Higher Engineering Mathematics, Sixth Edition

Revision Test

This Revision Test covers the material contained in Chapters 1 to 4.The marks for each question are shown in
brackets at the end of each question.

1. Factorisex^3 + 4 x^2 +x−6 using the factor theo-
   rem. Hence solve the equation
   x^3 + 4 x^2 +x− 6 =0(6)


2. Use the remainder theorem to find the remainder
   when 2x^3 +x^2 − 7 x−6isdividedby
   (a) (x− 2 ) (b) (x+ 1 )

Hence factorise the cubic expression (7)

3. Simplify

6 x^2 + 7 x− 5

2 x− 1

by dividing out (4)

4. Resolve the following into partial fractions

(a)

x− 11

x^2 −x− 2

(b)

3 −x

(x^2 + 3 )(x+ 3 )

(c)

x^3 − 6 x+ 9

x^2 +x− 2

(24)

5. Evaluate, correct to 3 decimal places,

5e−^0.^982

3ln0. 0173

(2)

6. Solve the following equations, each correct to 4
   significant figures:
   (a) lnx= 2 .40 (b) 3x−^1 = 5 x−^2

(c) 5= 8 ( 1 −e−

x

(^2) ) (10)

7. (a) The pressurepat heighthabove groundlevel is
   given by:p=p 0 e−khwherep 0 is the pressure
   at ground level andkis a constant. Whenp 0
   is 101 kilopascals and the pressure at a height
   of 1500m is 100 kilopascals, determine the
   value ofk.
   (b) Sketch a graph ofpagainsth(pthe vertical
   axis andhthe horizontal axis) for values of
   height from zero to 12000m whenp 0 is 101
   kilopascals.
   (c) If pressurep=95kPa, ground level pressure
   p 0 =101kPa, constant k= 5 × 10 −^6 , deter-
   mine the height above ground level,h,in
   kilometres correct to 2 decimal places. (13)


8. Solve the following equations:
   (a) log

(

x^2 + 8

)

−log( 2 x)=log3

(b) lnx+ln(x–3)=ln6x–ln(x–2) (13)

9. If θf−θi=

R

J

ln

(

U 2

U 1

)

find the value of U 2

given thatθf= 3. 5 ,θi= 2. 5 ,R= 0. 315 ,J= 0. 4 ,

U 1 = 50 (6)

10. Solve, correct to 4 significant figures:
    (a) 13e^2 x−^1 =7ex

(b) ln(x+ 1 )^2 =ln(x+ 1 )–ln(x+ 2 )+2 (15)