The angle $A$ of $\triangle A B C$ is found by measurement to be $67 \frac{1^{\circ}}{2}$ and the area of $\triangle A B C$ is calculated from the measurements of $b, c, A$. In measuring $A$, an error of 9 min is made then the percentage error in the area of the triangle is

Let $f: R \rightarrow R$ be a bijection. A curve represented by $y=f(x)$ is such that $f^{\prime}(x)>0 \forall x \in \mathbf{R}$. The tangent and normal drawn at $P(\alpha, 1)$ on the curve cuts the $X$-axis at $A, B$ respectively and $C$ is the foot of the perpendicular from $P$ onto the $X$-axis. If $P(\alpha, 1)$ is such a point that $A C+C B$ is minimum, then the tangent at $P$ is parallel to the line

The $x$-coordinate changes on the curve $y=3 x^5+15 x-8$ at the rate of $\frac{1}{5}$ units/sec. $A\left(x_1, y_1\right), B\left(x_2, y_2\right)$ are the points on the curve at which the $y$-coordinate changes at the rate of 6 units/sec, then the slope of $A B=$

In $\triangle A B C, \angle B=90^{\circ}$ and $(b+a)$ is always a constant. In order that $\triangle A B C$ encloses the maximum area, $\angle C=$