Question

Let $\left[\mathrm{t}\right]$ denote the greatest integer$\le \mathrm{t}$. Then the equation in $\mathrm{x}, {\left[\mathrm{x}\right]}^2+2\left[\mathrm{x}+2\right]-7=0$ has :

JEE Main 2020 (04 Sep Shift 1)

Options

• A: exactly two solutions
• B: exactly four integral solutions
• C: no integral solution
• D: infinitely many solutions

Question

Let $S$, be the set of all real roots of the equation, $3^x\left(3^x-1\right)+2=\left|3^x-1\right|+\left|3^x-2\right|$, then

JEE Main 2020 (08 Jan Shift 2)

Options

• A: contains exactly two elements.
• B: is a singleton.
• C: is an empty set.
• D: contains at least four elements.

Question

If $\alpha$ and $\beta$ are the distinct roots of the equation $x^2+{\left(3\right)}^{\frac{1}{4}}x+3^{\frac{1}{2}}=0$, then the value of ${\alpha }^{96}\left({\alpha }^{12}-1\right)+{\beta }^{96}\left({\beta }^{12}-1\right)$ is equal to:

JEE Main 2021 (20 Jul Shift 1)

Options

• A: $56\times 3^{25}$
• B: $56\times 3^{24}$
• C: $52\times 3^{24}$
• D: $28\times 3^{25}$

Question

The set of all values of $k>-1$, for which the equation ${\left(3x^2+4x+3\right)}^2-(k+1)\left(3x^2+4x+3\right)\left(3x^2+4x+2\right)+$ $k{\left(3x^2+4x+2\right)}^2=0$ has real roots, is:

JEE Main 2021 (27 Aug Shift 2)

Options

• A: $\left[-\frac{1}{2},1\right)$
• B: $\left(1,\frac{5}{2}\right]$
• C: $\left(\frac{1}{2},\frac{3}{2}\right]-\left\{1\right\}$
• D: $[2,3)$

Question

If $\alpha ,\beta$ are roots of the equation $x^2+5\left(\sqrt{2}\right)x+10=0,\alpha >\beta$ and $P_n={\alpha }^n-{\beta }^n$ for each positive integer $n,$ then the value of $\left(\frac{P_{17}{P}_{20}+5\sqrt{2}{P}_{17}{P}_{19}}{P_{18}{P}_{19}+5\sqrt{2}{P}_{18}^2}\right)$ is equal to

JEE Main 2021 (25 Jul Shift 1)

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Question

The sum of ${162}^{th}$ power of the roots of the equation $x^3-2x^2+2x-1=0$ is ______.

JEE Main 2021 (26 Feb Shift 1)

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Question

Let $a,b$ be two non-zero real numbers. If $p$ and $r$ are the roots of the equation $x^2-8ax+2a=0$ and $q$ and $s$ are the roots of the equation $x^2+12bx+6b=0$, such that $\frac{1}{p},\frac{1}{q},\frac{1}{r},\frac{1}{ s}$ are in A.P., then $a^{-1}-b^{-1}$ is equal to _____ .

JEE Main 2022 (25 Jul Shift 1)

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Question

Let $f\left(x\right)$ be a quadratic polynomial with leading coefficient $1$ such that $f\left(0\right)=p,p\ne 0$, and $f\left(1\right)=\frac{1}{3}$. If the equations $f\left(x\right)=0$ and $fofofof\left(x\right)=0$ have a common real root, then $f\left(-3\right)$ is equal to ______.

JEE Main 2022 (25 Jul Shift 2)

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Question

The sum of all real values of $x$ for which $\frac{3x^2-9x+17}{x^2+3x+10}=\frac{5x^2-7x+19}{3x^2+5x+12}$ is equal to

JEE Main 2022 (28 Jul Shift 1)

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Question

Let $\lambda \ne 0$ be a real number. Let $\alpha ,\beta$ be the roots of the equation $14x^2-31x+3\lambda =0$ and $\alpha ,\gamma$ be the roots of the equation $35x^2-53x+4\lambda =0$. Then $\frac{3\alpha }{\beta }$ and $\frac{4\alpha }{\gamma }$ are the roots of the equation :

JEE Main 2023 (29 Jan Shift 1)

Options

• A: $7x^2+245x-250=0$
• B: $7x^2-245x+250=0$
• C: $49x^2-245x+250=0$
• D: $49x^2+245x+250=0$

Question

The number of real roots of the equation $\sqrt{x^2-4x+3}+\sqrt{x^2-9}=\sqrt{4x^2-14x+6}$, is:

JEE Main 2023 (31 Jan Shift 1)

Options

• A: $0$
• B: $1$
• C: $3$
• D: $2$

Question

The equation $e^{4x}+8e^{3x}+13e^{2x}-8e^x+1=0,x\in R$ has :

JEE Main 2023 (31 Jan Shift 2)

Options

• A: four solutions two of which are negative
• B: two solutions and both are negative
• C: no solution
• D: two solutions and only one of them is negative

Question

The number of real solutions of the equation $x|x+5|+2|x+7|-2=0$ is_________

JEE Main 2024 (05 Apr Shift 2)

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Question

Let $\alpha, \beta$ be the distinct roots of the equation $x^2-\left(t^2-5 t+6\right) x+1=0, t \in \mathbb{R}$ and $a_n=\alpha^n+\beta^n$. Then the minimum value of $\frac{a_{2023}+a_{2025}}{a_{2024}}$ is

JEE Main 2024 (06 Apr Shift 1)

Options

• A: $-1 / 4$
• B: $-1 / 4$
• C: $-1 / 2$
• D: $1 / 4$

Question

Let $x_1, x_2, x_3, x_4$ be the solution of the equation $4 x^4+8 x^3-17 x^2-12 x+9=0$ and $\left(4+x_1^2\right)\left(4+x_2^2\right)\left(4+x_3^2\right)\left(4+x_4^2\right)=\frac{125}{16} m$. Then the value of $m$ is

JEE Main 2024 (06 Apr Shift 1)

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Question

The number of distinct real roots of the equation $|x+1||x+3|-4|x+2|+5=0$, is

JEE Main 2024 (08 Apr Shift 2)

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Question

If $\alpha ,\beta$ are the roots of the equation, ${\mathrm{x}}^2-\mathrm{x}-1=0$ and ${\mathrm{S}}_{\mathrm{n}}=2023{\alpha }^{\mathrm{n}}+2024{\beta }^{\mathrm{n}}$, then

JEE Main 2024 (27 Jan Shift 2)

Options

• A: $2 {\mathrm{S}}_{12}={\mathrm{S}}_{11}+{\mathrm{S}}_{10}$
• B: ${\mathrm{S}}_{12}={\mathrm{S}}_{11}+{\mathrm{S}}_{10}$
• C: $2 {\mathrm{S}}_{11}={\mathrm{S}}_{12}+{\mathrm{S}}_{10}$
• D: ${\mathrm{S}}_{11}={\mathrm{S}}_{10}+{\mathrm{S}}_{12}$

Question

Let $a,b,c$ be the length of three sides of a triangle satisfying the condition $\left(a^2+b^2\right)x^2-2b\left(a+c\right)$ $x+\left(b^2+c^2\right)=0$. If the set of all possible values of $x$ is in the interval $\left(\alpha ,\beta \right),$ then $12\left({\alpha }^2+{\beta }^2\right)$ is equal to _______.

JEE Main 2024 (31 Jan Shift 2)

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Question

Let $\alpha$ and $\beta$ be the roots of the equation $p{x}^2+qx-r=0$, where $p\ne 0$. If $p, q$ and $r$ be the consecutive terms of a non-constant G.P and $\frac{1}{\alpha }+\frac{1}{\beta }=\frac{3}{4}$, then the value of ${\left(\alpha -\beta \right)}^2$ is:

JEE Main 2024 (01 Feb Shift 2)

Options

• A: $\frac{80}{9}$
• B: $9$
• C: $\frac{20}{3}$
• D: $8$

Question

The number of solutions, of the equation $e^{\sin x}-2e^{-\sin x}=2$ is

JEE Main 2024 (31 Jan Shift 2)

Options

• A: $2$
• B: more than $2$
• C: $1$
• D: $0$