NTA JEE Main 2025 April 4th Shift 1

$$1 + 3 + 5^2 + 7 + 9^2 + 11 + 13^2 + \ldots$$ upto 40 terms is equal to

In the expansion of $$\left(\sqrt[3]{2} + \dfrac{1}{\sqrt[3]{3}}\right)^n$$, $$n \in \mathbb{N}$$, if the ratio of $$15^{\text{th}}$$ term from the end to the $$15^{\text{th}}$$ term from the beginning is $$\dfrac{1}{6}$$, then the value of $${}^nC_3$$ is:

Considering the principal values of the inverse trigonometric functions, $$\sin^{-1}\left(\dfrac{\sqrt{3}}{2} x + \dfrac{1}{2}\sqrt{1 - x^2}\right)$$, $$-\dfrac{1}{2} \lt x \lt \dfrac{1}{\sqrt{2}}$$, is equal to

Consider two vectors $$\vec{u} = 3\hat{i} - \hat{j}$$ and $$\vec{v} = 2\hat{i} + \hat{j} - \lambda\hat{k}$$, $$\lambda \gt 0$$. The angle between them is given by $$\cos^{-1}\left(\dfrac{\sqrt{5}}{2\sqrt{7}}\right)$$. Let $$\vec{v} = \vec{v}_1 + \vec{v}_2$$, where $$\vec{v}_1$$ is parallel to $$\vec{u}$$ and $$\vec{v}_2$$ is perpendicular to $$\vec{u}$$. Then the value $$|\vec{v}_1|^2 + |\vec{v}_2|^2$$ is equal to

Let the three sides of a triangle are on the lines $$4x - 7y + 10 = 0$$, $$x + y = 5$$ and $$7x + 4y = 15$$. Then the distance of its orthocentre from the orthocentre of the triangle formed by the lines $$x = 0$$, $$y = 0$$ and $$x + y = 1$$ is

The value of $$\displaystyle\int_{-1}^{1} \dfrac{\left(1 + \sqrt{|x| - x}\right)e^{-x} + \left(\sqrt{|x| - x}\right)e^{-x}}{e^{x} + e^{-x}} \, dx$$ is equal to

The length of the latus-rectum of the ellipse, whose foci are $$(2, 5)$$ and $$(2, -3)$$ and eccentricity is $$\dfrac{4}{5}$$, is

Consider the equation $$x^2 + 4x - n = 0$$, where $$n \in [20, 100]$$ is a natural number. Then the number of all distinct values of n, for which the given equation has integral roots, is equal to

A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let X denote the number of defective pens. Then the variance of X is

If $$10\sin^4\theta + 15\cos^4\theta = 6$$, then the value of $$\dfrac{27\csc^6\theta + 8\sec^6\theta}{16\sec^8\theta}$$ is: