Basic Maths

If the set of all solutions of \(|{x}^{2}+x-9|=|x|+|{x}^{2}-9|\)is \([\alpha ,\beta ]\cup [\gamma ,\infty ),\) then \(\left({\alpha }^{2}+{\beta }^{2}+{\gamma }^{2}\right)\) is equal to:

[JEE Main 2026, 4 Apr (Shift 1)]

The sum of the squares of the roots of \(|x-2{|}^{2}+|x-2|-2=0\) and the squares of the roots of \({x}^{2}-2|x-3|-5=0\), is

[JEE Main 2025, 8 Apr (Shift 1)]

The number of real roots of the equation \(\mathrm{x}|\mathrm{x}-2|+3|\mathrm{x}-3|+1=0\) is :

[JEE Main 2025, 7 Apr (Shift 2)]

The sum of all the real solutions of the equation \(\log _{(x+3)}\left(6 x^2+28 x+30\right)=5-2 \log _{(6 x+10)}\left(x^2+6 x+9\right)\) is equal to

[JEE Main 2026, 23 Jan (Shift 2)]

The sum of the squares of the roots of \(|x-2{|}^{2}+|x-2|-2=0\) and the squares of the roots of \({x}^{2}-2|x-3|-5=0\), is

[JEE Main 2025, 8 Apr (Shift 1)]

The product of all real solutions of the equation \({e}^{5{\left({\log }_{e}x\right)}^{2}+3}={x}^{8},x>0,\) is:

[JEE Main 2025, 22 Jan (Shift 1)]

\(e^{5(\ln x)^2+3}=x^8 \text {. Product of all real values of } \mathrm{x} \text {. }\)

If \({e}^{5\text{ }{\left(\ln \text{ }x\right)}^{2}+3}={x}^{8}\) then product of all real values of \(x.\)

If the domain of the function \(f\left(x\right)={\text{log}}_{\left(10{x}^{2}−17x+7\right)}\left(18{x}^{2}−11x+1\right)\) is \(\left(−∞,a\right)\cup \left(b,c\right)\cup \left(d,∞\right)−\left\{e\right\}\), then \(90\left(a+b+c+d+e\right)\) equals:

[JEE Main 2026, 24 Jan (Shift 1)]

The number of distinct real solutions of the equation \(x\left|x+4\right|+3\left|x+2\right|+10=0\) is

[JEE Main 2026, 22 Jan (Shift 1)]

The sum of the squares of the roots of \(|x-2{|}^{2}+|x-2|-2=0\) and the squares of the roots of \({x}^{2}-2|x-3|-5=0\), is

[JEE Main 2025, 8 Apr (Shift 1)]

The sum of the squares of the roots of \(|x-2{|}^{2}+|x-2|-2=0\) and the squares of the roots of \({x}^{2}-2|x-3|-5=0\),

JEE MAINS [2025]

The number of real roots of the equation \(x|x-2|+3|x-3|+1=0\) is:

[JEE Main 2025, 7 Apr (Shift 2)]

The number \( \log _{2} 7 \) is

JEE MAINS [2023]

The sum of the roots of the equation, \(x+1-2{\log }_{2}\left(3+{2}^{x}\right)+2{\log }_{4}\left(10-{2}^{-x}\right)=0\), is:

[JEE Main 2021, 31 Aug (Shift 2)]

Solution of \(2^x+2^{|x|} \geq 2 \sqrt{2}\) is :

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 \( S \):

[JEE Main 2020, 8 Jan (Shift 2)]

The number of integral solutions x of \({\log }_{\left(x+\frac{7}{2}\right)}{\left(\frac{x-7}{2x-3}\right)}^{2}\geq 0\) is ________ .

[JEE Main 2023, 11 Apr (Shift 1)]

The number of elements in the set \( \{x \in \mathbb{R}:(|x|-3)|x+4|=6\} \) is equal to

[JEE Main 2021, 16 Mar (Shift 1)]

The number \( \log _{2} 7 \) is

If \(\left|3x−5\right|\text{ }\leq \text{ }2,\) then

The number of real roots of the equation \(x|x|-5|x+2|\) \(+6=0\), is

[JEE Main 2023, 15 Apr (Shift 1)]

The number of integral solutions of

\( \log _{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2 x-3}\right)^{2} \geq 0 \)

[JEE Main 2023, 11 Apr (Shift 1)]

The number of real solutions of the equation, \({x}^{2}-|x|-12=0\text{ }\)

[JEE Main 2021, 25 Jul (Shift 2)]

The sum of the roots of the equation \(x+1-2{\log }_{2}\left(3+{2}^{x}\right)+2{\log }_{4}\left(10-{2}^{-x}\right)=0\) is-

[JEE Main 2021, 31 Aug (Shift 2)]