NTA JEE Main 23rd April 2013 Online

If $$f(x) = \sin(\sin x)$$ and $$f''(x) + \tan x \cdot f'(x) + g(x) = 0$$, then $$g(x)$$ is :

If the curves $$\frac{x^2}{\alpha} + \frac{y^2}{4} = 1$$ and $$y^3 = 16x$$ intersect at right angles, then a value of $$\alpha$$ is :

The cost of running a bus from A to B is Rs. $$(av + b/v)$$ where v km/h is the average speed of the bus. When the bus travels at 30 km/h, the cost comes out to be Rs. 75 while at 40 km/h, it is Rs. 65. Then the most economical speed (in km/h) of the bus is :

If a curve passes through the point $$(2, \frac{7}{2})$$ and has slope $$\left(1 - \frac{1}{x^2}\right)$$ at any point $$(x, y)$$ on it, then the ordinate of the point on the curve whose abscissa is -2 is :

The integral $$\int \frac{x \ dx}{2-x^2+\sqrt{2-x^2}}$$ equals :

The value of $$\int_{-\pi/2}^{\pi/2} \frac{\sin^2 x}{1+2^x} dx$$ is :

The area under the curve $$y = |\cos x - \sin x|$$, $$0 \leq x \leq \frac{\pi}{2}$$, and above x-axis is :

If $$\vec{a}$$ and $$\vec{b}$$ are non-collinear vectors, then the value of $$\alpha$$ for which the vectors $$\vec{u} = (\alpha - 2)\vec{a} + \vec{b}$$ and $$\vec{v} = (2 + 3\alpha)\vec{a} - 3\vec{b}$$ are collinear is :

If the projections of a line segment on the x, y and z-axes in 3-dimensional space are 2, 3 and 6 respectively, then the length of the line segment is :

A, B, C try to hit a target simultaneously but independently. Their respective probabilities of hitting the targets are $$\frac{3}{4}$$, $$\frac{1}{2}$$, $$\frac{5}{8}$$. The probability that the target is hit by A or B but not by C is :