Quadratic Equations

16 solved problems from JEE previous year papers.

Quadratic Equations

If the Set of All a ∈ R for Which 2x² + (a−5)x + 15 = 3a Has No Real Root Is the Interval (α, β), and X = {x ∈ Z : x Lies Between α and β}, Then the Sum of x² Over X Is Equal To

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…
Quadratic Equations

If k > 0 and the Product of the Roots of the Equation x² − 3kx + 2e^(2 log k) − 1 = 0 Is 7, Then the Sum of the Roots Is

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…
Quadratic Equations

Ratio of Roots of ax²+2bx+c=0 Equals Ratio of Roots of px²+2qx+r=0 — Find b²/q²

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…
Quadratic Equations

If the Roots of the Equation (a² + b²)x² − 2b(a + c)x + (b² + c²) = 0 Are Equal, Then a, b, c Are in

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…
Quadratic Equations

If √(a + √b) = 3 + √2, Then Find √(a² − b)

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…
Quadratic Equations

The Sum of the Squares of All the Roots of the Equation x² + |2x − 3| − 4 = 0 Is

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…
Quadratic Equations

If the Equation a(b−c)x² + b(c−a)x + c(a−b) = 0 Has Equal Roots, and a + c = 5 and b = 16/5, Then Find the Value of a² + c²

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…
Quadratic Equations

Find the Value of √(12 + √(12 + √(12 + ⋯)))

Find the value of the infinite nested radical √(12+√(12+√(12+…))) by setting it equal to x and squaring, reducing it to a…
Theory of Equations

Let α, β Be the Roots of x² − ax − b = 0 With Im(α) < Im(β) and Pₙ = αⁿ − βⁿ. If P₃ = −5√7 i, P₄ = −3√7 i, P₅ = 11√7 i and P₆ = 45√7 i, Then |α⁴ + β⁴| Is Equal To

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…
Theory of Equations

Let α and β Be the Roots of the Equation x² − 6x − 2 = 0. If aₙ = αⁿ − βⁿ for n ≥ 1, Then the Value of (a₁₀ − 2a₈)/(2a₉) Is Equal To

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.
Theory of Equations

Let α and β Be the Distinct Roots of 2x² + (cos θ)x − 1 = 0, θ ∈ (0, 2π). If m and M Are the Minimum and Maximum Values of α⁴ + β⁴, Then 16(M + m) Equals

Roots α, β of 2x²+cosθ·x−1=0 depend on θ∈(0,2π). Find 16(M+m) where m, M are the minimum and maximum values of α⁴+β⁴ over all θ.
Theory of Equations

Let p and q Be Two Positive Numbers Such That p + q = 2 and p⁴ + q⁴ = 272. Then p and q Are Roots of the Equation

Two positive numbers p, q satisfy p+q=2 and p^4+q^4=272. Find pq using power-sum identities, then write the quadratic having p, q…
Theory of Equations

Sum of Cubes of All Roots of x⁴−3x³−2x²+3x+1=0

Find α³+β³+γ³+δ³ for the roots of x⁴−3x³−2x²+3x+1=0 by factoring the quartic into two quadratics, avoiding solving for the roots…
Quadratic Equations

The Product of All the Rational Roots of the Equation (x² − 9x + 11)² − (x − 4)(x − 5) = 3 Is Equal To

Find the product of all rational roots of (x²−9x+11)²−(x−4)(x−5)=3 by substituting y=x²−9x+11 to reduce it to a simple quadratic…
Quadratic Equations

Find the Positive Square Root of 3+√5

Find the positive square root of 3+√5 by expressing it as √x+√y, splitting the equation into a rational and irrational part, and…
Quadratic Equations

Roots of x²−√2x−√3=0, Pn=αⁿ−βⁿ, Find (11√3−10√2)P10+(11√2+10)P11−11P12

Let α, β (α>β) be roots of x²−√2x−√3=0 and Pn=αⁿ−βⁿ. Find (11√3−10√2)P10+(11√2+10)P11−11P12 using the Newton recurrence for power…