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JEE Quadratic Equation PYQs with Solutions

517 previous year questions from Quadratic Equation (JEE Main & Advanced Mathematics) with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

517 questionsMathematics2002–2026Answers + solutions

About Quadratic Equation PYQs

Quadratic Equation has 517 previous year questions in the Quantrex Academy JEE Main & Advanced Mathematics bank. The 220 questions in this list come from JEE Main (199), JEE Advanced (19) and NTA Abhyas JEE Main (2), spanning 2002 to 2026. Each question opens on its own page with the options, the correct answer and the worked solution where available.

Questions by year (this list)

  • 2026: 21
  • 2025: 14
  • 2024: 20
  • 2023: 19
  • 2022: 18
  • 2021: 24
  • 2020: 21
  • 2019: 19
  • 2018: 5
  • 2017: 5
  • 2016: 4
  • 2015: 3

Quadratic Equation questions with solutions

1

Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I List-II (P) If α and β are the distinct roots of the equation x^2 + x + 1 = 0

JEE Advanced 2026 Solution View → 2

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)

JEE Advanced 2026 Solution View → 3

Consider the quadratic equation (n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0 , n ∈ ℝ . Let α be the minimum value of the product of its roots and β be the maximum value of the

JEE Main 2026 Solution View → 4

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:

JEE Main 2026 Solution View → 5

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

JEE Main 2026 Solution View → 6

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 :

JEE Main 2026 Solution View → 7

Let a, b ∈ ℂ . Let α, β be the roots of the equation x^2 + ax + b = 0 . If β - α = 11 and β^2 - α^2 = 3i 11 , then (β^3 - α^3)^2 is equal to:

JEE Main 2026 Solution View → 8

Let A, B , where A, B ∈ (-(π)/(2), (π)/(2)) , be the roots of the quadratic equation x^2 - 2x - 5 = 0 . Then 20 ^2((A+B)/(2)) is equal to:

JEE Main 2026 Solution View → 9

If α=1 and β=1+i 2 , where i= -1 are two roots of the equation x^3+ax^2+bx+c=0 , a,b,c ∈ ℝ , then ∫_ -1 ^ 1 (x^3+ax^2+bx+c)dx is equal to:

JEE Main 2026 Solution View → 10

If the quadratic equation (λ+2)x^2-3λ x+4λ=0 , λ ≠ -2 , has two positive roots, then the number of possible integral values of λ is:

JEE Main 2026 Solution View → 11

Let α, β be the roots of the equation x^2 - 3x + r = 0 , and (α)/(2), 2β be the roots of the equation x^2 + 3x + r = 0 . If the roots of the equation x^2 + 6x = m are 2α + β + 2r

JEE Main 2026 Solution View → 12

Let a, b, c ∈ \ 1, 2, 3, 4\ . If the probability, that ax^2 + 2 2 \,bx + c > 0 for all x ∈ ℝ , is (m)/(n) , (m, n) = 1 , then m + n is equal to _______.

JEE Main 2026 Solution View → 13

Let S=\ x^ 3 +a x^ 2 +b x+c: a, b, c ∈ ~N. and .a, b, c ≤ 20\ be a set of polynomials. Then the number of polynomials in S, which are divisible by x^ 2 +2 , is

JEE Main 2026 Solution View → 14

If α, β , where α<β , are the roots of the equation λ x^ 2 -(λ+3) x+3=0 such that (1)/(α)-(1)/(β)=(1)/(3) , then the sum of all possible values of λ is

JEE Main 2026 Solution View → 15

The smallest positive integral value of a , for which all the roots of x^ 4 -a x^ 2 +9=0 are real and distinct, is equal to

JEE Main 2026 Solution View → 16

The sum of all the real solutions of the equation _ (x+3) (6 x^ 2 +28 x+30)=5-2 _ (6 x+10) (x^ 2 +6 x+9) is equal to

JEE Main 2026 Solution View → 17

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.

JEE Main 2026 Solution View → 18

If α and β(α<β) are the roots of the equation (-2+ 3 )(| x -3|)+(x-6 x )+(9-2 3 )=0, x 0 , then (β)/(α) + α β is equal to :

JEE Main 2026 Solution View → 19

Let α, β be the roots of the quadratic equation 12 x^ 2 -20 x+3 λ=0, λ ∈ Z . If (1)/(2) |β-α| (3)/(2) , then the sum of all possible values of λ is :

JEE Main 2026 Solution View → 20

Let α and β be the roots of the equation x^ 2 +2 a x+(3 a+10)=0 such that α<1<β . Then the set of all possible values of a is :

JEE Main 2026 Solution View → 21

The sum of all the roots of the equation (x-1)^ 2 -5|x-1|+6=0 , is :

JEE Main 2026 Solution View → 22

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 Solution View → 23

For t -1 , let α_t and β_t be the roots of the equation ((t+2)^ (1)/(7) -1) x^2+((t+2)^ (1)/(6) -1) x+((t+2)^ (1)/(21) -1)=0 If _ t arrow-1^ + α_t=a and _ t arrow-1^ + β_t=b

JEE Main 2025 Solution View → 24

The number of real roots of the equation x|x-2|+3|x-3|+1=0 is :

JEE Main 2025 Solution View → 25

Let the set of all values of p ∈ ℝ , for which both the roots of the equation x^2-(p+2) x+(2 p+9)=0 are negative real numbers, be the interval (α, β] . Then β-2 α is equal to

JEE Main 2025 Solution View → 26

Consider the equation x^2+4 x-n=0 , where n ∈[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

JEE Main 2025 Solution View → 27

Let α and β be the roots of x^2+ 3 x -16=0 , and γ and δ be the roots of x^2+3 x-1=0 . If P_n=α^n+β^n and Q_n=γ^n+δ^n , then P_ 25 + 3 P_ 24 2 P_ 23 + Q_ 25 -Q_ 23 Q_ 24 is equal

JEE Main 2025 Solution View → 28

If the set of all a ∈ R-\ 1\ , for which the roots of the equation (1-a) x^2+2(a-3) x+9=0 are positive is (-∞,-α] ∪[β, γ) , then 2 α+β+γ is equal to _______ .

JEE Main 2025 Solution View → 29

Let P_n=α^n+β^n, n ∈ N . If P_ 10 =123, P_9=76 , P_8=47 and P_1=1 , then the quadratic equation having roots (1)/(α) and (1)/(β) is :

JEE Main 2025 Solution View → 30

If the set of all a ∈ 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 ∈ Z: α x β\ , then ∑_ x ∈ X x^2 is equal to :

JEE Main 2025 Solution View → 31

The number of solutions of the equation \(((9)/(x)- 9 x +2)((2)/(x)- 7 x +3)=0\) is:

JEE Main 2025 Solution View → 32

The sum, of the squares of all the roots of the equation x^2+|2 x-3|-4=0 , is

JEE Main 2025 Solution View → 33

The product of all the rational roots of the equation (x^2-9 x+11)^2-(x-4)(x-5)=3 , is equal to

JEE Main 2025 Solution View → 34

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=(36)/(5) , then a^2+c^2 is equal to

JEE Main 2025 Solution View → 35

Let α_θ and β_θ be the distinct roots of 2 x^2+( θ) x-1=0, θ ∈(0,2 π) . If m and M are the minimum and the maximum values of α_θ^4+β_θ^4 , then 16(M+m) equals :

JEE Main 2025 Solution View → 36

Let f(x)=x^4+a x^3+b x^2+c be a polynomial with real coefficients such that f(1)=-9 . Suppose that i 3 is a root of the equation 4 x^3+3 a x^2+2 b x=0 , where i= -1 . If α_1, α_2

JEE Advanced 2024 Solution View → 37

Let α, β ; α>β , be the roots of the equation x^2- 2 x- 3 =0 . Let P_n=α^n-β^n, n ∈ N . Then (11 3 -10 2 ) P_ 10 +(11 2 +10) P_ 11 -11 P_ 12 is equal to

JEE Main 2024 Solution View → 38

Let α, β be the roots of the equation x^2+2 2 x-1=0 . The quadratic equation, whose roots are α^4+β^4 and (1)/(10)(α^6+β^6) , is :

JEE Main 2024 Solution View → 39

The number of distinct real roots of the equation |x+1||x+3|-4|x+2|+5=0 , is

JEE Main 2024 Solution View → 40

The sum of all the solutions of the equation (8)^ 2 x -16 ·(8)^x+48=0 is :

JEE Main 2024 Solution View → 41

Let α, β be roots of x^2+ 2 x-8=0 . If U_ n =α^ n +β^n , then U_ 10 + 2 U_9 2 U_8 is equal to______

JEE Main 2024 Solution View → 42

Let α, β be the distinct roots of the equation x^2-(t^2-5 t+6) x+1=0, t ∈ ℝ and a_n=α^n+β^n . Then the minimum value of a_ 2023 +a_ 2025 a_ 2024 is

JEE Main 2024 Solution View → 43

Let x_1, x_2, x_3, x_4 be the solution of the equation 4 x^4+8 x^3-17 x^2-12 x+9=0 and (4+x_1^2)(4+x_2^2)(4+x_3^2)(4+x_4^2)=(125)/(16) m . Then the value of m is

JEE Main 2024 Solution View → 44

The number of real solutions of the equation x|x+5|+2|x+7|-2=0 is_________

JEE Main 2024 Solution View → 45

If 2 and 6 are the roots of the equation a x^2+b x+1=0 , then the quadratic equation, whose roots are (1)/(2 a+b) and (1)/(6 a+b) , is :

JEE Main 2024 Solution View → 46

Let α and β be the roots of the equation p x ^ 2 + qx - r = 0 , where p ≠ 0 . If p, q and r be the consecutive terms of a non-constant G.P and (1)/(α) + (1)/(β) = (3)/(4) , then

JEE Main 2024 Solution View → 47

Let S = \ x ∈ R: ( 3 + 2 ) ^ x + ( 3 - 2 ) ^ x = 10\ . Then the number of elements in S is:

JEE Main 2024 Solution View → 48

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

JEE Main 2024 Solution View → 49

The number of solutions, of the equation e ^ sinx - 2 e ^ - sinx = 2 is

JEE Main 2024 Solution View → 50

Let S be the set of positive integral values of a for which a x ^ 2 + 2(a + 1)x + 9a + 4 x ^ 2 - 8x + 32 < 0, ∀ x ∈ ℝ . Then, the number of elements in S is:

JEE Main 2024 Solution View → 51

For 0 < c < b < a , let ( a + b – 2c ) x ^ 2 + ( b + c – 2a ) x + ( c + a – 2b ) = 0 and α ≠ 1 be one of its root. Then, among the two statements (I) If α ∈ (- 1, 0) , then b

JEE Main 2024 Solution View → 52

The number of real solutions of the equation \(x(x^2+3|x|+5|x-1|+6|x-2|)=0\) is ______.

JEE Main 2024 Solution View → 53

Let \(α, β ∈\) be roots of equation \(x^2-70 x+λ=0\), where \((λ)/(2), (λ)/(3) ∉\). If \(λ\) assumes the minimum possible value, then \( ( α-1 + β-1 )(λ+35) |α-β| \) is equal to :

JEE Main 2024 Solution View → 54

Let the set C = \ ( x, y ) ∣ x ^ 2 - 2 ^ y = 2023, x, y ∈ ℕ\ . Then ∑ _ ( x, y ) ∈ C ( x + y ) is equal to _______.

JEE Main 2024 Solution View → 55

If α, β are the roots of the equation, x ^ 2 - x - 1 = 0 and S _ n = 2023 α ^ n + 2024 β ^ n , then

JEE Main 2024 Solution View → 56

The number of real roots of the equation x|x| - 5|x + 2| + 6 = 0 , is

JEE Main 2023 Solution View → 57

Let α, β be the roots of the equation x ^ 2 - 2x + 2 = 0 Then α ^ 14 + β ^ 14 is equal to

JEE Main 2023 Solution View → 58

Let α, β be the roots of the quadratic equation x ^ 2 + 6 x + 3 = 0 . Then α ^ 23 + β ^ 23 + α ^ 14 + β ^ 14 α ^ 15 + β ^ 15 + α ^ 10 + β ^ 10 is equal to

JEE Main 2023 Solution View → 59

The number of points, where the curve f ( x ) = e ^ 8x - e ^ 6x - 3 e ^ 4x - e ^ 2x + 1, x ∈ ℝ cuts x -axis, is equal to............

JEE Main 2023 Solution View → 60

If a and b are the roots of the equation x ^ 2 - 7x - 1 = 0 , then the value of a ^ 21 + b ^ 21 + a ^ 17 + b ^ 17 a ^ 19 + b ^ 19 is equal to

JEE Main 2023 Solution View → 61

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

JEE Main 2023 Solution View → 62

Let α, β, γ be the three roots of the equation x ^ 3 + bx + c = 0 if βγ = 1 = - α then b ^ 3 + 2 c ^ 3 - 3 α ^ 3 - 6 β ^ 3 - 8 γ ^ 3 is equal to

JEE Main 2023 Solution View → 63

The sum of all the roots of the equation | x ^ 2 - 8x + 15| - 2x + 7 = 0 is

JEE Main 2023 Solution View → 64

The number of integral values of k , for which one root of the equation 2 x ^ 2 - 8x + k = 0 lies in the interval (1, 2) and its other root lies in the interval (2, 3) , is :

JEE Main 2023 Solution View → 65

Let S = \ x: x ∈ ℝand ( 3 + 2 ) ^ x ^ 2 - 4 + ( 3 - 2 ) ^ x ^ 2 - 4 = 10\ . Then n(S) is equal to

JEE Main 2023 Solution View → 66

The equation e ^ 4x + 8 e ^ 3x + 13 e ^ 2x - 8 e ^ x + 1 = 0, x ∈ R has :

JEE Main 2023 Solution View → 67

The number of real roots of the equation x ^ 2 - 4x + 3 + x ^ 2 - 9 = 4 x ^ 2 - 14x + 6 , is:

JEE Main 2023 Solution View → 68

If the value of real number α > 0 for which x ^ 2 - 5 + 1 = 0 and x ^ 2 - - 5 = 0 have a common real roots is 3 2β then β is equal to ________

JEE Main 2023 Solution View → 69

Let α _ 1 , α _ 2 , … , α _ 7 α _ 1 , α _ 2 , … , α _ 7 be the roots of the equation x ^ 7 + 3 x ^ 5 - 13 x ^ 3 - 15x = 0 and | α _ 1 | ≥ | α _ 2 | ≥ … ≥ | α _ 7 | . Then, α _ 1 α

JEE Main 2023 Solution View → 70

Let λ ≠ 0 be a real number. Let α, β be the roots of the equation 14 x ^ 2 - 31x + 3λ = 0 and α, γ be the roots of the equation 35 x ^ 2 - 53x + 4λ = 0 . Then (3α)/(β) and

JEE Main 2023 Solution View → 71

Let a ∈ R and let α, β be the roots of the equation x ^ 2 + 60(1)/(4)x + a = 0 . If α ^ 4 + β ^ 4 = - 30 , then the product of all possible values of a is _____ .

JEE Main 2023 Solution View → 72

Let S = \ α: _ 2 ( 9 ^ 2α - 4 + 13) - _ 2 ((5)/(2) · 3 ^ 2α - 4 + 1) = 2\ . Then the maximum value of β for which the equation x ^ 2 - 2 ( ∑ _ α ∈ s α) ^ 2 x + ∑ _ a ∈ s (α + 1) ^

JEE Main 2023 Solution View → 73

The number of real solutions of the equation 3( x ^ 2 + 1 x ^ 2 ) - 2(x + (1)/(x)) + 5 = 0 , is

JEE Main 2023 Solution View → 74

Let λ ∈ ℝ and let the equation E be | x | ^ 2 - 2 | x | + | λ - 3 | = 0 . Then the largest element in the set S = x + λ: x is an integer solution of E is ______

JEE Main 2023 Solution View → 75

Let α, β(α > β) be the roots of the quadratic equation x ^ 2 - x - 4 = 0 . If P _ n = α ^ n - β ^ n , n ∈ ℕ , then P _ 15 P _ 16 - P _ 14 P _ 16 - P _ 15 ^ 2 + P _ 14 P _ 15 P _

JEE Main 2022 Solution View → 76

Let α, β be the roots of the equation x ^ 2 - 2 x + 6 = 0 and 1 α ^ 2 + 1, 1 β ^ 2 + 1 be the roots of the equation x ^ 2 + ax + b = 0 . Then the roots of the equation x ^ 2 - (a

JEE Main 2022 Solution View → 77

The sum of all real values of x for which 3 x ^ 2 - 9x + 17 x ^ 2 + 3x + 10 = 5 x ^ 2 - 7x + 19 3 x ^ 2 + 5x + 12 is equal to

JEE Main 2022 Solution View → 78

If α, β are the roots of the equation x ^ 2 - (5 + 3 ^ _ 3 5 - 5 ^ _ 5 3 )x + 3( 3 ^ ( _ 3 5)(1)/(3) - 5 ^ ( _ 5 3)(2)/(3) - 1) = 0 then the equation, whose roots are α + (1)/(β)

JEE Main 2022 Solution View → 79

The minimum value of the sum of the squares of the roots of x ^ 2 + (3 - a)x = 2a - 1 is

JEE Main 2022 Solution View → 80

If for some p, q, r ∈ R , all have positive 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 +

JEE Main 2022 Solution View → 81

Let f(x) be a quadratic polynomial with leading coefficient 1 such that f(0) = p, p ≠ 0 , and f(1) = (1)/(3) . If the equations f(x) = 0 and fofofof(x) = 0 have a common real

JEE Main 2022 Solution View → 82

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

JEE Main 2022 Solution View → 83

Let α be a root of the equation 1 + x ^ 2 + x ^ 4 = 0 . Then the value of α ^ 1011 + α ^ 2022 - α ^ 3033 is equal to:

JEE Main 2022 Solution View → 84

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

JEE Main 2022 Solution View → 85

The number of real solutions of the equation e ^ 4x + 4 e ^ 3x - 58 e ^ 2x + 4 e ^ x + 1 = 0 is _____.

JEE Main 2022 Solution View → 86

Let α, β be the roots of the equation x ^ 2 - 4 + 5 = 0 and α, γ be the roots of the equation x ^ 2 - (3 2 + 2 3 )x + 7 + 3λ 3 = 0 . If β + γ = 3 2 , then (α + 2β + γ) ^ 2 is

JEE Main 2022 Solution View → 87

If p and q are real number such that p + q = 3, p ^ 4 + q ^ 4 = 369 , then the value of ((1)/(p) + (1)/(q)) ^ - 2 is equal to

JEE Main 2022 Solution View → 88

The sum of the cubes of all the roots of the equation x ^ 4 - 3 x ^ 3 - 2 x ^ 2 + 3x + 1 = 0 is _____.

JEE Main 2022 Solution View → 89

Let a, b ∈ R be such that the equation a x ^ 2 - 2bx + 15 = 0 has repeated root α and if α and β are the roots of the equation x ^ 2 - 2bx + 21 = 0 , then α ^ 2 + β ^ 2 is equal

JEE Main 2022 Solution View → 90

For a natural number n , let α _ n = 19 ^ n - 12 ^ n . Then, the value of 31 α _ 9 - α _ 10 57 α _ 8 is ______

JEE Main 2022 Solution View → 91

The sum of all real roots of equation ( e ^ 2x - 4)(6 e ^ 2x - 5 e ^ x + 1) = 0 is

JEE Main 2022 Solution View → 92

If the sum of the squares of the reciprocals of the roots α and β of the equation 3 x ^ 2 + - 1 = 0 is 15 , then 6 ( α ^ 3 + β ^ 3 ) ^ 2 is equal to

JEE Main 2022 Solution View → 93

For x ∈ R , the number of real roots of the equation 3 x ^ 2 - 4| x ^ 2 - 1| + x - 1 = 0 is

JEE Advanced 2021 Solution View → 94

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

JEE Main 2021 Solution View → 95

The number of solutions of the equation 32 ^ ^ 2 x + 32 ^ sec ^ 2 x = 81, 0 ≤ x ≤ (π)/(4) is :

JEE Main 2021 Solution View → 96

The set of all values of k > - 1 , for which the equation (3 x ^ 2 + 4x + 3) ^ 2 - ( k + 1 ) (3 x ^ 2 + 4x + 3)(3 x ^ 2 + 4x + 2) + k (3 x ^ 2 + 4x + 2) ^ 2 = 0 has real roots, is:

JEE Main 2021 Solution View → 97

If x ^ 2 + 9 y ^ 2 - 4x + 3 = 0, x, y ∈ R, then x and y respectively lie in the intervals

JEE Main 2021 Solution View → 98

Let λ ≠ 0 be in R . If α and β are the roots of the equation x ^ 2 - x + 2λ = 0, and α and γ are the roots of the equation 3 x ^ 2 - 10x + 27λ = 0, then (βγ)/(λ) is equal to

JEE Main 2021 Solution View → 99

The sum of all integral values of k ( k ≠ 0 ) for which the equation (2)/(x - 1) - (1)/(x - 2) = (2)/(k) in x has no real roots, is_____.

JEE Main 2021 Solution View → 100

The number of real roots of the equation e ^ 4x - e ^ 3x - 4 e ^ 2x - e ^ x + 1 = 0 is equal to

JEE Main 2021 Solution View → 101

Let α = max _ x ∈ R \ 8 ^ 2sin3x · 4 ^ 4cos3x \ and β = min _ x ∈ R \ 8 ^ 2sin3x · 4 ^ 4cos3x \ . If 8 x ^ 2 + bx + c = 0 is a quadratic equation whose roots are α ^ 1 / 5 and β ^

JEE Main 2021 Solution View → 102

Let α, β be two roots of the equation x ^ 2 + (20) ^ 1 / 4 x + (5) ^ 1 / 2 = 0 . Then α ^ 8 + β ^ 8 is equal to

JEE Main 2021 Solution View → 103

The number of real solutions of the equation, x ^ 2 - |x| - 12 = 0 is:

JEE Main 2021 Solution View → 104

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:

JEE Main 2021 Solution View → 105

If α, β are roots of the equation x ^ 2 + 5( 2 )x + 10 = 0, α > β and P _ n = α ^ n - β ^ n for each positive integer n, then the value of ( P _ 17 P _ 20 + 5 2 P _ 17 P _ 19 P _

JEE Main 2021 Solution View → 106

If α and β are the distinct roots of the equation x ^ 2 + (3)(1)/(4)x + 3(1)/(2) = 0 , then the value of α ^ 96 ( α ^ 12 - 1) + β ^ 96 ( β ^ 12 - 1) is equal to:

JEE Main 2021 Solution View → 107

The probability of selecting integers a ∈ [- 5, 30] such that x ^ 2 + 2(a + 4)x - 5a + 64 > 0 , for all x ∈ R , is:

JEE Main 2021 Solution View → 108

The value of 3 + (1)/(4 + (1)/(3 + (1)/(4 + (1)/(3 + … ∞)))) is equal to

JEE Main 2021 Solution View → 109

The value of 4 + (1)/(5 + (1)/(4 + (1)/(5 + (1)/(4 + … … ∞)))) is:

JEE Main 2021 Solution View → 110

Let α and β be two real numbers such that α + β = 1 and αβ = - 1 . Let p _ n = (α) ^ n + (β) ^ n , p _ n - 1 = 11 and p _ n + 1 = 29 for some integer n ⩾ 1 . Then, the value of p

JEE Main 2021 Solution View → 111

The sum of 162 ^ th power of the roots of the equation x ^ 3 - 2 x ^ 2 + 2x - 1 = 0 is ______.

JEE Main 2021 Solution View → 112

Let α and β be the roots of x ^ 2 - 6x - 2 = 0 . If a _ n = α ^ n - β ^ n for n ⩾ 1 , then the value of a _ 10 - 2 a _ 8 3 a _ 9 is:

JEE Main 2021 Solution View → 113

If α, β ∈ R are such that 1 - 2i (here i ^ 2 = - 1 ) is a root of z ^ 2 + + β = 0 , then (α - β) is equal to:

JEE Main 2021 Solution View → 114

The integer k, for which the inequality x ^ 2 - 2(3k - 1)x + 8 k ^ 2 - 7 > 0 is valid for every x in R is:

JEE Main 2021 Solution View → 115

The number of the real roots of the equation (x + 1) ^ 2 + | x - 5 | = (27)/(4) is ________.

JEE Main 2021 Solution View → 116

Let p and q be two positive numbers such that p + q = 2 and p ^ 4 + q ^ 4 = 272 . Then p and q are roots of the equation:

JEE Main 2021 Solution View → 117

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

JEE Advanced 2020 Solution View → 118

If α and β are the roots of the equation 2x(2x + 1) = 1 , then β is equal to :

JEE Main 2020 Solution View → 119

If α and β be two roots of the equation x ^ 2 - 64x + 256 = 0 . Then the value of ( α ^ 3 β ^ 5 )(1)/(8) + ( β ^ 3 α ^ 5 )(1)/(8) is :

JEE Main 2020 Solution View → 120

If α and β are the roots of the equation, 7 x ^ 2 - 3x - 2 = 0, then the value of α 1 - α ^ 2 + β 1 - β ^ 2 is equal to:

JEE Main 2020 Solution View → 121

The product of the roots of the equation 9 x ^ 2 - 18 | x | + 5 = 0 is :

JEE Main 2020 Solution View → 122

Let λ ≠ 0 be in R . If α and β are the roots of the equation, x ^ 2 - x + 2λ = 0 and α and γ are the roots of the equation, 3 x ^ 2 - 10x + 27λ = 0, then (βγ)/(λ) is equal to:

JEE Main 2020 Solution View → 123

Let [t] denote the greatest integer ≤ t . Then the equation in x, [x] ^ 2 + 2[x + 2] - 7 = 0 has :

JEE Main 2020 Solution View → 124

Let α and β be the roots of x ^ 2 - 3x + p = 0 and γ and δ be the roots of x ^ 2 - 6x + q = 0 . If α, β, γ, δ from a geometric progression. Then ratio ( 2q + p ) : ( 2q - p ) is

JEE Main 2020 Solution View → 125

The set of all real values of λ for which the quadratic equation ( λ ^ 2 + 1) x ^ 2 - 4 + 2 = 0 always have exactly one root in the interval ( 0, 1 ) is :

JEE Main 2020 Solution View → 126

Consider the two sets: A = m ∈ R: 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?

JEE Main 2020 Solution View → 127

If α and β are the roots of the equation x ^ 2 + px + 2 = 0 and (1)/(α) and (1)/(β) are the roots of the equation 2 x ^ 2 + 2qx + 1 = 0, then (α - (1)/(α))(β - (1)/(β))(α +

JEE Main 2020 Solution View → 128

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

JEE Main 2020 Solution View → 129

Let α and β be the roots of the equation, 5 x ^ 2 + 6x - 2 = 0 . If S _ n = α ^ n + β ^ n , n = 1, 2, 3, . . . . , then

JEE Main 2020 Solution View → 130

Let a, b ∈ R, a ≠ 0 be such that the equation, a x ^ 2 - 2bx + 5 = 0 has a repeated root α, which is also a root of the equation, x ^ 2 - 2bx - 10 = 0 . If β is the other root of

JEE Main 2020 Solution View → 131

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

JEE Main 2020 Solution View → 132

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

JEE Main 2020 Solution View → 133

The least positive value of &lsquo; a &rsquo; for which the equation, 2 x ^ 2 + (a - 10)x + (33)/(2) = 2a has real roots is ___________.

JEE Main 2020 Solution View → 134

Let α and β be the roots of the equation x ^ 2 - x - 1 = 0 . If p _ k = (α) ^ k + (β) ^ k , k ≥ 1, then which one of the following statements is not true?

JEE Main 2020 Solution View → 135

Let α and β be two real roots of the equation (k + 1) ^ 2 x - 2 ⋅ ⁡ x = (1 - k), where k(≠ - 1) and λ are real numbers. If ^ 2 (α + β) = 50, then a value of λ is

JEE Main 2020 Solution View → 136

If the roots of the equation ax ^ 2 + bx + c = 0 are in the ratio m: n , then

NTA Abhyas JEE Main 2020 Solution View → 137

If f (x) = ∏ _ k = 1 ^ 999 (x^ 2 - 47x + k) , then product of all real roots of f (x) =0 is

NTA Abhyas JEE Main 2020 Solution View → 138

Let α and β be the roots of x ^ 2 - x - 1 = 0, with α > β . For all positive integers n, define a _ n = α ^ n - β ^ n α - β , n ≥ 1 b _ 1 = 1 and b _ n = a _ n - 1 + a _ n + 1 , n

JEE Advanced 2019 Solution View → 139

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

JEE Main 2019 Solution View → 140

If α and β are the roots of the equation 375 x ^ 2 - 25x - 2 = 0, then _ n → ∞ ∑ _ r = 1 ^ n α ^ r + _ n → ∞ ∑ _ r = 1 ^ n β ^ r is equal to:

JEE Main 2019 Solution View → 141

The number of real roots of the equation 5 + | 2 ^ x - 1| = 2 ^ x ( 2 ^ x - 2) is :

JEE Main 2019 Solution View → 142

If α and β are the roots of the quadratic equation x ^ 2 + xsinθ - 2sinθ = 0, θ ∈ (0, (π)/(2)) , then α ^ 12 + β ^ 12 ( α ^ - 12 + β ^ - 12 ) . ( α - β ) ^ 24 is equal to :

JEE Main 2019 Solution View → 143

If m is chosen in the quadratic equation ( m ^ 2 + 1) x ^ 2 - 3x + ( m ^ 2 + 1) ^ 2 = 0 such that the sum of its roots is greatest, then the absolute difference of the cubes of

JEE Main 2019 Solution View → 144

Let p, q ∈ Q . If 2 - 3 is a root of the quadratic equation x ^ 2 + px + q = 0, then

JEE Main 2019 Solution View → 145

If three distinct numbers a, b, c are in G.P. and the equations a x ^ 2 + 2bx + c = 0 and d x ^ 2 + 2ex + f = 0 have a common root, then which one of the following statements is

JEE Main 2019 Solution View → 146

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

JEE Main 2019 Solution View → 147

The sum of the solutions of the equation | x - 2| + x ( x - 4) + 2 = 0, (x > 0) is equal to

JEE Main 2019 Solution View → 148

The number of integral values of m for which the quadratic expression (1 + 2m) x ^ 2 - 2(1 + 3m)x + 4(1 + m), x ∈ R is always positive, is

JEE Main 2019 Solution View → 149

If λ 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 λ + (1)/(λ) = 1 , is :

JEE Main 2019 Solution View → 150

Let α and β be the roots of the quadratic equation x^ 2 θ-x( θ θ+1)+ θ=0(0 < θ < 45^ ∘ ), and α < β . Then ∑_ n=0 ^ ∞ (α^ n + (-1)^ n β^ n ) is equal to :

JEE Main 2019 Solution View → 151

If one real root of the quadratic equation 81 x^ 2 +k x+256=0 is cube of the other root, then a value of k is :

JEE Main 2019 Solution View → 152

The value of λ such that sum of the squares of the roots of the quadratic equation, x ^ 2 + (3 - λ)x + 2 = λ has the least value is:

JEE Main 2019 Solution View → 153

Consider the quadratic equation (c - 5) x ^ 2 - 2cx + (c - 4) = 0, c ≠ 5 . Let S be the set of all integral values of c for which one root of the equation lies in the interval (0

JEE Main 2019 Solution View → 154

The number of all possible positive integral value of α for which the roots of the quadratic equation 6 x ^ 2 - 11x + α = 0 are rational numbers is:

JEE Main 2019 Solution View → 155

If both the roots of the quadratic equation x ^ 2 - mx + 4 = 0 are real and distinct and they lie in the interval (1, 5), then m lies in the interval: Note: In the actual JEE

JEE Main 2019 Solution View → 156

Let α and β be the roots of the equation x ^ 2 + 2x + 2 = 0, then α ^ 15 + β ^ 15 is equal to

JEE Main 2019 Solution View → 157

Let p, q and r be real numbers (p ≠ q, r ≠ 0) , such that the roots of the equation (1)/(x + p) + (1)/(x + q) = (1)/(r) are equal in magnitude but opposite in sign, then the sum

JEE Main 2018 Solution View → 158

If an angle A of a satisfies 5cosA + 3 = 0 , then the roots of the quadratic equation 9 x ^ 2 + 27x + 20 = 0 are

JEE Main 2018 Solution View → 159

If λ ∈ R is such that the sum of the cubes of the roots of the equation, x^2+(2-λ) x+(10-λ)=0 is minimum, then the magnitude of the difference of the roots of this equation is

JEE Main 2018 Solution View → 160

If λ ∈ R is such that the sum of the cubes of the roots of the equation x ^ 2 + (2 - λ)x + (10 - λ) = 0 is minimum, then the magnitude of the difference of the roots of this

JEE Main 2018 Solution View → 161

If tan ⁡ A and tan ⁡ B are the roots of the quadratic equation 3 x ^ 2 - 10x - 25 = 0 , then the value of 3 ^ 2 ⁡ (A + B) - 10sin ⁡ (A + B)cos ⁡ (A + B) - 25 ^ 2 ⁡ (A + B) is :

JEE Main 2018 Solution View → 162

Paragraph: Let p, q be integers and let α, β be the roots of the equation, x^ 2 -x-1=0 , where α ≠ β . For n=0,1,2, … , let a_ n =p α^ n +q β^ n FACT: If a and b are rational

JEE Advanced 2017 Solution View → 163

Paragraph: Let p, q be integers and let α, β be the roots of the equation, x^ 2 -x-1=0 , where α ≠ β . For n=0,1,2, … , let a_ n =p α^ n +q β^ n FACT: If a and b are rational

JEE Advanced 2017 Solution View → 164

The sum of all the real values of x Satisfying the equation 2 ^ (x - 1)( x ^ 2 + 5x - 50) = 1 is:

JEE Main 2017 Solution View → 165

Let p(x) be a quadratic polynomial such that p(0) = 1 . If p(x) leaves remainder 4 when divided by x - 1 and it leaves remainder 6 when divided by x + 1 then:

JEE Main 2017 Solution View → 166

If, for a positive integer n , the quadratic equation, x(x + 1) + (x + 1)(x + 2) + . .. + (x + ¯ n - 1 )(x + n) = 10n has two consecutive integral solutions, then n is equal to:

JEE Main 2017 Solution View → 167

Let - (π)/(6) β _ 1 and α _ 2 > β _ 2 , then α _ 1 + β _ 2 equals

JEE Advanced 2016 Solution View → 168

If x is a solution of the equation 2x + 1 - 2x - 1 = 1, (x ≥ (1)/(2)) , then 4 x ^ 2 - 1 is equal to :

JEE Main 2016 Solution View → 169

If the equations x ^ 2 + bx - 1 = 0 and x ^ 2 + x + b = 0 have a common root different from - 1, then |b| is equal to :

JEE Main 2016 Solution View → 170

The sum of all real values of x satisfying the equation ( x ^ 2 - 5x + 5) ^ x ^ 2 + 4x - 60 = 1 is

JEE Main 2016 Solution View → 171

Let S be the set of all non - zero real numbers α such that the quadratic equation α x ^ 2 - x + α = 0 has two distinct real roots x _ 1 and x _ 2 satisfying the inequality | x _

JEE Advanced 2015 Solution View → 172

If the two roots of the equation, (a - 1)( x ^ 4 + x ^ 2 + 1) + (a + 1) ( x ^ 2 + x + 1) ^ 2 = 0 are real and distinct, then the set of all values of a is equal to

JEE Main 2015 Solution View → 173

Let α and β be the roots of equation x ^ 2 - 6x - 2 = 0 . If a _ n = α ^ n - β ^ n , ∀ n ≥ 1, then the value of a _ 10 - 2 a _ 8 2 a _ 9 is equal to

JEE Main 2015 Solution View → 174

The quadratic equation p ( x ) = 0 with real coefficients has purely imaginary roots. Then the equation p ( p ( x ) ) = 0 has

JEE Advanced 2014 Solution View → 175

The equation 3 x ^ 2 + x + 5 = x - 3 , where x ⁡ is real, has

JEE Main 2014 Solution View → 176

The sum of the roots of the equation, x^2+|2 x-3|-4=0 , is:

JEE Main 2014 Solution View → 177

If α and β are roots of the equation, x^2-4 2 k x+2 e^ 4 k -1=0 for some k , and α^2+β^2=66 , then α^3+β^3 is equal to:

JEE Main 2014 Solution View → 178

If 1 α , 1 β are the roots of the equation a x ^ 2 + bx + 1 = 0, (a ≠ 0, a, b ∈ R) , then the equation x(x + b ^ 3 ) + ( a ^ 3 - 3abx) = 0 has roots:

JEE Main 2014 Solution View → 179

If equations a x ^ 2 + bx + c = 0, (a, b, c ∈ R, a ≠ 0) and 2 x ^ 2 + 3x + 4 = 0 have a common root, then a: b: c equals :

JEE Main 2014 Solution View → 180

If a ∈ R and the equation - 3 ( x - [ x ] ) ^ 2 + 2 ( x - [ x ] ) + a ^ 2 = 0 (where [ x ] denotes the greatest integer ≤ x ) has no integral solution, then all possible values of

JEE Main 2014 Solution View → 181

Let α and β be the roots of equation p x ^ 2 + qx + r = 0 , p ≠ 0 . If p, q, r are in A.P. and (1)/(α) + (1)/(β) = 4 , then the value of |α - β| is

JEE Main 2014 Solution View → 182

If p and q are non-zero real numbers and α^3+β^3=-p, α β=q , then a quadratic equation whose roots are (α^2)/(β), (β^2)/(α) is :

JEE Main 2013 Solution View → 183

If α and β are roots of the equation x^2+p x+(3 p)/(4)=0 , such that |α-β|= 10 , then p belongs to the set :

JEE Main 2013 Solution View → 184

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 :

JEE Main 2013 Solution View → 185

The real number k for which the equation, 2 x ^ 3 + 3x + k = 0 has two distinct real roots in [ 0, 1 ] belongs to

JEE Main 2013 Solution View → 186

If the equations x ^ 2 + 2 x + 3 = 0 and a x ^ 2 + bx + c = 0, a, b, c ∈ R, have a common root, then a: b: c is:

JEE Main 2013 Solution View → 187

The value of 6+ _ (3)/(2) ( 1 3 2 4- 1 3 2 4- 1 3 2 4- 1 3 2 ⋯ ) is

JEE Advanced 2012 Solution View → 188

If a, b, c ∈ 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 ≠ 0 intersect x -axis at

JEE Main 2012 Solution View → 189

Let p, q, r ∈ R and r>p>0 . If the quadratic equation p x^2+q x+r=0 has two complex roots α and β , then |α|+|β| is

JEE Main 2012 Solution View → 190

Consider a quadratic equation a x^2+b x+c=0 , where 2 a+3 b+6 c=0 and let g(x)=a (x^3)/(3)+b (x^2)/(2)+c x . Statement 1: The quadratic equation has at least one root in the

JEE Main 2012 Solution View → 191

If a, b, c, d and p are distinct real numbers such that (a^2+b^2+c^2) p^2-2 p(a b+b c+c d)+(b^2+. .c^2+d^2) ≤ 0 , then

JEE Main 2012 Solution View → 192

If the sum of the square of the roots of the equation x^2-( α-2) x-(1+ α)=0 is least, then α is equal to

JEE Main 2012 Solution View → 193

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

JEE Main 2012 Solution View → 194

The value of b for which the equations x^2+b x-1=0, x^2+x+b=0 have one root in common is

JEE Advanced 2011 Solution View → 195

Let α and β be the roots of x^2-6 x-2=0 with α>β . If a_n=α^n-β^n for n ≥ 1 , then the value of a_ 10 -2 a_8 2 a_9 is

JEE Advanced 2011 Solution View → 196

Let p and q be real numbers such that p ≠ 0, p^3 ≠ q and p^3 ≠-q . If α and β are non-zero complex numbers satisfying α+β=-p and α^3+β^3=q , then a quadratic equation having

JEE Advanced 2010 Solution View → 197

If α and β are the roots of the equation x^2-x+1=0 , then α^ 2009 +β^ 2009 =

JEE Main 2010 Solution View → 198

The smallest value of k , for which both the roots of the equation x^2-8 k x+16(k^2-k+1)=0 are real, distinct and have values atleast 4 , is

JEE Advanced 2009 Solution View → 199

If the roots of the equation b x^2+c x+a=0 be imaginary, then for all real values of x , the expression 3 b^2 x^2+6 b c x+2 c^2 is

JEE Main 2009 Solution View → 200

Let a, b, c, p, q be real numbers. Suppose, α, β are the roots of the equation x^2+2 p x+q=0 and α, (1)/(β) are the roots of the equation a x^2+2 b x+c=0 , where β^2 ∉\ -1,0,1\ .

JEE Advanced 2008 Solution View →
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Related JEE Mathematics chapters

Application of Derivatives (795)Three Dimensional Geometry (756)Sequences and Series (748)Functions (700)Definite Integration (692)Vector Algebra (616)Differential Equations (586)Continuity and Differentiability (576)Probability (561)Complex Number (551)Binomial Theorem (550)Determinants (508)

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