NTA JEE Mains 4th April Shift 2 2026

Let $$f:[1,\infty) \to [1,\infty)$$ be defined by $$f(x) = (x-1)^4 + 1$$. among the two statements:
(I) The Set $$S = \{x \in [1,\infty) : f(x) = f^{-1}(x)\}$$ contains  exactly two elements and 
(II) The Set $$S = \{x \in [1,\infty) : f(x) = f^{-1}(x+1)\}$$ is an empty set,

Let $$S = \{z \in \mathbb{C} : z^2 + 4z + 16 = 0\}$$. Then $$\displaystyle\sum_{z \in S} |z + \sqrt{3}\,i|^2$$ is equal to :

If the system of equations $$x + y + z = 5$$, $$x + 2y + 3z = 9$$, $$x + 3y + \lambda z = \mu$$ has infinitely many solutions, then the value of $$\lambda + \mu$$ is :

If $$\alpha = 1$$ and $$\beta = 1 + i\sqrt{2}$$ (where $$i = \sqrt{-1}$$) are two roots of $$x^3 + ax^2 + bx + c = 0$$, where $$a, b, c \in \mathbb{R}$$, then $$\displaystyle\int_{-1}^{1}(x^3 + ax^2 + bx + c)\,dx$$ is equal to :

If the quadratic equation $$(\lambda + 2)x^2 - 3\lambda x + 4\lambda = 0$$, $$\lambda \neq -2$$, has two positive roots, then the number of possible integral values of $$\lambda$$ is :

Let $$A = \begin{bmatrix} 1 & 2 & 7 \\ 4 & -2 & 8 \\ 3 & 8 & -7 \end{bmatrix}$$ and $$\det(A - \alpha I) = 0$$. where $$\alpha$$ is a real number.if the  largest possible value of $$\alpha$$ is $$p$$, then the circle $$(x - p)^2 + (y - 2p)^2 = 320$$ intersects the coordinate axes at :

Let $$\alpha = \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots \infty$$ and $$\beta = \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots \infty$$. Then the value of$$(0.2)^{\log_{\sqrt{5}}(\alpha)} + (0.04)^{\log_5(\beta)}$$ is equal to :

For 10 observations $$x_1, x_2, \ldots, x_{10}$$, $$\displaystyle\sum_{i=1}^{10}(x_i + 2)^2 = 180$$ and $$\displaystyle\sum_{i=1}^{10}(x_i - 1)^2 = 90$$. Then their  standard deviation is :

In the expansion of $$\left(9x - \frac{1}{3\sqrt{x}}\right)^{18}$$, $$x > 0$$, the term independent of $$x$$ is $${221} k$$. Then $$k$$ is equal to :

Let $$P(3\cos\alpha, 2\sin\alpha)$$, $$\alpha \neq 0$$, be a point on the ellipse $$\frac{x^2}{9} + \frac{y^2}{4} = 1$$. Let $$Q$$ be a point on the circle $$x^2 + y^2 - 14x - 14y + 82 = 0$$, and $$R$$ be a point on the line $$x + y = 5$$. such that  the centroid of the $$\triangle PQR$$ is $$\left(2 + \cos\alpha,\; 3 + \frac{2}{3}\sin\alpha\right)$$, then the sum of the ordinates of all possible points $$R$$ is :