engineering mathematics 1 semester 1 2011-12 final approved (1)

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NAZARBAYEV UNIVERSITY

School of Engineering

Examination for Internal Students

Academic Session 2011/12

ENGINEERING MATHEMATICS 1

Date: Monday 12 December 2011 Time: 10.00

Time Allowed ± 3 hours

Section A ± Attempt all questions

Section B ± Attempt ONE question from two

All questions carry 20 marks, you can obtain a maximum of 120 marks in this

examination.

The use of calculators approved by the School of Engineering is permitted.

DO NOT WRITE ON THIS EXAMINATION QUESTION PAPER. THIS

PAPER WILL BE COLLECTED AFTER THE EXAMINATION HAS

ENDED.



ENGINEERING MATHS 1, Semester 1, 2011/12

1

SECTION A: Answer all questions.

1.

(a) Find x

y

d

dfor

i.

¹ º

¸©ª

¨

1

2cosh=

2 x

x y

(3 marks)

ii.

2=)(cos 222 xe xy x

(3 marks)

(b) Where does the curve x x y = have a turning point? Is it a maximum, minimum,

or point of inflection? Justify your answer.

(6 marks)

(c) A curve is given by t te y = and 2= t x . Find x y d/d and 22 d/d x y by parametric

differentiation. Where does the curve have a turning point? Is it a maximum,

minimum, or point of inflection? Justify your answer.(8 marks)

2.

(a) Verify that

,sin=),( ¹¹ º

¸©©ª

¨

y

x y y x g

satisfies the equation

0.=22

22

2

2

22

y

g y

y x

g xy

x

g x

x

x

xx

x

x

x

(6 marks)

(b) Let x y tan= 1 . Findd x

d y. Find also

.dtan1 x x

´ (6 marks)

(Question continued on next page)




2

(c) Define xcosh in terms of exponential functions and sketch the curve x y cosh= .

Show that,

1

1=1

2

2

y y y y

and use this to show that

.1l=cosh21 s y yn y

(8 marks)

3. Evaluate

(a)

,d3tan x x´

(5 marks)

(b)

,d23

153

x x x

x

´

(5 marks)

(c)

.d2cose x x x

´

(5 marks)

(d) Use an appropriate substitution followed by a hyperbolic substitution to evaluate

.)ln(1

d

2

e

1 y y

y

´

(5 marks)




3

4.

(a) Solve

1.=(2),

1)(

1)(=

d

d y

x x

y y

x

y

(6 marks)

(b) Solve

1/4.=/2)(,cotcos=d

dT y x y x

x

y

(6 marks)

(c) For the equation

0=9dd4

dd 2

2

y x y

x y

show that the auxiliary equation has complex roots and these give a basis of r eal

solutions. Express your answer in the form

).sincos(e= x B x A yx F FE

(8 marks)

5.

(a) i. Two unit vectors cÖ and dÖ are perpendicular. Find ( c)dc ÖÖÖ vv and ( c)dc Ö.ÖÖv

(3 marks)

ii. The three points (1, 2, 1), (0, 2, 1) and ( 1, 1, 2) form the vertices of a

triangle. Use vector methods to find the angle between the two sides of the

triangle which meet at (0, 2, 1). Find, also using vector methods, the area of

the triangle.(3 marks)

(b) Find the length of arc of the spiral U3e6=r from 0= U to T U 2= .

(6 marks)





4

(c) i. A batch of 1500 components is examined and 1411 are found to be acceptable.

Five components are picked at random from the production line. Calculate the

probability that

A. all are acceptable

B. four are acceptable

C. two are unacceptable.

(4 marks)

ii. Define the Poisson probability distribution with mean Q . On average 0.2% of

the nails produced by a factory are defective. What is the probability that no more

than 2 defective nails will be found in a box of 1200?

(4 marks)




5

SECTION B: Answer one question only.

6.

(a) Consider the system

1= z y x

2= z y x

3.= z y x

i Transform the augmented matrix of the system to reduced row-echelon form.

ii Hence find the solution of the system.

(6 marks)

(b)

¹¹ º

¸©©ª

¨h g

f e=A

i. Use row reduction to find the inverse (if it exists) of the matrix A . Explainclearly each step.

ii. When will the matrix A not have an inverse?

iii. Use the result from part (i) to show that T T )(=)( 1 1AA , where T indicates

the transpose.

(8 marks)

(c)

i. Two 12v matrices, s and t , are perpendicular if 0== st t s T T . Show this to

be true for (1,1)=T s and 1,1)(= T t .

ii. Find the set of all 12v matrices, ),(= y xT [ that are perpendicular to

(1,1)=T

s .

(6 marks)




6

7.

(a) i. Use Simpson's Rule with four subintervals to find an approximation to

.de1

0 x x x

´

ii. The error bound for Simpson's Rule is given by

\

4)(180

)(|| x

abk E S (

e

with

].,[|)(|(4)

ba xk x f e

and k a positive constant. Here, a and b are the limits of integration and x( is

the width of the subinterval. Find the upper bound for the error in part (i).

(6 marks)

(b) A standard pendulum of length L swinging under only the influence of gravity

(i.e. no resistance) has a period, T , of

J

J

T

T

sin1

d4=

22

/2

0 k g

LT

´

where /2)(sin= 022 Jk , 22sm9.8

} g is the acceleration due to gravity, and0J is the

initial angle from which the pendulum is released (in radians.) Use the Trapezoidal

Rule to approximate the period of the pendulum with 1= L when it is released from an

angle of /4=0 TJ rad. Use four subdivisions in your numerical integration.

(6 marks)





7

(c) Given the initial value problem

,=)(),,(= 00 y x y y x f yd

use the Runge-Kutta fourth order method to solve

.=),( y x y x f

Compute for one step with a step size of 0.2=h .

Note:

),(=1 nn y xh f k

)2

1,

2

1(= 12 k yh xh f k nn

)2

1,

2


)2

1,

2


h x x nn =1

)22(6

1= 43211 k k k k y y nn

(8 marks)

THIS IS THE END OF THE EXAMINATION

engineering mathematics 1 semester 1 2011-12 final approved (1)

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