JEE MAIN 2023 Previous Year Questions (PYQs) – Page 2 of 34

For the system of linear equations $\alpha x+y+z=1,\quad x+\alpha y+z=1,\quad x+y+\alpha z=\beta$, which one of the following statements is **NOT** correct?

The value of the integral $\displaystyle \int_{-\pi/4}^{\pi/4}\frac{x+\pi/4}{\,2-\cos 2x\,}\,dx$ is:

If $y(x)=x^{x},\ x>0$, then $y''(2)-2y'(2)$ is equal to:

The area of the region given by $\{(x,y):\, xy\le 8,\ 1\le y\le x^{2}\}$ is:

The number of integral values of $k$ for which one root of the equation $2x^{2}-8x+k=0$ lies in the interval $(1,2)$ and its other root lies in the interval $(2,3)$ is:

Let $a,b$ be two real numbers such that $ab<0$. If the complex number $\dfrac{1+ai}{\,b+i\,}$ is of unit modulus and $a+ib$ lies on the circle $|z-1|=|2z|$, then a possible value of $\dfrac{1+[a]}{4b}$, where $[\,\cdot\,]$ is the greatest integer function, is:

If $A=\dfrac12\begin{bmatrix}1 & \sqrt{3}\\ -\sqrt{3} & 1\end{bmatrix}$, then:

Two dice are thrown independently. Let \(A\) be the event that the number on the \(1^{\text{st}}\) die is less than the number on the \(2^{\text{nd}}\) die; \(B\) be the event that the number on the \(1^{\text{st}}\) die is even and that on the \(2^{\text{nd}}\) die is odd; and \(C\) be the event that the number on the \(1^{\text{st}}\) die is odd and that on the \(2^{\text{nd}}\) die is even. Then:

Let \(\vec a = 2\hat i - 7\hat j + 5\hat k\), \(\vec b = \hat i + \hat k\) and \(\vec c = \hat i + 2\hat j - 3\hat k\) be three given vectors. If \(\vec r\) is a vector such that \(\vec r \times \vec a = \vec c \times \vec a\) and \(\vec r \cdot \vec b = 0\), then \(|\vec r|\) is equal to:

Let $\alpha x = \exp(x^{\beta} y^{\gamma})$ be the solution of the differential equation $2x^{2}y\,dy - (1 - xy^{2})\,dx = 0,\ x>0,\ y(2)=\sqrt{\log_{e}2}.$ Then $\alpha + \beta - \gamma$ equals: