NTA JEE Mains 24th Jan 2025 Shift 1

Let $$ S=\{p_1,p_2,....,p_{10}\} $$ be the set of first ten prime numbers. Let $$ A=S\cup P, $$ where $$ P $$ is the set of all possible products of distinct elements of $$ S. $$ Then the number of all ordered pairs $$ (x,y),\; x\in S,\; y\in A, $$ such that $$ x $$ divides $$ y,$$ is: $$ \underline{ \hspace{2cm} } $$

$$ \text{If for some } \alpha,\beta;\; \alpha\le\beta,\; \alpha+\beta=8$$ and  $$\sec^2(\tan^{-1}\alpha)+\cosec^2(\cot^{-1}\beta)=36,$$ $$\alpha^2+\beta^2$$  is:_______

$$ \text{Let } A \text{ be a } 3\times 3 \text{ matrix such that } X^TAX=0 \text{ for all nonzero } 3\times1 \text{ matrices } X=\begin{bmatrix}x\\y\\z\end{bmatrix}. \text{ If } A\begin{bmatrix}1\\1\\1\end{bmatrix} = \begin{bmatrix}1\\4\\-5\end{bmatrix}, \; A\begin{bmatrix}1\\2\\1\end{bmatrix} = \begin{bmatrix}0\\4\\-8\end{bmatrix}, \text{ and } \det(\operatorname{adj}(2(A+I)))=2^\alpha 3^\beta 5^\gamma, \; \alpha,\beta,\gamma\in\mathbb{N}, \text{ then } \alpha^2+\beta^2+\gamma^2 \text{ is:} \underline{\hspace{2cm}}$$

$$ \text{Let } f \text{ be a differentiable function such that } 2(x+2)^2f(x)-3(x+2)^2 = 10\int_0^x (t+2)f(t)\,dt,\quad x\ge0. \text{ Then } f(2) \text{ is equal to:}\underline{\hspace{1cm}} $$

The number of 3-digit numbers that are divisible by 2 and 3, but not divisible by 4 and 9, is_______

During the transition of electron from state A to state C of a Bohr atom,  the wavelength of emitted radiation is  2000 A˚  and it becomes  6000 A˚  when the electron jumps from state B to state C.  Then the wavelength of the radiation emitted during the transition of electrons  from state A to state B is:

Consider the following statements: A. The junction area of solar cell is made very narrow compared to a photo diode. B. Solar cells are not connected with any external bias. C. LED is made of lightly doped p-n junction. D. Increase of forward current results in continuous increase of LED light intensity. E. LEDs have to be connected in forward bias for emission of light. Choose the correct answer from the options given below:

$$ \text{An alternating current is given by } I=I_A\sin\omega t + I_B\cos\omega t. \text{ The r.m.s. current will be:} $$

A car of mass  $$m$$ moves on a banked road having radius  $$' r '$$ and banking angle  $$\theta.$$ To avoid slipping from the banked road, the maximum permissible speed of the car is  $$v_0.$$  The coefficient of friction $$\mu$$ between the wheels of the car and the banked road is:

A satellite is launched into a circular orbit of radius $$R$$ around the earth. A second satellite is launched into an orbit of radius $$1.03R.$$ The time period of revolution of the second satellite is larger than the first one approximately by: