JEE (Advanced) 2010 Paper-1

The value(s) of $$\int_{0}^{1}\frac{x^4(1-x)^4}{1+x^2}dx $$ is(are)

The number of A in $$T_p$$, such that A is either symmetric or skew-symmetric or both, and det (A) divisible by p is

The number of A in $$T_p$$, suchthat the trace of A is not divisible by p but det (A) is divisible by p is
[Note : The trace of a matrix is the sumofits diagonal entries.]

The number of A in $$T_p$$, such that det (A) is not divisible byp is

Equation of a common tangent with positive slope to the circle as well as to the hyperbola is

Equation of the circle with AB as its diameter is

The numberof values of $$\theta$$ in the interval $$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$ such that $$\theta \neq \frac{n \pi}{5}$$ for $$n = 0, \pm 1, \pm 2$$ and $$\tan \theta = \cot 5 \theta$$ as well as $$\sin 2 \theta = \cos 4 \theta$$ is

The maximum value of the expression $$\frac{1}{\sin^2 \theta + 3 \sin \theta \cos \theta + 5 \cos^2 \theta}$$ is

If $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$ are vectors in space given by $$\overrightarrow{a} = \frac{\hat{i} - 2 \hat{j}}{\sqrt{5}}$$ and $$\overrightarrow{b} = \frac{2\hat{i} + \hat{j} + 3\hat{k}}{\sqrt{14}}$$, then the value of $$\left(2\overrightarrow{a} + \overrightarrow{b}\right).\left[\left(\overrightarrow{a} \times \overrightarrow{b}\right) \times \left(\overrightarrow{a} - 2\overrightarrow{b}\right)\right]$$ is

The line 2x + y = 1 is tangent to the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$. If this line passes through the point of intersection of the nearest directrix and the x-axis, then the eccentricity of the hyperbola is