Quadratic Equations

22 JEE Main questions · Maths

MathsJEE Main 21 Jan 2026, Shift 1·Q19·MCQ·Easy–moderate
Open
Equations involving modulus
The sum of all the roots of the equation $(x-1)^2 - 5\lvert x - 1\rvert + 6 = 0$ , is:
1. (1)4
2. (2)3
3. (3)1
4. (4)5
MathsJEE Main 21 Jan 2026, Shift 2·Q9·MCQ·Easy–moderate
Open
Location of roots
Let $\alpha$ and $\beta$ be the roots of equation $x^2 + 2ax + (3a + 10) = 0$ such that $\alpha < 1 < \beta$ . Then the set of all possible values of $a$ is:
1. (1) $\left(-\infty, \dfrac{-11}{5}\right) \cup (5, \infty)$
2. (2) $(-\infty, -2) \cup (5, \infty)$
3. (3) $(-\infty, -3)$
4. (4) $\left(-\infty, \dfrac{-11}{5}\right)$
MathsJEE Main 22 Jan 2026, Shift 1·Q8·MCQ·Easy–moderate
Open
Equations involving modulus
The number of distinct real solutions of the equation $x\lvert x + 4\rvert + 3\lvert x + 2\rvert + 10 = 0$ is
1. (1)3
2. (2)1
3. (3)0
4. (4)2
MathsJEE Main 22 Jan 2026, Shift 2·Q13·MCQ·Easy–moderate
Open
Difference of roots
Let $\alpha, \beta$ be the roots of the quadratic equation $12x^2 - 20x + 3\lambda = 0$ , $\lambda \in \mathbb{Z}$ . If $\dfrac12 \le \lvert\beta - \alpha\rvert \le \dfrac32$ , then the sum of all possible values of $\lambda$ is:
1. (1)6
2. (2)1
3. (3)3
4. (4)4
MathsJEE Main 23 Jan 2026, Shift 1·Q8·MCQ·Moderate
Open
Equations reducible to quadratic with modulus
If $\alpha$ and $\beta$ $(\alpha < \beta)$ are the roots of the equation $\left(-2 + \sqrt3\right)\left(\left\lvert\sqrt x - 3\right\rvert\right) + \left(x - 6\sqrt x\right) + \left(9 - 2\sqrt3\right) = 0,$ $x \ge 0$ , then $\sqrt{\dfrac{\beta}{\alpha}} + \sqrt{\alpha\beta}$ is equal to:
1. (1)8
2. (2)9
3. (3)10
4. (4)11
MathsJEE Main 23 Jan 2026, Shift 1·Q19·MCQ·Easy–moderate
Open
Word problem leading to quadratic
A building construction work can be completed by two masons $A$ and $B$ together in 22.5 days. Mason $A$ alone can complete the construction work in 24 days less than mason $B$ alone. Then mason $A$ alone will complete the construction work in:
1. (1)24 days
2. (2)42 days
3. (3)30 days
4. (4)36 days
MathsJEE Main 23 Jan 2026, Shift 2·Q7·MCQ·Moderate
Open
Logarithmic equations
The sum of all the real solutions of the equation $\log_{(x+3)}(6x^2 + 28x + 30) = 5 - 2\log_{(6x+10)}(x^2 + 6x + 9)$ is equal to:
1. (1)2
2. (2)1
3. (3)0
4. (4)4
MathsJEE Main 24 Jan 2026, Shift 1·Q20·MCQ·Moderate
Open
Equations involving modulus
The number of the real solutions of the equation : $x\lvert x + 3\rvert + \lvert x - 1\rvert - 2 = 0$ is
1. (1)3
2. (2)2
3. (3)5
4. (4)4
MathsJEE Main 24 Jan 2026, Shift 2·Q18·MCQ·Moderate
Open
Location of roots
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
1. (1)9
2. (2)3
3. (3)4
4. (4)7
MathsJEE Main 28 Jan 2026, Shift 1·Q8·MCQ·Easy–moderate
Open
Polynomial divisibility
Let $S = \{x^3 + ax^2 + bx + c : a, b, c \in \mathbb{N} \text{ and } a, b, c \le 20\}$ be a set of polynomials. Then the number of polynomials in $S$ , which are divisible by $x^2 + 2$ , is
1. (1)20
2. (2)6
3. (3)120
4. (4)10
MathsJEE Main 28 Jan 2026, Shift 1·Q10·MCQ·Easy–moderate
Open
Roots and coefficients
If $\alpha, \beta$ , where $\alpha < \beta$ , are the roots of the equation $\lambda x^2 - (\lambda + 3)x + 3 = 0$ such that $\dfrac{1}{\alpha} - \dfrac{1}{\beta} = \dfrac{1}{3}$ , then the sum of all possible values of $\lambda$ is:
1. (1)6
2. (2)2
3. (3)4
4. (4)8
MathsJEE Main 28 Jan 2026, Shift 2·Q7·MCQ·Moderate
Open
Location of roots
Let the arithmetic mean of $\dfrac1a$ and $\dfrac1b$ be $\dfrac{5}{16}$ , $a > 2$ . If $\alpha$ is such that $a, 4, \alpha, b$ are in A.P., then the equation $\alpha x^2 - ax + 2(\alpha - 2b) = 0$ has :
1. (1)One root in $(1, 4)$ and another in $(-2, 0)$
2. (2)One root in $(0, 2)$ and another in $(-4, -2)$
3. (3)Complex roots of magnitude less than 2
4. (4)Both roots in the interval $(-2, 0)$
MathsJEE Main 2 Apr 2026, Shift 1·Q1·MCQ·Moderate
Open
Roots differing by a constant
Let $\alpha, \alpha + 2$ , $\alpha \in \mathbb{Z}$ , be the roots of the quadratic equation $x(x+2) + (x+1)(x+3) + (x+2)(x+4) + \ldots + (x+n-1)(x+n+1) = 4n$ for some $n \in \mathbb{N}$ . Then $(n + \alpha)$ is equal to:
1. (1)0
2. (2)1
3. (3)2
4. (4)3
MathsJEE Main 2 Apr 2026, Shift 2·Q1·MCQ·Moderate
Open
Relation between roots and coefficients
Let $\alpha, \beta$ be the roots of the equation $x^2 - 3x + r = 0$ , and $\dfrac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2 + 3x + r = 0$ . If the roots of the equation $x^2 + 6x = m$ are $2\alpha + \beta + 2r$ and $\alpha - 2\beta - \dfrac{r}{2}$ , then $m$ is equal to:
1. (1) $-135$
2. (2) $-567$
3. (3)135
4. (4)567
MathsJEE Main 2 Apr 2026, Shift 2·Q12·MCQ·Easy–moderate
Open
Vertex of quadratic expression
Let the parabola $y = x^2 + px + q$ passing through the point $(1, -1)$ be such that the distance between its vertex and the $x$ -axis is minimum. Then the value of $p^2 + q^2$ is:
1. (1)2
2. (2)4
3. (3)5
4. (4)8
MathsJEE Main 4 Apr 2026, Shift 1·Q2·MCQ·Moderate
Open
Modulus equations and inequalities
If the set of all solutions of $\lvert x^2 + x - 9\rvert = \lvert x\rvert + \lvert x^2 - 9\rvert$ is $[\alpha, \beta] \cup [\gamma, \infty)$ , then $(\alpha^2 + \beta^2 + \gamma^2)$ is equal to:
1. (1)9
2. (2)18
3. (3)36
4. (4)72
MathsJEE Main 4 Apr 2026, Shift 2·Q4·MCQ·Moderate
Open
Cubic with complex roots
If $\alpha = 1$ and $\beta = 1 + i\sqrt2$ , where $i = \sqrt{-1}$ are two roots of the equation $x^3 + ax^2 + bx + c = 0$ , $a, b, c \in \mathbb{R}$ , then $\displaystyle\int_{-1}^{1}\left(x^3 + ax^2 + bx + c\right)dx$ is equal to:
1. (1) $-2$
2. (2) $-4$
3. (3) $-8$
4. (4) $-10$
MathsJEE Main 4 Apr 2026, Shift 2·Q5·MCQ·Moderate
Open
Location of roots
If the quadratic equation $(\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0$ , $\lambda \neq -2$ , has two positive roots, then the number of possible integral values of $\lambda$ is:
1. (1)1
2. (2)2
3. (3)3
4. (4)4
MathsJEE Main 5 Apr 2026, Shift 1·Q1·MCQ·Moderate
Open
Symmetric functions of roots
Let $a, b \in \mathbb{C}$ . Let $\alpha, \beta$ be the roots of the equation $x^2 + ax + b = 0$ . If $\beta - \alpha = \sqrt{11}$ and $\beta^2 - \alpha^2 = 3i\sqrt{11}$ , then $(\beta^3 - \alpha^3)^2$ is equal to:
1. (1)160
2. (2)176
3. (3)194
4. (4)187
MathsJEE Main 5 Apr 2026, Shift 2·Q1·MCQ·Moderate
Open
Roots in geometric progression
Let $\alpha, \beta$ be the roots of the equation $x^2 - x + p = 0$ and $\gamma, \delta$ be the roots of the equation $x^2 - 4x + q = 0$ ; $p, q \in \mathbb{Z}$ . If $\alpha, \beta, \gamma, \delta$ are in G.P., then $\lvert p + q\rvert$ equals:
1. (1)16
2. (2)32
3. (3)34
4. (4)38
We corrected a printing error in the original paper
Source printed “be the roots the equation $x^2 - 4x + q = 0$ ” (missing “of”).
MathsJEE Main 6 Apr 2026, Shift 1·Q2·MCQ·Moderate
Open
Relation between roots and coefficients
Let one root of the quadratic equation in $x$ : $(k^2 - 15k + 27)x^2 + 9(k - 1)x + 18 = 0$ be twice the other. Then the length of the latus rectum of the parabola $y^2 = 6kx$ is equal to:
1. (1)4
2. (2)6
3. (3)8
4. (4)12
MathsJEE Main 8 Apr 2026, Shift 2·Q21·Numerical·Moderate
Open
Logarithmic equations
The sum of squares of all real solutions of the equation $\log_{(x+1)}(2x^2 + 5x + 3) = 4 - \log_{(2x+3)}(x^2 + 2x + 1)$ is equal to .
We corrected a printing error in the original paper
Source printed “all real solution” (singular).