code no: r21051 r10 ii b. tech i semester supplementary examinations, june - 2015 cse-all.pdf ·...

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Code No: R21051

II B. Tech I Semester Supplementary Examinations, June - 2015

DATA STRUCTURES (Comm. to CSE,IT,ECC)

Time: 3 hours Max. Marks: 75

Answer any FIVE Questions

All Questions carry Equal Marks

~~~~~~~~~~~~~~~~~~~~~~~~~

1 a) Differentiate between linear search and binary search and also give examples for

each.

[7M]

b) Write an algorithm’s for linear search and binary search.

[8M]

2 a) Explain about the sorting technique which uses selection concept. [5M]

b) Write an algorithm for insertion sort and also explain with one example.

[10M]

3 a) What is FIFO? How to represent Stack? Explain. [8M]

b) Write an algorithm for evaluating arithmetic expression.

[7M]

4 a) Differentiate between doubly linked list and circular linked list. [5M]

b) Write an algorithm for creating a singly linked list and perform the insertion and

deletion operations on it.

[10M]

5 a) Explain about the Binary tree Traversal with examples. [7M]

b) Write an algorithm for creation of Binary tree.

[8M]

6 a) What is threaded binary tree? Explain. [5M]

b) Write an algorithm for inserting an element into a Binary search tree.

[10M]

7 a) What is BFS? Discuss with example. [7M]

b) Write any one algorithm for minimum cost spanning tree.

[8M]

8 a) What is Abstract data type? Discuss with example. [6M]

b) Write an ADT for Stack and perform operations on it. [9M]

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Code No: R21051

II B. Tech I Semester Supplementary Examinations, June - 2015 DATA STRUCTURES

(Comm. to CSE,IT,ECC)




~~~~~~~~~~~~~~~~~~~~~~~~~

1 a) What is Fibonacci search? Explain with examples. [8M]

b) Write an algorithm for Fibonacci search.

[7M]

2 a) Explain about the sorting technique which uses distribution concept. [5M]

b) Write an algorithm for Quick sort and also explain with one example.

[10M]

3 a) What is LIFO? How to represent Queue? Explain. [6M]

b) Write an algorithm for infix to postfix conversion expression.

[9M]

4 a) What are the applications of the singly linked list. [5M]

b) Write an algorithm for creating a singly linked list and perform the insertion and

deletion operations on it.

[10M]

5 a) Explain about the properties of the Binary tree. [6M]

b) Write an algorithm for pre-order traversal of a binary tree.

[9M]

6 a) What is binary search tree? Explain. [5M]

b) Write an algorithm for deleting an element from a Binary search tree.

[10M]

7 a) What is DFS? Discuss with example. [8M]

b) Write an algorithm for warshall’s algorithm.

[7M]

8 a) What is Abstraction? Discuss with example. [5M]

b) Write an ADT for Queue and perform operations on it. [10M]

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1 a) What do you mean by linear and binary recursion? Give examples [8M]

b) Write an algorithm for GCD computation using recursion

[7M]

2 a) Is merge sort is stable sort? Discuss. [5M]

b) Write an algorithm for merge sort and also explain with one example.

[10M]

3 a) What is enqueue and dequeue? What are the applications of Queue? Explain. [7M]

b) Write an algorithm for performing queue operations.

[8M]

4 a) What is singly linked list? How to represent it? Discuss. [7M]

b) Write an algorithm for merging two singly linked lists.

[8M]

5 a) What is Binary Tree? What are the operations of Binary tree? Discuss. [6M]

b) Write an algorithm for post-order traversal of a binary tree.

[9M]

6 a) What are the applications of the Balanced binary tree? Explain. [5M]

b) Write an algorithm for pre-order traversal without using recursion.

[10M]

7 a) How to represent graphs? Discuss. [5M]

b) Write an algorithm for minimum cost spanning tree using kruskal’s

[10M]

8 a) What is set? How to perform operations on it? discuss [6M]

b) Write an ADT for Stack and perform operations on it. [9M]

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1 a) What is algorithm? How to analyze the performance of an algorithm? Discuss. [7M]

b) Explain about the Towers of Hanoi problem and also write algorithm for it.

[8M]

2 a) Which sorting technique is efficient? Discuss. [5M]

b) Write an algorithm for heap sort and also explain with one example.

[10M]

3 a) What is priority Queue? Explain [5M]

b) Write an algorithm for infix to postfix conversion,

[10M]

4 a) What are the advantages and disadvantages of singly linked list? Explain. [8M]

b) Write an algorithm for reversing a singly linked list. [7M]

5 a) How to represent Binary trees? Discuss. [6M]

b) Write an algorithm for in-order traversal of a binary tree.

[9M]

6 a) What is balanced binary tree? Explain. [5M]

b) Write an algorithm for post-order traversal without using recursion.

[10M]

7 a) What are the applications of Graphs? Discuss. [5M]

b) Write an algorithm for minimum cost spanning tree using prim’s

[10M]

8 a) What are the applications of set? Discuss. [5M]

b) Write an ADT for Queue and perform operations on it. [10M]

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Code No: R21054

II B. Tech I Semester Supplementary Examinations, June - 2015 DIGITAL LOGIC DESIGN

(Com. to CSE, IT)




~~~~~~~~~~~~~~~~~~~~~~~~~

1 a) Convert the number (17.125)16 to base 10, base 4 , base 5 and base 2.

b) Show that gray code is both reflective and unit distance code .

2 a) Explain any four basic theorems of algebra with necessary proofs.

b) Find the dual complement of + . + .

3 a) Draw 3-variable and 4-variable k-map and define pair , quad and octet.

b) Simplify the following function using k map

F(w,x,y,z) = ∑ (0,1,2,5,6,8,9) + ∑d(3,10,11,15)

4 Design a combinational circuit to convert BCD to gray code.

5 a) What is an encoder? Design octal to binary encoder.

b) Realize F(w,x,y,z)= Ʃ (1,4,6,7,8,9,10,11,15) using 4 to 1 MUX.

6 a) Using PROM realize the following expression .

F1(a,b,c)=Ʃ (0,1,3,5,7)

F2(a,b,c)= Ʃ(1,2,5,6)

b) Derive the PLA programming table for the combinational circuit that squares a 3-bit

number

7 a) Convert D flip-flop into T, JK and SR flip-flop.

b) Explain the operation of JK flip flop with the help of input output waveforms.

8 a) Show that a BCD ripple counter can be constructed using a 4-bit binary ripple

counter with asynchronous clear and a NAND gate that detects the occurrence of

count 1010.

b) Explain with the help of neat diagram, the operation of 3-bit bidirectional shift

register.

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1 a)

b)

Convert the following to decimal and then to binary

i)(2311)16 ii)(A44D)16 iii)(7444)8 iv) (7667)8 v) (158)10

Explain about weighted and non-weighted codes with example

2 a) Find the dual of the following expressions

i) + ( + ) ii) vwx+vwyz+wxy+vxyz

b) Explain the truth tables of X-OR , NAND,NOR gates.

3 a) Express the following equation in its minimal SOP from and realize it using

2 input NAND gate only = + ( ⊕ ⊕ )

b) Find the minimal POS from using K-map for the following expression

F(w,x,y,z) =π (0,1,2,6,7,9) +πd(3,4,8,9)

4 Design a combinational logic circuit that has 3 inputs. The output is required to go

HIGH whenever the number of inputs have even number of 1’s. Draw the truth

table. Minimize the Boolean function using K-map. Draw the circuit diagram.

5 a) Explain the implementation of 4-input priority encoder with truth table, k-maps,

Boolean function and schemantic diagrams.

b) Design a 4 to 1 MUX using a 2 to 4 decoder and basic gates.

6 a) Implement binary to excess 3 code converter using ROM

b) What is PLA? Explain the programming table of PLA. How is the size of a PLA

specified

7 a) Draw the circuit diagram of clocked D-flip flop with NAND gates and explain its

operation using truth table. Give its timing diagram.

b) Construct a JK flip-flop using a D flip-flop, a 2 to 1-line multiplexer and an

inverters.

8 a) Explain Ring counter operation and its applications using a diagram

b) Explain with the help of neat diagram the operation of 4-bit universal shift register.

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1 a) Convert the number(127.75)8 to base 10,base 3 , base 16 and base 2.

b) Explain subtraction in the 2`s complement system with suitable example.

2 a) Prove the identity of the following equations

i) + + = + ) + + + = 1

b) Implement the INVERTER gate, OR gate and AND gate using NAND gate

,NOR gate.

3 a) Simplify the following function using K-map.

(, , , ) = Ʃ(1,3,4,5,6,11,13,14,15)

b) Find the prime implicants for the following and determine which are essential.

F(w,x,y,z)=Ʃ(0,2,4,5,6,7,8,10,13,15) using K-map

4 a) What is half subtractor? Write its logic diagram and truth table

b) Explain carry propagation in parallel adder with a neat diagram.

5 a) Draw the circuit for 3 to 8 decoder and explain.

b) Implement the given function using multiplexer F(x,y,z) = Ʃ (0,2,6,7).

6 a) Given a 32x8 ROM chip with an enable input, show the external connection

necessary to construct a 128x8 ROM with four chips and a decoder

b) How programmable logic array is advantageous over ROMs ? What is meant by an

LSI device.

7 a) What do you mean by triggering? Explain the various triggering modes with

examples.

b) Draw the logic diagram and write functional table of an SR latch using NAND gates.

Explain the operation

8 a) Draw and explain the 4-bit shift register with necessary example.

b) Design a serial adder using JK flip-flop , shift register and logic gates.

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1 a) Determine the value of b for the following :

i)(292)10 =(1204)b ii) (16)10 = (100)b

b) Explain the binary to Gray conversion with the help of examples.

2 a) Simplify the following Boolean functions to minimum number of literals,

) = ! + !" + ! ) = + ![" + ( + ") ]

b) Implement ! = ′ + + (′ + ′′) using NAND gates.

3 a) Simplify the following function using map method.

= + + + + + ′′′′

b) For the given function &(', , , ) = Ʃ(0,1,5,7,8,10,14,15)

i) Find all prime implicates and indicate which are essential.

ii) Find a minimal expression using k map and realize using basic gates.

4 Design an 8-bit BCD adder using 4-bit binary adder.

5 a) Explain how decoder can be converted into a demultiplexer with a neat block

diagram.

b) Implement a full adder with two multiplexer.

6 a) Draw the block diagram of a ROM. Define address and word. Relate the

number of output lines with number of bits in a word. How an output word can

be selected

b) Give the block diagram of PLA. Which are the terms programmable? How

inverter is useful in the PLA construction at the output.

7 a) Draw the logic diagram of an SR latch with control input using NAND gates

b) What is race around condition? How is it rectified in master-slave JK flip-

flop?

8 a) What is a shift register classified? Explain about the following mode of

operation i) shift right ii) shift left iii) bidirectional in a four bit shift

register.

b) Draw the logic diagram of a 4-bit binary ripple counter using positive edge

triggering.

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Code No: R21026


ELECTRONIC DEVICES AND CIRCUITS (Com. to EEE, ECE, EIE, ECC, CSE, IT, BME)




~~~~~~~~~~~~~~~~~~~~~~~~

1. a) Explain in detail about Magnetic Deflection in cathode ray tube.

b) The electron beam in a CRT is displaced vertically by a magnetic field of flux density

2 x 10-4

Wb/m2 .The length of the magnetic field along the tube axis is the same as that

of electrostatic deflection plates. The final anode voltage is 800v.Calcualte the voltage

whichshould be applied to the Y- deflection plates 1cm apart , to return the spot back to

centre of screen. (7M+8M)

2. a) Prove that the conductivity of a semiconductor is given by σ= q( p µp + n µp)

b) Define the terms conductivity and mobility in a semiconductor. (7M+8M)

3. a) Draw the V-I Characteristic of Zener diode and explain its operation.

b) Explain Avalanche breakdown and Zener break down . (7M+8M)

4. a) Obtain the ripple factor of a Full wave rectifier with shunt capacitor filter.

b) Compare the performance of inductive L-section and π-section filters. (7M+8M)

5. a) Define α, β and γ of a transistor show how are they related to each other?

b) Explain how a transistor is used an amplifier. (7M+8M)

6. a) Explain the characteristic parameters of the JFET

b) A JFET has a driven current of 4mA.If Dss = 8mA and Vgs(off)= - 6V. find the values of

Vgs and Vp. (10M+5M)

7. a) Explain in detail about Bias compensation method

b) Explain in detail about collector to base bias. (7M+8M)

8. a) Derive the network parameters for Two port devices.

b) What are salient features of Hybrid Parameters? (7M+8M)

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1. a) Explain in detail about electrostatic deflection in cathode ray tube.

b) An electrostatic CRT has a final anode voltage of 600V. The deflection plates are 3.5cm

long and 0.8 cm apart. The screen is at a distance of 20cm from the center of plates. A

voltage of 20V is applied to the defection plates.

Calculate i) Velocity of electron on reaching the field.

ii) Acceleration due to deflection field.

iii) Deflection produced on the screen in cm.

iv) Deflection sensitivity in cm/v. (5M+10M)

2. a) Explain the difference between intrinsic and extrinsic semiconductor

b) What is meant by p-type impurity in a semiconductor? (8M+7M)

3. a) Draw the V-I Characteristic of PN junction diode and explain its operation.

b) Show that the Zener diode can be used as a voltage regulator. (8M+7M)

4. A Half wave rectifier has a load of 3.5kΩ , If the diode resistance and secondary coil resistance

together have a resistance of 800Ω and input voltage has a signal voltage of peak value 240V

calculate i) Peak , average and rms valve of current flowing ii) dc power output

iii) ac power input iv) efficiency of the rectifier. (15M)

5. a) Explain the input and output Characteristic of a transistor in Common Emitter Configuration

b) Explain about punch through. (10M+5M)

6. a) With the help of suitable diagrams explain the working of different types of MOSFET.

b) Compare the MOSFET with JFET. (10M+5M)

7. a) Explain in detail about operating point and Basic stability

b) Explain in detail about compensation against variation in VBE , ICO. (8M+7M)

8. a) Why Hybrid parameters are called so? Define them.

b) A CE amplifier has the h-parameters given by hie = 1000Ω,hre = 2 x 10-4

, hfe =50

and hoe =25µ mho. If both the load and source resistance are 1k Ω, Determine the

i) Current gain ii) Voltage gain (7M+8M)

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1. a) Explain about in magnetic field.

b) A charged particle having charge thrice that of an electron and mass twice that of an electron

is accelerated through a potential difference of 50V before it enters a uniform magnetic

field flux density of 0.02 Wb/m2 at an angle of 25

o with field.

Calculate i).The velocity of the charged particle before entering the field.

ii). Radius of the helical path

iii) Time of revolution. (5M+10M)

2. a) Explain about Fermi Level in between intrinsic and extrinsic semiconductor.

b) define Drift and diffusion currents in semiconductor. (8M+7M)

3. a) Explain the principle behind the Varactor diode and list out its applications.

b) Explain the Construction of a PIN diode and give the applications of PIN diode. (7M+8M)

4. a) Explain about series and shunt voltage regulators.

b) Derive an expression for the ripple factor in a full wave rectifier using inductor filter.

(7M+8M)

5. a) A transistor has IB = 100µA and IC = 2µA Find

i) β of the transistor ii) α of the transistor iii) Emitter current IE

iv) if IB changes by +25µA and IC changes by +0.6mA. Find the new valve of β?

b) Explain about Photo Transistor. (5M+10M)

6. a) Describe the working principle of an SCR with V-I Characteristics.

b) Briefly describe some applications of JFET. (10M+5M)

7. a) Explain in detail about Stabilization factors.

b) Explain about Thermistor and Sensistor compensation. (7M+8M)

8. a) Explain about Conversion formulas for the parameters of three transistor configuration

b) Explain in detail about Measurement of h-parameters (7M+8M)

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1. a) Explain about Two – Dimensional motion of electron.

b) Explain in detail about Magnetic Focusing. (7M+8M)

2. a) Discuss the following with respect to semiconductor

i) doping ii) Dopant iii) donor iv) acceptor.

b) Explain “Majority and minority carriers” in semiconductors. (8M+7M)

3. a) What is tunneling? Form the energy band diagram explain the V-I characteristic of a

tunnel diode.

b) List out the applications of tunnel diode and mention its advantages and disadvantages.

(10M+5M)

4. a) Draw the circuit diagram of an Half wave rectifier and explain its operation

b) Show that a Full wave rectifier is twice as efficiency as a Half wave rectifier. (8M+7M)

5. a) Explain the input and output Characteristic of a transistor in Common Base Configuration.

b) The Common Base d.c current gain of a transistor is 0.967 .If emitter current is 10mA,

What is the value of base current? (10M+5M)

6. a) Describe the working principle of an UJT with V-I Characteristics.

b) Explain why an SCR is operated only in the forward biased condition. (8M+7M)

7. a) Explain about Self Bias Amplifiers.

b) Explain about Thermal runaway and Thermal stability. (8M+7M)

8. a) Explain in detail about analysis of a transistor amplifier circuit biasing h - parameters.

b) Explain about Transistor Amplifier configurations. (7M+8M)

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Code No: R21022


MANAGERIAL ECONOMICS AND FINANCIAL ANALYSIS (Com. to EEE, ME, ECE, EIE, CSE, IT, ECC, BME)




~~~~~~~~~~~~~~~~~~~~~~~~~~

1. a) “Managerial Economics is the integration of economic theory with business practice for

the purpose of facilitating decision making and forward planning by management”.

Explain?

b) Define demand and describe its determinants with suitable examples?

2. a) Differentiate between the following:

i) Point Price elasticity of demand and Arc Price elasticity of demand

ii) Elasticity of demand and elasticity of supply

b) Write briefly about trend projection method?

3. a) What is the relationship between firm’s short run production function and its fixed and

variable costs?

b) Distinguish between:

i) Implicit and Explicit costs

ii) Accounting and Economic costs

4. a) What are the features of Perfect competition?

b) Distinguish between Market skimming pricing and penetrating pricing.

5. a) What are the features of Joint stock Company?

b) Write short note on features of Business cycles.

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Code No: R21022

6. The following are the balances extracted from the books of Rakesh on 31st March 2015.

PARTICULARS AMOUNT (IN RUPEES)

Rakesh Capital 30,000

Rakesh Drawing 5,000

Furniture and Fixtures 2,600

Opening Stock 22,000

Debtors 18,000

Rent from tenants 1,000

Purchases 1,10,000

Sales 1,50,000

Electricity Charges 1,100

Sales return 2,000

Discounts (Dr) 1,600

Bank Overdraft 4,200

Creditors 13,800

Discount (Cr) 2,000

Taxes and Insurances 2,000

General Expenses 4,000

Salaries 9,000

Commission (Dr) 2,200

Carriage on purchases 1,800

Bad debts 800

Adjustments:

Closing stock at the end was Rs 20,060; Depreciate furniture and fixtures by Rs 550.

Prepare Trading, Profit and Loss account and Balance Sheet for the year ended 31st March

2015 after taking the above adjustments.

7. From the following particulars you are required to prepare a statement of proprietary fund:

Capital turnover ratio=2; Fixed Asset turnover ratio=4; Gross profit turnover ratio=20%; Stock

velocity=5; Debtors velocity= 3 months; Creditors velocity is 2.5 months. The gross profit is

Rs 50,000,Reserves and Surplus is Rs 15000 and closing stock was Rs 5000 lesser than

opening stock.

8. Explain briefly the various methods of capital budgeting.

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1. What do you understand by Elasticity of demand? How do you measure it? What is its

significance?

2. Explain and illustrate the following: and also mention why they arise: a) The Law of Constant

Returns b) The Law of increasing returns.

3. What is opportunity Cost? Give some examples of opportunity cost. How are these costs

relevant for managerial decisions?

4. What is a Market? Explain, in brief, the different Market structures.

5. Explain the merits and demerits of different forms of Business organization and their suitability

with different types of business Activities

6. What are the accounting concepts that govern accounting process? Explain in brief.

7. The following are the extracts from the financial statements of Marvel Ltd. as on 31st March

2014 and 31st March 2015 respectively

31st March 2014

( Amount in Rupees)

31st March 2015

( Amount in Rupees)

Stock 10,000 25,000

Debtors 20,000 20,000

Bills receivable 10,000 5,000

Cash 18,000 15,000

Bills payable 15,000 20,000

Bank Overdraft 2,000

9% Debenture 5,00,000 5,00,000

Sales 3,50,000 3,00,000

Gross Profit 70,000 50,000

Compute Current ratio, Acid test ratio and Stock Turnover ratio and also interpret the result.

8. Compare and contrast with illustration

Accounting rate of return and Payback period method

Net present Value and IRR method

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1. Define Managerial Economics. Explain the nature and scope of managerial economics?

2. What is meant by Elasticity of demand? What are the factors governing the potential demand

for a product as either Elastic or Inelastic?

3. Explain how production function is valuable to manufacturer?

4. What are the features of perfect competition? Explain with the help of a diagram how price

output is determined in perfect competition?

5. Define and evaluate partnership form of business organization.

6. What is working capital? What are the factors governing working capital requirement?

Illustrate.

7. Ram Enterprise is considering purchasing a CNC machine. The following are the earnings after

tax from the two alternative proposal under consideration each costing Rs 8,00,000. Select the

better proposal if the company wishes to operate @ 10% rate of return

Year 1 Year 2 Year 3 Year 4 Year 5

Proposal I 80,000 2,40,000 3,20,000 4,80,000 3,20,000

Proposal 2 2,40,000 3,20,000 4,00,000 2,40,000 1,60,000

Present value of Rs 1 @10% 0.909 0.826 0.751 0.683 0.620

8. Journalize the following transactions and post them to ledger.

i) Ram invested Rs 10,000 in cash

ii) He bought goods worth Rs 2000 from Shyam

iii) He bought a machine for Rs 5000 from Lakshman on account

iv) He paid to Lakshman Rs. 2000

v) He sold goods for cash Rs. 3000

vi) He sold goods to A on account for Rs 4000

vii) He paid to Shyam Rs. 1000

viii)He received amount from A Rs. 2000.

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1. a) What do you understand by demand? What are the different types of demand

b) State the law of demand what its exception are

2. How do you forecast demand for product and services? Explain the method of least squares as a

method of forecasting demand.

3. Do you think monopoly is present in the current business environment? Explain how price

output is determined in monopoly in short run

4. Rahim sells 500 kg of sweets per hour at a rate of Rs 100 per kg. The fixed overhead is Rs 7000

and the variable cost is Rs 25 per kg. There is a proposal to reduce the price by 10%.Calculate

the present PV and present BEP both in units and in Rupees; present level of profit and future

PV ratio and BEP both in units and in Rupees. How many kilograms must be sold to earn

present level of profit?

5. Explain any four pricing methods based on strategy.

6. Based on the following information of the financial ratios prepare Balance sheet of Premier

India Ltd as on 31st March 2015.

Current ratio= 2.5; Liquidity ratio= 1.5; Net working Capital= 6,00,000; Stock turnover ratio=

5; Gross profit to sales =20%. Turnover ratio to net fixed assets (on cost of sales) =2; Average

debt collection period = 2.4 months; Fixed asset to net worth =0.8; Long term debt to capital

and Reserves =7:25.

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Code No: R21022

7. From the following prepare Trading, Profit and loss and Balance Sheet as on 31st March 2015.

Particulars Debit Credit

Capital 20000

Purchases 29000

Sales 55000

Carriage Inwards 5000

Wages outstanding 2000

Plant 20000

Depreciation on plant 4000

Rent received 1000

Salaries and wages 3000

Reserve for bad and doubtful debt 1000

Bad debts 2000

Interest 5000

Premises 20000

Interest paid 5000

Creditors 6000

Opening stock 25000

Loans 38000

Debtors 15000

Adjustment

Closing stock at the end: Rs 40,000

Depreciate building at the rate of 15% per annum.

8. A ltd is considering to purchase a new machine costing Rs 5,85,000. An additional investment

will be required for installation costing Rs 15,000 and for working capital Rs 1,00,000. The

machine has a working life of 5 years and salvage value will be Rs 1,00,000. The working

capital will be released after 5 years. The estimated cash inflows before depreciation and tax are

estimated as follows

Years Cash Inflows (Rupees)

1 1,00,000

2 1,80,000

3 2,50,000

4 2,00,000

5 1,50,000

You can assume straight line method for charging depreciation, cost of capital of 15% and

corporate tax rate of 50%. Should the company purchase the machine?

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Code No: R21053

II B. Tech I Semester Supplementary Examinations, June

MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE AND ENGINEERING

Time: 3 hours

1 a) Write down the following statements in symbolic form using

(i) Every real number has an additive inverse.

(ii) The set of real numbers has a multiplicative identity.

b) Define conjunctive normal form

(p → ( r → q ) ) ↔ ( ( p

2 a) Explain the basic properties of

b) Prove that by principle of mathematical induction

P(n) = 1.2+2.22+3.2

3……….+n.2

3 a) In a class of 70 students 27 are studying

studying both languages.

languages? ii) How many are studying neither of these languages?

b) Let Dn denote the positive devisers of’ n’ ordered by

diagram of D30, D72.

4 a) Show that if a bipartite graph is regular, both of its bipartites must have the same

number of vertices.

b) Define isomorphism. Show that the following graphs are isomorphic or not?

5 a) Explain Breadth First Search algorithm to find spanning tree of a graph

example?

b) Explain kruskal’s algorithm and find mi

suitable example


MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE AND ENGINEERING(Com. to CSE, IT, ECC)



~~~~~~~~~~~~~~~~~~~~~~~~~

Write down the following statements in symbolic form using quantifiers

real number has an additive inverse.

set of real numbers has a multiplicative identity.

efine conjunctive normal form.Find conjunctive normal form of

( ( p Λ r ) → q )

Explain the basic properties of integers with suitable examples

Prove that by principle of mathematical induction

……….+n.2n

= (n-1) 2n+1

+2

In a class of 70 students 27 are studying Hindi, 35 are studying English, 12

studying both languages. i) How many in the class are studying at least one of the

How many are studying neither of these languages?

denote the positive devisers of’ n’ ordered by divisibility. Draw hasse

bipartite graph is regular, both of its bipartites must have the same

Show that the following graphs are isomorphic or not?

earch algorithm to find spanning tree of a graph with suitable

s algorithm and find minimal spanning tree of the graph with

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quantifiers. 7

8

7

8

English, 12 are

one of the

8

hasse 7

bipartite graph is regular, both of its bipartites must have the same 7

Show that the following graphs are isomorphic or not? 8

with suitable 7

nimal spanning tree of the graph with 8

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Code No: R21053

6 a) State and prove Lagrange’s theorem 8

b) (i) Show that every cyclic group is abelian?

(ii) Show that inverse of any element in a group is unique?

7

7 a) A student visits a sports club every day from Monday to Friday after school hours

and plays one of the three games cricket, tennis, football. In how many ways can he

play each of the three games at least once during a week (from Monday to Friday)

8

b) State and prove pigeonhole principle. Explain with example?

7

8 a) Solve the following recurrence relation an =5 an-1 + 6 an-2 with initial conditions

a0 =a1 = 3 and n>=2

8

b) Solve the following recurrence relation an + an-1 – 8 an-2 -12 an-3 =0 for n>=3, given

that a0 =1 and a1 =5,a2 =1?

7

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Time: 3 hours

1 a) Explain the converse, inverse

b) Describe disjunctive normal form and find disjunctive normal form of

( P→ ( Q∨ R ) ) Λ (¬Q)

2 a) State and prove fundamental

b) Prove that by principle of mathematical

for all positive integral value of n including zero?

3 a) Among the first 500 positive integers

(i) Determine the integers which are neither

(ii)Determine the integers which are divisible by 5 but not by 7

b) Show that by using set representation

4 a) Show that a connected graph with

b) Define Hamiltonian and

Hamiltonian but not Elurian and (ii)

5 a) Explain Depth First Search algorithm

example?

b) Explain graph coloring problem.

number of the given graph

Semester Supplementary Examinations, June - 2015




~~~~~~~~~~~~~~~~~~~~~~~~~

converse, inverse and contra positive with suitable examples?

Describe disjunctive normal form and find disjunctive normal form of

Q) Λ (¬ R) Λ P

State and prove fundamental theorem of arithmetic?

of mathematical induction 34n+2

+ 52n+1

is a multiple of 14

for all positive integral value of n including zero?

Among the first 500 positive integers

etermine the integers which are neither divisible by 5,7,nor 9

(ii)Determine the integers which are divisible by 5 but not by 7 and 9.

Show that by using set representation A ∩ ( B - C) = (A ∩ B) – (A ∩ C)

Show that a connected graph with n vertices has at least n-1 edges?

iltonian and Eulrian graphs (i) Draw a graph with six vertices which is

lurian and (ii) Elurian but not hamiltonian?

earch algorithm to find spanning tree of a graph with

Explain graph coloring problem. Define chromatic number and find chromatic

number of the given graph

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and contra positive with suitable examples? 7

8

7

is a multiple of 14 8

8

C) 7

7

raw a graph with six vertices which is 8

to find spanning tree of a graph with suitable 8

Define chromatic number and find chromatic 7

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Code No: R21053

6 a) Let ( L, ≤ ) be a laattice and a,b,c ∈ L then the following results hold

(i) If a ≤ b and a ≤ c then a ≤ b ∨ c

(ii) If a ≤ b and a ≤ c then a ≤ b Λ c

8

b) Prove that the complement of elements in a distributive lattice is unique

7

7 a) Find the total number of positive integers that can be formed from the digits

1,2,3,4,5, if no digit is repeated in any one integer?

8

b) ABC is an equilateral triangle whose sides are of length 1CM each .if we select five

points inside the triangle ,prove that at least one of these points are such that the

distance between them is less than 1/2 CM.

7

8 a) Solve the following recurrence relation an – 7 an-2 +6an-3 =0 with initial conditions

a0=8, a1=6 and a2=22?

8

b) Evaluate the sum 12+2

2+3

2+……………+r

2 using generating function? 7

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Time: 3 hours

1 a) Explain Principle Conjunctive

normal of (¬ P → R) Λ ( Q

b) Show that ( (p∨ q) Λ ¬ (

is a tautology?

2 a) Prove that √ is not a rational number for any prime p?

b) Prove that by principle of mathematical induction 10

by 9?

3 a) In a group of 40 persons 13 are musicians ,8 are poets , 4 are musicians as well

as poets ,5 are poets as well as dramatists ,3 are dramatists as well as musicians

1and one person having all the characteristics. h

dramatists? How many of them are only musicians?

b) Show that by using set representation A

4 a) Suppose G1 and G2 are isomorphic .

also connected.

b) Explain matrix representation of graphs with suitable examples?

5 a) Explain Kruskal’s algorithm and find minimal spanning tree of the given

graph?

b) Define planar graph. Check whether k

graphs?





~~~~~~~~~~~~~~~~~~~~~~~~~

onjunctive Normal Form ?obtain Principle conjunctive

( Q ↔ R)

( ¬ p Λ ( ¬ q ∨ ¬ r ) ) ) ∨ ( ¬ p Λ ¬ q ) ∨ ( ¬ p Λ

is not a rational number for any prime p?

Prove that by principle of mathematical induction 10n + 3* 4

n+2 + 5 is divisible

In a group of 40 persons 13 are musicians ,8 are poets , 4 are musicians as well


1and one person having all the characteristics. how many of them are

dramatists? How many of them are only musicians?

Show that by using set representation A∪ (B – C) ≠ (A ∪ B) – (A ∪ C)

Suppose G1 and G2 are isomorphic .Prove that if G1 is connected then G2 is

Explain matrix representation of graphs with suitable examples?

s algorithm and find minimal spanning tree of the given

heck whether k5 and k3,3 are planar or non planar

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orm ?obtain Principle conjunctive 8

Λ ¬ r) 7

7

+ 5 is divisible 8

In a group of 40 persons 13 are musicians ,8 are poets , 4 are musicians as well


8

7

rove that if G1 is connected then G2 is 8

7

s algorithm and find minimal spanning tree of the given 8

are planar or non planar 7

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Code No: R21053

6 a) Prove that the complement of an element ‘a’ in bounded distributive lattice ,if

it exists ,is unique?

8

b) Let (S,*) be a commutative semigroup show that if ( a * a) = a, (b * b )= b then

( a * b) * (a * b ) = ( a * b )

7

7 a) Find the least number of ways of choosing 3 different numbers from 1 to 10 so

that all choices have the same sum?

8

b) State and prove multinomial theorem

7

8 Find the explicit formula for the sequence defined by cn = 3cn-1 – 2cn-1 with

initial conditions c1=5, c2 =3 by using the following approaches

(i) Characteristic Equation

(ii) Generating Function

15

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(Com. to CSE, IT, ECC)




~~~~~~~~~~~~~~~~~~~~~~~~~

1 a) Describe disjunctive normal form and find disjunctive normal form of

( P→ ( Q∨ R ) ) Λ (¬Q ) Λ (¬ R) Λ P

8

b) Write down the following statements in symbolic form using quantifires.

(i)every real number has an additive inverse.

(ii) the set of real numbers has a multiplicative identity.

7

2 a) State and prove division theorem? 7

b) Prove that by principle of mathematical induction 102n-1

+ 1 is divisible by 11 for

each natural number ?

8

3 a) Let X=1,2,3 and f,g,h be functions X to X given by

g =<1,2>,<2,1>,<3,3>, f = <1,2>,<2,3>,<3,1>, h = <1,1>,<2,2>,<3,3> find fo

g o h and h o g o f.

7

b) Let A = 1,2,3, define A→A such that f=(1,2),(2,1),(3,3) find f-1

, f2 and f

3.

8

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Code No: R21053

4 a) In a graph G ,if the intersection of two paths is a disconnected sub graph ,show that

the union of the two paths has at least one cycle?

7

b) Define isomorphism? Show that the following graphs are isomorphic or not?

8

5 a) Define Peterson graph. Perform the Breadth First Search on Peterson graph 7

b) What is the chromatic number of the following graphs

i)Cn ii) Kn

iii) K m,n

iv) tree with n vertices

8

6 a) State and prove Lagrange’s theorem ? 7

b) (i) Show that every cyclic group is abelian?

(ii) Show that inverse of any element in a group is unique?

8

7 a) In how many ways can 7 men and 7 women be seated in a row

(i) if any person may sit next to any other

(ii) If men and women must occupy alternate seats?

8

b) A multiple choice test has 15 questions and 4 choices for each answer. How many

ways the 15 questions can be answered so that 3 answers are correct?

7

8 a) Solve the following recurrence relation an – 7 an-2 +6an-3 =0 with initial conditions

a0=8, a1=6 and a2=22?

8

b) Solve the following recurrence relation un = 3 un-1

, n≥ 1 using generating function

7

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Code No: R21052


PROBABILITY AND STATISTICS (Com. to CSE, IT)




~~~~~~~~~~~~~~~~~~~~~~

1. a) Among 100 students 50 are studying Maths,30 are studying Physics, and 20 are

studying Maths and Physics. If a student is chosen at random find the probability that the

student is (i) studying Maths or Physics (ii) studying neither Physics nor Maths.

b) Companies 1B , 2B and 3B produce 30%,45% and 25% of the cars respectively. It is

known that 2%,3% and 2% of these cars produced are defective. (i) What is the probability

that a car purchased is defective. (ii) If a car purchased is found to be defective , what is the

probability that this car is produced by the company

2. a) If X is a continuous random variable with p .d .f f(x) = 2x , 0≤ x ≤1,= 0, elsewhere.

If p(a ≤ x ≤ 1) =19

81 ,find the value of ‘a’.

b) A random variable X has the following probability function

x 0 1 2 3 4 5 6 7

p(x) 0 k 2k 2 k 3k k 2 2k 2 7k 2 +k

Determine (i) k , (ii) P(0 ≤ x ≤ 4), (iii) the minimum value of ‘x’ such that P(X ≤ x ) > 1

2

3. a) A hospital switch board receives an average of 4 emergency calls in a 10 minute interval.

What is the probability that (i) there are at most 2 emergency calls in a 10 minute interval,

(ii) there are exactly 3 emergency calls in a 10 minute interval.

b) If X is normally distributed with mean 2 and variance 0.1, then find P(|X-2| ≥ 0.01) ?

4. a) The average marks scored by 32 boys is 72 with a S.D. of 8. While that for 36 girls is 70

with a S.D. of 6. Does this indicate that the boys perform better than girls at level of

significance 0.05?

b) An electrical firms manufactures light bulbs that have a length of life is approximately

normal distribution with a standard deviation of 100 hours, prior experience leads us to

believe that µ is a value of normal random variable what mean µ0 = 800 hours and standard

deviation 0σ = 10 hrs. If a random sample of 25 bulbs has an average life of 780 hours. Find

95% Bayesian interval for µ.

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Code No: R21052

5. A random sample from a company’s very extensive files show that orders for a certain piece

of machinery were filled, respectively in 10,12,19,14,15,18,11 and 13 days. Use the level of

significance α = 0.01to test the claim that on the average such orders are filled in 10.5 days.

Choose the alternative hypothesis so that rejection of the null hypothesis µ = 10.5 days implies

that it takes longer than indicated.

6. Fit a Poisson distribution to the following data and for its goodness of fit at level of

significance 0.05?

0 1 2 3 4

419 352 154 56 19

7. Twenty-five successive samples of 200 switches, each taken from a production line,

contained, respectively 6,7,13,7,0,9,4,6,0,4,5,11,6,18,1,4,9,8,2,17,9,12,10,5 and 4 defectives.

If the fraction of defectives is to be maintained at 0.02,construct a p chart for these data and

state whether or not this standard is being met.

8. A telephone booth with Poisson arrivals spaced 10 minutes apart on the average, and

exponential call lengths averaging 3 minutes. What is the probability that an arrival will have

to wait more than 10 minutes before the phone is free? What is the probability that it will take

him more than 10 minutes altogether to wait for phone and complete his call?

Estimate the fraction of a day that the phone will be in use.

Find the average number of units in the system.

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Code No: R21052






~~~~~~~~~~~~~~~~~~~~~~

1. a) If P( A C ) = 3/8 , P ( B C ) = ½ and P ( A I B )= ¼, find

P( A B ) and P ( B A ) and P( A B C ).

b) What is the probability that a leap year selected at random will contain 53 Sundays?

2. a) If a random variable has the probability density function

f(x) = k( 2x -1), -1 ≤x ≤ 3,

= 0, else where

Find the value of ‘k’ and p(1

2≤x ≤

5

2).

b) Let X denote the sum of the two numbers that appear when a pair of fair dice is tossed.

Determine the (i) Distribution function, (ii) mean and (iii) variance.

3. a) Fit a Poisson distribution to the following data

x 0 1 2 3 4 5 6 7

f 305 365 210 80 28 9 2 1

b) In a normal population with mean 15 and SD 3.5, it is found that 647 observations

exceed 16.25. What is the total number of observations in the population?

4 a) In a test given to two groups of students, marks obtained are as follows

Group I: 18 20 36 50 49 36 34 49 41

Group II: 29 28 26 35 30 44 44

Examine the significance of the difference between the means of the marks secured by

students of the above groups?

b) Determine 99% confidence interval for the mean of contents of soft drink bottles if

contents of 7 such soft drink bottles are 10.0, 10.4, 9.8, 10, 9.8, 10.2,9.6 ml.

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Code No: R21052

5 a) A sample of cam shafts intended for use in gasoline engines has an average eccentricity

of 1.02 and a standard deviation of 0.044 inch. Assuming the data may be treated a random

sample from a normal population, determine a 95% confidence interval for the actual mean

eccentricity of a cam shaft?

b) 20 people were attacked by a disease and only 18 survived will you reject the hypothesis

that the survival rate if attacked by this disease in 85% in favour of the hypothesis that is

more at 5% level.

6. Two random samples are drawn from two normal populations as follows:

A 17 27 18 25 27 29 13 17

B 16 16 20 27 26 25 21

Test whether the samples are drawn from the same normal population. Use a 0.05 level of

significance.

7. Explain, from the perspective of quality improvement programs, why the , , and fraction

defective charts should be used to listen to the process and observe its natural variability at

any stage, rather than for the long-run control of the process.

8. In a railway marshalling yard, goods trains arrive at a rate of 30 trains per day. Assuming

that the inter-arrival time follows an exponential distribution and the service time ( the time

taken to hump a train) distribution is also exponential with an average 36 minutes.

Calculate the following.

(i) The average number of trains in the queues.

(ii) The probability that the queue size exceeds 10.

If the input of trains increases to an average 33 per day, what will be change in (i)

and (ii)?

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Code No: R21052






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1. a) An integer is chosen at random from the first 100 positive integers . What is the

probability that the integer chosen is divisible by 2 or 5?

b)Three machines A,B and C produce respectively 60%, 30% and 10% of the total number

of items of a factory. The percentages of defective output of these machines are respectively

2%, 3% and 4%. An item is selected at random and is found defective. Find the probability

that the item was produced by machine C.

2. a) The cumulative distribution function for a continuous random variable X is

2( ) 1 , 0xF x e x

−

= − ≥

= 0 , x<0

Find (i) the density function f(x), (ii) mean and (iii) variance of the density function.

b)A sample of 3 items is selected at random from a box containing 10 items of which 4 are

defective. Find the expected number of defective items?

3. a) If the marks obtained by a number of students in a certain subject are approximately

normally distributed with mean 65 and standard deviation 5. If three students are selected at

random from this group what is the probability that at least one of them would have scored

above 75?

b) A part of an air pollution survey an inspector decides to examine the exhaust of 6 of a

company’s 24 trucks. If the 4 of the company trucks emit excessive amounts of pollutants,

what is the probability that none of them will be included in the inspector's sample?

4. a) The mean life of a sample of 10 electric bulbs was found to be 1456 hours with S.D. of

423 hours. A second sample of 17 bulbs chosen from a different batch showed a mean life

of 1280 hours with S.D. of 398 hours. Is there a significant difference between the means

of two batches?

b) Find 95% confidence limits for the mean of a normality distributed population from

which the following sample was taken15,17,10,18,16,9,7,11,13,14.

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Code No: R21052

5 a) In a study of an automobile insurance a random sample 0f 80 body repair costs had a

mean of Rs.472.36 and the S.D. of Rs.65.35. It −

X is used as a point estimate to the true

average repair costs, with what confidence we can assert that the maximum error doesn’t

exceed Rs. 10?

b) In a random sample of 400 industrial accidents, it was found that 231 were due at least

partially to unsafe working conditions construct a 99% confidence interval for the

corresponding true proportion.

6. a) The IQ s (intelligence quotients) of 16 students from one area of a city showed a mean of

107 with a standard deviation of 10, while the IQs of 14 students from another area of the

city showed a mean of 112 with a standard deviation of8. Is there a significant difference

between the IQs of the two groups at a 0.05 level of significance?

b) An instructor has two classes A and B, in a particular subject. Class A has 16 students

while class B has 25 students. On the same examination, although there was no significant

difference in mean grades, class A has a standard deviation of 9 while class B had a

standard deviation of 12. Can conclude at the 0.01 level of significance that the variability

of class B is greater than that of A?

7. A process for manufacturer of 4-by-8 foot woodgrained panels has performed in the past

with an average of 2.7 imperfections per 100 panels. Construct a chart to be used in the

inspection of the panels and discuss the control if 25 successive 100-panel lots contained,

respectively 4,1,0,3,5,3,5,4,1,4,0,1,4,2,3,7,4,2,1,3,0,2,6,1 and 3 imperfections.

8. Arrivals at a telephone booth are considered to be Poisson, with an average time of 10

between on arrival and the next. The length of a phone call assumed to be distributed

exponentially with mean 3 minutes, then

(i) What is the probability that a person arriving at the booth will have to wait?

(ii) What is the average length of the queues that form from time to time?

The telephone department will install a second booth when convinced that an arrival would

expect to have waited at least three minutes for the phone. By how much must the flow of

arrivals be increased in order to justify a second booth?

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Code No: R21052






~~~~~~~~~~~~~~~~~~~~~~

1. a) An integer is chosen at random from the first 100 positive integers . What is the probability

that the integer chosen is divisible by 2 or 5?

b) Three machines A,B and C produce respectively 60%, 30% and 10% of the total number of

items of a factory. The percentages of defective output of these machines are respectively

2%, 3% and 4%. An item is selected at random and is found defective. Find the probability

that the item was produced by machine C.

2. a) The cumulative distribution function for a continuous random variable X is

2( ) 1 , 0xF x e x

−

= − ≥ = 0 , x<0Find (i) the density function f(x), (ii) mean and (iii)

variance of the density function.

b) A sample of 3 items is selected at random from a box containing 10 items of which 4 are

defective. Find the expected number of defective items?

3. a) If the marks obtained by a number of students in a certain subject are approximately

normally distributed with mean 65 and standard deviation 5. If three students are selected at

random from this group what is the probability that at least one of them would have scored

above 75?

b) A part of an air pollution survey an inspector decides to examine the exhaust of 6 of a

company’s 24 trucks. If the 4 of the company trucks emit excessive amounts of pollutants,

what is the probability that none of them will be included in the inspector's sample?

4. a) The mean life of a sample of 10 electric bulbs was found to be 1456 hours with S.D. of 423

hours. A second sample of 17 bulbs chosen from a different batch showed a mean life of

1280 hours with S.D. of 398 hours. Is there a significant difference between the means of

two batches?

b) Find 95% confidence limits for the mean of a normality distributed population from which

the following sample was taken15,17,10,18,16,9,7,11,13,14.

5. a) In a study of an automobile insurance a random sample 0f 80 body repair costs had a mean

of Rs.472.36 and the S.D. of Rs.65.35. It −

X is used as a point estimate to the true average

repair costs, with what confidence we can assert that the maximum error doesn’t exceed

Rs. 10?

b) In a random sample of 400 industrial accidents, it was found that 231 were due at least

partially to unsafe working conditions construct a 99% confidence interval for the

corresponding true proportion.

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6. a) The IQ s (intelligence quotients) of 16 students from one area of a city showed a mean of

107 with a standard deviation of 10, while the IQs of 14 students from another area of the

city showed a mean of 112 with a standard deviation of8. Is there a significant difference

between the IQs of the two groups at a 0.05 level of significance?

b) An instructor has two classes A and B, in a particular subject. Class A has 16 students while

class B has 25 students. On the same examination, although there was no significant

difference in mean grades, class A has a standard deviation of 9 while class B had a

standard deviation of 12. Can conclude at the 0.01 level of significance that the variability

of class B is greater than that of A?

7. A process for manufacturer of 4-by-8 foot wood grained panels has performed in the past with

an average of 2.7 imperfections per 100 panels. Construct a chart to be used in the inspection

of the panels and discuss the control if 25 successive 100-panel lots contained, respectively

4,1,0,3,5,3,5,4,1,4,0,1,4,2,3,7,4,2,1,3,0,2,6,1 and 3 imperfections.

8. Arrivals at a telephone booth are considered to be Poisson, with an average time of 10

between on arrival and the next. The length of a phone call assumed to be distributed

exponentially with mean 3 minutes, then

What is the probability that a person arriving at the booth will have to wait?

What is the average length of the queues that form from time to time?

The telephone department will install a second booth when convinced that an arrival would

expect to have waited at least three minutes for the phone. By how much must the flow of

arrivals be increased in order to justify a second booth?

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