NTA JEE Mains 23rd Jan 2025 Shift 1

Let the circle touch the line x-y+1=0, have the centre on the positive x -axis, and cut off a chord of length $$\frac{4}{\sqrt{13}}$$ along the line -3x+2y=1. Let H be the hyperbola $$\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1$$ , whose one of the foci is the centre of C and the length of the transverse axis is the diameter of C. Then $$2\alpha^{2}+3\beta^{2}$$ is equal to ______

If the equation $$a(b-c)x^{2}+b(c-a)x+c(a-b)=0$$ has equal roots, where a+c=15 and $$b=\frac{36}{5}$$, then $$a^{2}+c^{2}$$ is equal to

If the set of all values of a, for which the equation $$5x^{3}-15x-a=0$$ has three distinct real roots, is the interval $$(\alpha, \beta)$$, then $$\beta-2\alpha $$ is equal to ______

The sum of all rational terms in the expansion of $$(1+2^{1/2}+3^{1/2})^{6}$$ is equal to

If the area of the larger portion bounded between the curves $$x^{2}+y^{2}=25$$ and y=|x-1| is $$\frac{1}{4}(b\pi+c),b,c\in N$$, then b+c is equal to

A point particle of charge Q is located at P along the axis of an electric dipole 1 at a distance r as shown in the figure. The point P is also on the equatorial plane of a second electric dipole 2 at a distance r . The dipoles are made of opposite charge q separated by a distance 2a. For the charge particle at P not to experience any net

force, which of the following correctly describes the situation?

A spherical surface of radius of curvature R, separates air from glass (refractive index = 1.5). The centre of curvature is in the glass medium. A point object $$'O'$$ placed in air on the optic axis of the surface, so that its real image is formed at $$'I'$$ inside glass. The line $$OI$$ intersects the spherical surface at $$P$$ and $$PO=PI$$ . The distance $$PO$$ equals to

The position of a particle moving on x-axis is given by $$x(t)=A\sin t+B\cos^{2}t+Ct^{2}+D$$, where is time. The dimension of $$\frac{ABC}{D}$$ is

Given a thin convex lens (refractive index $$\mu_{2}$$), kept in a liquid (refractive index $$\mu_{1},\mu_{1}<\mu_{2}$$) having radii of curvatures $$|R_{1}|$$ and $$|R_{2}|$$ . Its second surface is silver polished. Where should an object be placed on the optic axis so that a real and inverted image is formed at the same place?

Refer to the circuit diagram given in the figure. which of the following observations are correct? A. Total resistance of circuit is $$6\Omega$$ B. Current in Ammeter is 1 A C. Potential across AB is 4 Volts. D. Potential across CD is 4 Volts E. Total resistance of the circuit is $$8\Omega$$ . Choose the correct answer from the options given below: