Theory of Equations46 questions5 PYQ

Theory of Equations — JEE Maths Practice Questions & Solutions

46 questions on Theory of Equations with full step-by-step solutions, including past-year (PYQ) problems. Free to practice.

hardPYQ · JEE Main 2026
Let α,β\alpha,\beta be the roots of the equation x2−3x+r=0x^2-3x+r=0, and α2, 2β\dfrac{\alpha}{2},\ 2\beta be the roots of the equation x2+3x+r=0x^2+3x+r=0. If the roots of the equation x2+6x=mx^2+6x=m are 2α+β+2r2\alpha+\beta+2r and α−2β−r2\alpha-2\beta-\dfrac{r}{2}, then mm is equal to
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mediumPYQ · JEE Main 2026
If the quadratic equation (λ+2)x2−3λx+4λ=0(\lambda+2)x^2-3\lambda x+4\lambda=0, λ≠−2\lambda\ne-2, has two positive roots, then the number of possible integral values of λ\lambda is
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mediumPYQ · JEE Main 2026
Let tan⁡A,tan⁡B\tan A,\tan B, where A,B∈(−π2,π2)A,B\in\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right), be the roots of x2−2x−5=0x^2-2x-5=0. Then 20sin⁡2(A+B2)20\sin^2\left(\dfrac{A+B}{2}\right) is equal to
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mediumPYQ · JEE Main 2026
Let α, α+2, α∈Z\alpha,\ \alpha+2,\ \alpha\in\mathbb{Z}, be the roots of the quadratic equation x(x+2)+(x+1)(x+3)+(x+2)(x+4)+⋯+(x+n−1)(x+n+1)=4nx(x+2)+(x+1)(x+3)+(x+2)(x+4)+\cdots+(x+n-1)(x+n+1)=4n for some n∈Nn\in\mathbb{N}. Then (n+α)(n+\alpha) is equal to
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mediumPYQ · JEE Main 2026
Let one root of the quadratic equation (k2−15k+27)x2+9(k−1)x+18=0(k^2-15k+27)x^2+9(k-1)x+18=0 be twice the other. Then the length of the latus rectum of the parabola y2=6kxy^2=6kx is equal to
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expert
If roots of the equation a(x−ℓ)(x−m)+n=0a(x-\ell)(x-m) + n = 0 are α,β\alpha, \beta, then roots of the equation a ⁣(xx−k−α) ⁣(xx−k−β)−n=0a\!\left(\dfrac{x}{x-k} - \alpha\right)\!\left(\dfrac{x}{x-k} - \beta\right) - n = 0 are
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hard
Consider ax2−bx+c=0ax^2-bx+c=0 with a,b,c∈Na, b, c\in\mathbb{N}. If it has two distinct real roots in (1,2)(1,2), then
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hard
If a∈(0,10)a \in (0, 10) is such that the roots of the equation
x2−4x−a2+2a+3=0x^{2} - 4x - a^{2} + 2a + 3 = 0
 lie between the roots of the equation
x2−4x+4−a2=0,x^{2} - 4x + 4 - a^{2} = 0 ,
 then the number of integral values of aa is
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hard
If α\alpha and β\beta are real roots of
(log⁡3x)2−6log⁡3x+k=0,k∈R,\left(\log_3 x\right)^{2} - 6\log_3 x + k = 0, \qquad k \in \mathbb{R},
 then the minimum integral value of (α+β)(\alpha + \beta) is
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hard
If a∈Ra \in \mathbb{R} and the equation (a−2)(x−[x])2+2(x−[x])+a2=0(a-2)(x-[x])^2 + 2(x-[x]) + a^2 = 0 (where [x][x] denotes GIF) has no integral solution and has exactly one solution in (2,3)(2, 3), then aa lies in the interval
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hard
Let α\alpha be a root of ax2+bx+c=0ax^2 + bx + c = 0 and β\beta be a root of −ax2+bx+c=0-ax^2 + bx + c = 0, where a,b,ca, b, c are real and a≠0a \neq 0. Then the equation a2x2+bx+c=0\dfrac{a}{2}x^2 + bx + c = 0 has a root γ\gamma such that
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hard
If the equations
ax2−(a+b)x+b=0andbx2+(b−c)x−c=0ax^{2} - (a+b)x + b = 0 \qquad \text{and} \qquad bx^{2} + (b-c)x - c = 0
 have exactly one root in common (a,b,c≠0)\left(a, b, c \ne 0\right), then which of the following can be correct?
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hard
Let f(x)=2x2−2(2λ+1)x+λ(λ+1)=0f(x) = 2x^{2} - 2(2\lambda+1)x + \lambda(\lambda+1) = 0, where λ\lambda is a parameter. If both roots of f(x)=0f(x) = 0 belong to (−1,1)(-1, 1), then the greatest integral value of ∣10λ∣\left|10\lambda\right| is
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hard
Let α,β\alpha, \beta be the roots of x2+ax+b=0x^{2} + ax + b = 0 and γ,δ\gamma, \delta be the roots of x2−ax+b−2=0x^{2} - ax + b - 2 = 0. If
1α+1β+1γ+1δ=512andαβγδ=24,\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma} + \frac{1}{\delta} = \frac{5}{12} \qquad \text{and} \qquad \alpha\beta\gamma\delta = 24 ,
 then find the value of aa.
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hard
If the roots of the quadratic equation
(x−α)(x−β)+(x−γ)(x−δ)=0(x-\alpha)(x-\beta) + (x-\gamma)(x-\delta) = 0
 are mm and nn respectively, then the roots of the equation
2(x−m)(x−n)−(x−α)(x−β)=02(x-m)(x-n) - (x-\alpha)(x-\beta) = 0
 will be
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hard
Let α,β,γ,δ\alpha, \beta, \gamma, \delta be the roots of x4−x3−x2−1=0x^{4} - x^{3} - x^{2} - 1 = 0. Also consider
p(x)=x6−x5−x3−x2−x.p(x) = x^{6} - x^{5} - x^{3} - x^{2} - x .
 Then the value of p(α)+p(β)+p(γ)+p(δ)p(\alpha) + p(\beta) + p(\gamma) + p(\delta) cannot be
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medium
If α\alpha and β\beta are the roots of the equation x2−2x−5=0x^2 - 2x - 5 = 0, then the equation whose roots are αα−1\dfrac{\alpha}{\alpha-1} and ββ−1\dfrac{\beta}{\beta-1} is
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medium
Find the only integer which does not lie in the range of the function
f(x)=x2−3x+2x2+x−6.f(x) = \frac{x^{2}-3x+2}{x^{2}+x-6} .
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medium
Let a1,a2,…,ana_1, a_2, \ldots, a_n be a sequence such that a1=2a_1 = 2 and
an−an−1=n2∀ n≥2.a_n - a_{n-1} = n^{2} \quad \forall\ n \ge 2 .
 If ak=205a_k = 205, then kk is equal to
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medium
If one of the roots of ax2+ax+a+1=0ax^2 + ax + a + 1 = 0 is less than 11 and the other is greater than 11, then the complete set of values of aa is
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medium
If a,b,c,da, b, c, d are non-zero real numbers such that cc and dd are the roots of x2+ax+b=0x^2 + ax + b = 0 and aa and bb are the roots of x2+cx+d=0x^2 + cx + d = 0, then the absolute value of a+2b+3c+4da + 2b + 3c + 4d is
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medium
If α,β,γ\alpha, \beta, \gamma be the roots of the equation x3+ax+a=0x^3 + ax + a = 0 (where a∈Ra \in \mathbb{R}, a≠0a \neq 0) satisfying α2β+β2γ+γ2α=−8\dfrac{\alpha^2}{\beta} + \dfrac{\beta^2}{\gamma} + \dfrac{\gamma^2}{\alpha} = -8, then aa is
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medium
If α+1α\alpha + \dfrac{1}{\alpha} and β+1β\beta + \dfrac{1}{\beta} are the zeroes of f(x)=x2−5x−af(x) = x^2 - 5x - a where α,β∈(0,∞)\alpha, \beta \in (0, \infty), then the set of values of aa is
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medium
If α\alpha and β\beta are the roots of the equation x2−x+11=0x^2 - x + 11 = 0, then the value of 3α3−3α2+2β3−2β2+11α3\alpha^3 - 3\alpha^2 + 2\beta^3 - 2\beta^2 + 11\alpha is
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medium
If the quadratic equations 3x2+ax+1=03x^2 + ax + 1 = 0 and 2x2+bx+1=02x^2 + bx + 1 = 0 have a common root, then the value of ∣2a2−5ab+3b2∣|2a^2 - 5ab + 3b^2| is, where 2a≠3b2a \neq 3b
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medium
If (α,β)(\alpha, \beta), (β,γ)(\beta, \gamma), (γ,α)(\gamma, \alpha) are respectively the roots of x2−2px+2=0x^2 - 2px + 2 = 0; x2−2qx+3=0x^2 - 2qx + 3 = 0; x2−2rx+6=0x^2 - 2rx + 6 = 0 where α,β,γ\alpha, \beta, \gamma are all positive, then the value of p+q+rp + q + r is
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medium
A value of bb for which the equations x2+bx−1=0x^2 + bx - 1 = 0 and x2+x+b=0x^2 + x + b = 0 have one root in common is
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medium
Let α,β\alpha, \beta be the roots of the equation (x−a)(x−b)=c(x-a)(x-b) = c, c≠0c \neq 0. Then the roots of the equation (x−α)(x−β)+c=0(x-\alpha)(x-\beta) + c = 0 are
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medium
The set of values of aa for which (a−1)x2−(a+1)x+(a−1)≥0(a-1)x^2 - (a+1)x + (a-1) \geq 0 holds true for all x≥2x \geq 2 is
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medium
If a,b,c,da, b, c, d are distinct positive real numbers such that aa and bb are the roots of x2−10cx−11d=0x^2 - 10cx - 11d = 0 and cc and dd are the roots of x2−10ax−11b=0x^2 - 10ax - 11b = 0, then the value of a+b+c+da + b + c + d is
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medium
The number of integral values of aa so that the graph of y=16x2+8(a+5)x−7a−5y = 16x^2 + 8(a+5)x - 7a - 5 is always above the xx-axis is
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medium
If α\alpha and β\beta are the roots of the equation ax2+bx+c=0ax^2 + bx + c = 0, then 1aα+b+1aβ+b\dfrac{1}{a\alpha + b} + \dfrac{1}{a\beta + b} is equal to
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medium
Let α,β,γ,δ\alpha, \beta, \gamma, \delta be the roots (real or non-real) of x4−3x+1=0x^4 - 3x + 1 = 0. The value of α3+β3+γ3+δ3\alpha^3 + \beta^3 + \gamma^3 + \delta^3 is equal to
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medium
If tan⁡2 ⁣3π8\tan^2\!\dfrac{3\pi}{8} is a root of the equation 2x2−3ax+4b=02x^2 - 3ax + 4b = 0, where a,b∈Qa, b \in \mathbb{Q}, then the value of a+4ba + 4b can be equal to
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medium
Let P(x)=x2−(2−p)x+p−2P(x) = x^2 - (2-p)x + p - 2. If P(x)P(x) assumes both positive and negative values ∀x∈R\forall x \in \mathbb{R}, then the range of pp is
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medium
If α,β,γ\alpha, \beta, \gamma are roots of the equation 111x3−11x−1=0111x^3 - 11x - 1 = 0, then (αβ)−2+(βγ)−2+(γα)−2(\alpha\beta)^{-2} + (\beta\gamma)^{-2} + (\gamma\alpha)^{-2} equals
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medium
If the equation x3+2x2−4x+5=0x^3 + 2x^2 - 4x + 5 = 0 has roots α,β,γ\alpha, \beta, \gamma, then the value of (α3+5)(β3+5)(γ3+5)13αβγ\dfrac{(\alpha^3+5)(\beta^3+5)(\gamma^3+5)}{13\alpha\beta\gamma} is equal to
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medium
If c2=4dc^2 = 4d and the two equations x2−ax+b=0x^2 - ax + b = 0 and x2−cx+d=0x^2 - cx + d = 0 have one common root, then the value of 2(b+d)2(b+d) is equal to
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medium
The number of real solutions of the equation
(x2024+1)(1+x2+x4+⋯+x2022)=2024 x2023\left(x^{2024}+1\right)\left(1+x^2+x^4+\cdots+x^{2022}\right) = 2024\,x^{2023}
 is
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medium
If α,β\alpha, \beta are the integral roots of the quadratic equation
x2−αβ x+α2+β=0,x^{2} - \alpha\beta\,x + \alpha^{2} + \beta = 0 ,
 then the number of possible different quadratic equation(s) is
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easy
The values of α\alpha and β\beta for which the quadratic equation x2+2x+2+e2α−cos⁡β=0x^2 + 2x + 2 + e^{2\alpha} - \cos\beta = 0 has a real solution is
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easy
If the vertex of the quadratic expression y=2x2−4x+6y = 2x^2 - 4x + 6 is (m,n)(m, n), then the quadratic equation whose vertex is (2m,n2)\left(2m, \dfrac{n}{2}\right) is
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easy
If α,β\alpha, \beta are the roots of x2−p(x+1)−c=0x^2 - p(x+1) - c = 0, c≠1c \neq 1, then the value of (α+1)(β+1)(\alpha+1)(\beta+1) is equal to
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easy
Let α\alpha and β\beta be the roots of the equation x2−(p−2)x−(p−1)=0x^2 - (p-2)x - (p-1) = 0, p∈Rp \in \mathbb{R}. If α2+β2\alpha^2 + \beta^2 is least, then pp equals
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easy
If α,β,γ\alpha, \beta, \gamma are roots of the equation x3−2x2−1=0x^3 - 2x^2 - 1 = 0 and Tn=αn+βn+γnT_n = \alpha^n + \beta^n + \gamma^n, then the value of T11−T8T10\dfrac{T_{11} - T_8}{T_{10}} is equal to
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easy
If the quadratic polynomial P(x)=4x2+(k+8)x+9P(x) = 4x^2 + (k+8)x + 9, k∈Ik \in \mathbb{I}, is positive for every real xx, then the greatest absolute value of kk is
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