NTA JEE Mains 2nd April Shift 1 2026

If the domain of the function $$f(x)=\sqrt{\log_{0.6}\left(\left|\frac{2x-5}{x^2-4}\right|\right)}$$ is $$(-\infty, a] \cup \{b\} \cup [c, d) \cup (e, \infty)$$, then the value of $$a + b + c + d + e$$ is __________.

If $$\sum_{k=1}^{n} a_k = 6n^3$$, then $$\sum_{k=1}^{6} \left(\frac{a_{k+1} - a_k}{36}\right)^2$$ is equal to __________.

Let $$a, b, c \in \{1, 2, 3, 4\}$$. If the probability, that $$ax^2 + 2\sqrt{2}bx + c > 0$$ for all $$x \in \mathbb{R}$$, is $$\frac{m}{n}$$, $$\gcd(m, n) = 1$$, then $$m + n$$ is equal to __________.

Let a circle C have its centre in the first quadrant, intersect the coordinate axes at exactly three points and cut off equal intercepts from the coordinate axes. If the length of the chord of C on the line $$x + y = 1$$ is $$\sqrt{14}$$, then the square of the radius of C is __________.

If $$\alpha = \int_0^{2\sqrt{3}} \log_2(x^2 + 4) dx + \int_2^4 \sqrt{2^x - 4} \, dx$$, then $$\alpha^2$$ is equal to __________.

The dimensional formula of $$\frac{1}{2} \epsilon_0 E^2$$ ($$\epsilon_0$$ = permittivity of vacuum and E = electric field) is $$M^a L^b T^c$$. The value of $$2a - b + c$$ = __________.

The diameter of a wire measured by a screw gauge of least count 0.001 cm is 0.08 cm. The length measured by a scale of least count 0.1 cm is 150 cm. When a weight of 100 N is applied to the wire, the extension in length is 0.5 cm, measured by a micrometer of least count 0.001 cm. The error in the measured Young's modulus is $$\alpha \times 10^9$$ N/m$$^2$$. The value of $$\alpha$$ is __________ . (Ignore the contribution of the load to Young's modulus error calculation)

The velocity of a particle is given as $$\vec{v} = -x\hat{i} + 2y\hat{j} - z\hat{k}$$ m/s. The magnitude of acceleration at point $$(1, 2, 4)$$ is __________ m/s$$^2$$.

The position of an object having mass 0.1 kg as a function of time t is given as $$\vec{r} = \left(10t^2 \hat{i} + 5t^3 \hat{j}\right)$$ m. At $$t = 1$$ s, which of the following statements are correct? A. The linear momentum $$\vec{p} = \left(2\hat{i} + 1.5\hat{j}\right)$$ kg·m/s. B. The force acting on the object $$\vec{F} = \left(2\hat{i} + 3\hat{j}\right)$$ N. C. The angular momentum of the object about its origin $$\vec{L} = 15 \hat{k}$$ J·s. D. The torque acting on the object about its origin $$\vec{\tau} = 20 \hat{k}$$ N·m. Choose the correct answer from the options given below :

A planet (P$$_1$$) is moving around the star of mass 2M in the orbit of radius R. Another planet (P$$_2$$) is moving around another star of mass 4M in a orbit of radius 2R. Ratio of time periods of revolution of P$$_2$$ and P$$_1$$ is __________.