NTA JEE Mains 4th April Shift 1 2026

A coin is tossed 8 times. If the probability that exactly 4 heads appear in the first six tosses and exactly 3 heads appear in the last five tosses is $$p$$, then $$96p$$ is equal to _____.

Consider the parabola $$P: y^2 = 4kx$$ and the ellipse $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$. Let the line segment joining the points of intersection of $$P$$ and $$E$$, be their latus rectums. If the eccentricity of $$E$$ is $$e$$, then $$e^2 + 2\sqrt{2}$$ is equal to _____.

If $$A = \frac{\sin 3°}{\cos 9°} + \frac{\sin 9°}{\cos 27°} + \frac{\sin 27°}{\cos 81°}$$ and $$B = \tan 81° - \tan 3°$$, then $$\frac{B}{A}$$ is equal to _____.

Let $$\vec{a_k} = (\tan\theta_k)\hat{i} + \hat{j}$$ and $$\vec{b_k} = \hat{i} - (\cot\theta_k)\hat{j}$$, where $$\theta_k = \frac{2^{k-1}\pi}{2^n + 1}$$, for some $$n \in \mathbb{N}$$, $$n > 5$$. Then the value of $$\frac{\sum_{k=1}^{n}|\vec{a_k}|^2}{\sum_{k=1}^{n}|\vec{b_k}|^2}$$ is _____.

The number of points, at which the function $$f(x) = \max\{6x, 2 + 3x^2\} + |x - 1|\cos\left|x^2 - \frac{1}{4}\right|$$, $$x \in (-\pi, \pi)$$, is not differentiable, is _____.

In a screw gauge when the circular scale is given five complete rotations it moves linearly by 2.5 mm. If the circular scale has 100 divisions, the least count of screw gauge is _____ mm.

The increase in the pressure required to decrease the volume $$(\Delta V)$$ of water is $$6.3 \times 10^7$$ N/m². The percentage decrease in the volume is _____. (Bulk modulus of water = $$2.1 \times 10^9$$ N/m².)

The time taken by a block of mass $$m$$ to slide down from the highest point to the lowest point on a rough inclined plane is 50% more compared to the time taken by the same block on identical inclined smooth plane. Both inclined planes are at $$45°$$ with the horizontal. The coefficient of kinetic friction between the rough inclined surface and block is :

Two nuclei of mass number 3 combine with another nucleus of mass number 4 to yield a nucleus of mass number 10. If the binding energy per nucleon for the mass numbers 3, 4 and 10 are 5.6 MeV, 7.4 MeV and 6.1 MeV, respectively, then in the process, $$\Delta Mc^2$$ = _____ MeV.

A solid sphere of mass $$M$$ and radius $$R$$ is divided into two unequal parts. The smaller part having mass $$\frac{M}{8}$$ is converted to a sphere of radius $$r$$ and the larger part is converted into a circular disc of thickness $$t$$ and radius $$2R$$. If $$I_1$$ is moment of inertia of a sphere having radius $$r$$ about an axis through its centre and $$I_2$$ is the moment of inertia of a disc about its diameter, the ratio of their moment of inertia $$\frac{I_2}{I_1}$$ = :