NTA JEE Mains 4th April Shift 2 2026

From a month of 31 days, 3 dates are selected at random. If the  probability that these dates  are in an increasing A.P.  is equal to $$\frac{a}{b}$$, where $$ a , b \in N$$ and  $$\gcd(a, b) = 1$$. Then $$a + b$$ is equal to :

Let $$f(x) = \begin{cases} e^{x-1}, & x < 0 \\ x^2 - 5x + 6, & x \geq 0 \end{cases}$$ and $$g(x) = f(|x|) + |f(x)|$$. If the number of points where $$g$$ is not continuous and is not differentiable are $$\alpha$$ and $$\beta$$ respectively, then $$\alpha + \beta$$ is equal to :

Let $$A$$ and $$B$$ be points on the two half-lines $$x - \sqrt{3}|y| = \alpha$$, $$\alpha > 0$$, at distance of $$\alpha$$ from the point of intersection $$P$$. The line $$AB$$ meets the angle bisector of the given half-lines at the point $$Q$$. If $$PQ = \frac{9}{2}$$ and $$R$$ is the radius of the circumcircle  of $$\triangle PAB$$, then $$\frac{\alpha^2}{R}$$ is equal to :

Let $$A, B,$$ and $$C$$ be vertices of a variable right-angled triangle inscribed in the parabola $$y^2 = 16x$$.Let the vertex $$B$$ containing the right angle be $$(4, 8)$$ and  the locus of the centroid of $$\triangle ABC$$ be a  conic $$C_0$$, then three times the length  of latus rectum of  $$C_0)$$ is :

Let $$f$$ be twice differentiable function such that $$f(x) = \displaystyle\int_0^x \tan(t - x)\,dt - \int_0^x f(t)\tan t\,dt$$, $$x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$. Then $$f''\!\left(\frac{\pi}{6}\right) + 12f'\!\left(-\frac{\pi}{6}\right) + f\!\left(\frac{\pi}{6}\right)$$ is equal to :

Match the List-I with List II

Choose the correct answer from the options given below:  

Two cars A and B move in the same direction along a straight line with speed $$100 km/h$$ and $$80 km/h$$, respectively such that  Car $$A$$ is moving ahead of Car $$B$$. A person in car B throws a stone with a speed $$v$$ so that it hit car $$A$$ with a speed of $$5$$ m/s. The value of  $$v$$ is ________ $$km/h$$ :

At $$t = 0$$, a body of mass $$100$$ g starts moving under the influence of a force $$(5\hat{i} + 10\hat{j})$$ N. After $$2$$ s, its  position is $$(2x\hat{i} + 5y\hat{j})$$ m. The ratio $$x : y$$ is ______.

If x and y coordinates of a projectile as a function of time (t) are given as $$24t$$ and $$43.6t - 4.9t^2$$, respectively, then the angle (in degrees) made by the projectile with horizontal when $$t = 2$$ s is ______.

The height in terms of radius of the earth (R), at which the acceleration due to gravity becomes $$\frac{g}{9}$$, where g is acceleration due to gravity on earth's surface, is ______.