JEE MAIN Numerical Questions

Let $A=\{2,3,4,5,6\}$. Let $R$ be a relation on the set $A\times A$ given by $(x,y)R(z,w)$ if and only if $x$ divides $z$ and $y=w$. Then the number of elements in $R$ is ______.

If the domain of the function $f(x)=\sqrt{{\log }_{0.6}\left(\left|\frac{2x-5}{x^2-4}\right|\right)}$ is $(-\infty ,a]\cup \{b\}\cup [c,d)\cup (e,\infty )$, then the value of $a+b+c+d+e$ is ____.

If ${\sum }_{k=1}^n{a}_k=6n^3$, then ${\left(\frac{{\sum }_{k=1}^6(a_{k+1}-a_k)}{36}\right)}^2$ is equal to ______.

Let $a,b,c\in \{1,2,3,4\}$. If the probability that $a{x}^2+2\sqrt{2}bx+c>0$ for all $x\in \mathbb{R}$ is $\frac{m}{n}$ with $\gcd (m,n)=1$, then $m+n$ is equal to ______.

Let a circle $C$ have its centre in the first quadrant, intersect the coordinate axes at exactly three points, and cut off equal intercepts from the coordinate axes. If the length of the chord of $C$ on the line $x+y=1$ is $\sqrt{14}$, then the square of the radius of $C$ is ______.

Consider the matrices $A=\left[\begin{array}{cc}2& -2\\ 4& -2\end{array}\right]$ and $B=\left[\begin{array}{cc}3& 9\\ 1& 3\end{array}\right]$. If matrices $P$ and $Q$ are such that $PA=B$ and $AQ=B$, then the absolute value of the sum of the diagonal elements of $2(P+Q)$ is ____.

Let $A$ be the point $(3,0)$ and circles with variable diameter $AB$ touch the circle $x^2+y^2=36$ internally. Let the curve $C$ be the locus of the point $B$. If the eccentricity of $C$ is $e$, then $72e^2$ is equal to ____.

If the area of the region bounded by $16x^2-9y^2=144$ and $8x-3y=24$ is $A$, then $3(A+6{\log }_e(3))$ is equal to ____.

The number of points in the interval $[2,4]$, at which the function $f(x)=[x^2-x-\frac{1}{2}]$, where $[\cdot ]$ denotes the greatest integer function, is discontinuous, is ____.

The surface of sodium metal is irradiated with radiation of wavelength $x$ nm. The kinetic energy of ejected electrons is $2.8\times 10^{-20}$ J. The work function of sodium is $2.3$ eV. The value of $x$ is ______ $\times 10^2$ nm. (Nearest integer)

(Given: $h=6.6\times 10^{-34}$ J s; $1\mathrm{eV}=1.6\times 10^{-19}$ J; $c=3.0\times 10^8$ m s${}^{-1}$)

Consider the following gas phase reaction being carried out in a closed vessel at 25C:

$2A(g)\to 4B(g)+C(g)$

The total pressure of the system at time 30 minutes is 300 mm Hg and at infinite time (reaction completion) is 600 mm Hg. The pressure of C(g) at 30 minutes time interval would be ________ mm Hg. (nearest integer)

Consider the following two half-cell reactions along with the standard reduction potential given:

$\mathrm{C}{\mathrm{O}}_2+6{\mathrm{H}}^{+}+6{\mathrm{e}}^{-}⟶\mathrm{C}{\mathrm{H}}_3\mathrm{OH}+{\mathrm{H}}_2\mathrm{O}$ $E_{red}^{\circ }=0.02\mathrm{V}$

$\frac{1}{2}{\mathrm{O}}_2+2{\mathrm{H}}^{+}+2{\mathrm{e}}^{-}⟶{\mathrm{H}}_2\mathrm{O}$ $E_{red}^{\circ }=1.23\mathrm{V}$

A fuel cell was set up using the above two reactions such that the cell operates under the standard condition of 1 bar pressure and 298 K temperature. The fuel cell works with 80% efficiency. If the work derived from the cell using 1 mol of $C{H}_{3}OH$ is used to compress an ideal gas isothermally against a constant pressure of 1 kPa, then the change in the volume of the gas, $\Delta V=$ ______ m${}^3$ (nearest integer).

Given: $F=96500\mathrm{C}\mathrm{mo}{\mathrm{l}}^{-1}$

Number of paramagnetic ions among the following d- and f-block metal ions is _______.

$\mathrm{M}{\mathrm{n}}^{2+},\mathrm{C}{\mathrm{u}}^{2+},\mathrm{Z}{\mathrm{n}}^{2+},\mathrm{Y}{\mathrm{b}}^{2+},\mathrm{S}{\mathrm{c}}^{3+},\mathrm{L}{\mathrm{a}}^{3+},\mathrm{G}{\mathrm{d}}^{3+},\mathrm{L}{\mathrm{u}}^{3+},\mathrm{T}{\mathrm{i}}^{4+},\mathrm{C}{\mathrm{e}}^{4+}$

(Atomic number of Mn = 25, Cu = 29, Zn = 30, Yb = 70, Sc = 21, La = 57, Gd = 64, Lu = 71, Ti = 22, Ce = 58)

Consider the following reaction sequence starting from p-nitrotoluene (para-nitrotoluene): (Given : molar mass in g $mo{l}^{-1}$ C : 12, H : 1, O : 16, N : 14, Br : 80, Ag : 108)

At the transition temperature $T$, $A\rightleftharpoons B$ and $\Delta G^{\circ }=105-35\log T$ where A and B are two states of substance X. The transition temperature in ${}^{\circ }\mathrm{C}$ when pressure is 1 atm is (Nearest integer).