B.TECH - Semester 5 engineering mathematics 4 Question Paper 2020 (feb)
Practice authentic previous year university questions for better exam preparation.
Sample Questions
- (a) Find the image of $|z-3 i|=3$ under $w=1 / z$
- Show that the function $f(z)=\bar{z}$ is nowhere differentiable.
- If $f(z)=(x-y)^{2}+2 i(x-y)$. Show that the C-R equations are satisfied along the curve $x-y=1$.
- Evaluate $\int_{0}^{1+i}(x-y+i x)^{2} d z$ along the line from $z=0$ to $z=1+i$.
- Show that $H=\left\{(x, y, z) \varepsilon R^{3}: x-y=z\right\}$ is a subspace of $R^{3}$.
- If a $6 \times 3$ matrix $A$ has rank 3 . Find $\operatorname{dimNulA,~} \operatorname{dimRowA}$ and rankA ${ }^{\top}$. PART - B Answer any one full question from each Module. Each full question carries 20 marks.
- (a) Show that $e^{x}(x \cos y-y \sin y)$ is a harmonic function. Find the analytic function for which $e^{x}(x \cos y-y \sin y)$ is the imaginary part.
- (b) Prove that an analytic function with a constant argument is a constant.
- (c) Find the constant $a$ and $b$ for which the fun $x^{2}+a y^{2}-2 x y+i\left(b x^{2}-y^{2}+2 x y\right)$ is analytic. P.T.O.
- (b) Show that a bilinear transformation preserves cross ratio.
- (c) Show that the transformation $w=\frac{2 z+3}{z-4}$ maps the circle $x^{2}+y^{2}-4 x=0$ into the straight line $4 u+3=0$.
- (a) Evaluate $\int_{0}^{2+i} \bar{z}^{2} d z$ along the line $y=\frac{x}{2}$.
- (b) Evaluate $\int_{c} \frac{3 z-1}{z^{3}-z}$ where C is (i) $|z|=\frac{1}{2}$ (ii) $|z|=2$.
- (c) Evaluate $\int_{C} \frac{d z}{\left(z^{2}+4\right)^{2}}$ where C is the circle $|z-i|=2$ using residue theorem.
- (a) Evaluate $\int_{0}^{2 \pi} \frac{d \theta}{2+\cos \theta}$
- (b) Evaluate $\int_{-\infty}^{\infty} \frac{d x}{\left(x^{2}+1\right)^{3}}$.
- (a) If $V=R^{4}$ and $W$ is a subspace generated by $(1,-2,5,-3),(2,3,1,-4)$ and $(3,8,-3,-5)$. Find (i) a basis and $\operatorname{dim} W$. (ii) Extend the basis of $W$ to a basis of $R^{4}$.
- (b) If $A=\left[\begin{array}{ccc}8 & -2 & -9 \\ 6 & 4 & 8 \\ 4 & 0 & 4\end{array}\right]$ and $W=\left[\begin{array}{c}2 \\ 1 \\ -2\end{array}\right]$. Determine if $W$ is in $C o l A$ and if $W$ is in NulA.
- (a) Define a subspace of a vector space. Describe the subspaces of $R^{3}$.
- (b) If $B=\left[b_{1}, b_{2}\right]$ and $C=\left[c_{1}, c_{2}\right]$ are the bases for $R^{2}$. Find the change of coordinate matrix from $B$ to $C$ and the change of coordinate matrix from $C$ to $B$ if $b_{1}=\left[\begin{array}{l}7 \\ 5\end{arra...
- (a) Find a least square solution of $A x=b$ for $A=\left[\begin{array}{cc}1 & -2 \\ -1 & 2 \\ 0 & 3 \\ 2 & 5\end{array}\right] b=\left[\begin{array}{c}3 \\ 1 \\ -4 \\ 2\end{array}\right]$.
- (b) Find an orthonormal basis for $\mathbb{R}^{3}$ for the basis $\{(1,1,1),(0,1,1),(0,0,1)\}$.
- (a) Find the singular value decomposition of $A=\left[\begin{array}{ccc}3 & 1 & 1 \\ -1 & 3 & 1\end{array}\right]$.
- (b) Show that $q=3 x^{2}+3 y^{2}+3 z^{2}+2 x y-2 y z+2 x z$ is positive definite.