Consider the following system of equations \[ \begin{cases} \alpha x+2y+z=1,\\ 2\alpha x+3y+z=1,\\ 3x+\alpha y+2z=\beta \end{cases} \] for some $\alpha,\beta\in\mathbb{R}$. Then which of the following is NOT correct?

1

Let $$A=\{(x,y)\in\mathbb{R}^{2}:\ y\ge 0,\ 2x\le y\le \sqrt{4-(x-1)^{2}}\}$$ and $$B=\{(x,y)\in\mathbb{R}\times\math…

Topic: JEE Main 2023 (29 January Morning Shift)

2

Let $[x]$ denote the greatest integer $\le x$. Consider the function $$f(x)=\max\{x^{2},\,1+[x]\}.$$ Then the value o…

Topic: JEE Main 2023 (29 January Morning Shift)

3

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $$f(x)=\frac{x^{2}+2x+1}{x^{2}+1}.$$ Then:

Topic: JEE Main 2023 (29 January Morning Shift)

4

The domain of $$f(x)=\frac{\log_{(x+1)}(x-2)}{e^{2\log_e x}-(2x+3)},\quad x\in\mathbb{R}$$ is:

Topic: JEE Main 2023 (29 January Morning Shift)

5

Let $\Delta$ be the area of the region $\{(x,y)\in\mathbb{R}^{2}:\ x^{2}+y^{2}\le 21,\ y^{2}\le 4x,\ x\ge 1\}$. Then …

Topic: JEE Main 2023 (29 January Morning Shift)

6

For two non-zero complex numbers $z_{1}$ and $z_{2}$, if $\operatorname{Re}(z_{1}z_{2})=0$ and $\operatorname{Re}(z_{1}…

Topic: JEE Main 2023 (29 January Morning Shift)

7

A light ray emits from the origin making an angle $30^\circ$ with the positive $x$-axis. After getting reflected by t…

Topic: JEE Main 2023 (29 January Morning Shift)

8

Let $\alpha$ and $\beta$ be real numbers. Consider a $3\times 3$ matrix $A$ such that $A^{2}=3A+\alpha I$. If $A^{4}=…

Topic: JEE Main 2023 (29 January Morning Shift)

9

Let $f(x)=x+\dfrac{a}{\pi^{2}-4}\sin x+\dfrac{b}{\pi^{2}-4}\cos x,\ x\in\mathbb{R}$ be a function which satisfies $\d…

Topic: JEE Main 2023 (29 January Morning Shift)

10

Let the tangents at the points $A(4,-11)$ and $B(8,-5)$ on the circle $x^{2}+y^{2}-3x+10y-15=0$, intersect at the point…

Topic: JEE Main 2023 (29 January Morning Shift)

1

If the vectors $\vec a=\lambda\,\hat i+\mu\,\hat j+4\,\hat k$, $\vec b=-2\,\hat i+4\,\hat j-2\,\hat k$ and $\vec c=2\…

Topic: JEE Main 2023 (29 January Morning Shift)

2

Let $x=2$ be a root of the equation $x^{2}+px+q=0$ and define \[ f(x)= \begin{cases} \dfrac{1-\cos\!\big(x^{2}-4px+q^{2…

Topic: JEE Main 2023 (29 January Morning Shift)