1. \[\mathop {\lim }\limits_{{\text{x}} \to \infty } {\left( {1 + \frac{1}{{\text{x}}}} \right)^{2{\text{x}}}}\] is equal to
2. \[\mathop {\lim }\limits_{{\text{x}} \to \infty } \frac{{{\text{x}} - \sin {\text{x}}}}{{{\text{x}} + \cos {\text{x}}}}\] equals
3. For the spherical surface x2 + y2 + z2 = 1, the unit outward normal vector at the point \[\left( {\frac{1}{{\sqrt 2 }},\,\frac{1}{{\sqrt 2 }},\,0} \right)\] is given by
4. Divergence of the vector field \[{{\rm{x}}^2}{\rm{z\hat i}} + {\rm{xy\hat j}} - {\rm{y}}{{\rm{z}}^2}{\rm{\hat k}}\] at (1, -1, 1) is
5. \[\mathop {\lim }\limits_{{\text{x}} \to 0} \left( {\frac{{1 - \cos {\text{x}}}}{{{{\text{x}}^2}}}} \right)\] is
6. As x varies from -1 to +3, which one of the following describes the behaviour of the function f(x) = x3 - 3x2 + 1?
7. The following plot shows a function y which varies linearly with x. The value of the integral \[{\text{I}} = \int\limits_1^2 {{\text{y dx}}} \] is
8. Given:
\[{\text{x}}\left( {\text{t}} \right) = 3\sin \left( {1000\pi {\text{t}}} \right)\] and \[{\text{y}}\left( {\text{t}} \right) = 5\cos \left( {1000\pi {\text{t}} + \frac{\pi }{4}} \right)\]
The X-Y plot will be
\[{\text{x}}\left( {\text{t}} \right) = 3\sin \left( {1000\pi {\text{t}}} \right)\] and \[{\text{y}}\left( {\text{t}} \right) = 5\cos \left( {1000\pi {\text{t}} + \frac{\pi }{4}} \right)\]
The X-Y plot will be
9. The infinite series \[1 + {\text{x}} + \frac{{{{\text{x}}^2}}}{{2!}} + \frac{{{{\text{x}}^3}}}{{3!}} + \frac{{{{\text{x}}^4}}}{{4!}} + \,...\] corresponds to
10. If \[\phi = 2{{\text{x}}^3}{{\text{y}}^2}{{\text{z}}^4}\] then \[{\nabla ^2}\phi \] is
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