JEE Main 2025 PYQ: find the interval (α,β) of a for which 2x²+(a-5)x+15=3a has no real root, then sum the squares of every integer strictly between α and β.
Find the sum of the roots of x^2-3kx+2e^(2logk)-1=0 given k>0 and the product of the roots is 7, by simplifying e^(2logk) to k^2 and using Vieta’s formulas.
If the ratio of the roots of ax²+2bx+c=0 equals the ratio of the roots of px²+2qx+r=0, find b²/q² using componendo-dividendo on the equal root ratios.
Show that a, b, c are in geometric progression given the equation (a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0 has equal roots, by setting the discriminant to zero.
Solve sqrt(a + sqrt(b)) = 3 + sqrt(2) for a and b by squaring and matching rational and irrational parts, then find sqrt(a^2 – b).
Find the sum of the squares of all roots of x²+|2x−3|−4=0, by splitting into two cases at the modulus’s sign-change point and checking each candidate root against its case’s domain.
Given a(b−c)x²+b(c−a)x+c(a−b)=0 has equal roots, with a+c=5 and b=16/5, find a²+c² by recognising x=1 as a root and using the equal-roots condition to relate a, b, c.
Find the value of the infinite nested radical √(12+√(12+√(12+…))) by setting it equal to x and squaring, reducing it to a quadratic equation.
Given Pn=α^n−β^n for roots of x²−ax−b=0, and values of P3 through P6, find a and b using a linear recurrence, then compute |α⁴+β⁴|.
If α, β are roots of x²−6x−2=0 and an=α^n−β^n, find (a10−2a8)/(2a9) using Newton’s recurrence formula for power sums.