NTA JEE Main 8th April 2019 Shift 1

If $$S_1$$ and $$S_2$$ are respectively the sets of local minimum and local maximum points of the function, $$f(x) = 9x^{4} + 12x^{3} - 36x^{2} + 25$$, $$x \in R$$, then:

Let $$f: [0, 2] \rightarrow R$$ be a twice differentiable function such that $$f''(x) > 0$$, for all $$x \in [0, 2]$$. If $$\phi(x) = f(x) + f(2 - x)$$, then $$\phi$$ is:

$$\int \frac{\sin\frac{5x}{2}}{\sin\frac{x}{2}} dx$$ is equal to:

If $$f(x) = \frac{2 - x\cos x}{2 + x\cos x}$$ and $$g(x) = \log_e x$$, then the value of the integral $$\int_{-\pi/4}^{\pi/4} g(f(x)) \, dx$$ is:

The area (in sq. units) of the region $$A = \{(x, y) \in R \times R \mid 0 \le x \le 3, 0 \le y \le 4, y \le x^{2} + 3x\}$$ is:

Let $$y = y(x)$$ be the solution of the differential equation, $$(x^{2} + 1)^{2}\frac{dy}{dx} + 2x(x^{2} + 1)y = 1$$ such that $$y(0) = 0$$. If $$\sqrt{a} \; y(1) = \frac{\pi}{32}$$, then the value of $$a$$ is:

The magnitude of the projection of the vector $$2\hat{i} + 3\hat{j} + \hat{k}$$ on the vector perpendicular to the plane containing the vectors $$\hat{i} + \hat{j} + \hat{k}$$ and $$\hat{i} + 2\hat{j} + 3\hat{k}$$, is:

The length of the perpendicular from the point (2, -1, 4) on the straight line $$\frac{x + 3}{10} = \frac{y - 2}{-7} = \frac{z}{1}$$ is:

The equation of a plane containing the line of intersection of the planes $$2x - y - 4 = 0$$ and $$y + 2z - 4 = 0$$ and passing through the point (1, 1, 0) is:

Let $$A$$ and $$B$$ be two non-null events such that $$A \subset B$$. Then, which of the following statements is always correct?