Let $D$ be the domain of the function $f(x)=\sin^{-1}\!\left(\log_{3x}\!\left(\dfrac{6+2\log_{3}x}{-5x}\right)\right)$. If the range of the function $g: D \to \mathbb{R}$ defined by $g(x)=x-[x]$ (where $[x]$ is the greatest integer function) is $(\alpha,\beta)$, then $\alpha^{2}+\dfrac{5}{\beta}$ is equal to:
Previous 10 Questions — JEE Main 2023 (12 April Morning Shift)
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$ \text{The area of the region enclosed by the curve } y=x^{3} \text{ and its tangent at the point } (-1,-1) \text{ is:…
Topic: JEE Main 2023 (12 April Morning Shift)
If the local maximum value of the function $f(x)=\left(\dfrac{\sqrt{3}e}{2\sin x}\right)^{\sin^{2}x},; x\in\left(0,\dfr…
Topic: JEE Main 2023 (12 April Morning Shift)
Let $\mathrm{P}\left(\dfrac{2\sqrt{3}}{\sqrt{7}}, \dfrac{6}{\sqrt{7}}\right), \mathrm{Q}, \mathrm{R}$ and $\mathrm{S}$ …
Topic: JEE Main 2023 (12 April Morning Shift)
Let $\alpha, \beta$ be the roots of the quadratic equation $x^{2}+\sqrt{6}x+3=0$. Then $\dfrac{\alpha^{23}+\beta^{23}+\…
Topic: JEE Main 2023 (12 April Morning Shift)
Let $\alpha, \beta$ be the roots of the quadratic equation $x^{2}+\sqrt{6}x+3=0$. Then $\dfrac{\alpha^{23}+\beta^{23}+\…
Topic: JEE Main 2023 (12 April Morning Shift)
If the point $(\alpha, \dfrac{7\sqrt{3}}{3})$ lies on the curve traced by the mid-points of the line segments of the li…
Topic: JEE Main 2023 (12 April Morning Shift)
Let $C$ be the circle in the complex plane with centre $z_0=\tfrac{1}{2}(1+3i)$ and radius $r=1$.
Let $z_1=1+i$ and t…
Topic: JEE Main 2023 (12 April Morning Shift)
The number of five digit numbers, greater than 40000 and divisible by 5 , which can be formed using the digits 1, 3, 7 …
Topic: JEE Main 2023 (12 April Morning Shift)
Let the lines
\[
\ell_1:\ \frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha}{-2}
\quad\text{and}\quad
\ell_2:\ 3x+2y+z-2=0\;=\…
Topic: JEE Main 2023 (12 April Morning Shift)
Let $y=y(x),\ y>0$, be a solution curve of the differential equation
\[
(1+x^2)\,dy = y(x-y)\,dx.
\]
If $y(0)=1$ and $y…
Topic: JEE Main 2023 (12 April Morning Shift)
