NTA JEE Main 8th April 2019 Shift 2

If $$f(1) = 1, f'(1) = 3$$, then the derivative of $$f(f(f(x))) + (f(x))^{2}$$ at $$x = 1$$ is:

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is:

Given that the slope of the tangent to a curve $$y = y(x)$$ at any point $$(x, y)$$ is $$\frac{2y}{x^{2}}$$. If the curve passes through the centre of the circle $$x^{2} + y^{2} - 2x - 2y = 0$$, then its equation is:

If $$\int \frac{dx}{x^3(1 + x^6)^{2/3}} = xf(x)(1 + x^6)^{1/3} + C$$, where C is a constant of integration, then the function $$f(x)$$ is equal to:

Let $$f(x) = \int_0^x g(t) \, dt$$, where $$g$$ is a non-zero even function. If $$f(x + 5) = g(x)$$, then $$\int_0^x f(t) \, dt$$ equals:

Let $$S(\alpha) = \{(x, y): y^{2} \le x, 0 \le x \le \alpha\}$$ and $$A(\alpha)$$ is area of the region $$S(\alpha)$$. If for a $$\lambda$$, $$0 < \lambda < 4$$, $$A(\lambda):A(4) = 2:5$$, then $$\lambda$$ equals:

Let $$\vec{a} = 3\hat{i} + 2\hat{j} + x\hat{k}$$ and $$\vec{b} = \hat{i} - \hat{j} + \hat{k}$$, for some real $$x$$. Then the condition for $$\vec{a} \times \vec{b} = r$$ to follow is:

The vector equation of the plane through the line of intersection of the planes $$x + y + z = 1$$ and $$2x + 3y + 4z = 5$$ which is perpendicular to the plane $$x - y + z = 0$$ is:

If a point $$R(4, y, z)$$ lies on the line segment joining the points $$P(2, -3, 4)$$ and $$Q(8, 0, 10)$$, then the distance of R from the origin is:

The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least 90% is: