NTA JEE Mains 6th April Shift 2 2026

Let $$R = \{(x, y) \in \mathbb{N} \times \mathbb{N} : \log_e(x + y) \le 2\}$$. Then the minimum number of elements, required to be added in R to make it a transitive relation, is __________.

If $$(1 - x^3)^{10} = \displaystyle\sum_{r=0}^{10} a_r x^r (1 - x)^{30 - 2r}$$, then $$\dfrac{9a_9}{a_{10}}$$ is equal to __________.

Let the line $$x - y = 4$$ intersect the circle $$C: (x - 4)^2 + (y + 3)^2 = 9$$ at the points Q and R. If $$P(\alpha, \beta)$$ is a point on C such that $$PQ = PR$$, then $$(6\alpha + 8\beta)^2$$ is equal to __________.

Let the image of the point $$P(0, -5, 0)$$ in the line $$\dfrac{x - 1}{2} = \dfrac{y}{1} = \dfrac{z + 1}{-2}$$ be the point R and the image of the point $$Q\left(0, \dfrac{-1}{2}, 0\right)$$ in the line $$\dfrac{x - 1}{-1} = \dfrac{y + 9}{4} = \dfrac{z + 1}{1}$$ be the point S. Then the square of the area of the parallelogram PQRS is __________.

Let $$f(x) = \begin{cases} x^3 + 8, & x < 0 \\ x^2 - 4, & x \ge 0 \end{cases}$$ and $$g(x) = \begin{cases} (x - 8)^{1/3}, & x < 0 \\ (x + 4)^{1/2}, & x \ge 0 \end{cases}$$. Then the number of points, where the function $$g \circ f$$ is discontinuous, is __________.

The percentage error in the calculated volume of a sphere, if there is 2% error in its diameter measurement, is __________.

Match List - I with List - II. 

Choose the correct answer from the options given below :

A solid sphere (A) of mass $$5m$$ and a spherical shell (B) of mass $$m$$, both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of A and B, they start rolling without slipping with an acceleration of $$a_A$$ and $$a_B$$, respectively. The ratio of $$a_A$$ and $$a_B$$ is __________.

A body of mass 1 kg moves along a straight line with a velocity $$v = 2x^2$$. The work done by the body during displacement from $$x = 0$$ to 5 m is __________ J.

A cylinder with adiabatic walls is closed at both ends and is divided into two compartments by a frictionless adiabatic piston. Ideal gas is filled in both (left and right) the compartments at same P, V, T. Heating is started from left side until pressure changes to $$27 P/8$$. If initial volume of each compartment was 9 litres then the final volume in right-hand side compartment is __________ litres. (for this ideal gas $$C_P/C_V = 1.5$$)