JEE (Advanced) 2010 Paper-1

Let $$\omega$$ be a complex cube root of unity with $$\omega\neq1$$. A fair die is thrownthree times. If $$r_{1},r_{2}$$ and $$r_{3}$$ are the numbers obtained on the die, then the probability that is $$\omega^{r_{1}}+\omega^{r_{2}}+\omega^{r_{3}}=0$$

Let P, Q, R and S be the points on the plane with position vectors $$-2\hat{i}-\hat{j},4\hat{i},3\hat{i}+3\hat{j}$$ and $$-3\hat{i}+2\hat{j}$$ respectively. The quadrilateral PQRS must be a

The number of $$3\times3$$ matrices A whose entries are either O or 1 and for which the system $$A\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\0\\0\end{bmatrix}$$ has exactly two distinct solutions, is

The value of $$\lim_{x \rightarrow 0}\frac{1}{x^{3}}\int_{0}^{x} \frac{t \ln(1+t)}{t^{4}+4}dt$$ is

Let p and q be real numbers such that $$p\neq 0,p^{3}\neq q$$ and $$p^{3}\neq -q$$. If $$\alpha$$ and $$\beta$$ are nonzero complex numbers satisfying $$\alpha+\beta=-p$$ and $$\alpha^{3}+\beta^{3}=q$$, then a quadratic equation having $$\frac{\alpha}{\beta}$$ and $$\frac{\beta}{\alpha}$$ as its roots is

Let f, g and h be real-valued functions defined on the interval [0,1] by $$f(x) = e^{x^2} + e^{-x^2}, g(x) = xe^{x^2} + e^{-x^2}$$ and $$h(x) = x^2 e^{x^2} + e^{-x^2}$$. If a, b and c denote, respectively, the absolute maximum of f, g and h on [0, 1], Then

Let A and B be two distinct points on the parabola $$y^2 = 4x$$. If the axis of the parabola touches a circle of radius r having AB as its diameter, then the slope of the line joining A and B can be

Let ABC be a triangle such that $$\angle ACB = \frac{\pi}{6}$$ and let a, b and c denote the lengths of the sides opposite to A, B and C respectively. The value(s) of x for which $$a = x^2 + x + 1, b = x^2 - 1$$ and $$c = 2x + 1$$ is (are)

Let $$z_1$$ and $$z_2$$ be two district complex numbers and let $$z = (1 - t)z_1 + tz_2$$ for some real number t with 0 < t 1. If Arg(w) denotes the principal argument of a nonzero complex number w, then

Let f be a real-valued function defined on the interval $$(0, \infty)$$ by $$f(x) = \ln x + \int_{0}^{x}\sqrt{1 + \sin t}$$ dt. Then which of the following statement(s) is (are) true ?