B.TECH - Semester 5 advanced mathematics and queuing models Question Paper 2021 (dec)
Practice authentic previous year university questions for better exam preparation.
- (a) Find a least square solution to the syslem $\mathrm{AX}=\mathrm{B}$ where $\mathrm{A}=\left[\begin{array}{ll}1 & 2 \\ 1 & 5 \\ 1 & 7 \\ 1 & \mathrm{~B}\end{array}\right]$ $$ B=\left[\begin{array}{l} 1 \\ 2 \\ 3 \\ 3 \end{array}\right] \quad X=\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\ri...
- Find the co-urdinales of $v=\left[\begin{array}{l}1 \\ 2 \\ 3\end{array}\right]$, relative to the basis $v_{1}=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right], v_{2}=\left[\begin{array}{l}1 \\ 1 \\ 0\end{array}\right] v_{1}=\left[\begin{array}{l}1 \\ 0 \\ 0\end{array}\right]$ in $\mathbf{R}^{1}$
- Give formulas for: (a) Average queue length
- Give formulas for: (b) Probability that number of cuslomers in the system exceeds $K$ for the M/M/1 : $\infty$ /FIFO madel.
- Arrivals at a telephone booth is a Poisson process with average lime of 12 minules belween one arrival and the next. The length of a phone call is exponentially distribuled with mean a minules. Find the probability that a person arriving at the booth will have lo wait in the queue? ( $10 \times 4=40...
- (a) Solve by simplex melhod the LPP Minimize $z=x_{1}-3 x_{2}+2 x_{3}$ subject to the constraints $3 x_{1}-x_{2}+2 x_{3} \leq 7$, $-2 x_{1} \div 4 x_{2} \leq 12 ;-4 x_{1}+3 x_{2}+8 x_{3} \leq 10, x_{1}, x_{2}, x_{3} \geq 0$.
- (b) A project schedule has the following characteristics | Activily: | $1-3$ | $1-2$ | $2-4$ | $3-4$ | $3-5$ | $4-9$ | $5-6$ | $5-7$ | $6-8$ | $7-8$ | $8-10$ | $9-10$ | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | Time : | 4 | 1 | 1 | 1 ...
- (i) Construct a network diagram. (ii) Find the critical palth and project duration. OR
- (a) Solve by Big M melhod the LPP, Maximize $z=6 x_{1}-3 x_{2}+2 x_{3}$ subject to the constraints $2 x_{1}+x_{2}+x_{3} \leq 16$; $3 x_{1}+2 x_{2}+x_{3} \leq 18: x_{2}-2 x_{3} \geq 8 . x_{1}, x_{2}, x_{3} \geq 0$
- (b) The following are the fime estimates for various activities of a project. The lime eslimale are in days. | Aclivily: | $1-2$ | $1-6$ | $2-3$ | $2-4$ | $3-5$ | $4-5$ | $5-8$ | $6-7$ | $7-8$ | | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | Least lime: | 3 | 2 |...
- (i) Draw the project network. (ii) Find the critical path. (iii) Find the probability that the project is completed in 31 days. Module - II 13 (a) Solve by Lu decomposition, the system $\mathrm{AX}=\mathrm{B}$ where $A=\left[\begin{array}{cccc}1 & -2 & -4 & -3 \\ 2 & -7 & -7 & -6 \\ -1 & 2 & 6 & 4...
- (b) Find whether $u=\left[\begin{array}{c}1 \\ -1 \\ 2\end{array}\right]$ is in the column space of $A=\left[\begin{array}{ccc}1 & -1 & 2 \\ 2 & 1 & 0 \\ 2 & 2 & 1\end{array}\right]$.
- (b) Find an orthonomal basis of the subspace spanned by $$ u_{1}=\left[\begin{array}{c} 2 \\ -5 \\ 1 \end{array}\right] \text { and } u_{2}=\left[\begin{array}{c} 4 \\ -1 \\ 2 \end{array}\right] $$
- Find all basic solutions to $x_{1} \div x_{2}-x_{2}=3: x_{1}-x_{2}+2 x_{1}=2$.
- Obtain a BFS to the following LPP after converling it in standard form. Maximize $z=x_{1}+2 x_{2}: x_{1}+x_{2} \leq 4, x_{1}-x_{2} \geq 0, x_{1} \geq 0, x_{2} \geq 0$.
- Eriefly merilion the diference in the PEIST and CPM teslmiques.
- Give a basis for the vector space of all $2 \times 2$ real malrices.
- Is $\left[V_{1}=\left[\begin{array}{c}1 \\ -1\end{array}\right] V_{2}=\left[\begin{array}{l}1 \\ 1\end{array}\right]\right\}$ an orthogonal basis of $\mathbf{R}^{2} ?$ Why?
- Explain least square preblem, Give a formula to find a least square solution to the linear system: $A \mathrm{X}=\mathrm{B}$.
- Explain the characteristics of a Queuing system.
- Customers arrive al a barbar shop according to a Poisson process with a mean inter arrivals lime of 12 minules. Customers spend an average of 10 minules in the barbers chair. (a) What is the expected number of customers in the barber shop and in the queue?
- Customers arrive al a barbar shop according to a Poisson process with a mean inter arrivals lime of 12 minules. Customers spend an average of 10 minules in the barbers chair. (b) Calculate the percentage of time a person can walk straight into the barbers chair withoul having to wail.
- Customers arrive al a barbar shop according to a Poisson process with a mean inter arrivals lime of 12 minules. Customers spend an average of 10 minules in the barbers chair. (c) What is the probability that the waiting time in the system is greater than 30 minules?
- Customers arrive al a barbar shop according to a Poisson process with a mean inter arrivals lime of 12 minules. Customers spend an average of 10 minules in the barbers chair. (d) What is the probability that more than 3 cuslomers are in the syslem?
- A super market has two girls attending to sales al counters. It the service time for each oustomer is exponential wiln mean 4 minutes and if people arrive in Poisson fashion al the rale of 10 per hour, (a) Find the probability that a cuslomer has lo wail for service.
- A super market has two girls attending to sales al counters. It the service time for each oustomer is exponential wiln mean 4 minutes and if people arrive in Poisson fashion al the rale of 10 per hour, (b) What is the expected percentage of idle time for each girf?
- A super market has two girls attending to sales al counters. It the service time for each oustomer is exponential wiln mean 4 minutes and if people arrive in Poisson fashion al the rale of 10 per hour, (c) If a customer has to wait in the queve, what is the expected length of his wailing lime? ( $3 ...
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