NTA JEE Mains 24th Jan 2025 Shift 2

Let  $$P$$  be the image of the point  $$Q(7,-2,5)$$ in the line  $$L:\;\frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4},$$ and  $$R(5,p,q)$$  be a point on $$L.$$ Then the square of the area of  $$\triangle PQR$$ is  $$\underline{\hspace{2cm}}.$$

$$\text{If }\int \frac{2x^2+5x+9}{\sqrt{x^2+x+1}}\,dx=x\sqrt{x^2+x+1}+\alpha\sqrt{x^2+x+1}+\beta\log_e\!\left|x+\frac12+\sqrt{x^2+x+1}\right|+C,\text{where } C \text{ is the constant of integration,}\text{ then } \alpha+2\beta \text{ is equal to }\underline{\hspace{2cm}}.$$

$$\text{Let } y=y(x) \text{ be the solution of the differential equation }2\cos x\,\frac{dy}{dx}= \sin 2x - 4y\sin x,x\in\left(0,\frac{\pi}{2}\right).\text{If } y\!\left(\frac{\pi}{3}\right)=0,\text{ then } y'\!\left(\frac{\pi}{4}\right)+ y\!\left(\frac{\pi}{4}\right)\text{ is equal to }\underline{\hspace{2cm}}.$$

Number of functions $$ f:\{1,2,\ldots,100\}\to\{0,1\} $$ that assign 1 to exactly one of the positive integers less than or equal to 98 is equal to______________

Let $$H_1:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$$  and $$H_2:-\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$$ be two hyperbolas having length of latus rectums $$15\sqrt{2}$$ and $$12\sqrt{5}$$ respectively. Let their eccentricities be $$e_1=\sqrt{\frac{5}{2}}$$  and  $$e_2$$ respectively. If the product of the lengths of  their transverse axes is  $$100\sqrt{10},$$ then  $$25e_2^2$$  is equal to $$\underline{\hspace{2cm}}.$$

Which of the following figure represents the relation between Celsius and Fahrenheit temperatures?

$$\text{The position vector of a moving body at any instant of time is given as }\vec r=(5t^2\hat i-5t\hat j)\,m.\text{The magnitude and direction of velocity at } t=2\,s \text{ is:}$$

The output of the circuit is low (zero) for:

$$(A)\; X=0,\,Y=0 \text{ }(B)\; X=0,\,Y=1 \text{ }(C)\; X=1,\,Y=0 \text{ }(D)\; X=1,\,Y=1$$ Choose the correct answer from the options given below:

Young's double slit inteference apparatus is immersed in a liquid of refractive index 1.44. It has slit separation of 1.5 mm . The slits are illuminated by a parallel beam of light whose wavelength in air is 690 nm . The fringe-width on a screen placed behind the plane of slits at a distance of 0.72 m , will be :

The magnitude of heat exchanged by a system for the given cyclic process ABCA (as shown in figure) is (in SI unit) :