81. If \[{\text{f}}\left( {\text{x}} \right) = \sin \left| {\text{x}} \right|\] then value of \[\frac{{{\text{df}}}}{{{\text{dx}}}}\] at \[{\text{x}} = \frac{{ - \pi }}{4}\] is
82. The value of integral \[\mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}\limits_{\text{S}}
{\overrightarrow {\text{r}} \cdot \overrightarrow {\text{n}} {\text{ds}}} \] over the closed surface S bounding a volume, where \[\overrightarrow {\rm{r}} = {\rm{x\hat i}} + {\rm{y\hat j}} + {\rm{z\hat k}}\] is the position vector and \[{{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}}
\over n} }}}\] is the normal to the surface S, is
83. Which of the following statements are correct regarding dot product of vectors?
1. Dot product is less than or equal to the product of magnitudes of two vectors.
2. When two vectors are perpendicular to each other, then their dot product is non-zero.
3. Dot product of two vectors is positive or negative depending whether the angle between the vectors is less than or greater than \[\frac{\pi }{2}\].
4. Dot product is equal to the product of one vector and the projection of the vector on the first one.
Select the correct answer:
1. Dot product is less than or equal to the product of magnitudes of two vectors.
2. When two vectors are perpendicular to each other, then their dot product is non-zero.
3. Dot product of two vectors is positive or negative depending whether the angle between the vectors is less than or greater than \[\frac{\pi }{2}\].
4. Dot product is equal to the product of one vector and the projection of the vector on the first one.
Select the correct answer:
84. The volume of the solid surrounded by the surface \[{\left( {\frac{{\rm{x}}}{{\rm{a}}}} \right)^{\frac{2}{3}}} + {\left( {\frac{{\rm{y}}}{{\rm{b}}}} \right)^{\frac{2}{3}}} + {\left( {\frac{{\rm{z}}}{{\rm{c}}}} \right)^{\frac{2}{3}}} = 1\] is
85. For the function, f(x, y) = x2 - y2 defined on R2, the point (0, 0) is
86. A path AB in the form of one quarter of a circle of unit radius is shown in the figure. Integration of (x + y)2 on path AB traversed in a counterclockwise sense is
87. The minimum value of the function f(x) = x3 - 3x2 - 24x + 100 in the interval [-3, 3] is
88. The directional derivative of f(x, y, z) = 2x2 + 3y2 + z2 at the point P(2, 1, 3) in the direction of the vector a = i - 2k is
89. What is the value of \[\mathop {\lim }\limits_{{\rm{n}} \to \infty } {\left( {1 - \frac{1}{{\rm{n}}}} \right)^{2{\rm{n}}}}?\]
90. The line integral of function F = yzi, in the counter-clockwise direction, along the circle x2 + y2 = 1 at z = 1 is
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