Revision Mathematics For University Freshers

General mathematics for university freshers

GENERAL MATHEMATICS 1 (MAT 101)

 

 

 

 

QUESTION 1​​ 

  • Given the sets:  ​​​​ U=x:0x10,  A=1,3,5,6,8,10,  B=1,2,4,5,7,9

  where AU and BU, obtain

iAB          iiAB iiiBA iv AcBcv U(AcBc)

  • Find the greatest common divisor (gcd) of 144 and 90, obtain the constants​​ u,vz, such that​​ d=144u+90v​​ 

ANSWER

  • iAB=1,5=intersection;what is common to both sides

iiAB=3,6,8,10             [AB;what is in A but not in B]

iiiBA=2,4,7,9BA;what is in B but not in A

ivfrom the universal setAc=0,2,4,7,9,10Bc=0,3,5,6,8,10 ,   AcBc=0,2,3,4,5,6,7,8,9,10

vUAcBc={1}

  • G.C.D is the greatest common divisor

144=90×1+54

90=54×1+36

54=36×1+18

36=18×2+0

thegcdof 144 and 90=18​​ 

18=54-36×1

18=54-90-54×1×1

18=54-90+54

18=542-90

18=144-90×1×2-90

18=1442-90×2-90

18=1442-903

d=144u+90v  ;       u=2 ,  v=-3

 

 

 

QUESTION 2​​ 

Let​​ Z=3-4i , obtain​​ 

ithe modulus r, of Z    iithe amplitude θ, of Z     iiiwrite Z in polar form     and

ivrepresent Z in an argand diagram

 

ANSWER

ithe modulus r=(3)2+(-4)2=9+16=25

​​        r=5

 

ii Z=3-4i lies in the fourth quadrant

 amplitude θ=270+α

α=tan-1xy=tan-13-4=36.87°

amplitude θ=270+36.87=306.87°

 

iii polar form=rCisθ=5Cis306.87°

 

 

(iv) ​​​​ 

 

 

 

 

 

 

 

 

 

 

QUESTION 3

The sum of two numbers is 12 and the sum of their squares is 212​​ times their product. Find the numbers.

ANSWER

Let the two numbers be​​ x​​ and​​ y,​​ such that

x+y=12                                                                 1

x2+y2=2.5xy                                                   2

x+y=12 ,        x=12-y                         substitute into equa 2

(12-y)2+y2=2.5[12-yy]

12-y12-y+y2=2.512y-y2

144-12y-12y+y2+y2=30y-2.5y2

144-24y+2y2=30y-2.5y2

144-54y+4.5y2=0

4.5y2-54y+144=0

y=-b±b2-4ac2aa=4.5,  b=-54,     c=144

y=--54±-542-4×144×4.52×4.5

y=54-189     or        54+189

y=8   or 4

the two numbers are 8 and 4

 

 

QUESTION 4

If the​​ x2-21+3ax+73+2a=0 ​​​​ has equal roots, find the value of a.

ANSWER

b2-4ac=0     conditon for quadratic equation with equal roots

a=1,  b=-21+3a=-2-6a,  c=73+2a=21+14a

(-2-6a)2-4121+14a=0

(-2)2-2-2-6a+36a2-421+14a=0

4-24a+36a2-84+56a=0

36a2+32a-80=0

solving quadratically,  a=-2     or      109

 

 

QUESTION 5

The sum of the first 10 terms of a G.P is 33 times the sum of its first 5 terms. Find the common ratio.

ANSWER

sum of G.P=arn-1r-1

sum of 10 terms=33×sum of 5 terms

        a(r10-1)r-1=33×a(r5-1)r-1

              r10-1=33r5-1

              r10-1=33r5-33

 

 

              r10-33r5+32=0                                         but,  r10=(r5)2

             (r5)2-33r5+32=0                                      let,  r5=x

                 x2-33x+32=0

solving quadratically,  x=32   or    1

but,  x=r5

 r5=32    or   1

r=325=2   or    15=1

  common ratio r=2      (since common ratio cannot be 1)         

 

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