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Quadratic Equation — JEE Main & Advanced Mathematics PYQs

269 previous year questions from Quadratic Equation with answers and solutions. Numbered list, year tags, and one-tap solutions — built for serious JEE / NEET practice.

269 questionsMathematicsSolutions on every page
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 = 4a - 2b + c, = 9a + 3b + c, = 9a

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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 , then pq i

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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 ( ) lie in the interv

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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 ( ) lie in the interva

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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 is :

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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) + f(1) =

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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 real root

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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 which t

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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 x₁ and min

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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 . x₁ x₂ + x₁ x₃ + x₂ x₄ + x₃ x₄ =

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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 . x₂^3 + x₄^3 =

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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 :

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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 equal to :

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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 equal

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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 ( 2 )^2 + ( 2 )^2 + ( 2 )

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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 in [0, 2 ] for whi

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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 of P(x)

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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, ) , then t

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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 real and

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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 real and

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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 distinc

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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 and neg

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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 then t lie

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

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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 :

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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 roots

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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 :

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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 =

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

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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| =

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

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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 :

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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.

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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 :

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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 = .....

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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.

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

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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 :

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

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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 (- , ) .

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When x¹⁰⁰ is divided by x^2 - 3x + 2 , the remainder is (2^ k+1 - 1)x - 2(2^k - 1) , then k =

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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 :

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

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

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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

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

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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 :

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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 :

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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, 2 ] is :

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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

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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 :

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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 :

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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 .

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Consider the equation x^3 - ax^2 + bx - c = 0 , where a, b, c are rational number, a 1 . It is given that x₁, x₂ and x₁x₂ are the real roots of the equation. If (b+c) = 2(a+1) , th

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

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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 relati

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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 :

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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 ?

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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 th

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

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

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Let the inequality ^2 x + a x + a^2 1 + x is satisfied x R , for a (- , k₁] [k₂, ) , then |k₁| + |k₂| =

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

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

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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) =

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Let , 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 + 1 , + 1 such that sum of its roots is equal to product of ro

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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 .

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

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**(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 + 2) =

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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

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**(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)** T

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**(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 8 + cosec 2 8 + cosec 3 8

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

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

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

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

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

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

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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 ?

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

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

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

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

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

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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 :

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

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

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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?

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

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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 values of

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

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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 :

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

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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 state

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

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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 4a+2b+c

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

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

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

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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?

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

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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 :

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If , , and are the roots of the equation x^4 - bx - 3 = 0 ; then an equation whose roots are + + ^2 , + + ^2 , + + ^2 and + + ^2 is :

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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 :

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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?

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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?

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

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

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

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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 :

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

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

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

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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 :

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

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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

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

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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 :

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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 :

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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 equa

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