JEE (Advanced) 2009 Paper-1

Tangents drawn from the point P(1, 8) to the circle $$x^{2}+y^{2}-6x-4y-11=0$$ touch the circle at the points A and B. The equation of the circumcircle of the triangle PAB is

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is

Let P(3, 2, 6) be a point in space and Q be a point on the line $$\overrightarrow{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu(-3\hat{i}+\hat{j}+5\hat{k})$$ Then the value of $$\mu$$ for which the vector $$\overrightarrow{PQ}$$ is parallel to the plane $$x-4y+3z=1$$ is

Let $$Z=\cos\theta+i\ \sin\theta$$. Then the value of $$\sum_{m=1}^{15}Im(z^{2m-1})$$ at $$\theta=2^{0}$$ is

Let $$z=x+iy$$ be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation $$Z\overline{z}^{3}+\overline{z}Z^{3}=350$$ is

If $$\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$$ and $$\overrightarrow{d}$$ are unit vectors such that $$(\overrightarrow{a}\times\overrightarrow{b}).(\overrightarrow{c}\times\overrightarrow{d})=1$$ and $$\overrightarrow{a}.\overrightarrow{c}=\frac{1}{2}$$ then

Let f be a non-negative function defined on the interval $$[0,1]$$. If
$$\int_{0}^{x}\sqrt{1 - (f'(t))^2} dt = \int_{0}^{x} f(t)dt, 0 \leq x \leq 1$$, and f(0) = 0, then

The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse $$x^2 + 9y^2 = 9$$ meets its auxiliary circle at the point M. Then the area of the triangle with vertices at A, M and the origin O is

In a triangle ABC with fixed base BC, the vertex A moves such that
$$\cos B + \cos C = 4 \sin^2 \frac{A}{2}$$
If a, b and c denote the lengths of the sides of the triangle opposite to the angles A, B and C, respectively, then

Area of the region bounded by the curve $$y = e^x$$ and line x = 0 and y = e is