Let $(a,b)\subset(0,2\pi)$ be the largest interval for which $\sin^{-1}(\sin\theta)-\cos^{-1}(\sin\theta)>0,\ \theta\in(0,2\pi)$, holds. If $\alpha x^{2}+\beta x+\sin^{-1}(x^{2}-6x+10)+\cos^{-1}(x^{2}-6x+10)=0$ and $\alpha-\beta=b-a$, then $\alpha$ is equal to:

1

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

Topic: JEE Main 2023 (31 January Evening Shift)

2

Let $f:\mathbb{R}-\{2,6\}\to\mathbb{R}$ be the real-valued function defined as $f(x)=\dfrac{x^{2}+2x+1}{x^{2}-8x+12}$. …

Topic: JEE Main 2023 (31 January Evening Shift)

3

Let $\vec a=\hat{\imath}+2\hat{\jmath}+3\hat{k}$, $\vec b=\hat{\imath}-\hat{\jmath}+2\hat{k}$ and $\vec c=5\hat{\imath}…

Topic: JEE Main 2023 (31 January Evening Shift)

1

Let $\alpha>0$. If $\displaystyle \int_{0}^{\alpha}\frac{x}{\sqrt{x+\alpha}-\sqrt{x}}\,dx=\dfrac{16+20\sqrt{2}}{15}$, t…

Topic: JEE Main 2023 (31 January Evening Shift)

2

Let the mean and standard deviation of marks of class $A$ of $100$ students be respectively $40$ and $\alpha\ (>\,0)$, …

Topic: JEE Main 2023 (31 January Evening Shift)

3

Let $a_{1},a_{2},a_{3},\ldots$ be an A.P. If $a_{7}=3$, the product $a_{1}a_{4}$ is minimum and the sum of its first $n…

Topic: JEE Main 2023 (31 January Evening Shift)

4

$ \text{The complex number } z=\dfrac{i-1}{\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}} \text{ is equal to:} $

Topic: JEE Main 2023 (31 January Evening Shift)

5

Among the relations $S=\{(a,b):\,a,b\in\mathbb{R}\setminus\{0\},\ 2+\frac{a}{b}>0\}$ and $T=\{(a,b):\,a,b\in\mathbb{R},…

Topic: JEE Main 2023 (31 January Evening Shift)

6

The absolute minimum value of the function $f(x)=\lvert x^{2}-x+1\rvert+\lfloor x^{2}-x+1\rfloor$, where $\lfloor t\rfl…

Topic: JEE Main 2023 (31 January Evening Shift)

7

If $\phi(x)=\dfrac{1}{\sqrt{x}}\displaystyle\int_{\pi/4}^{x}\!\big(4\sqrt{2}\sin t-3\phi'(t)\big)\,dt,\ x>0,$ then $\ph…

Topic: JEE Main 2023 (31 January Evening Shift)

8

Let $\mathcal{H}$ be the hyperbola whose foci are $(1\pm\sqrt{2},\,0)$ and whose eccentricity is $\sqrt{2}$. Then the l…

Topic: JEE Main 2023 (31 January Evening Shift)

9

$ \displaystyle \lim_{x\to\infty} \left\{ \frac{\big(\sqrt{3x+1}+\sqrt{3x-1}\big)^{6}+\big(\sqrt{3x+1}-\sqrt{3x-1}\big)…

Topic: JEE Main 2023 (31 January Evening Shift)

10

Let $y=y(x)$ be the solution of the differential equation $(3y^{2}-5x^{2})\,y\,dx+2x\,(x^{2}-y^{2})\,dy=0$ such that $y…

Topic: JEE Main 2023 (31 January Evening Shift)