NTA JEE Mains 2nd April Shift 1 2026

If $$\sin\left(\frac{\pi}{18}\right) \sin\left(\frac{5\pi}{18}\right) \sin\left(\frac{7\pi}{18}\right) = K$$, then the value of $$\sin\left(\frac{10K\pi}{3}\right)$$ is :

Let $$S = \{x \in [-\pi, \pi] : \sin x(\sin x + \cos x) = a, a \in \mathbb{Z}\}$$. Then $$n(S)$$ is equal to :

If the point of intersection of the lines $$\frac{x+1}{3} = \frac{y+a}{5} = \frac{z+b+1}{7}$$ and $$\frac{x-2}{1} = \frac{y-b}{4} = \frac{z-2a}{7}$$ lies on the $$xy$$-plane, then the value of $$a + b$$ is :

If $$\vec{a}$$ and $$\vec{b}$$ are two vectors such that $$|\vec{a}| = 2$$ and $$|\vec{b}| = 3$$, then the maximum value of $$3|3\vec{a} + 2\vec{b}| + 4|3\vec{a} - 2\vec{b}|$$ is :

Let a line L passing through the point $$(1, 1, 1)$$ be perpendicular to both the vectors $$2\hat{i} + 2\hat{j} + \hat{k}$$ and $$\hat{i} + 2\hat{j} + 2\hat{k}$$. If P(a, b, c) is the foot of perpendicular from the origin on the line L, then the value of $$34(a + b + c)$$ is :

If $$\lim_{x \to 2} \frac{\sin(x^3 - 5x^2 + ax + b)}{(\sqrt{x-1} - 1) \log_e(x-1)} = m$$, then $$a + b + m$$ is equal to :

If the curve $$y = f(x)$$ passes through the point $$(1, e)$$ and satisfies the differential equation $$dy = y(2 + \log_e x) dx$$, $$x > 0$$, then $$f(e)$$ is equal to :

The number of critical points of the function $$f(x) = \begin{cases} \left|\frac{\sin x}{x}\right|, & x \neq 0 \\ 1, & x = 0 \end{cases}$$ in the interval $$(-2\pi, 2\pi)$$ is equal to :

Let $$[\cdot]$$ denote the greatest integer function. Then the value of $$\int_0^3 \left(\frac{e^x + e^{-x}}{[x]!}\right) dx$$ is :

Let $$y = y(x)$$ be the solution curve of the differential equation $$(1 + \sin x)\frac{dy}{dx} + (y+1)\cos x = 0$$, $$y(0) = 0$$. If the curve $$y = y(x)$$ passes through the point $$\left(\alpha, \frac{-1}{2}\right)$$, then a value of $$\alpha$$ is :