AIEEE 2005 Maths Question Paper Mock Test

$\int {\left\{\frac{(\log x-1)}{1+(\log x{)}^2}\right\}}^{2}dx$ is equal to:

Let $f:\mathbb{R}\to \mathbb{R}$ be a differentiable function having $f(2)=6,f^{\prime }(2)=\frac{1}{48}$. Then ${\lim }_{x\to 2}\frac{{\int }_6^{f(x)}4t^{3}dt}{x-2}$ equals:

Let $f(x)$ be a non-negative continuous function such that the area bounded by the curve $y=f(x)$, x-axis and the ordinates $x=\pi /4$ and $x=\beta >\pi /4$ is $\beta \sin \beta +\frac{\pi }{4}\cos \beta +\sqrt{2}\beta$. Then $f(\pi /2)$ is:

The locus of a point $P(\alpha ,\beta )$ moving under the condition that the line $y=\alpha x+\beta$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is:

If the angle $\theta$ between the line $\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}$ and the plane $2x-y+\lambda z+4=0$ is such that $\sin \theta =1/3$, then the value of $\lambda$ is: