JEE MAIN 2023 Previous Year Questions (PYQs) – Page 15 of 34

If the system of equations $x + 2y + 3z = 3$ $4x + 3y - 4z = 4$ $8x + 4y - \lambda z = 9 + \mu$ has infinitely many solutions, then the ordered pair $(\lambda,\mu)$ is equal to:

For $a \in \mathbb{C}$, let $A = \{\, z \in \mathbb{C} : \Re(a + \bar z) > \Im(\bar a + z) \,\}$ and $B = \{\, z \in \mathbb{C} : \Re(a + \bar z) < \Im(\bar a + z) \,\}$. Then among the two statements: (S1): If $\Re(a), \Im(a) > 0$, then the set $A$ contains all the real numbers. (S2): If $\Re(a), \Im(a) < 0$, then the set $B$ contains all the real numbers.

The set of all values of $a$ for which $\displaystyle \lim_{x\to a}\big([x-5]-[2x+2]\big)=0$, where $[\alpha]$ denotes the greatest integer less than or equal to $\alpha$, is equal to:

Let $a,b,c,d$ be positive real numbers such that $a+b+c+d=11$. If the maximum value of $a^5 b^3 c^2 d$ is $3750\beta$, then the value of $\beta$ is:

The locus of the mid-points of the chords of the circle $C_{1} : (x-4)^{2}+(y-5)^{2}=4$ which subtend an angle $\theta_{i}$ at the centre of the circle $C_{1}$, is a circle of radius $r_{i}$. If $\theta_{1}=\dfrac{\pi}{3}$, $\theta_{3}=\dfrac{2\pi}{3}$ and $r_{1}^{2}=r_{2}^{2}+r_{3}^{2}$, then $\theta_{2}$ is equal to:

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is :

Let $y=y(x)$ be the solution of the differential equation $(x^{2}-3y^{2})\,dx+3xy\,dy=0$, with $y(1)=1$. Then $6y^{2}(e)$ is equal to:

Let $A=\{1,3,4,6,9\}$ and $B=\{2,4,5,8,10\}$. Let $R$ be a relation defined on $A\times B$ such that $R=\{\,((a_1,b_1),(a_2,b_2)) : a_1 \le b_2 \text{ and } b_1 \le a_2 \,\}$. Then the number of elements in the set $R$ is:

If $f(x)=\dfrac{2^{2x}}{2^{2x}+2},\ x\in\mathbb{R}$, then $f\!\left(\dfrac{1}{2023}\right)+f\!\left(\dfrac{2}{2023}\right)+\cdots+f\!\left(\dfrac{2022}{2023}\right)$ is equal to:

If the $1011^{\text{th}}$ term from the end in the binomial expansion of \(\left(\dfrac{4x}{5}-\dfrac{5}{2x}\right)^{2022}\) is \(1024\) times the $1011^{\text{th}}$ term from the beginning, then \(|x|\) is equal to: