NTA JEE Mains 22nd Jan 2025 Shift 1

Let $$ A $$ be a square matrix of order 3 such that $$det(A)=-2 \text{ and }det(3adj(-6adj(3A)))=2^{m+n}\cdot3^{mn}$$, $$m>n. \text{ Then } 4m+2n\text{ is equal to } $$_______

$$ \text{If } \sum_{r=0}^5 \frac{^{11}C_{2r+1}}{2r+2}=\frac{m}{n},gcd(m, n)=1, \text{ then }m - n \text{ is equal to } $$ _______

$$ \text{Let }\overrightarrow{c} \text{ be the projection vector of }\overrightarrow{b}=\lambda\widehat{i}+4\widehat{k}, \lambda > 0 , \text{ on the vector } \overrightarrow{a}=\widehat{i}+2\widehat{j}+2\widehat{k}. \text{ If } \mid \overrightarrow{a}+ \overrightarrow{c}\mid= 7, \text{ then the area of the parallelogram formed by the vectors }\overrightarrow{b} \text{ and }\overrightarrow{c} \text{ is } $$______

Let the function, $$f(x)=\begin{cases}-3ax^{2}-2, & x < 1\\a^{2}+bx, & x \geq 0\end{cases}$$ be differentiable for all  $$x \in R,  $$ where $$ a>1, b \in R$$.  If the area of the region enclosed by $$ y=f(x) \text{and the line } y= -20 \text{ is } \alpha+\beta\sqrt{3},\alpha, \beta \in Z$$,  then the value of  $$\alpha + \beta \text{ is } $$_______

$$ \text{Let } L_1:\frac{x-1}{3}=\frac{y-1}{-1}=\frac{z+1}{0} \text{ and } L_2:\frac{x-2}{2}=\frac{y}{0}=\frac{z+4}{\alpha},\alpha \in R$$, be two lines, which intersect at the point $$ B. \text{ If } P $$ is the foot of perpendicular from the point $$A(1,1,-1) \text{ on } ,L_2 \text{ then the value of }26\alpha(PB)^{2} \text{ is } $$______

An electron is made to enter symmetrically between two parallel and equally but oppositely charged metal plates, each of 10 cm length. The electron emerges out of the electric field region with a horizontal component of velocity $$ 10^{6} m/s$$ . If the magnitude of the electric field between the plates is $$ 9.1\text{ } V/cm $$, then the vertical component of velocity of electron is (mass of electron $$=9.1\times 10^{31}kg $$ and charge of electron $$ =1.6\times 10^{-19}C) $$

Given below are two statements :
Statement-I : The equivalent emf of two nonideal batteries connected in parallel is smaller than either of the two emfs.
Statement-II : The equivalent internal resistance of two nonideal batteries connected in parallel is smaller than the internal resistance of either of the two batteries. In the light of the above statements, choose the correct answer from the options given below.

A uniform circular disc of radius ' R ' and mass ' M ' is rotating about an axis perpendicular to its plane and passing through its centre. A small circular part of radius R/2 is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above.

An amount of ice of mass $$10^{-3}kg \text{ and temperature } -10^{o}C$$ is transformed to vapour of temperature $$110^{o}C$$ by applying heat. The total amount of work required for this conversion is, (Take, specific heat of ice $$= 2100Jkg^{-1}K^{-1},$$ specific heat of water $$ 4180Jkg^{-1}K^{-1},$$  specific heat of steam  $$=1920Jkg^{-1}K^{-1},$$  Latent heat of ice $$=3.35\times10^{5}Jkg^{-1} $$ and Latent heat of steam $$ = 2.25\times10^{6}Jkg^{-1})$$

$$ \text{An electron in the ground state of the hydrogen atom has the orbital radius of }5.3\times 10^{-11} m \text{ while that for the electron in third excited state is } 8.48\times 10^{-10}m. \text{ The ratio of the de Broglie wavelengths of electron in the excited state to that in the ground state is }$$