Nature of Roots & Formation of Equations: JEE Questions & Solutions

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

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

If, where <, are the roots of the equation x^2-( +3)x+3=0 such that 1 - 1 = 13, then the sum of all possible values of is:

Let 1 and 1 be the distinct roots of 2x^2 + ()x - 1 = 0, (0, 2). If m and M are the minimum and the maximum values of 1 + 1, then 16(M + m)…

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 = 365, then a^2 + c^2 is equal to.

Let, be the roots of the equation x^2 - ax - b = 0 with Im() < Im(). Let Pn = ^n - ^n. If P3 = -5 7, P4 = -3 7, P5 = 11 7, P6 = 45 7, then…

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

If for some,;, + = 8 and ^2( ^-1) + ^2( ^-1) = 36, then ^2 + is:

Let f: R 0 (-, 1) be a polynomial of degree 2, satisfying f(x)f ( 1x) = f(x) + f ( 1x). If f(K) = -2K, then the sum of squares of all…

If the set of all a R, for which the equation 2x^2 + (a - 5)x + 15 = 3a has no real root, is the interval (,), and X = x Z: < x <, then x X…

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 (-, - ] [, ], then 2 + + is…

Let and be the roots of x^2 + 3x - 16 = 0, and and be the roots of x^2 + 3x - 1 = 0. If Pn = ^n + ^n and Qn = ^n + ^n, then P25 + 3P242P23…

Consider the equation x^2 + 4x - n = 0, where n [20, 100] is a natural number. Then the number of all distinct values of n, for which the…

Let the set of all values of p R, for which both the roots of the equation x^2 - (p + 2)x + (2p + 9) = 0 are negative real numbers, be the…

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

Let, N be roots of the equation x^2 - 70x + = 0, where /2, /3 N. If assumes the minimum possible value, then ( -1+ -1)( +35)| - | is equal…

For 0 < c < b < a, let (a + b - 2c)x^2 + (b + c - 2a)x + (c + a - 2b) = 0, and = -1 be one of its roots. Which of the following statements…

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

Let and be the roots of the equation px^2 + qx - r = 0, where p 0. If p, q, r are consecutive terms of a non-constant G.P. and 1 + 1 = 34,…

Let,;, be the roots of the equation x^2 - 2x - 3 = 0. Let Pn = ^n - ^n, n N. Then (11 3 - 10 2)P10 + (11 2 + 10)P11 - 11P12 is equal to:

Let a R and let, be the roots of the equation x^2 + 60^ 14 x + a = 0. If ^4 + ^4 = -30, then the product of all possible values of a is:

Let 0 be a real number. Let, be the roots of the equation 14x^2 - 3 x + 3 = 0 and, be the roots of the equation 35x^2 - 53x + 4 = 0. Then 3…

Let 1, 2,, 7 be the roots of the equation x^7 + 3x^5 - 13x^3 - 15x = 0 and | 1| | 2| | 7|. Then 1 2 - 3 4 + 5 6 is equal to:

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

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

Let,, be the three roots of the equation x^3 + bx + c = 0. If = 1 = -, then b^3 + 2c^3 - 3 ^3 - 6 ^3 - 8 ^3 is equal to:

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

Let, be the roots of the equation x^2 - 5x + 2 = 0. Then ^4 + ^4 is equal to: