IAS PYQs 4
2008
1) Show that
\(\oint_{\mathrm{S}} \dfrac{\mathrm{ds}}{\sqrt{\mathrm{a}^{2} \mathrm{x}^{2}+\mathrm{b}^{2} \mathrm{y}^{2}+\mathrm{c}^{2} \mathrm{z}^{2}}}=\dfrac{4 \pi}{\sqrt{\mathrm{abc}}}\), where \(\mathrm{s}\) is the surface of the ellipsoid \(a x^{2}+b y^{2}+c z^{2}=1\).
2) Find the unit vector along the normal to the surface \(z=x^{2}+y^{2}\) at the point \((-1,-2,5)\).
[10M]
3) Prove that the necessary and sufficient condition for the vector function \(\overrightarrow{\mathrm{V}}\) of the scalar variable \(t\) to have constant magnitude is \(\overrightarrow{\mathrm{V}} \cdot \dfrac{\mathrm{d} \overrightarrow{\mathrm{v}}}{\mathrm{dt}}=0\).
[10M]
4) Prove that the shortest distances between a diagonal of a rectangular parallelopiped whose sides are of lengths a, \(\mathrm{b}\) and \(\mathrm{c}\) and the edges not meeting it are
\[\dfrac{b c}{\sqrt{b^{2}+c^{2}}}, \dfrac{c a}{\sqrt{c^{2}+a^{2}}}, \dfrac{a b}{\sqrt{a^{2}+b^{2}}}\][10M]
5) If \(\vec{F}=2 x^{2} \hat{i}-4 y z \hat{j}+z x \hat{k}\), evaluate
\(\iint_{5} \vec{F} \cdot \hat{n} d s\),
where \(s\) is the surface of the cube bounded by the planes \(x=0\), \(x=1\), \(y=0\), \(y=1\), \(z=0\), \(z=1\).
[10M]
2007
1) Evaluate
\[\int_{0} \bar{F} \overline{d r}\]where \(\bar{F}=c\left[-3 a \sin ^{2} \theta \cos \theta \bar{i}+a\left(2 \sin \theta-3 \sin ^{2} \theta\right) \hat{j}+b \sin 2 \theta \vec{k}\right]\) and the curve \(\mathrm{C}\) is given by \(\vec{r}=a \cos \theta \vec{i}+a \sin \theta j+b \theta \bar{k}\) \(\theta\) varying from \(\dfrac{\pi}{4}\) to \(\dfrac{\pi}{2}\).
2) Show that
\[Curl \left(\dfrac{\overrightarrow{\boldsymbol{a}} \times \overrightarrow{\boldsymbol{r}}}{\boldsymbol{r}^{3}}\right)=-\dfrac{\dot{\boldsymbol{a}}}{r^{3}}+\dfrac{3 \boldsymbol{r}}{r^{3}}(\overrightarrow{\boldsymbol{a}}, \overrightarrow{\boldsymbol{r}})\]Where \(\vec{a}\) is a constant vector and
\[\vec{r}=x \vec{i}+y \vec{j}+z \vec{k}\][10M]
3) Find the curvature and torsion at any point of the curve
\(x=a \cos 2 t, y=a \sin 2 t, z=2 a \sin t\)
[10M]
4) Evaluate the surface integral \(\int_{S}(y z \bar{i}+z x \bar{j}+x y \bar{k}). d \vec{a}\) Where \(\mathrm{S}\) is the surface of the sphere \(x^{2}+y^{2}+z^{2}=1\) in the first octant.
[10M]
5) Apply Stokes’ theorem to prove that
\[\int_{n}(y d x+z d y+x d z)=-2 \sqrt{2} \pi a^{2}\]Where \(C\) is curve given by
\[x^{2}+y^{2}+z^{2}-2 a x+2 a y=0, x+y=2 a\][10M]
2006
1) If \(\vec{f}=3 x y i-y^{2} j\), determine the value of \(\int_{C} \vec{f}.d \vec{r}\), where \(C\) is the curve \(y=2 x^{2}\) in the \(xy\)-plane from \((0,0)\) to \((1,2)\).
[10M]
2) If \(u \vec{f}=\vec{\nabla} v\), where \(u\), \(y\) are scalar fields and \(\vec{f}\) is a vector field, find the value of \(\vec{f} \cdot\) curt \(\vec{f}\).
[10M]
3) If \(O\) be the origin, \(A, B,\) two fixed points and \(P(x, y, z)\) a vaniable point, show that \(\operatorname{curl}(\vec{P} A \times \overrightarrow{P B})=2 \overrightarrow{A B}\)
[10M]
4) Using Stokes’ theorem, determine the value of the integral
\[\int_{\Gamma}(y d x+z d y+x d z)\]where \(\Gamma\) is the curve defined by
\[x^{2}+y^{2}+z^{2}=a^{2} y+z=a\][10M]
5) Prove that the cylindrical co-ordinate system is orthogonal.
[10M]
2005
1) For the curve \(\) \vec{r}=a\left(3 t-t^{3}\right) \vec{i}+3 a t^{2}\vec{j}+a\left(3 t+b^{3}\right) \vec{k} \(\) being \(a\) constant. Show that the radius of curvature is equal to its radius of torsion.
2) Find \(f(r)\) if \(f(x)=6\) both solenoidal and irrotational.
[10M]
3) Evaluate \(\iint_S \bar{F} \cdot d \bar{s}\), where \(\bar{F}=yz \bar{i} + zx \hat{j} + xy \bar{k}\) and \(S\) is the part of the sphere \(x^2+y^2+z^2=1\) that lies in the first octant.
[10M]
4) Verify the divergence theorem for \(\bar{F}=4 x \bar{i}-2 y^2 \bar{j}+z^{2} \bar{k}\) taken over the region bounded by \(x^{2}+y^{2}=4\), \(z=0\) and \(z=3\).
[10M]
5) By using vector methods, find an equation for the tangent plane to the surface \(z=x^{2}+y^{2}\) at the point \((1,-1,2)\).
[10M]