Mathematical Calculations for Science

APPLIED SCIENCE
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Mathematical Calculations for Science

APPLIED SCIENCE
Contents
Logarithms and Indices –.................................................................................................................3
Task-1 Solution............................................................................................................................3
Task-2 Solution............................................................................................................................5
Logarithms and Algebra-.................................................................................................................6
Task-1, Solution...........................................................................................................................6
Task-2, Solution...........................................................................................................................9
Differentiation and integration-.....................................................................................................10
Task-1, Solution.........................................................................................................................10
Task-2, Solution.........................................................................................................................13
Circular measure and trigonometry –............................................................................................15
Task-1 Solution..........................................................................................................................15
Task-2 Solution..........................................................................................................................16
Task-3- Solution........................................................................................................................17
Task 4- Solution.........................................................................................................................18
References......................................................................................................................................20
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APPLIED SCIENCE
Logarithms and Indices –
Task-1 Solution
1a) C2 x C3 (On multiplication power are added for same base)
⇒C2+3 =C5 Ans
1b) b3 x b4 xb5 (On multiplication power are added for same base)
b3+4 +5=b9 Ans
1c) ( a2 )
3 (On power of the power is multiplied)
( a2 x 3 )=a5 Ans
1d) m7
m3 (Power is subtracted on division)
( m7−3 ) =m4 Ans
1e) ( 5 a3 b c4 )
2 (On power of the power is multiplied)
25 a3 x 2 b1 x2 c4 x2 =25 a6 b2 c8 Ans
1f) p−3 q2 (The negative power goes to the denominator)
q2
p3 Ans
1g) ( b2
c4 )
3
(On power of the power is multiplied)
b2 x3
c4 x3 = b6
c12 =b6 . c−12 Ans
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APPLIED SCIENCE
2. considering base 10 for given logarithm
2a) log8 + log6
Log is multiplied when it is in addition form
Log8 +log6 = log48 = 1.68124. Ans
2b) log8 -log6
Log is divided when it is in subtraction form.
Log8 – log6 ¿ log ( 8
6 ) = log1.33 = 0.12483 Ans
2c) log (8)c =
Log is multiplied when it is in power form.
log (8)c = c.log (8) = 0.90308C Ans
3) Solution
3a) 102x = 320.2 (Taking log on both side)
log102x = log320.2
then, 2x = 2.5054, then x = 1.2527 Ans
3b) b.3.1x = 80 where b = 8
Taking log from both side
Logb.3.1x = log80
Logb + xlog3.1 = log 80 (putting the value of b and taking log value of 8 and 80)
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APPLIED SCIENCE
Log8 + xlog3.1 = log80
X x 0.491 = 1.9030 – 0.9030
X = 2.036 Ans
3c) logx3 = 9
As per logarithm rule,
Logx3 = 9
⟹ 3 logx=9
⟹ logx=3
⟹ x=103 or x = 1000 Ans
Task-2 Solution
1) As given in question,
T = 27.3 days, G = 6.67 x 10-11 Nm2kg-2, M = 5.975 x 1024 kg
R =?
As per given formula,
T = √ 4 π2 r3
GM (Giving power on both side by 2
T 2= 4 π2 r3
GM
⇒ T 2 GM
4 π 2 =r3 or, ⇒ r3= ( 27.3 x 24 ) 2 x 6.67 x 10−11 x 50975 x 1024
4 x 3.14 x 3.14
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APPLIED SCIENCE
⇒ r3= ( 27.3∗24∗60∗60 )2 x 6.67 x 10−11 x 50975 x 1024
4 x 3.14 x 3.14
⇒ r3=5.622 x 1025 , ⇒ r= 3
√5.622 x 1025= 383088467.7 m = 3.83 x 108 m Ans
2) As given in question,
ρ=1 x 105, V = 7x10-5 m3, m = 6.6 x 10-27 and N = 2 x 1022 atoms
a) As per given equation,
pV =⅓ Nm c2
3 pV
Nm =c2
Putting the value in above equation
c2= 3 x 105 x 7 x 10−5
2 x 1022 x 6.6 x 10−27
⇒ c2= 21
0.00132 =159090.909 m2 s−2
Mean square speed = 159090.909 m2 s−2
b) As given in question,
r . m. s . speed = √c2 Putting the value from previous solution
r . m. s . speed = √159090.909=398.862 m/s Ans
Logarithms and Algebra-
Task-1, Solution
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