2008

2007

1) Show that the series \(\sum(-1)^{n}\left[\sqrt{n^{2}+1}-n\right]\) Is conditionally convergent.

[10M]

2) Applying Cauchy’s criterion for convergence, show that the sequence \((s_{n})\) defined by \(s_{n}=1+\dfrac{1}{2}+\dfrac{1}{3}+\ldots\) is not convergent.

[10M]

3) Show that \(\iint_{D} \dfrac{(x-y)}{(x+y)^{3}} d x d y\) does not exist, where \(D=\left\{(x, y) \in R^{2} \mid 0 \leq x \leq 1,0 \leq y \leq 1\right\}\)

[10M]

4) If \(f: R^{2} \rightarrow R\) such that \(f(x , y)=\left\{\begin{array}{c} \dfrac{x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}},(x, y) \neq(0,0) \\ 0 \quad,(x, y) =(0,0) \end{array}\right.\)
Then show that \(f_{x y} \neq f_{y x}\)

[10M]

2006

1) Evaluate the double integral \(\iint_{R} x^{2} d x d y\) where \(R\) is the region bounded by the line \(y = x\) and tha curve \(y = x ^{2}\)z

[10M]

2) Show that the function \(f (x)\) defined by \(f(x)=\dfrac{1}{x},\; x \in[1, \infty)\) is uniformly continuous on \([1, \infty)\)

[10M]

3) Show that the series \(x^{4}+\dfrac{x^{4}}{1+x^{4}}+\dfrac{x^{4}}{\left(1+x^{4}\right)^{2}}+\ldots\) is not uniformly convergent on [0,1).

[10M]

4) Examine the following function for extrema: \(f\left(x_{1}, x_{2}\right)=x_{1}^{3}-6 x_{1} x_{2}+3 x_{2}^{2}-24 x_{1}+4\).

[10M]

5) Show that (i) \(h(x)=\sqrt{x+\sqrt{x}} ,\;x \geq 0\) is continuous on \([0, \infty)\) (ii) \(h(x)=e^{\sin x}\) is continuous on \(R\)

[10M]

6) If \(f(x, y)=\left\{\begin{array}{cc}\dfrac{x^{2}-y^{2}}{x^{2}+y^{2}}, & \text { where }, x, y \neq 0 \\ 0, & \text { where } x=y=0\end{array}\right.\) show that at (0,0)\(\dfrac{\partial^{2} f}{\partial x \partial y} \dfrac{\partial^{2} f}{\partial v \partial y}\)

[10M]

2005

1) Evaluate the double integral \(\iint_{R} x^{2} d x d y\) where \(R\) is the region bounded by the line \(y = x\) and tha curve \(y = x ^{2}\)

[10M]

2) Show that the function \(f (x)\) defined by \(f(x)=\dfrac{1}{x},\; x \in[1, \infty)\) is uniformly continuous on \([1, \infty)\)

[10M]

3) Show that (i) \(h(x)=\sqrt{x+\sqrt{x}}, x \geq 0\) is continuous on \([0, \infty)\) (ii) \(h(x)=e^{\sin x}\) is continuous on \(R\).

[10M]

4) If \(f(x, y)=x y \dfrac{x^{2}-y^{2}}{x^{2}+y^{2}},\) when \((x, y) \neq(0,0)\) and \(f(0,0)=0,\) show that at (0,0) \(\dfrac{\partial^{2} f}{\partial x \partial y} \neq \dfrac{\partial^{2} f}{\partial y \partial x}\)

[10M]