JEE Main 2024 Solutions
JEE Main Mathematics (2024)

Let $S = \{x \in \mathbb{R} : (\sqrt{3}+\sqrt{2})^x + (\sqrt{3}-\sqrt{2})^x = 10\}$. Then the number of elements in $S$ is

If $z = x+iy$, $xy \ne 0$, satisfies the equation $z^2 + i\bar{z} = 0$, then $|z^2|$ is equal to

If $S = \{z \in \mathbb{C} : |z-i| = |z+i| = |z-1|\}$, then $n(S)$ is

If $\alpha$ satisfies the equation $x^2+x+1=0$ and $(1+\alpha)^7 = A + B\alpha + C\alpha^2$, $A, B, C \ge 0$, then $5(3A-2B-C)$ is equal to ____.

Let $S = \{z \in \mathbb{C} : |z-1|=1\}$ and $(\sqrt{2}-1)(z+\bar{z})-i(z-\bar{z})=2\sqrt{2}$. Let $z_1, z_2 \in S$ be such that $|z_1|=\max_{z \in S}|z|$ and $|z_2|=\min_{z \in S}|z|$. Then $|\sqrt{2}z_1-z_2|^2$ equals

Let $\alpha, \beta \in \mathbb{N}$ be roots of the equation $x^2-70x+\lambda=0$, where $\dfrac{\lambda}{2}, \dfrac{\lambda}{3} \notin \mathbb{N}$. If $\lambda$ assumes the minimum possible value, then $\dfrac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}$ is equal to ____.

If the domain of the function $f(x) = \cos^{-1}\!\left(\dfrac{2-|x|}{4}\right) + \{\log_e(3-x)\}^{-1}$ is $[-\alpha, \beta) - \{\gamma\}$, then $\alpha + \beta + \gamma$ is equal to

Let $\{x\}$ denote the fractional part of $x$ and $f(x) = \dfrac{\cos^{-1}(1-\{x\}^2)\sin^{-1}(1-\{x\})}{\{x\} - \{x\}^3}$, $x \ne 0$. If $L$ and $R$ respectively denote the left hand limit and the right hand limit of $f(x)$ at $x = 0$, then $\dfrac{32}{\pi^2}(L^2 + R^2)$ is equal to

Let $P = \{z \in \mathbb{C} : |z+2-3i| \le 1\}$ and $Q = \{z \in \mathbb{C} : z(1+i)+\bar{z}(1-i)+\le -8\}$. Let in $P \cap Q$, $|z-3+2i|$ be maximum and minimum at $z_1$ and $z_2$ respectively. If $|z_1|^2+2|z_2|^2 = \alpha+\beta\sqrt{2}$, where $\alpha, \beta$ are integers, then $\alpha+\beta$ equals ____.

If $A$ denotes the sum of all the coefficients in the expansion of $(1-3x+10x^2)^n$ and $B$ denotes the sum of all the coefficients in the expansion of $(1+x^2)^n$, then

Number of integral terms in the expansion of $\left\{7^{(1/2)}+11^{(1/6)}\right\}^{824}$ is equal to ____.

If the coefficient of $x^{30}$ in the expansion of $\left(1+\dfrac{1}{x}\right)^6(1+x^2)^7(1-x^3)^8$; $x \ne 0$ is $\alpha$, then $|\alpha|$ equals ____.

${}^{n-1}C_r = (k^2-8)\,{}^nC_{r+1}$ if and only if

Let $A = \{1, 2, 3, \ldots, 7\}$ and let $P(A)$ denote the power set of $A$. If the number of functions $f: A \to P(A)$ such that $a \in f(a)$, $\forall\, a \in A$ is $m^n$, $m$ and $n \in \mathbb{N}$ and $m$ is least, then $m+n$ is equal to ____.

The number of elements in the set $S = \{(x, y, z) : x, y, z \in \mathbb{Z},\; x+2y+3z=42,\; x, y, z \ge 0\}$ equals ____.

If $n$ is the number of ways five different employees can sit into four indistinguishable offices where any office may have any number of persons including zero, then $n$ is equal to

Let the set of all $a\in R$ such that the equation $\cos 2x+a\sin x=2a-7$ has a solution be $[p, q]$ and $r=\tan 9^\circ-\tan 27^\circ-\frac{1}{\cot 63^\circ}+\tan 81^\circ$, then $pqr$ is equal to ________.

If $2\sin^3x+\sin 2x\cos x+4\sin x-4=0$ has exactly 3 solutions in the interval $\left[0, \frac{n\pi}{2}\right], n\in N$, then the roots of the equation $x^2+nx+(n-3)=0$ belong to:

If $\tan A=\frac{1}{\sqrt{x(x^2+x+1)}}, \tan B=\frac{\sqrt{x}}{\sqrt{x^2+x+1}}$ and $\tan C=(x^{-3}+x^{-2}+x^{-1})^{\frac{1}{2}}, 0<A,B,C<\frac{\pi}{2}$, then $A+B$ is equal to: