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JEE Quadratic Equation PYQs with Solutions

269 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.

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About Quadratic Equation PYQs

Quadratic Equation has 269 previous year questions in the Quantrex Academy JEE Main & Advanced Mathematics bank. The 220 questions in this list come from Black Book Advanced Problems in Mathematics (220). Each question opens on its own page with the options, the correct answer and the worked solution where available.

Quadratic Equation questions with solutions

1

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . Let α = 4a - 2b + c, β = 9a +

Black Book Advanced Problems in Mathematics Solution View → 2

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . Let p = b - 4a, q = 2a + b

Black Book Advanced Problems in Mathematics Solution View → 3

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . The smaller root (α) lie in the

Black Book Advanced Problems in Mathematics Solution View → 4

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . The larger root (β) lie in the

Black Book Advanced Problems in Mathematics Solution View → 5

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . The least value of f(x) ∀ x ∈ R

Black Book Advanced Problems in Mathematics Solution View → 6

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . The value of f(-2) + f(0) +

Black Book Advanced Problems in Mathematics Solution View → 7

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . If f(x) = a has two distinct

Black Book Advanced Problems in Mathematics Solution View → 8

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . The set of all values of m for

Black Book Advanced Problems in Mathematics Solution View → 9

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . The sum of maximum value of x_1

Black Book Advanced Problems in Mathematics Solution View → 10

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . x_1 x_2 + x_1 x_3 + x_2 x_4 +

Black Book Advanced Problems in Mathematics Solution View → 11

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . x_2^3 + x_4^3 =

Black Book Advanced Problems in Mathematics Solution View → 12

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . f(6) is equal to :

Black Book Advanced Problems in Mathematics Solution View → 13

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . Sum of the roots of f(x) is

Black Book Advanced Problems in Mathematics Solution View → 14

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . Product of the roots of f(x) is

Black Book Advanced Problems in Mathematics Solution View → 15

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . The value of ((α)/(2))^2 +

Black Book Advanced Problems in Mathematics Solution View → 16

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . Number of values of θ in [0

Black Book Advanced Problems in Mathematics Solution View → 17

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . The sum of all the coefficient

Black Book Advanced Problems in Mathematics Solution View → 18

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . If the range of P(x) is [m, ∞)

Black Book Advanced Problems in Mathematics Solution View → 19

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . If the above cubic has three

Black Book Advanced Problems in Mathematics Solution View → 20

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . If the cubic has exactly two

Black Book Advanced Problems in Mathematics Solution View → 21

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . If the cubic has four real and

Black Book Advanced Problems in Mathematics Solution View → 22

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . If f(x) can take both positive

Black Book Advanced Problems in Mathematics Solution View → 23

Paragraph: **** Let f(x) = ax^2 + bx + c, a ≠ 0 , such that f(-1-x) = f(-1+x) ∀ x ∈ R . Also given that f(x) = 0 has no real roots and 4a + b > 0 . If f(x) is non-negative ∀ x ≥ 0

Black Book Advanced Problems in Mathematics Solution View → 24

Let f(x) = ax^2 + bx + c where a, b, c are integers. If (π)/(7) · (3π)/(7) + (3π)/(7) · (5π)/(7) + (5π)/(7) · (π)/(7) = f( (π)/(7)) then find the value of f(2) :

Black Book Advanced Problems in Mathematics Solution View → 25

Let a, b, c, d be distinct integers such that the equation (x-a)(x-b)(x-c)(x-d) - 9 = 0 has an integer root ' r ', then the value of a+b+c+d-4r is equal to :

Black Book Advanced Problems in Mathematics Solution View → 26

Consider the equation (x^2+x+1)^2 - (m-3)(x^2+x+1) + m = 0 , where m is a real parameter. The number of positive integral values of m for which equation has two distinct real

Black Book Advanced Problems in Mathematics Solution View → 27

The number of positive integral values of m, m ≤ 16 for which the equation given in the above questions has 4 distinct real root is :

Black Book Advanced Problems in Mathematics Solution View → 28

If the equation (m^2-12)x^4 - 8x^2 - 4 = 0 has no real roots, then the largest value of m is p q where p, q are coprime natural numbers, then p+q =

Black Book Advanced Problems in Mathematics Solution View → 29

The least positive integral value of ' x ' satisfying (e^x - 2) ( (x + (π)/(4)))(x - _e 2)( x - x) < 0 is :

Black Book Advanced Problems in Mathematics Solution View → 30

The integral values of x for which x^2 + 17x + 71 is perfect square of a rational number are a and b , then |a - b| =

Black Book Advanced Problems in Mathematics Solution View → 31

Let P(x) = x^6 - x^5 - x^3 - x^2 - x and α, β, γ, δ are the roots of the equation x^4 - x^3 - x^2 - 1 = 0 , then P(α) + P(β) + P(γ) + P(δ) =

Black Book Advanced Problems in Mathematics Solution View → 32

The number of real values of ' a ' for which the largest value of the function f(x) = x^2 + ax + 2 in the interval [-2, 4] is 6 will be :

Black Book Advanced Problems in Mathematics Solution View → 33

The number of all values of n , (where n is a whole number) for which the equation (x-8)/(n-10) = (n)/(x) has no solution.

Black Book Advanced Problems in Mathematics Solution View → 34

The number of negative integral values of m for which the expression x^2 + 2(m-1)x + m + 5 is positive ∀ x > 1 is :

Black Book Advanced Problems in Mathematics Solution View → 35

If the expression ax^4 + bx^3 - x^2 + 2x + 3 has the remainder 4x + 3 when divided by x^2 + x - 2 , then a + 4b = .....

Black Book Advanced Problems in Mathematics Solution View → 36

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

Black Book Advanced Problems in Mathematics Solution View → 37

If x^2 - 3x + 2 is a factor of x^4 - px^2 + q = 0 , then p + q =

Black Book Advanced Problems in Mathematics Solution View → 38

The sum of all real values of k for which the expression x^2 + 2xy + ky^2 + 2x + k = 0 can be resolved into linear factors is :

Black Book Advanced Problems in Mathematics Solution View → 39

The curve y = (a + 1)x^2 + 2 meets the curve y = ax + 3, a ≠ -1 in exactly one point, then a^2 =

Black Book Advanced Problems in Mathematics Solution View → 40

Find the number of integral values of ' a ' for which the range of function f(x) = (x^2-ax+1)/(x^2-3x+2) is (-∞, ∞) .

Black Book Advanced Problems in Mathematics Solution View → 41

When x^ 100 is divided by x^2 - 3x + 2 , the remainder is (2^ k+1 - 1)x - 2(2^k - 1) , then k =

Black Book Advanced Problems in Mathematics Solution View → 42

Let P(x) be a polynomial equation of least possible degree, with rational coefficients, having 7 + 49 as one of its roots. Then the product of all the roots of P(x) = 0 is :

Black Book Advanced Problems in Mathematics Solution View → 43

The range of values k for which the equation 2 ^4 x - ^4 x + k = 0 has atleast one solution is [λ, μ] . Find the value of (9μ + δ) .

Black Book Advanced Problems in Mathematics Solution View → 44

Let P(x) be a polynomial with real coefficient and P(x) - P'(x) = x^2 + 2x + 1 . Find P(1) .

Black Book Advanced Problems in Mathematics Solution View → 45

Find the smallest positive integral value of a for which the greater root, of the equation x^2 - (a^2 + a + 1)x + a(a^2 + 1) = 0 lies between the roots of the equation x^2 - a^2 x

Black Book Advanced Problems in Mathematics Solution View → 46

If the equation x^4 + kx^2 + k = 0 has exactly two distinct real roots, then the smallest integral value of |k| is :

Black Book Advanced Problems in Mathematics Solution View → 47

Let a, b, c, d be the roots of x^4 - x^3 - x^2 - 1 = 0 . Also consider P(x) = x^6 - x^5 - x^3 - x^2 - x , then the value of P(a) + P(b) + P(c) + P(d) is equal to :

Black Book Advanced Problems in Mathematics Solution View → 48

The number of integral values of a, a ∈ [-5, 5] for which the equation x^2 + 2(a - 1)x + a + 5 = 0 has one root smaller than 1 and the other root greater than 3 is :

Black Book Advanced Problems in Mathematics Solution View → 49

The number of non-negative integral values of n, n ≤ 10 so that a root of the equation n^2 ^2 x - 2 x - (2n + 1) = 0 lies in interval [0, (π)/(2)] is :

Black Book Advanced Problems in Mathematics Solution View → 50

Let f(x) = ax^2 + bx + c , where a, b, c are integers and a > 1 . If f(x) takes the value p , a prime for two distinct integer values of x , then the number of integer values of x

Black Book Advanced Problems in Mathematics Solution View → 51

If x and y are real numbers connected by the equation 9x^2 + 2xy + y^2 - 92x - 20y + 244 = 0 , then the sum of maximum value of x and the minimum value of y is :

Black Book Advanced Problems in Mathematics Solution View → 52

Consider two numbers a, b , sum of which is 3 and the sum of their cubes is 7 . Then sum of all possible distinct values of a is :

Black Book Advanced Problems in Mathematics Solution View → 53

If y^2(y^2 - 6) + x^2 - 8x + 24 = 0 and the minimum value of x^2 + y^4 is m and maximum value is M ; then find the value of M - 2m .

Black Book Advanced Problems in Mathematics Solution View → 54

Consider the equation x^3 - ax^2 + bx - c = 0 , where a, b, c are rational number, a ≠ 1 . It is given that x_1, x_2 and x_1x_2 are the real roots of the equation. If (b+c) =

Black Book Advanced Problems in Mathematics Solution View → 55

Let α satisfy the equation x^3 + 3x^2 + 4x + 5 = 0 and β satisfy the equation x^3 - 3x^2 + 4x - 5 = 0, α, β ∈ R , then α + β =

Black Book Advanced Problems in Mathematics Solution View → 56

Let x, y and z are positive reals and x^2 + xy + y^2 = 2 ; y^2 + yz + z^2 = 1 and z^2 + zx + x^2 = 3 . If the value of xy + yz + zx can be expressed as p q where p and q are

Black Book Advanced Problems in Mathematics Solution View → 57

The number of ordered pairs (a, b) , where a, b are integers satisfying the inequality (x^2 + (a-b)x + (1-a-b)) > (-x^2 + (a+b)x - (1+a+b)) ∀ x ∈ R , is :

Black Book Advanced Problems in Mathematics Solution View → 58

The real value of x satisfying 20x + 20x+13 = 13 can be expressed as (a)/(b) where a and b are relatively prime positive integers. Find the value of b ?

Black Book Advanced Problems in Mathematics Solution View → 59

If the range of the values of a for which the roots of the equation x^2 - 2x - a^2 + 1 = 0 lie between the roots of the equation x^2 - 2(a+1)x + a(a-1) = 0 is (p, q) , then find

Black Book Advanced Problems in Mathematics Solution View → 60

Find the number of positive integers satisfying the inequality x^2 - 10x + 16 < 0 .

Black Book Advanced Problems in Mathematics Solution View → 61

If θ and θ are the roots of the quadratic equation ax^2 + bx + c = 0 (ac ≠ 0) . Then find the value of (b^2 - a^2)/(ac) .

Black Book Advanced Problems in Mathematics Solution View → 62

Let the inequality ^2 x + a x + a^2 ≥ 1 + x is satisfied ∀ x ∈ R , for a ∈ (-∞, k_1] ∪ [k_2, ∞) , then |k_1| + |k_2| =

Black Book Advanced Problems in Mathematics Solution View → 63

α and β are roots of the equation 2x^2 - 35x + 2 = 0 . Find the value of (2α - 35)^3(2β - 35)^3

Black Book Advanced Problems in Mathematics Solution View → 64

The sum of all integral values of ' a ' for which the equation 2x^2 - (1+2a)x + 1 + a = 0 has a integral root.

Black Book Advanced Problems in Mathematics Solution View → 65

Let f(x) be a polynomial of degree 8 such that F(r) = (1)/(r), r = 1, 2, 3, ....., 8, 9 , then (1)/(F(10)) =

Black Book Advanced Problems in Mathematics Solution View → 66

Let α, β are two real roots of equation x^2 + px + q = 0, p, q ∈ ℝ, q ≠ 0 . If the quadratic equation g(x) = 0 has two roots α + (1)/(α), β + (1)/(β) such that sum of its roots is

Black Book Advanced Problems in Mathematics Solution View → 67

If A, B and C are the roots of cubic x^3 + ax^2 + bx + c = 0 , where A, B, C are the angles of a triangle then find the value of a^2 - 2b - 2c .

Black Book Advanced Problems in Mathematics Solution View → 68

Find the number of positive integral values of k for which kx^2 + (k-3)x + 1 < 0 for atleast one positive x .

Black Book Advanced Problems in Mathematics Solution View → 69

**(A)** The least positive integer x , for which (2x - 1)/(2x^3 + 3x^2 + x) is positive, is equal to **(P)** 4/3 **(B)** If the quadratic equation 3x^2 + 2(a^2 + 1)x + (a^2 - 3a +

Black Book Advanced Problems in Mathematics Solution View → 70

Given the inequality ax + k^2 > 0 . The complete set of values of ' a ' so that **(A)** The inequality is valid for all values of x and k is **(P)** R **(B)** There exists a value

Black Book Advanced Problems in Mathematics Solution View → 71

**(A)** The real root (s) of the equation x^4 - 8x^2 - 9 = 0 are **(P)** No real roots **(B)** The real root (s) of the equation x^ 2/3 + x^ 1/3 - 2 = 0 are **(Q)** -3, 3 **(C)**

Black Book Advanced Problems in Mathematics Solution View → 72

**(A)** If a, b are the roots of equation x^2 + ax + b = 0 ( a, b ∈ R ), then the number of ordered pairs (a, b) is equal to **(P)** 1 **(B)** If P = cosec(π)/(8) + cosec(2π)/(8)

Black Book Advanced Problems in Mathematics Solution View → 73

Let S is the set of all real x such that (2x-1)/(2x^3+3x^2+x) is positive, then S contains :

Black Book Advanced Problems in Mathematics Solution View → 74

If kx^2 - 4x + 3k + 1 > 0 for atleast one x > 0 , then if k ∈ S , then S contains:

Black Book Advanced Problems in Mathematics Solution View → 75

The equation |x^2 - x - 6| = x + 2 has:

Black Book Advanced Problems in Mathematics Solution View → 76

If the roots of the equation x^2 - ax - b = 0 ( a, b ∈ R ) are both lying between -2 and 2 , then :

Black Book Advanced Problems in Mathematics Solution View → 77

Consider the equation in real number x and a real parameter λ, |x - 1| - |x - 2| + |x - 4| = λ . Then for λ ≥ 1 , the number of solutions, the equation can have is/are :

Black Book Advanced Problems in Mathematics Solution View → 78

If a and b are two distinct non-zero real numbers such that a - b = (a)/(b) - (1)/(b) - (1)/(a) , then :

Black Book Advanced Problems in Mathematics Solution View → 79

Let f(x) = ax^2 + bx + c, a > 0 and f(2-x) = f(2+x) ∀ x ∈ R and f(x) = 0 has 2 distinct real roots, then which of the following is true ?

Black Book Advanced Problems in Mathematics Solution View → 80

In the above problem, if roots of equation f(x) = 0 are non-real complex, then which of the following is false ?

Black Book Advanced Problems in Mathematics Solution View → 81

If exactly two integers lie between the roots of equation x^2 + ax - 1 = 0 . Then integral value(s) of ' a ' is/are :

Black Book Advanced Problems in Mathematics Solution View → 82

If the minimum value of the quadratic expression y = ax^2 + bx + c is negative attained at negative value of x , then :

Black Book Advanced Problems in Mathematics Solution View → 83

The quadratic expression ax^2 + bx + c > 0 ∀ x ∈ R , then:

Black Book Advanced Problems in Mathematics Solution View → 84

The possible positive integral value of ' k ' for which 5x^2 - 2kx + 1 < 0 has exactly one integral solution may be divisible by :

Black Book Advanced Problems in Mathematics Solution View → 85

If the equation x^2 + px + q = 0 , the coefficient of x was incorrectly written as 17 instead of 13. Then roots were found to be -2 and -15 . The correct roots are :

Black Book Advanced Problems in Mathematics Solution View → 86

If x^2 - 3x + 2 > 0 and x^2 - 3x - 4 ≤ 0 , then:

Black Book Advanced Problems in Mathematics Solution View → 87

If 5^x + (2 3 )^ 2x - 169 ≤ 0 is true for x lying in the interval :

Black Book Advanced Problems in Mathematics Solution View → 88

Let f(x) = x^2 + ax + b and g(x) = x^2 + cx + d be two quadratic polynomials with real coefficients and satisfy ac = 2(b + d) . Then which of the following is (are) correct?

Black Book Advanced Problems in Mathematics Solution View → 89

The expression 1 x+2 x-1 + 1 x-2 x-1 simplifies to :

Black Book Advanced Problems in Mathematics Solution View → 90

If all values of x which satisfies the inequality _ 1/3 (x^2 + 2px + p^2 + 1) ≥ 0 also satisfy the inequality kx^2 + kx - k^2 ≤ 0 for all real values of k , then all possible

Black Book Advanced Problems in Mathematics Solution View → 91

Which of the following statement(s) is/are correct?

Black Book Advanced Problems in Mathematics Solution View → 92

If (a, 0) is a point on a diameter inside the circle x^2 + y^2 = 4 . Then x^2 - 4x - a^2 = 0 has :

Black Book Advanced Problems in Mathematics Solution View → 93

Let x^2 - px + q = 0 where p ∈ ℝ, q ∈ ℝ, pq ≠ 0 have the roots α, β such that α + 2β = 0 , then :

Black Book Advanced Problems in Mathematics Solution View → 94

If a, b, c are rational numbers ( a > b > c > 0 ) and quadratic equation (a+b-2c)x^2 + (b+c-2a)x + (c+a-2b) = 0 has a root in the interval (-1, 0) then which of the following

Black Book Advanced Problems in Mathematics Solution View → 95

For the quadratic polynomial f(x) = 4x^2 - 8ax + a , the statements(s) which hold good is/are :

Black Book Advanced Problems in Mathematics Solution View → 96

Given a, b, c are three distinct real numbers satisfying the inequality a - 2b + 4c > 0 and the equation ax^2 + bx + c = 0 has no real roots. Then the possible value(s) of

Black Book Advanced Problems in Mathematics Solution View → 97

Let f(x) = x^2 - 4x + c ∀ x ∈ R , where c is a real constant, then which of the following is/are true ?

Black Book Advanced Problems in Mathematics Solution View → 98

If 0 < a < b < c and the roots α, β of the equation ax^2 + bx + c = 0 are imaginary, then :

Black Book Advanced Problems in Mathematics Solution View → 99

If x satisfies |x - 1| + |x - 2| + |x - 3| > 6 , then :

Black Book Advanced Problems in Mathematics Solution View → 100

If both roots of the quadratic equation ax^2 + x + b - a = 0 are non real and b > -1 , then which of the following is/are correct?

Black Book Advanced Problems in Mathematics Solution View → 101

If a, b are two numbers such that a^2 + b^2 = 7 and a^3 + b^3 = 10 , then :

Black Book Advanced Problems in Mathematics Solution View → 102

The number of non-negative integral ordered pair(s) (x, y) for which (xy - 7)^2 = x^2 + y^2 holds is greater than or equal to :

Black Book Advanced Problems in Mathematics Solution View → 103

If α, β, γ and δ are the roots of the equation x^4 - bx - 3 = 0 ; then an equation whose roots are (α+β+γ)/(δ^2), (α+β+δ)/(γ^2), (α+γ+δ)/(β^2) and (β+γ+δ)/(α^2) is :

Black Book Advanced Problems in Mathematics Solution View → 104

The value of k for which both roots of the equation 4x^2 - 2x + k = 0 are completely in (-1, 1) may be equal to :

Black Book Advanced Problems in Mathematics Solution View → 105

If a, b, c ∈ R , then for which of the following graphs of the quadratic polynomial y = ax^2 - 2bx + c (a ≠ 0) ; the product (abc) is negative?

Black Book Advanced Problems in Mathematics Solution View → 106

If the equation ax^2 + bx + c = 0; a, b, c ∈ R and a ≠ 0 has no real roots then which of the following is/are always correct?

Black Book Advanced Problems in Mathematics Solution View → 107

If α and β are the roots of the equation ax^2 + bx + c = 0; a, b, c ∈ ℝ; a ≠ 0 then which is (are) correct:

Black Book Advanced Problems in Mathematics Solution View → 108

The equation ^2 x - x + λ = 0, x ∈ (0, π/2) has roots then value(s) of λ can be equal to :

Black Book Advanced Problems in Mathematics Solution View → 109

If the equation (x^2 + 5x) - (x + a + 3) = 0 has exactly one solution for x , then possible integral value of a is :

Black Book Advanced Problems in Mathematics Solution View → 110

The number of non-negative integral ordered pair(s) (x, y) for which (xy - 7)^2 = x^2 + y^2 holds is greater than or equal to :

Black Book Advanced Problems in Mathematics Solution View → 111

If a < 0 , then the value of x satisfying x^2 - 2a|x - a| - 3a^2 = 0 is/are :

Black Book Advanced Problems in Mathematics Solution View → 112

If 0 < a < b < c and the roots α, β of the equation ax^2 + bx + c = 0 are imaginary, then

Black Book Advanced Problems in Mathematics Solution View → 113

If x satisfies |x - 1| + |x - 2| + |x - 3| > 6 , then

Black Book Advanced Problems in Mathematics Solution View → 114

The value of k for which both roots of the equation 4x^2 - 2x + k = 0 are completely in (-1, 1) , may be equal to :

Black Book Advanced Problems in Mathematics Solution View → 115

Let α, β, γ, δ are roots of x^4 - 12x^3 + λ x^2 - 54x + 14 = 0 . If α + β = γ + δ , then

Black Book Advanced Problems in Mathematics Solution View → 116

If (a^3)/(a-1), (a^2-3)/(a-1); (b^3)/(b-1), (b^2-3)/(b-1); (c^3)/(c-1), (c^2-3)/(c-1) lie on L: lx + my + n = 0 ; where a, b, c are real numbers different from 1 ; then

Black Book Advanced Problems in Mathematics Solution View → 117

Sum of values of x and y satisfying the equation 3^x - 4^y = 77; 3^ x/2 - 2^y = 7 is :

Black Book Advanced Problems in Mathematics Solution View → 118

If f(x) = ∏_ i=1 ^3 (x-a_i) + ∑_ i=1 ^3 a_i - 3x where a_i < a_ i+1 for i = 1, 2 , then f(x) = 0 has :

Black Book Advanced Problems in Mathematics Solution View → 119

Complete set of real values of 'a' for which the equation x^4 - 2ax^2 + x + a^2 - a = 0 has all its roots real :

Black Book Advanced Problems in Mathematics Solution View → 120

The cubic polynomial with leading coefficient unity all whose roots are 3 units less than the roots of the equation x^3 - 3x^2 - 4x + 12 = 0 is denoted as f(x) , then f'(x) is

Black Book Advanced Problems in Mathematics Solution View → 121

The set of values of k (k ∈ R) for which the equation x^2 - 4|x| + 3 - |k - 1| = 0 will have exactly four real roots, is :

Black Book Advanced Problems in Mathematics Solution View → 122

The number of integers satisfying the inequality (x)/(x+6) ≤ (1)/(x) is :

Black Book Advanced Problems in Mathematics Solution View → 123

The product of uncommon real roots of the two polynomials p(x) = x^4 + 2x^3 - 8x^2 - 6x + 15 and q(x) = x^3 + 4x^2 - x - 10 is:

Black Book Advanced Problems in Mathematics Solution View → 124

If λ_1, λ_2 (λ_1 > λ_2) are two values of λ for which the expression f(x, y) = x^2 + λ xy + y^2 - 5x - 7y + 6 can be resolved as a product of two linear factors, then the value of

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Let α, β be the roots of the quadratic equation ax^2 + bx + c = 0 , then the roots of the equation a(x+1)^2 + b(x+1)(x-2) + c(x-2)^2 = 0 are :

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If a, b ∈ R distinct numbers satisfying |a-1| + |b-1| = |a| + |b| = |a+1| + |b+1| , then the minimum value of |a-b| is :

Black Book Advanced Problems in Mathematics Solution View → 127

The smallest positive integer p for which expression x^2 - 2px + 3p + 4 is negative for atleast one real x is :

Black Book Advanced Problems in Mathematics Solution View → 128

For x ∈ R , the expression (x^2+2x+c)/(x^2+4x+3c) can take all real values if c ∈ :

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If 2 lies between the roots of the equation t^2 - mt + 2 = 0 , (m ∈ R) then the value of [(3|x|)/(9+x^2)]^m is : (where [.] denotes greatest integer function)

Black Book Advanced Problems in Mathematics Solution View → 130

The number of integral roots of the equation x^8 - 24x^7 - 18x^5 + 39x^2 + 1155 = 0 is :

Black Book Advanced Problems in Mathematics Solution View → 131

If the value of m^4 + (1)/(m^4) = 119 , then the value of |m^3 - (1)/(m^3)| =

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If the equation ax^2 + 2bx + c = 0 and ax^2 + 2cx + b = 0, a ≠ 0, b ≠ c , have a common root, then their other roots are the roots of the quadratic equation:

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If α, β and γ are the roots of the equation 9x^3 - 9x^2 - x + 1 = 0 ; α, β, γ ∈ [0, π] then the radius of the circle whose centre is (Σα, Σ α) and passing through (2 ^ -1 ( π/4)

Black Book Advanced Problems in Mathematics Solution View → 134

For real values of x, the value of expression (11x^2-12x-6)/(x^2+4x+2) :

Black Book Advanced Problems in Mathematics Solution View → 135

(x+3)/(x^2-x-2) ≥ (1)/(x-4) holds for all x satisfying:

Black Book Advanced Problems in Mathematics Solution View → 136

If x = 4 + 3i (where i = -1 ), then the value of x^3 - 4x^2 - 7x + 12 equals:

Black Book Advanced Problems in Mathematics Solution View → 137

Let f(x) = (x^2+x-1)/(x^2-x+1) , then the largest value of f(x) ∀ x ∈ [-1, 3] is:

Black Book Advanced Problems in Mathematics Solution View → 138

In above problem, the range of f(x) ∀ x ∈ [-1, 1] is:

Black Book Advanced Problems in Mathematics Solution View → 139

If the roots of the equation (1)/(x+p) + (1)/(x+q) = (1)/(r) are equal in magnitude but opposite in sign, then the product of the roots is:

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If a root of the equation a_1x^2 + b_1x + c_1 = 0 is the reciprocal of a root of the equation a_2x^2 + b_2x + c_2 = 0 , then :

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If α ≠ β but α^2 = 5α - 3 and β^2 = 5β - 3 , then the equation with roots α/β, β/α is:

Black Book Advanced Problems in Mathematics Solution View → 142

If the difference between the roots of x^2 + ax + b = 0 is same as that of x^2 + bx + a = 0, a ≠ b , then:

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If θ_i ; i=1,2,3,4 are the roots of equation x^4 - x^3 2β + x^2 2β - x β - β = 0 , then (θ_1+θ_2+θ_3+θ_4) =

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Let a, b, c, d are positive real numbers such that a/b ≠ c/d , then the roots of the equation: (a^2+b^2)x^2 + 2x(ac+bd) + (c^2+d^2) = 0 are:

Black Book Advanced Problems in Mathematics Solution View → 145

If α, β are the roots of ax^2 + bx + c = 0 , then the equation whose roots are 2+α, 2+β , is:

Black Book Advanced Problems in Mathematics Solution View → 146

Minimum possible number of positive root of the quadratic equation x^2 - (1+λ)x + λ - 2 = 0, λ ∈ R :

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Let α, β be real roots of the quadratic equation x^2 + kx + (k^2+2k-4) = 0 , then the minimum value of α^2 + β^2 is equal to :

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Polynomial P(x) = x^2 - ax + 5 and Q(x) = 2x^3 + 5x - (a-3) when divided by x - 2 have same remainders, then 'a' is equal to:

Black Book Advanced Problems in Mathematics Solution View → 149

If a and b are non-zero distinct roots of x^2 + ax + b = 0 , then the least value of x^2 + ax + b is equal to :

Black Book Advanced Problems in Mathematics Solution View → 150

Let α, β be the roots of the equation ax^2 + bx + c = 0 . A root of the equation a^3x^2 + abcx + c^3 = 0 is :

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Let a, b, c be the lengths of the sides of a triangle (no two of them are equal) and k ∈ R . If the roots of the equation x^2 + 2(a+b+c)x + 6k(ab+bc+ca) = 0 are real, then:

Black Book Advanced Problems in Mathematics Solution View → 152

Root(s) of the equation 9x^2 - 18|x| + 5 = 0 belonging to the domain of definition of the function f(x) = (x^2-x-2) is/are:

Black Book Advanced Problems in Mathematics Solution View → 153

If β + ^2α, β + ^2α are the roots of x^2 + 2bx + c = 0 and γ + ^4α, γ + ^4α are the roots of x^2 + 2Bx + C = 0 , then:

Black Book Advanced Problems in Mathematics Solution View → 154

Minimum value of |x-p| + |x-15| + |x-p-15| . If p ≤ x ≤ 15 and 0 < p < 15 :

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If the quadratic equation 4x^2 - 2x - m = 0 and 4p(q-r)x^2 - 2q(r-p)x + r(p-q) = 0 have a common root such that second equation has equal roots then the value of m will be :

Black Book Advanced Problems in Mathematics Solution View → 156

The range of k for which the inequality k ^2x - k x + 1 ≥ 0 ∀ x ∈ (-∞, ∞) is :

Black Book Advanced Problems in Mathematics Solution View → 157

If (1+α)/(1-α), (1+β)/(1-β), (1+γ)/(1-γ) are roots of the cubic equation f(x) = 0 where α, β, γ are the roots of the cubic equation 3x^3 - 2x + 5 = 0 , then the number of negative

Black Book Advanced Problems in Mathematics Solution View → 158

The sum of all integral values of λ for which (λ^2+λ-2)x^2 + (λ+2)x < 1 ∀ x ∈ R , is :

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If α, β, γ, δ ∈ R satisfy ((α+1)^2+(β+1)^2+(γ+1)^2+(δ+1)^2)/(α+β+γ+δ) = 4 . If biquadratic equation a_0x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 = 0 has the roots (α+(1)/(β)-1)

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If the complete set of value of x satisfying |x-1| + |x-2| + |x-3| ≥ 6 is (-∞, a] ∪ [b, ∞) , then a+b= :

Black Book Advanced Problems in Mathematics Solution View → 161

If exactly one root of the quadratic equation x^2 - (a+1)x + 2a = 0 lies in the interval (0, 3) , then the set of value 'a' is given by:

Black Book Advanced Problems in Mathematics Solution View → 162

The condition that the root of x^3 + 3px^2 + 3qx + r = 0 are in H.P. is :

Black Book Advanced Problems in Mathematics Solution View → 163

If x is real and 4y^2 + 4xy + x + 6 = 0 , then the complete set of values of x for which y is real, is:

Black Book Advanced Problems in Mathematics Solution View → 164

The solution of the equation _ x 2(3-2x) < _ x 2(2x-1) is :

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If the roots α, β of the equation px^2 + qx + r = 0 are real and of opposite sign (where p, q, r are real coefficient), then the roots of the equation α(x-β)^2 + β(x-α)^2 = 0 are:

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Let a, b and c be three distinct real roots of the cubic x^3 + 2x^2 - 4x - 4 = 0 . If the equation x^3 + qx^2 + rx + s = 0 has roots 1/a, 1/b and 1/c , then the value of (q+r+s)

Black Book Advanced Problems in Mathematics Solution View → 167

Solution set of the inequality, 2 - _2(x^2 + 3x) ≥ 0 is :

Black Book Advanced Problems in Mathematics Solution View → 168

For what least integral 'k' is the quadratic trinomial (k - 2)x^2 + 8x + (k + 4) is positive for all real values of x ?

Black Book Advanced Problems in Mathematics Solution View → 169

If roots of the equation (m - 2)x^2 - (8 - 2m)x - (8 - 3m) = 0 are opposite in sign, then number of integral values(s) of m is/are :

Black Book Advanced Problems in Mathematics Solution View → 170

If _ 0.6 ( _6 (x^2+x)/(x+4)) < 0 , then complete set of value of 'x' is :

Black Book Advanced Problems in Mathematics Solution View → 171

Two different real numbers α and β are the roots of the quadratic equation ax^2 + c = 0 with a, c ≠ 0 , then α^3 + β^3 is :

Black Book Advanced Problems in Mathematics Solution View → 172

The least integral value of 'k' for which (k - 1)x^2 - (k + 1)x + (k + 1) is positive for all real value of x is:

Black Book Advanced Problems in Mathematics Solution View → 173

If (-2, 7) is the highest point on the graph of y = -2x^2 - 4ax + k , then k equals:

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If a+b+c=0, a, b, c ∈ Q then roots of the equation (b+c-a)x^2 + (c+a-b)x + (a+b-c) = 0 are:

Black Book Advanced Problems in Mathematics Solution View → 175

If two roots of x^3 - ax^2 + bx - c = 0 are equal in magnitude but opposite in sign. Then:

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If α and β are the real roots of x^2 + px + q = 0 and α^4, β^4 are the roots of x^2 - rx + s = 0 . Then the equation x^2 - 4qx + 2q^2 - r = 0 has always ( α ≠ β, p ≠ 0, p, q, r, s

Black Book Advanced Problems in Mathematics Solution View → 177

If x^2 + px + 1 is a factor of ax^3 + bx + c , then:

Black Book Advanced Problems in Mathematics Solution View → 178

In a Δ ABC (A)/(2), (B)/(2), (C)/(2) are in H.P., then the value of (A)/(2) (C)/(2) is :

Black Book Advanced Problems in Mathematics Solution View → 179

Let f(x) = 10 - |x - 10| ∀ x ∈ [-9, 9] , if M and m be the maximum and minimum value of f(x) respectively, then :

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Solution of the quadratic equation (3|x| - 3)^2 = |x| + 7 , which belongs to the domain of the function y = (x - 4)x is :

Black Book Advanced Problems in Mathematics Solution View → 181

Number of real solutions of the equation x^2 + 3|x| + 2 = 0 is :

Black Book Advanced Problems in Mathematics Solution View → 182

If the roots of equation x^2 - bx + c = 0 be two consecutive integers, then b^2 - 4c =

Black Book Advanced Problems in Mathematics Solution View → 183

If x is real, then maximum value of (3x^2 + 9x + 17)/(3x^2 + 9x + 7) is:

Black Book Advanced Problems in Mathematics Solution View → 184

If (x^2 + 2x + 7)/(2x + 3) < 6, x ∈ R then:

Black Book Advanced Problems in Mathematics Solution View → 185

If x is real, then range of (3x-2)/(7x+5) is:

Black Book Advanced Problems in Mathematics Solution View → 186

Let A denotes the set of values of x for which (x+2)/(x-4) ≤ 0 and B denotes the set of values of x for which x^2 - ax - 4 ≤ 0 . If B is the subset of A, then a CAN NOT take the

Black Book Advanced Problems in Mathematics Solution View → 187

If the quadratic polynomial P(x) = (p - 3)x^2 - 2px + 3p - 6 ranges from [0, ∞) for every x ∈ R , then the value of p can be :

Black Book Advanced Problems in Mathematics Solution View → 188

If graph of the quadratic y = ax^2 + bx + c is given below :

Black Book Advanced Problems in Mathematics Solution View → 189

If quadratic equation ax^2 + bx + c = 0 does not have real roots, then which of the following may be false :

Black Book Advanced Problems in Mathematics Solution View → 190

Minimum value of y = x^2 - 3x + 5, x ∈ [-4, 1] is :

Black Book Advanced Problems in Mathematics Solution View → 191

If 3x^2 - 17x + 10 = 0 and x^2 - 5x + m = 0 has a common root, then sum of all possible real values of 'm' is :

Black Book Advanced Problems in Mathematics Solution View → 192

For real numbers x and y, if x^2 + xy - y^2 + 2x - y + 1 = 0 , then :

Black Book Advanced Problems in Mathematics Solution View → 193

If 3x^4 - 6x^3 + kx^2 - 8x - 12 is divisible by x - 3 , then it is also divisible by :

Black Book Advanced Problems in Mathematics Solution View → 194

The complete set of values of a so that equation ^4 x + a ^2 x + 4 = 0 has at least one real root is :

Black Book Advanced Problems in Mathematics Solution View → 195

Let r, s, t be the roots of the equation x^3 + ax^2 + bx + c = 0 , such that (rs)^2 + (st)^2 + (rt)^2 = b^2 - kac , then k =

Black Book Advanced Problems in Mathematics Solution View → 196

If the roots of the cubic x^3 + ax^2 + bx + c = 0 are three consecutive positive integers, then the value of (a^2)/(b+1) =

Black Book Advanced Problems in Mathematics Solution View → 197

Let 'k' be a real number. The minimum number of distinct real roots possible of the equation (3x^2 + kx + 3)(x^2 + kx - 1) = 0 is :

Black Book Advanced Problems in Mathematics Solution View → 198

If r and s are variables satisfying the equation (1)/(r+s) = (1)/(r) + (1)/(s) . The value of ((r)/(s))^3 is equal to:

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Let f(x) = x^2 + ax + b . If the maximum and the minimum values of f(x) are 3 and 2 respectively for 0 ≤ x ≤ 2 , then the possible ordered pair of (a, b) is :

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The roots of the equation |x^2 - x - 6| = x + 2 are given by :

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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)Quadratic Equation (517)