NTA JEE Main 2025 April 7th Shift 2

If the locus of $$z \in \mathbb{C}$$, such that $$\text{Re}\left(\frac{z-1}{2z+i}\right) + \text{Re}\left(\frac{\bar{z}-1}{2\bar{z}-i}\right) = 2$$, is a circle of radius $$r$$ and center $$(a, b)$$ then $$\frac{15ab}{r^2}$$ is equal to :

Let the length of a latus rectum of an ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ be 10. If its eccentricity is the minimum value of the function $$f(t) = t^2 + t + \frac{11}{12}$$, $$t \in \mathbf{R}$$, then $$a^2 + b^2$$ is equal to :

Let $$y = y(x)$$ be the solution of the differential equation $$(x^2 + 1)y' - 2xy = (x^4 + 2x^2 + 1)\cos x$$, $$y(0) = 1$$. Then $$\int_{-3}^{3} y(x) \, dx$$ is :

If the equation of the line passing through the point $$\left(0, -\frac{1}{2}, 0\right)$$ and perpendicular to the lines $$\vec{r} = \lambda(\hat{i} + a\hat{j} + b\hat{k})$$ and $$\vec{r} = (\hat{i} - \hat{j} - 6\hat{k}) + \mu(-b\hat{i} + a\hat{j} + 5\hat{k})$$ is $$\frac{x - 1}{-2} = \frac{y + 4}{d} = \frac{z - c}{-4}$$, then $$a + b + c + d$$ is equal to :

Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of P. If $$p + q = 126$$, then the eccentricity of the ellipse $$\frac{x^2}{16} + \frac{y^2}{n} = 1$$ is :

Consider the lines $$L_1 : x - 1 = y - 2 = z$$ and $$L_2 : x - 2 = y = z - 1$$. Let the feet of the perpendiculars from the point $$P(5, 1, -3)$$ on the lines $$L_1$$ and $$L_2$$ be Q and R respectively. If the area of the triangle PQR is A, then $$4A^2$$ is equal to :

The number of real roots of the equation $$x|x - 2| + 3|x - 3| + 1 = 0$$ is :

Let $$e_1$$ and $$e_2$$ be the eccentricities of the ellipse $$\frac{x^2}{b^2} + \frac{y^2}{25} = 1$$ and the hyperbola $$\frac{x^2}{16} - \frac{y^2}{b^2} = 1$$, respectively. If $$b < 5$$ and $$e_1 e_2 = 1$$, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is :

Let the system of equations $$x + 5y - z = 1$$, $$4x + 3y - 3z = 7$$, $$24x + y + \lambda z = \mu$$, $$\lambda, \mu \in \mathbf{R}$$, have infinitely many solutions. Then the number of the solutions of this system, if x, y, z are integers and satisfy $$7 \le x + y + z \le 77$$, is

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is :