Let  be three points in .  Find 

Let x e^r = cosθ, y = e^r sin θ and f be a continuously differentiable function of x and y, whose partial derivatives are also continuously differentiable. Show that

Let the function f be defined by

Show that f has directional derivatives in all directions at (0,0).

Find the following limits:

(i)  

(ii)  

(b) Find the third Taylor polynomial of the function f (x,y) = 1 + 5xy + 3²y at (1,2).

(c) Using only the definitions, find f_xy(0,0) and f_xy (,0,0) if they exists, for the function

State whether the following statements are true or false. Give reasons for your answers.

(i) 

(ii) A real-valued function of three variables which is continuous everywhere is differentiable.

(iii) The function  , defined by F(x, y) = ( y + ,2 x + y) at any (x, y) ∈ 

(iv)  ,defined by

is integrable.

(v) The function defined by  has an extremum at (0,0).

At a call omtre, callers have to wait till an operator is ready to take their call. To monitor this process, 5 calls were recorded every hour for the S-hour working day. The data below shows the waiting time in seconds:

i) Use the data to construct control charts for mean and variability and comments about the process. If process is out of control, then calculate the revised controllinis

ii) If the specification limits as the 812, then find the process capability. Does it appear that the process is capable of mecting the specification reques

The failure data for 40 electronic components is shown below:

Estimate the reliability, cumulative failure distribution, failure density and failure rate functions.

A system has seven independent components and reliability block diagram of it shown blow:

Find reliability of the system.

A two-person zero-sum game having the following payoff matrix for player A 

(i) Check whether saddle point exit or not.

(ii) If saddle point does not exit then determine optimal strategies for both the manufacturers and value of the game.

An office supply company ordered a lot of 400 printers. When the lot arrives the company inspector will randomly inspect 12 printers. If more than three printers in the sample are non-conforming, the lot will be rejected. If fewer than two printers are non-conforming the lot will be accepted. Otherwise, a second sample of size 5 will be taken. Suppose the inspector finds two non-conforming printers in the first sample and two in the second saraple. Also AQI. and LTPD at 0.05 and 0.10 respectively. Let incoming quality be 4%

i) What is the probability of accepting the lot at the first sample?

ii) What is the probability of accepting the lot at the second sample?

iii) Find AQL and ATI

A manufacturer of men's jeans purchases zippers in lots of 500. The jeans manufacturer uses single-sample acceptance sampling with a sample size of 10 to determine whether to accept the lot. The manufacturer uses 2 as the acceptance nuraber. Suppose 3% nonconforming zippers are acceptable to the manufacturer and 8% oncoming zippers are not acceptable. Let incoming quality be 4%

i) Construct an OC carve

ii) Average outing quality (AOQ), if the rejected lots are scromed and all defective zippers are replaced by non-defectives

i) Average total inspection (ATI).

Twenty saraples each of size 10 were inspected. The number of defectives detected in cach of them is given below: : 0, 1, 0, 3, 9, 2, 0, 7, 0, 1, 1, 0, 0, 3, 1, 0, 0, 2, 1, 0 Find the control limits for the number of defectives and establish quality standards for the future. Plot the graph and interpost.

State whether the following statements are True or False. Give reason in support of your

(a) Statistical quality control (SQC) is a technique of process control only.

(b) Twenty pieces of different length of cloth contained 2, 4, 1, 3, 5, 4, 2, 7, 3, 5, 2, 2, 4, 5,6, 4, 2, 1, 2, 4 defects respectively. To check the process is under control with respect to the number of defects, we should use np-chart.

(c) If the probabilities are not associated with the occurence of different states of nature, then the situation is known as decision making under risk.

(d) In single sampling plan, if we increase acceptance number then the OC curve will be steeper.

(e) A system has four components connected in parallel configuration with reliability  0.4, 0.5, 0.8 To improve the reliability of the system most, we have to replace the component which reliability is 0.2.

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