NTA JEE Mains 5th April Shift 1 2026

In an equilateral triangle $$PQR$$,let the vertex $$P = (3, 5)$$ and the side $$QR$$ be along the line $$x + y = 4$$. If the orthocentre of the triangle $$PQR$$ is $$(\alpha, \beta)$$, then $$9(\alpha + \beta)$$ is equal to :

The sum of all integral values of $$p$$ such that the equation $$3\sin^2 x + 12\cos x - 3 = p, x \in R,$$ has at least one solution is :

The square of the distance of the point $$P(5, 6, 7)$$ from the line $$\frac{x-2}{2} = \frac{y-5}{3} = \frac{z-2}{4}$$ is equal to:

Let $$\vec{a} = \sqrt{7}\hat{i} + \hat{j} - \hat{k}$$ and $$\vec{b} = \hat{j} + 2\hat{k}$$. If $$\vec{r}$$ is a vector such that $$\vec{r} \times \vec{a} + \vec{a} \times \vec{b} = \vec{0}$$ and $$\vec{r} \cdot \vec{a} = 0$$, then $$|3\vec{r}|^2$$ is equal to :

The square of the distance of the point of intersection of the lines $$\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(a\hat{i} - \hat{j})$$, $$a \neq 0$$ and $$\vec{r} = (4\hat{i} - \hat{k}) + \mu(2\hat{i} + a\hat{k})$$ from the origin is :

The area of the region $$R = \{(x, y) : xy \leq 27, \; 1 \leq y \leq x^2\}$$ is equal to:

The product of all values of $$\alpha$$, for which $$\displaystyle\lim_{x \to 0} \left(\frac{1 - \cos(\alpha x) \cos((\alpha+1)x) \cos((\alpha+2)x)}{\sin^2((\alpha+1)x)}\right) = 2$$ is :

The value of the integral $$\displaystyle\int_0^{\infty} \frac{\log_e (x)}{x^2 + 4} \, dx$$ is:

Let $$f: \mathbb{R} \to \mathbb{R}$$ be a differentiable function such that $$f\left(\frac{x+y}{3}\right) = \frac{f(x) + f(y)}{3}$$ for all $$x, y \in R$$ and $$f'(0) = 3$$. Then the minimum value of function $$g(x) = 3 + e^x f(x)$$ is :

The value of the integral $$\displaystyle\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \left(\frac{4 - \operatorname{cosec}^2 x}{\cos^4 x} \right) dx$$ is equal to :