+\beta^{96}\left(\beta^{12}-1\right)

is equal to :

(A) $56 \times 3^{25}$
(B) $28 \times 3^{25}$
(C) $52 \times 3^{24}$
(D) $56 \times 3^{24}$

Q130·MathematicsNumericalJEE Main 2021

The number of solutions of the equation

\log_{(x+1)}\left(2x^{2}+7x+5\right)+\log_{(2x+5)}(x+1)^{2}-4=0

,

x>0

, is………

Q131·MathematicsSingle correctJEE Main 2021

The probability of selecting integers

a \in [-5, 30]

such that

x^2+2(a+4)x-5a+64 > 0

, for all

x \in R

is :

(A) $\frac{2}{9}$
(B) $\frac{1}{6}$
(C) $\frac{7}{36}$
(D) $\frac{1}{4}$

Q132·MathematicsSingle correctJEE Main 2021

The value of

3 + \cfrac{1}{4 + \cfrac{1}{3 + \cfrac{1}{4 + \cfrac{1}{3 + ...\infty}}}}

is equal to

(A) $1.5 + \sqrt{3}$
(B) $2 + \sqrt{3}$
(C) $3 + 2\sqrt{3}$
(D) $4 + \sqrt{3}$

Q133·MathematicsSingle correctJEE Main 2021

The number of elements in the set {x ∈ ℝ : (|x| − 3) |x + 4| = 6} is equal to

(A)3
(B)2
(C)4
(D)1

Q134·MathematicsSingle correctJEE Main 2021

Let

P(x)=x^{2}+bx+c

be a quadratic polynomial with real coefficients such that

\int\limits_{0}^{1}P(x)dx=1

and P(x) leaves remainder 5 when it is divided by

(x - 2)

. Then the value of

9(b + c)

is equal to:

(A) $9$
(B) $15$
(C) $7$
(D) $11$

Q135·MathematicsNumericalJEE Main 2021

The number of solutions of the equation

\log_{4}(x-1)=\log_{2}(x-3)

is _______.

Q136·MathematicsNumericalJEE Main 2021

Let

\alpha

and

\beta

be two real numbers such that

\alpha + \beta = 1

and

\alpha\beta = -1

. Let

P_{n} = (\alpha)^{n} + (\beta)^{n}

,

P_{n-1} = 11

and

P_{n+1} = 29

for some integer

n \geq 1

. Then, the value of

P_{n}^{2}

is ________________.

Q137·MathematicsSingle correctJEE Main 2021

Let α and β be the roots of x

^{2}

– 6x – 2 = 0. If an = α

^{n}

– β

^{n}

for n ≥ 1, then the value of a

^{10}

– 2a

^{8}

3a

_{9}

is:

(A)4
(B)1
(C)2
(D)3

Q138·MathematicsSingle correctJEE Main 2021

The integer 'k', for which the inequality

x^{2} - 2(3k - 1)x + 8k^{2} - 7 > 0

is valid for every x in R is :

(A) $3$
(B) $2$
(C) $4$
(D) $0$

Q139·MathematicsSingle correctJEE Main 2021

Let a, b, c be in arithmetic progression. Let the centroid of the triangle with vertices

(a,c),(2,b)

and

(a,b)

be

\left(\frac{10}{3},\frac{7}{3}\right)

. If

\alpha,\beta

are the roots of the equation

ax^{2}+bx+1=0

, then the value of

\alpha^{2}+\beta^{2}-\alpha\beta

is:

(A) $\frac{71}{256}$
(B) $-\frac{69}{256}$
(C) $\frac{69}{256}$
(D) $-\frac{71}{256}$

Q140·MathematicsSingle correctJEE Main 2021

Let p and q be two positive number such that

p + q = 2

and

p^4 + q^4 = 272

. Then p and q are roots of the equation :

(A) $x^2 - 2x + 2 = 0$
(B) $x^2 - 2x + 8 = 0$
(C) $x^2 - 2x + 136 = 0$
(D) $x^2 - 2x + 16 = 0$

Q141·MathematicsNumericalJEE Main 2021

The number of the real roots of the equation

(x+1)^{2}+|x-5|=\frac{27}{4}

is________.

Q142·MathematicsSingle correctJEE Advanced 2020

Suppose a, b denote the distinct real roots of the quadratic polynomial

x^{2} + 20x - 2020

and suppose c,d denote the distinct complex roots of the quadratic polynomial

x^{2} - 20x + 2020

. Then the value of

ac(a - c) + ad(a - d) + bc(b - c) + bd(b - d)

is

(A) $0$
(B) $8000$
(C) $8080$
(D) $16000$

Q143·MathematicsSingle correctJEE Main 2020

If

\alpha

and

\beta

are the roots of the equation

2x(2x + 1) = 1

, then

\beta

is equal to:

(A) $2\alpha(\alpha + 1)$
(B) $-2\alpha(\alpha + 1)$
(C) $2\alpha(\alpha - 1)$
(D) $2\alpha^{2}$

Q144·MathematicsSingle correctJEE Main 2020

If

\alpha

and

\beta

are the roots of the equation,

7x^2-3x-2=0

, then the value of

\dfrac{\alpha}{1-\alpha^2}+\dfrac{\beta}{1-\beta^2}

is equal to:

(A) $\dfrac{27}{32}$
(B) $\dfrac{27}{16}$
(C) $\dfrac{3}{8}$
(D) $\dfrac{1}{24}$

Q145·MathematicsSingle correctJEE Main 2020

The product of the roots of the equation

9x^{2} - 18|x| + 5 = 0

, is:

(A) $\dfrac{5}{9}$
(B) $\dfrac{25}{9}$
(C) $\dfrac{5}{27}$
(D) $\dfrac{25}{81}$

Q146·MathematicsSingle correctJEE Main 2020

Let

\lambda \neq 0

be in R. If

\alpha

and

\beta

are the roots of the equation,

x^{2} - x + 2\lambda = 0

and

\alpha

and

\gamma

are the roots of the equation,

3x^{2} - 10x + 27\lambda = 0

, then

\frac{\beta\gamma}{\lambda}

is equal to:

(A)18
(B)36
(C)9
(D)27

Q147·MathematicsSingle correctJEE Main 2020

Let

\alpha

and

\beta

be roots of

x^{2} - 3x + p = 0

and

\gamma

and

\delta

be the roots of

x^{2} - 6x + q = 0

. If

\alpha

,

\beta

,

\gamma

,

\delta

, form a geometric progression. Then ratio

(2q + p) : (2q - p)

is:

(A) $33 : 31$
(B) $5 : 3$
(C) $3 : 1$
(D) $9 : 7$

Q148·MathematicsSingle correctJEE Main 2020

The set of all real values of

\lambda

for which the quadratic equations,

\left(\lambda^2 + 1\right)x^2 - 4\lambda x + 2 = 0

always have exactly one root in the interval (0, 1) is:

(A) $(0, 2)$
(B) $(2, 4]$
(C) $(-3, -1)$
(D) $(1, 3]$

Q149·MathematicsSingle correctJEE Main 2020

Let

a, b \in R, a \neq 0

be such that the equation

ax^{2} - 2bx + 5 = 0

has a repeated root

\alpha

, which is also a root of the equation,

x^{2} - 2bx - 10 = 0

. If

\beta

is the other root of this equation, then

\alpha^{2} + \beta^{2}

is equal to:

(A)28
(B)24
(C)26
(D)25

Q150·MathematicsSingle correctJEE Main 2020

The number of real roots of the equation,

e^{4x}+e^{3x}-4e^{2x}+e^{x}+1=0

is:

(A)2
(B)4
(C)3
(D)1

Q151·MathematicsNumericalJEE Main 2020

The least positive value of 'a' for which the equation,

2x^{2}+(a-10)x+\dfrac{33}{2}=2a

has real root is

Q152·MathematicsSingle correctJEE Main 2020

Let S be the set of all real roots of the equation,

3^{x}\left(3^{x}-1\right)+2=\left|3^{x}-1\right|+\left|3^{x}-2\right|

. Then S:

(A)contains at least four elements
(B)is a singleton
(C)is an empty set
(D)contains exactly two elements

Q153·MathematicsSingle correctJEE Main 2020

Let

\alpha

and

\beta

be two real roots of the equation

(k+1)\tan^{2}x - \sqrt{2}.\lambda \tan x = (1-k)

, where

k(\ne -1)

and

\lambda

are real numbers. If

\tan^{2}(\alpha+\beta) = 50

, then a value of

\lambda

is:

(A)10
(B)5
(C) $5\sqrt{2}$
(D) $10\sqrt{2}$

Q154·MathematicsSingle correctJEE Main 2020

Let

\alpha

and

\beta

be the roots of the equation

x^{2}-x-1=0

. If

P_{k}=(\alpha)^{k}+(\beta)^{k}, k \ge 1

, then which one of the following statements is not true?

(A) $p_{5}=11$
(B) $(p_{1}+p_{2}+p_{3}+p_{4}+p_{5})=26$
(C) $p_{3}=p_{5}-p_{4}$
(D) $p_{5}=p_{2}.p_{3}$

Q155·MathematicsSingle correctJEE Main 2019

If

\alpha

,

\beta

and

\gamma

are three consecutive terms of a non-constant G.P. such that the equations

\alpha x^{2} + 2\beta x + \gamma = 0

and

x^{2} + x - 1 = 0

have a common root, then

\alpha(\beta+\gamma)

is equal to :

(A) $\alpha\gamma$
(B)0
(C) $\alpha\beta$
(D) $\beta\gamma$

Q156·MathematicsSingle correctJEE Main 2019

If α and β are the roots of the quadratic equation, x

^{2}

+ x sinθ − 2sinθ = 0, θ ∈ (0,

\frac{\pi}{2}

) then

\frac{\alpha^{12}+\beta^{12}}{(\alpha^{-12}+\beta^{-12})(\alpha-\beta)^{24}}

is equal to:

(A) $\frac{2^{12}}{(\sin\theta+8)^{12}}$
(B) $\frac{2^{12}}{(\sin\theta-4)^{12}}$
(C) $\frac{2^{12}}{(\sin\theta-8)^{6}}$
(D) $\frac{2^{6}}{(\sin\theta+8)^{12}}$

Q157·MathematicsSingle correctJEE Main 2019

The number of real roots of the equation

5 + |2^{x} - 1| = 2^{x}(2^{x} - 2)

is

(A)4
(B)3
(C)2
(D)1

Q158·MathematicsSingle correctJEE Main 2019

If m is chosen in the quadratic equation

\left(m^{2}+1\right)x^{2}-3x+\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:

(A) $8\sqrt{3}$
(B) $4\sqrt{3}$
(C) $10\sqrt{5}$
(D) $8\sqrt{5}$

Q159·MathematicsSingle correctJEE Main 2019

The number of integral values of m for which the equation

\left(1+m^{2}\right)x^{2}-2(1+3m)x+(1+8m)=0

has no real root is:

(A)infinitely many
(B)2
(C)3
(D)1

Q160·MathematicsSingle correctJEE Main 2019

The sum of the solutions of the equation

\left|\sqrt{x}-2\right|+\sqrt{x}\left(\sqrt{x}-4\right)+2=0

,

(x>0)

is equal to:

(A) $9$
(B) $4$
(C) $10$
(D) $12$

Q161·MathematicsSingle correctJEE Main 2019

If

\lambda

be the ratio of the roots of the quadratic equation in x,

3m^{2}x^{2}+m(m-4)x+2=0

, then the least value of m for which

\lambda+\frac{1}{\lambda}=1

, is:

(A) $2-\sqrt{3}$
(B) $4-3\sqrt{2}$
(C) $-2+\sqrt{2}$
(D) $4-2\sqrt{3}$

Q162·MathematicsSingle correctJEE Main 2019

The number of integral values of m for which the quadratic expression,

(1 + 2m)x^{2} - 2(1 + 3m)x + 4(1 + m)

,

x \in R

, is always positive, is :

(A)3
(B)8
(C)7
(D)6

Q163·MathematicsSingle correctJEE Main 2019

If one real root of the quadratic equation

81x^{2}+kx+256=0

is cube of the other root, then a value of k is:

(A)− 81
(B)100
(C)144
(D)− 300

Q164·MathematicsSingle correctJEE Main 2019

Let α and β be the roots of the quadratic equation

x^{2}\sin\theta-x\left(\sin\theta\cos\theta+1\right)+\cos\theta=0\;\left(0<\theta<45^{0}\right)

, and α < β. Then

\displaystyle\sum_{n=0}^{\infty}\left(\alpha^{n}+\dfrac{(-1)^{n}}{\beta^{n}}\right)

is equal to:

(A) $\dfrac{1}{1-\cos\theta}-\dfrac{1}{1+\sin\theta}$
(B) $\dfrac{1}{1+\cos\theta}+\dfrac{1}{1-\sin\theta}$
(C) $\dfrac{1}{1-\cos\theta}+\dfrac{1}{1+\sin\theta}$
(D) $\dfrac{1}{1+\cos\theta}-\dfrac{1}{1-\sin\theta}$

Q165·MathematicsSingle correctJEE Main 2019

The number of all possible positive integral values of

\alpha

for which the roots of the quadratic equation,

6x^{2} - 11x + \alpha = 0

are rational numbers is:

(A)2
(B)5
(C)3
(D)4

Q166·MathematicsSingle correctJEE Advanced 2017

PARAGRAPH 2 Let

p

,

q

be integers and let

\alpha

,

\beta

be the roots of the equation

x^{2} - x - 1 = 0

, where

\alpha \ne \beta

. For

n = 0, 1, 2, \dots

, let

a_{n} = p\alpha^{n} + q\beta^{n}

. FACT: If

a

and

b

are rational numbers and

a + b\sqrt{5} = 0

, then

a = 0 = b

. If

a_{4} = 28

, then

p + 2q =

(A) $21$
(B) $14$
(C) $7$
(D) $12$

Q167·MathematicsSingle correctJEE Advanced 2016

Let

-\frac{\pi}{6} < \theta < -\frac{\pi}{12}

. Suppose

\alpha_1

and

\beta_1

are the roots of the equation

x^{2} - 2x\sec\theta + 1 = 0

and

\alpha_2

and

\beta_2

are the roots of the equation

x^{2} + 2x\tan\theta - 1 = 0

. If

\alpha_1 > \beta_1

and

\alpha_2 > \beta_2

, then

\alpha_1 + \beta_2

equals

(A) $2(\sec\theta - \tan\theta)$
(B) $2\sec\theta$
(C) $-2\tan\theta$
(D) $0$

Q168·MathematicsMultiple correctJEE Advanced 2015

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) $\left(-\dfrac{1}{2}, -\dfrac{1}{\sqrt{5}}\right)$
(B) $\left(-\dfrac{1}{\sqrt{5}}, 0\right)$
(C) $\left(0, \dfrac{1}{\sqrt{5}}\right)$
(D) $\left(\dfrac{1}{\sqrt{5}}, \dfrac{1}{2}\right)$

Q169·MathematicsMultiple correctJEE Advanced 2014

Let

a \in \mathbb{R}

and let

f: \mathbb{R} \to \mathbb{R}

be given by

f(x) = x^5 - 5x + a

, then

(A) $f(x)$ has three real roots if $a > 4$
(B) $f(x)$ has only one real roots if $a > 4$
(C) $f(x)$ has three real roots if $a < -4$
(D) $f(x)$ has three real roots if $-4 < a < 4$

Q170·MathematicsSingle correctJEE Advanced 2014

The quadratic equation

p(x) = 0

with real coefficients has purely imaginary roots. Then the equation

p(p(x)) = 0

has

(A)only purely imaginary roots
(B)all real roots
(C)two real and two purely imaginary roots
(D)neither real nor purely imaginary roots

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