If f(x) = |x – 10 – x2| evaluate f(-7)
Solution
f(x) = |x - 10 – x2|
f(-7) = |-7 - 10 – (-7)2|
f(-7) = |-17 – (-7)2| (since -7 - 10 = -17)
f(-7) = |-17 – 49| since
f(-7) = |-66| = 66 since any number out of absolute
signs is +ve.
Hence
f(-7) = 66
TRY THIS..................
If f(x) = | x – 60 – x2| evaluate f(-8)
If 5x + 13y= 8 + 6x; find the x intercept.
Solution
x-intercept
is when y=0.
5x
+ 13(0)= 8 + 6x.
5x
- 6x + 13(0)= 8
-x
+ 0 = 8
-x
= 8.
-x = 8
dividing by -1 both sides.
-1 -1
x =
-8
Hence x-intercept is (-8,0)
TRY
THIS………………………
If
11x - 3y= 33 + 13x; find the x intercept.
Rayan deposited the amount of 20,000/=
in a bank for 5 years and got a profit of 1500/=. Find the interest rate?
Solution
I = 1500/=, P = 20,000/=, T = 5 years,
R = ?
I = PRT
100
1500 = 20000 x R x 5
100
1500 = 20000 x R x 5
100
1500 = 200 x R x 5
1500 = 1000R
1.51500
= 1000R
1000
1000
Hence the interest rate was 1.5%
TRY THIS………….
Rayan deposited the amount of 6,000/=
in a bank for 5 years and got a profit of 4200/=. Find the interest rate?
A man got a profit of 7400/=
after selling an item. Find the buying price if the percentage profit was 10%.
Solution
%’ge profit = Profit X
100 where B. P. represents
Buying Price.
B.P
10% = 7400 X
100
B.P.
B.P. x 10% = 740,000 x B.P. [multiplying by B.P. both sides]
B.P.
B.P. x 10 = 740,000
B.P. x 101 = 740,000
110 10
B.P. =74,000
Hence Buying Price was 74,000/=
TRY THIS………………
A man got a profit of 90,500/=
after selling an item. Find the buying price if the percentage profit was 20%.
9x
=3x +72; find x
Solution
9x
=3x +72
9x
- 3x = 72 taking 3x on the
left side
9x
- 3x – 72 = 0. Taking 72 on
the left side as well
32x
- 3x – 72 = 0. Rewriting 9x
as 32x.
(3x)2
– (3x) – 72 = 0. Factoring 3x
out.
Let
u = 3x ……… (i)
(u)2
– (u) – 72 = 0.
u2
– 9u + 8u – 72 = 0 [Factorizing by
splitting the middle term].
(u2
– 9u) + (8u – 72) = 0
u(u
– 9) + 8(u – 9) =0
(u
– 9)(u + 8) =0
u – 9 = 0 or
u + 8 =0
u
= 9 or u =-8
We
neglect the –ve answer and we stick with the positive one.
Now from (i) u = 3x
9
= 3x
32
= 3x equal bases they cancel
out.
x = 2
TRY
THIS………….
9x =3x + 111/4;
find x
4
Log327 + Logx =8. Find x
Solution
Log327
+ Logx =8
Log333
+ Logx =8
3Log33
+ Logx =8
(3
x 1) + Logx =8
3
+ Logx =8
Logx
=8 - 3
Logx
=5
Log10x
=5 [Since
any log without a base it is a base 10 logarithm]
x
= 105 [after
changing logarithmic form into exponential form]
x
= 100,000 answer.
TRY
THIS………….
Log5625
+ Logu =9. Find u
There are 37 villagers at the meeting. 12 are farmers and 18 are workers and 8 are both farmers and workers. How many villagers are
i)
Either
farmers or workers.
ii)
Neither
farmers nor workers
Solution
n(U)=37
n(F)=12
n(W)=18
n(FnW)=8
n(FuW)=?
n(FuW)’=?
i) n(FuW) = n(F) + n(W) - n(FnW)
n(FuW)
= 12 + 18 - 8
n(FuW)
= 30 – 8
n(FuW)
= 22
----------------------------------------
n(U)
= n(FuW) + n(FuW)’
37
= 22+ n(FuW)’
37
- 22=n(FuW)’
15=n(FuW)’
Hence those who are neither farmers nor workers are 15
TRY THIS.....................
There are 40 villagers at the meeting. 12 are farmers and 18 are workers and 6 are both farmers and workers. How many villagers are
i) either farmers or workers.
ii) neither farmers nor workers
=========================
Expand
y(y – 4)(y – 10)
Solution
=
y(y – 4) (y – 10)
=
y[y (y – 10) – 4 (y – 10)]
=
y[y2 – 10y – 4y + 40]
=
y3 – 14y2 + 44y
=
y3 – 14y2 + 44y answer
TRY THIS………..
Expand
w(w – 9) (w – 10)
Jemson
bought a radio at 50,000/= and after one year he sold it at a loss of 14%. Find
the price of a radio after a year.
Solution
=
50,000 - (14% of 50,000)
=
50,000 - (0.14 of 50,000) after dividing
14% by 100
=
50,000 - 7,000
=
43000/=
Hence the new price
is 43000/=
TRY
THIS………….
Patson
bought a radio at 70,000/= and after one year he sold it at a loss of 24%. Find
the price of a radio after a year.
If Logax = 0.82,
find Loga(1/x7)
Solution
= Loga(1/x7)
= Logax-7
= -7 x Logax
= -7 x 0.82
= -5.74
TRY THIS………..
If Logac= 60, find
Loga(1/c5)
Expand
(y – 4)(y – 10)
Solution
=
(y – 4) (y – 10)
=
y (y – 10) – 4 (y – 10)
=
y2 – 10y – 4y + 40 [Since -4 x -40 = +40]
=
y2 – 14y + 44 answer
TRY THIS………..
Expand
(w – 9) (w – 10)
If
a number is chosen at random from the integers 1, 2, 3, 4, …………….., 16. What
will be the probability that it is a multiple of three?
Solution
n(s)={1,
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}
n(s)=16
n(E)
= {3,6,9,12,15}
n(E)
=5
p(E)
= n(E)
n(s)
p(E)
= 5
16
probability that it
is a multiple of three is 5/16.
TRY THIS………
If
a number is chosen at random from the integers, 3, 4, …………….., 22. What will be
the probability that it is a multiple of four?
Kitakuli earns 320,000/= per month. Her boss has promised to offer her a salary increase of 13% per month. Calculate the amount to be added to her in a year.
Solution
Increase
per one month
=
13% x 320,000/=
=
13/100 x 320,000/=
=
13 x 3200
=
41,600/=
Increase
for the whole year
=increase per one month x 12
=41,600 x 12
=499200
Hence for the whole
year she will get 499,200/=
TRY THIS………
Kahotipoti
earns 270,000/= per month. Her boss has promised to offer her a salary increase
of 11% per month. Calculate the amount to be added to her in a year.
If
nLog5125= Log21024 ; Find n
Solution
nLog5125=
Log21024
nLog553=
Log2210 [Since 125=53
and 1024=210]
3nLog55=
10Log22
3n
x 1= 10 x 1 [since Logaa=1]
3n=
10
n = 10/3 answer
TRY THIS………
If
uLog5125= Log22567 ; Find u
Factorize the expression Cos4m – Sin4m
Solution
=
Cos4m – Sin4m
=
(Cos2m)2 – (Sin2m)2.
=
(Cos2m – Sin2m) (Cos2m – Sin2m).[Since a2 – b2
=(a-b)(a+b)]
= (Cos2m – Sin2m) (Cos2m –
Sin2m) answer
TRY
THIS………….
Factorize
the expression Cos8c – Sin8c
Evaluate the following giving
your answer in standard form.
1982.7 x 10-9
5 x 10-30
Solution
= 1982.7 x 10-9
5 x 10-30
= 1982.7
x 10-9
5
10-30
= 396.54 x 10-9 –(-30)
= 396.54x 10-9 + 30
= 396.54x 1021 [then we change 396.54 into
standard form as well]
= 3.9654x 102 x 1021 [when we have
exponents with same base, we add the
powers.]
= 3.9654 x 1023 [Since
102 x 1021
= 3.97 x 1023 [correct
to 2 d.p.]
Hence 1982.7
x 10-9 = 3.97 x 1023
5 x 10-20
TRY THIS………………..
Evaluate the following giving
your answer in standard form.
21765 x 10-9
2 x 10-30
Expand
(y – 4)(y – 10)
Solution
=
(y – 4) (y – 10)
=
y (y – 10) – 4 (y – 10)
=
y2 – 10y – 4y + 40 [Since -4 x -40 = +40]
=
y2 – 14y + 44 answer
TRY THIS………..
Expand
(w – 9) (w – 10)
Simplify 500
64-2/3
Solution
= 500
64-2/3
= 500 x 1
64-2/3
= 500 x 642/3
[Since 1/a-n= an]
= 500 x (641/3) 2 [Since
641/3= cube root of 64 = 4]
= 500 x (4)2
= 500 x 16
= 8000
Hence 500
= 8000
64-2/3
TRY THIS…………………………
Simplify 200
27-2/3
The sum of four consecutive numbers which are multiples of 6 is 204. Find the 4th number.
Solution
Let
the numbers be as shown in the table below
|
1st number |
2nd number |
3rd number |
4th number |
TOTAL |
|
n |
n+6 |
n+12 |
n+18 |
204 |
Then,
n + (n+6) + (n+12) + (n+18) = 204
4n
+ 6+12+18= 204
4n
+ 36 = 204
4n
= 204 – 36
4n
= 168
4n = 168
4 4
n
= 42
4th
number = n + 18
= 42 + 18
= 60
Hence the 4th number
is 60
TRY
THIS…………………..
The
sum of four consecutive numbers which are multiples of 7 is 406. Find the 4th
number
The first term of an AP is 13 and the last term is 57.If the
arithmetic progression consists of 40 terms, calculate the sum of all the
terms.
Solution
A1 = 13, n=40, An= 57
Sn = n(A1 + An)
2
S20 = 40(13 + 57)
2
S20 = 20 x 70
S20 = 1400
TRY THIS………….
The first term of an AP is 11 and the last term is 163.If the
arithmetic progression consists of 60 terms, calculate the sum of all the
terms.
