Physics ivs B VOCATIONAL PAPER OF SEMESTER 2

1. Unit-1
Topics: Error, zero , positive and negative error, propagation of errors(sum, difference, multiplication, division),
standard deviation
Plotting of graphs, scale of axis, dependent variable, independent variable, linear graph, slope of graph and
derivative with examples, area under the graph and integration with examples
Logarithm, types of logarithm, logarithm rules and properties, logarithms formulas, logarithmic examples and
applications, Antilogarithm
Area under the curve

2. Errors are differences between observed values and what is true in nature. Error causes results that are
inaccurate or misleading and can misrepresent nature.
ZERO ERROR-It is a type of error in which an instrument gives a reading when the true reading at that time is zero.
For example needle of ammeter failing to return to zero when no current flows through it.
If the experimental value is less than the accepted value, the error is negative. If the experimental value is larger
than the accepted value, the error is positive.
ERROR

3. The propagation of error has different formulas used for the following mathematical operations: addition, subtraction,
multiplication, division, and powers.
Propagation of Errors in Addition: Suppose a result x is obtained by addition of two quantities say
a and b
i.e. x = a + b
Let Δ a and Δ b are absolute errors in the measurement of a
and b and Δ x be the corresponding absolute error in x.
∴ x ± Δ x = ( a ± Δ a) + ( b ± Δ b)
∴ x ± Δ x = ( a + b ) ± ( Δ a + Δ b)
∴ x ± Δ x = x ± ( Δ a + Δ b)
∴ ± Δ x = ± ( Δ a + Δ b)
∴ Δ x = Δ a + Δ b
Thus maximum absolute error in x = maximum absolute error
in a + maximum absolute error in b
Thus, when a result involves the sum of two observed
quantities, the absolute error in the result is equal to the sum
of the absolute error in the observed quantities.

4. Propagation of Errors in Subtraction:
Suppose a result x is obtained by subtraction of two quantities
say a and b
i.e. x = a – b
Let Δ a and Δ b are absolute errors in the measurement of a
and b and Δ x be the corresponding absolute error in x.
∴ x ± Δ x = ( a ± Δ a) – ( b ± Δ b)
∴ x ± Δ x = ( a – b ) ± Δ a – + Δ b
∴ x ± Δ x = x ± ( Δ a + Δ b)
∴ ± Δ x = ± ( Δ a + Δ b)
∴ Δ x = Δ a + Δ b
Thus the maximum absolute error in x = maximum absolute
error in a + maximum absolute error in b.
Thus, when a result involves the difference of two observed
quantities, the absolute error in the result is equal to the sum
of the absolute error in the observed quantities.

5. Propagation of Errors in Product:
Suppose a result x is obtained by the product of two quantities
say a and b
i.e. x = a × b ……….. (1)
Let Δ a and Δ b are absolute errors in the measurement of a
and b and Δ x be the corresponding absolute error in x.
∴ x ± Δ x = ( a ± Δ a) x ( b ± Δ b)
∴ x ± Δ x = ab ± a Δ b ± b Δ a ± Δ aΔ b
∴ x ± Δ x = x ± a Δ b ± b Δ a ± Δ aΔ b
∴ ± Δ x = ± a Δ b ± b Δ a ± Δ aΔ b …… (2)
Dividing equation 2 by 1
The quantities Δa/a, Δb/b and Δx/x are called relative errors in the values of a, b and x respectively. The
product of relative errors in a and b i.e. Δa × Δb is very small hence is neglected.

6. Hence maximum relative error in x = maximum relative error in
a + maximum relative error in b
Thus maximum % error in x = maximum % error in a +
maximum % error in b
Thus, when a result involves the product of two observed
quantities, the relative error in the result is equal to the sum of
the relative error in the observed quantities.

7. Propagation of Errors in Quotient:
Suppose a result x is obtained by the quotient of two quantities say a
and b.
i.e. x = a / b ……….. (1)
Let Δ a and Δ b are absolute errors in the measurement of a and b
and Δ x be the corresponding absolute error in x.
The values of higher power of Δ b/b are very
small and hence can be neglected.

8. The quantities Δa/a, Δb/b and Δx/x are called relative errors in
the values of a, b and x respectively.
Hence maximum relative error in x = maximum relative error
in a + maximum relative error in b
Example – 01:
The lengths of the two rods are recorded as 25.2 ± 0.1 cm and 16.8 ± 0.1 cm. Find the sum of the lengths of the
two rods with the limit of errors.
Solution:
We know that in addition the errors get added up
The Sum of Lengths = (25.2 ± 0.1) + (16.8 ± 0.1) = (25.2 + 16.8) ± (0.1 + 0.1) = 42.0 ± 0.2 cm
Example – 02:
The initial temperature of liquid is recorded as 25.4 ± 0.1 °C and on heating its final temperature is recorded as 52.7 ± 0.1
°C. Find the increase in temperature.
Solution:
We know that in subtraction the errors get added up
The increase in temperature = (52.7 ± 0.1) – (25.4 ± 0.1) = (52.7 – 25.4) ± (0.1 + 0.1) = 27.3 ± 0.2 °C.

9. STANDARD DEVIATION
Standard deviation is a statistic that measure the
dispersion of dateset relative to its mean and is
calculated as square root of the variance
𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 =
σ𝑖=1
𝑛
𝑥𝑖 − ҧ
𝑥 2
𝑛 − 1
𝑥𝑖– value of the ith point in the data set
ഥ
𝑥 – the mean value of the data set
n – the number of data points in the data set

10. Example 1: There are 39 plants in the garden. A few plants were selected randomly and their heights in cm were
recorded as follows: 51, 38, 79, 46, 57. Calculate the standard deviation of their heights.
Solution:
n = 5
Sample mean ҧ
𝑥 = (51+38+79+46+57)/5 = 54.2
Since, sample data is given, we use the sample SD formula.
𝑆. 𝐷 =
σ𝑖=1
𝑛
𝑥𝑖 − ҧ
𝑥 2
𝑛 − 1
𝑆. 𝐷
=
51 − 54.2 2 + 38 − 54.2 2 + 79 − 54.2 2 + 46 − 54.2 2 + 57 − 54.2 2
4
= 15.5
Answer: Standard deviation for this data is 15.5

11. Example 2: In a class of 50, 4 students were selected at random and their total marks in the final
assessments are recorded, which are: 812, 836, 982, and 769. Find the variance and standard deviation of
their marks.
Solution:
n = 4
Sample Mean ( ҧ
𝑥) = (812+836+982+769)/4 = 849.75
𝑆. 𝐷 =
812−849.75 2+ 836−849.75 2+ 982−849.75 2+ 769−849.75 2
3
𝑆. 𝐷 = 8541.58 =92.4

12. A graph is a mathematical diagram which shows the relationship between
two or more sets of numbers or measurements.
The scale of an axis is the units into which the axis is divided. The units are
marked by ticks, labels, and grid lines. When you change an axis' scale, you
change how the ticks, labels, and grid lines will display.
Axes are the lines that are used to measure data on graphs and grids.
There are two types of axes: the vertical axis, and the horizontal axis.
These are also known as the x and y-axis.
GRAPH

13. Plotting a graph from its equation
We can find the coordinates of several points on a line by picking x values
and working out y values.
Example Question
A line has equation y = 2x + 1.
Using x values from –2 to +3, plot the graph of this equation.
The first stage is to draw up a table of x values and work
out the y values using the equation:
x –2 –1 0 1 2 3
y = 2x + 1 –3 –1 1 3 5 7
Next, each pair of x and y values can be plotted on the graph
as coordinates. In this case the coordinates are: ( –2 , –3 ) ,
( –1 , –1 ) , ( 0 , 1 ) , ( 1 , 3 ) , ( 2, 5 ) and ( 3 , 7 ).
Finally the points are joined with a straight line running all the
way across the graph:

14. A variable is an alphabet or term that represents an unknown number or unknown value or unknown quantity.
The variables are specially used in the case of algebraic expression or algebra. For example, x+9=4 is a linear
equation where x is a variable, where 9 and 4 are constants.
The variable is a quantity that can be changed or which is not fixed according to the mathematical operation
performed. Usually, in algebra, we express an unknown number using the term ‘x’ and ‘y’. But this is not particular,
and we can use any alphabet.
Examples of Variables
•x+2=8
•y+3=12
•5x-2=10
•4x/3=7
In the above examples, x and y are called variables.
Types of variables in Math
The variables can be classified into two categories, namely
•Dependent Variable
•Independent Variable

15. Dependent Variables
The dependent variable is characterized as the variable whose quality depends on the estimation of another
variable in its condition. That is, the estimation of the word variable is dependably said to be reliant on the free
variable of math condition.
For instance, consider the condition y = 4x + 3. In this condition, the estimation of the variable ‘y’ changes as per
the adjustments in the estimation of ‘x’. In this manner, the variable ‘y’ is said to be a reliant variable.
Independent Variables
In an algebraic equation, an independent variable describes a variable whose values are independent of changes. If
x and y are two variables in an algebraic equation and every value of x is linked with any other value of y, then ‘y’
value is said to be a function of x value known as an independent variable, and ‘y’ value is known as a dependent
variable.
Example: In the expression y = x2, x is an independent variable and y is a dependent variable.
A dependent variable is a variable in an expression that depends on the value of another variable. For example, if
y = 2x, then y depends on x. So, if x = 1, then y = 2.
An independent variable is a variable that does not depend on any other variable for its value. For example, in an
expression, 2y = 9x + 1, x is an independent variable. So, for each value of x, there will be a different value of y.

16. Linear and non linear functions:
Linear function Non linear function
A linear function is plotted as a straight line with no
curves
Non linear equation do not form a straight line, instead
they always have a curve
The degree of equation representing a linear function will
always equal to 1.
The degree of the equation for a nonlinear function will
always be greater than 1.
A linear equation will always form a straight line in the
XY- cartesian plane and the line can extend to any
direction depending upon the limits or constraints of the
equation.
Nonlinear functions will always form a curved graph. The
curve of the graph will depend upon the degree of the
function. The higher the degree, the higher the curvature
Linear functions or equations are written as 𝐲 = 𝐦𝐱 + 𝐛
Here, m is the slope, while b is the constant
An example of a nonlinear equations is 𝐚𝐱𝟐
+ 𝐛𝐱 = 𝐜
The degree of the equation is 2, so it is quadratic
equation . IF we increase the degree to 3, it will be cubic
equation
𝟑𝐱 + 𝐲 = 𝟒,
𝟒𝐱 + 𝟓 = 𝐲
𝟐𝐱𝟐
+ 𝟑𝐱 = 𝟖
𝟑𝐱𝟑
+ 𝟐𝐱𝟐
+ 𝟑𝐱 = 𝟒

19. Slope of the graph and derivative
The steepness of a hill is called a slope. The same goes for the steepness of a line. The slope is defined as the
ratio of the vertical change between two points, the rise, to the horizontal change between the same two points,
the run.
•Slope = (Change in y coordinate)/(Change in x coordinate)
•Slope = rise/run.
Slope = m = tan θ = (y2 – y1)/(x2 – x1)
The slope of a line is usually represented by the letter m. (x1, y1) represents the first point whereas (x2, y2)
represents the second point.

20. 𝑥1, 𝑦1 = −3, −2 𝑎𝑛𝑑 𝑥2, 𝑦2 = (2,2)
𝑚 =
𝑦2 − 𝑦1
𝑥2 − 𝑥1
=
2 − (−2)
2 − (−3)
=
4
5
𝑥1, 𝑦1 = −2,3 𝑎𝑛𝑑 𝑥2, 𝑦2 = (2, −1)
𝑚 =
𝑦2 − 𝑦1
𝑥2 − 𝑥1
=
−1 − 3
2 − (−2)
=
−4
4
= −1
A line with a positive slope (m > 0), as the line
above, rises from left to right whereas a line with a
negative slope (m < 0) falls from left to right.
𝑦 = 𝑚𝑥 + 𝑏

21. A derivative of a function is a representation of the rate of change of one variable in relation to another at a
given point on a function.
The slope describes the steepness of a line as a relationship between the change in y-values for a change in the
x-values.
Clearly, very similar ideas. But let’s look at the important differences. A function’s derivative is a function in and
of itself. It may be a constant (this will happen if our function is linear) but it may very well change between
values of x.
Let f(x) = x2. Our derivative f’(x) = 2x. If we take a look at the graph of x2, we can see that for each step we take
along the curve, the value of y changes more and more. Between x = 0 and x = 1, y only increases by 1. But
between x = 1 and x = 2, y increases by 3. If we keep going with this trend, between x = 2 and x = 3, y changes by
5. We don’t have a constant change between equally spaced values of x, but rather y changes by twice as much
each step.
A function does not have a general slope, but rather the slope of a tangent
line at any point. In our above example, since the derivative (2x) is not
constant, this tangent line increases the slope as we walk along the x-
axis. We cannot have a slope of y = x2 at x = 2, but what we can have is the
slope of the line tangent to this point, which has a slope of 4.

22. In Mathematics, logarithms are the other way of writing the exponents. A logarithm of a number with a base is
equal to another number. A logarithm is just the opposite function of exponentiation. For example, if 102 = 100
then log10 100 = 2.
Hence, we can conclude that,
Logb x = n or bn = x
Where b is the base of the logarithmic function.
This can be read as “Logarithm of x to the base b is equal to n”.
In this article, we are going to learn the definition of logarithms,
A logarithm is defined as the power to which a number must be raised to get some other values. It is the most
convenient way to express large numbers.
by= a ⇔logba=y
Where,
•“a” and “b” are two positive real numbers
•y is a real number
•“a” is called argument, which is inside the log
•“b” is called the base, which is at the bottom of the log.
In other words, the logarithm gives the answer to the question “How many times a number is multiplied to get the
other number?”.
LOGARITHM

23. xponents Logarithms
62 = 36 Log6 36 = 2
102 = 100 Log10 100 = 2
33 = 27 Log3 27 = 3
Logarithm Types
In most cases, we always deal with two different types of logarithms, namely
•Common Logarithm
•Natural Logarithm
Common Logarithm
The common logarithm is also called the base 10 logarithms. It is represented as log10 or simply log. For example, the
common logarithm of 1000 is written as a log (1000). The common logarithm defines how many times we have to
multiply the number 10, to get the required output.
For example, log (100) = 2
If we multiply the number 10 twice, we get the result 100.
Natural Logarithm
The natural logarithm is called the base e logarithm. The natural logarithm is represented as ln or loge. Here, “e”
represents the Euler’s constant which is approximately equal to 2.71828. For example, the natural logarithm of 78 is
written as ln 78. The natural logarithm defines how many we have to multiply “e” to get the required output.
For example, ln (78) = 4.357.
Thus, the base e logarithm of 78 is equal to 4.357.

25. Logarithmic Formulas
logb(mn) = logb(m) + logb(n)
logb(m/n) = logb (m) – logb (n)
Logb (xy) = y logb(x)
Logbm√n = logb n/m
m logb(x) + n logb(y) = logb(xmyn)
logb(m+n) = logb m + logb(1+nm)
logb(m – n) = logb m + logb (1-n/m)
Question: How about log3 81?
Answer: 4 (because 34 )
Question: Get the value of Log 1000?
Answer: 3
Explanation: When there is no base
value associated with Log, we assume
it’s log10
so log10 1000 = 103
Question: Solve this: log2 4*16 using Log Law.
The same question can also be written as log2 4 + log2 16
Answer:
log2 4*16
=> log2 4 + log2 16
=> log2 2 2 + log2 2 4
=> 2log2 2 + 4log2 2
=> 2 * 1 + 4 * 1
=> 2 + 4
=> 6
so log2 4*16 = 6
Question: Solve this log3(327)
This question can also be written as 27log33
Answer:
log3(327)
=> 27log33
=> 27 * 1
=> 27
Answer is 27.

26. Question: Solve log4 1024 – log4 16
Answer:
log4 1024 – log4 16
=> log4 (1024 / 16)
=> log4 64
=> log4 43
=> 3log4 4
=>3 * 1
=> 3
S0, log4 1024 – log4 16 = 3
Question: 1 / log2 128
Answer:
1 / log2 128
=> 1 / log2 27
=> 1 / 7 * 1
=> 1 / 7
=> 0.1429
Question: Solve this: log3 9 + log3 81 – log5 1250 + log5 2
Answer:
log3 9 + log3 81 + log5 1250 – log5 2
=> log3 (9 * 81) + log5 (1250 / 2)
=> log3 729 + log5 625
=> log3 36 + log5 54
=> 6log3 3+ 4log5 5
=> 6*1 + 4*1
=> 6+4
=> 10
So the value of log3 9 + log3 81 + log5 1250 – log5 2 = 10

27. Antilog Definition: The Antilog, which is also known as “Anti- Logarithms” of a number is the
inverse technique of finding the logarithm of the same number. Consider, if x is the
logarithm of a number y with base b, then we can say y is the antilog of x to the base b. It is
defined by
If logb y = x Then, y = antilog x
Both logarithm and antilog have their base as 2.7183. If the logarithm and antilogarithm
have their base 10, that should be converted into natural logarithm and antilog by
multiplying it by 2.303.

28. Sample Example
Question:
Find the antilog of 3.3010
Solution:
Given, antilog (3.3010)
Step 1: Characteristics part = 3 and mantissa part = 3010
Step 2: Use the antilog table for the row.30, then the column for 1, you get 2000.
Step 3: Find the value from the mean difference column for the row .30 and column 0, it gives the value 0
Step 4: Add the values obtained in step 2 and 3 , 2000 + 0 = 2000.
Step 5: Now insert the decimal place. We know that the characteristic part is 3 and we have to add it with 1.
Therefore, we get the value 4. Insert the decimal point after 4 places, and we get 2000.
Therefore, the solution of the antilog 3.3010 is 2000.
To know about antilog tables, register with BYJU’S to get a clear knowledge on solving problems in an easy and
engaging way.

29. integration formulas

30. Area under the curve

31. as Caleulali the aa ndy h
42
94+20 54
2
18 89.ani

32. Fod eue endue
A
I 2
4-X2xtR
A=
unib
classmate
Date
page
8) ds
2

33. os Find the aea under yrx
fm N0 to =4
A=
A=.X
3
A=
A=
4
A=ydx
4
Y toY=
0

34. A
AVaos ds
A
=4
pataboe
3
yass
d
2

35. The
the
cirele
Asea undes the cauecirel.
Cealon
change to
2
cassnate
Date
Page
A4ydx
a sin
Q
Rer
lun
19
Kut
B