NTA JEE Main 10th April 2016 Online

Let C be a curve given by $$y(x) = 1 + \sqrt{4x - 3}$$, $$x > \frac{3}{4}$$. If $$P$$ is a point on C, such that the tangent at $$P$$ has slope $$\frac{2}{3}$$, then a point through which the normal at $$P$$ passes, is:

Let $$f(x) = \sin^4 x + \cos^4 x$$. Then, $$f$$ is an increasing function in the interval:

The integral $$\int \frac{dx}{(1+\sqrt{x})\sqrt{x - x^2}}$$ is equal to

For $$x \in R$$, $$x \neq 0$$, if $$y(x)$$ is a differentiable function such that $$x\int_1^x y(t)dt = (x+1)\int_1^x ty(t)dt$$, then $$y(x)$$ equals (where C is a constant)

The value of the integral $$\int_4^{10} \frac{[x^2]}{[x^2 - 28x + 196] + [x^2]}dx$$, where $$[x]$$ denotes the greatest integer less than or equal to $$x$$, is

The solution of the differential equation $$\frac{dy}{dx} + \frac{y}{2}\sec x = \frac{\tan x}{2y}$$, where $$0 \leq x < \frac{\pi}{2}$$ and $$y(0) = 1$$, is given by

The number of distinct real values of $$\lambda$$, for which the lines $$\frac{x-1}{1} = \frac{y-2}{2} = \frac{z+3}{\lambda^2}$$ and $$\frac{x-3}{1} = \frac{y-2}{\lambda^2} = \frac{z-1}{2}$$, are coplanar is

$$ABC$$ is a triangle in a plane with vertices $$A(2, 3, 5)$$, $$B(-1, 3, 2)$$ and $$C(\lambda, 5, \mu)$$. If the median through $$A$$ is equally inclined to the coordinate axes, then the value of $$(\lambda^3 + \mu^3 + 5)$$ is

Let $$ABC$$ be a triangle whose circumcentre is at $$P$$. If the position vectors of $$A$$, $$B$$, $$C$$ and $$P$$ are $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ and $$\frac{\vec{a}+\vec{b}+\vec{c}}{4}$$ respectively, then the position vector of the orthocentre of this triangle, is:

An experiment succeeds twice as often as it fails. The probability of at least 5 successes in the six trials of this experiment is