1
JEE Main 2023 (Online) 30th January Evening Shift
Numerical
+4
-1
Iron oxide FeO, crystallises in a cubic lattice with a unit cell edge length of 5.0 Å. If density of the $\mathrm{FeO}$ in the crystal is $4.0 \mathrm{~g} \mathrm{~cm}^{-3}$, then the number of $\mathrm{FeO}$ units present per unit cell is __________. (Nearest integer)

Given: Molar mass of $\mathrm{Fe}$ and $\mathrm{O}$ is 56 and $16 \mathrm{~g} \mathrm{~mol}^{-1}$ respectively. $\mathrm{N}_{\mathrm{A}}=6.0 \times 10^{23} \mathrm{~mol}^{-1}$
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2
JEE Main 2023 (Online) 30th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
A vector $\vec{v}$ in the first octant is inclined to the $x$-axis at $60^{\circ}$, to the $y$-axis at 45 and to the $z$-axis at an acute angle. If a plane passing through the points $(\sqrt{2},-1,1)$ and $(a, b, c)$, is normal to $\vec{v}$, then :
A
$a+b+\sqrt{2} c=1$
B
$\sqrt{2} a+b+c=1$
C
$\sqrt{2} a-b+c=1$
D
$a+\sqrt{2} b+c=1$
3
JEE Main 2023 (Online) 30th January Evening Shift
MCQ (More than One Correct Answer)
+4
-1
Let $a_{1}=1, a_{2}, a_{3}, a_{4}, \ldots .$. be consecutive natural numbers.

Then $\tan ^{-1}\left(\frac{1}{1+a_{1} a_{2}}\right)+\tan ^{-1}\left(\frac{1}{1+a_{2} a_{3}}\right)+\ldots . .+\tan ^{-1}\left(\frac{1}{1+a_{2021} a_{2022}}\right)$ is equal to :
A
$\frac{\pi}{4}-\cot ^{-1}(2022)$
B
$\frac{\pi}{4}-\tan ^{-1}(2022)$
C
$\cot ^{-1}(2022)-\frac{\pi}{4}$
D
$\tan ^{-1}(2022)-\frac{\pi}{4}$
4
JEE Main 2023 (Online) 30th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
For $\alpha, \beta \in \mathbb{R}$, suppose the system of linear equations

\begin{aligned} & x-y+z=5 \\ & 2 x+2 y+\alpha z=8 \\ & 3 x-y+4 z=\beta \end{aligned}

has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of :
A
$x^2+18 x+56=0$
B
$x^2-10 x+16=0$
C
$x^2+14 x+24=0$
D
$x^2-18 x+56=0$
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