Category: Class 12

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Business Maths Guide Pdf Chapter 5 Numerical Methods Miscellaneous Problems Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Miscellaneous Problems

Question 1.
If f (x) = e^ax then show that f(0), Δf(0), Δ²f(0) are in G.P
Solution:
Given f(x) = e^ax
f(0) = e° = 1 ……… (1)
Δf(x) = e^a(x+h) – e^ax
= e ^ax+ah – e^ax
= e^ax. e^ah – e^ax
= e^ax (e^ah – 1)
Δf(0) = e° (e^ah – 1)
= (e^ah – 1) …….. (2)
Δ²f(x)= Δ [Δf(x)]
= Δ [e^a(x+h) – e^ax]
[e^a(x+h+h) – e^a(x+h)] – [e^a(x+h) – e^ax]
= e^a(x+2h) – e^a(x+h) – e^a(x+h) + e^ax
Δ²f(0) = Δ [Δf(x)]
= e^a(2h) – e^a(h) – e^a(h) + e⁰
= e^2ah – e^ah – e^ah + 1
= (e^ah)² – 2e^ah + 1
= [e^ah – 1]² ………… (3)
from (1), (2) & (3)
[t₂]² =[Δf(0)]² = (e^ah – 1)²
t₁ × t₃ = f(0) × Δ²f(0)
= (1)(e^ah – 1)² = (e^ah – 1)²
⇒ [Δf(0)]² = f(0) × Δ²f(0)
∴ f(0), Δf(0), Δ²f(0) an is G.P.

Question 2.
Prove that
(i) (1 + Δ) (1 – ∇) = 1
(ii) Δ∇ = Δ – ∇
(iii) EV = Δ = ∇E
Solution:
(i) LHS = (1 + Δ) (1 – ∇)
= (E) (E^-1) = E^1-1
= E° = 1
= RHS
Hence proved.

(ii) LHS = Δ∇
= (E – 1)(1 – E^-1)
= E – EE^-1 + E^-1
= E – 1 – 1 – E^-1
= E – 2 – E^-1 ………… (1)
RHS = Δ – ∇
= (E – 1) -(1 – E^-1)
= E – 1 – 1 + E^-1
= E – 2 + E^-1 ………. (2)
from (1) & (2) LHS = RHS
Hence proved.

(iii) E∇ = EE^-1Δ [∵ ∇ = E^-1Δ]
= Δ ……… (1)
∇E = E^-1 ΔE
= E^-1 EΔ
= Δ ………. (2)
from (1) (2)
E∇ = Δ = ∇E

Question 3.
A second degree polynomial passes though the point (1, -1) (2, -1) (3, 1) (4, 5). Find the polynomial.
Solution:
Points are (1, -1), (2, -1), (3, 1) and (4, 5)

we will use Newton’s backward interpolation formula to find the polynomial.

= 5 + (x – 4) (4) + (x – 4) (x – 3) + 0
= 5 + 4x – 16 + x² – 7x + 12
y(x) = x² – 3x + 1

Question 4.
Find the missing figures in the following table

Solution:
Here y₀ = 7; y₁ = 11; y₂ = ?; y₃ = 18; y₄ = ?; y₅ = 32
Since only four values of f(x) are given, the polynomial which fits the data is of degree three. Hence fourth differences are zeros.
Δ⁴y_k = 0
(ie) (E – 1)⁴ y_k = 0
(i.e) (E⁴ – 4E³ + 6E² – 4E + 1)y_k = 0 ……….. (1)
Put k = 0 in (1)
(E⁴ – 4E³ + 6E² – 4E + 1)y₀ = 0
E⁴ y₀ – 4E³ y₀ + 6E² y₀ – 4E y₀ + y₀ = 0
y₄ – 4y₃ + 6y₂ – 4y₁ + y₀ = 0
y₄ – 4(18) + 6y₂ – 4(11) + 7 = 0
y₄ – 72 + 6y₂ – 44 + 7 = 0
y₄ + 6y₂ = 109
(2)
Put k = 1 in (1)
(E⁴ – 4E³ + 6E² – 4E + 1)y₁ = 0
[E⁴ y₁ – 4E y₁ + 6E² y₁ – 4Ey₁ + y] = 0
y₅ – 4y₄ + 6y₃ – 4y₂ + y₁ = 0
32 – 4 (y₄) + 6(18) — 4(y₂) + 11 = 0
32 – 4y₄ + 108 – 4y₂ + 11 = 0
-4y₄ – 4y₂ + 151 = 0
4y₄ + 4y₂ = 151 ,……. (3)
Solving equation (1) & (2)

Substitute y₂ = 14.25 in eqn (1)
y₄ + 6(14.25) = 109
y₄ + 25.50 = 109
y₄ = 109 – 85.5
∴ y₄ = 23.5
∴ Required two missing values are 14.25 and 23.5.

Question 5.
Find f (0.5) if f(-1) = 202, f(0) = 175, f(1) = 82 and f(2) = 55
Solution:
From the given data

Here we have to apply Newton’s forward interpolation formula, since the value of f(x) is required near the beginning of the table.
y_{(x= x0+nh)} =f(x₀) + \(\frac { n }{1!}\) Δf(x₀) + \(\frac { n(n-1) }{2!}\) Δ²f(x₀) + \(\frac { n(n-1)(n-2) }{3!}\) Δ³f(x₀) + ………
Given:
x = 0.5 and h = 1
x₀ + nh = x
-1 + n(1) = 0.5
n = 1 + 0.5
∴ n = 1.5

= 202 – 40.5 – 24.75 – 8.25
= 202 – 73.5
f(0.5) = 128.5

Question 6.
From the following data find y at x = 43 and x = 84

Solution:
To find y at x = 43
Since the value of y is required near the beginning of the table, we use the Newton’s forward interpolation formula.

= 184 + (0.3) (20) + (0.3) (-0.7)
= 184 + 6.0 – 0.21
= 190 + 0.21
y_(x=43) = 189.79
To find y at x = 84
Since the value of y is required at the end of the table, we apply backward interpolation formula.

x_n + nh = x
90 + n(10) = 84
10n = 84 – 90
10n = -6
∴ n = -0.6
y_(x=84) = 304 + \(\frac { (0.6) }{1!}\) (28) + \(\frac {(0.6)(-0.6 + 1) }{2!}\)(2) +
= 304 + (0.6) (28) + \(\frac { (-0.6)(0.4) }{2}\) + 2
= 304 – 16.8 – 0.24
= 304 – 17.04
= 286.96

Question 7.
The area A of circle of diameter ‘d’ is given for the following values

Find the approximate values for the areas of circles of diameter 82 and 91 respectively.
Solution:
To find A at D = 82
Since the value of A is required near the beginning of the table. We use the Newton’s forward interpolation formula.

= 5026 + 259.2 – 4.8 – 0.128 – 0.1664
= 5285.2 – 5.0944
= 5280.1056
A = 5280.11
To find Δ at D = 91
Since the value of A is required near the beginning of the table. We use the Newton’s forward interpolation formula.

= 7854 – 1378.8 + 28.8 + 0.096 + 0.0576
= 7882.9536 – 1378.8
= 6504.1536
= 6504.15

Question 8.
If u₀ = 560, u₁ = 556, u₂ = 520, u₄ = 385, show that u₃ = 465
Solution:
U₀ = 560; U₁ = 556; U₂ = 520; U₄ = 385
Since only four values of U are given, the polynomial which fits the data is of degree three. Hence fourth differences are zeros.
Δ⁴U₀
(E – 1)⁴ U₀ = 0
⇒ (E⁴ – 4E³ + 6E² – 4E + 1) U₀ = 0
⇒ E⁴U₀ – 4E³U₀ + 6E²U₀ – 4EU₀ + U₀ = 0
U₄ – 4U₃ + 6U₂ – 4U₁ + U₀ = 0
385 – 4(U₃) + 6 (520) – 4 (556) + 560 = 0
385 – 4(U₃) + 3120 – 2224 + 560 = 0
1841 – 4U₃ = 0
4U₃ = 1841 ⇒ U₃ = \(\frac { 1841 }{4}\)
U₃ = 460.25

Question 9.
From the following table obtain a polynomial of degree y in x

Solution:
We will use Newton’s backward interpolation formula to find the polynomial.

To find y in terms of x
x_n + nh = x
5 + n(1) = x
∴ n = x – 5

= 1 + 2x – 10 + 2 (x² – 9x + 20) + \(\frac { 4 }{3}\) (x – 5) (x² – 7x + 12) + \(\frac { 2 }{3}\)(x² – 9x + 20)(x² – 5x + 6)
= 1 + 2x – 10 + 2x² – 18x + 40 + \(\frac { 4 }{3}\)
[x³ – 7x² + 12x – 5x² + 35x – 60] + \(\frac { 2 }{3}\) [x⁴ – 5x³ + 6x² – 9x³ + 45x² – 54x + 20x² – 100x + 120]

Question 10.
Using Lagrange’s interpolation formula find a polynominal which passes through the points (0, -12), (1, 0), (3, 6) and (4, 12).
Solution:
We can construct a table using the given points.

Here x₀ = 0; x₁ = 1; x₂ = 3; x₃ = 4,
y₀ = -12; y₁ = 0; y₂ = 6; y₃ = 12

= (x³ – 7x² + 12x – x² + 7x – 12) – (x³ – 5x² + 4x) + (x³ – 4x² + 3x)
= (x³ – 8x² + 19x – 12) – (x³ – 5x² + 4x) + (x³ – 4x² + 3x)
= x³ – 8x² + 19x – 12 – x³ + 5x² – 4x + x³ – 4x² + 3x
∴ y = x³ – 7x² + 18x – 12

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Business Maths Guide Pdf Chapter 4 Differential Equations Ex 4.5 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 4 Differential Equations Ex 4.5

Question 1.
\(\frac { d^2y }{dx^2}\) – 6\(\frac { dy }{dx}\) + 8y = 0
Solution:
Given (D² – 6D + 8) y = 0, D = \(\frac{d}{d x}\)
The auxiliary equations is
m² – 6m + 8 = 0
(m – 4)(m – 2) = 0
m = 4, 2
Roots are real and different
The complementary function (C.F) is (Ae^4x + Be^2x)
The general solution is y = Ae^4x + Be^2x

Question 2.
\(\frac { d^2y }{dx^2}\) – 4\(\frac { dy }{dx}\) + 4y = 0
Solution:
The auxiliary equations A.E is m² – 4m + 4 = 0
(m – 2)² = 0
m = 2, 2
Roots are real and equal
The complementary function (C.F) is (Ax + B) e^2x
The general solution is y = (Ax + B) e^2x

Question 3.
(D² + 2D + 3) y = 0
Solution:
The auxiliary equation is m² + 2m + 3 = 0
Here a = 1, b = 2, c = 3

The complementary function is
e^ax (Acosßx + Bsinßx)
∴ C.F = e^-x [Acos√2x + Bsin √2x]
∴ The general solution is
y = e^-x (Acos√2x + Bsin√2x)

Question 4.
\(\frac { d^2y }{dx^2}\) – 2k\(\frac { dy }{dx}\) + k²y = 0
Solution:
Given (D² – 2kD + k²)y = 0, D = \(\frac{d}{d x}\)
The auxiliary equations is m² – 2km + k = 0
⇒ (m – k)² = 0
⇒ m = k, k
Roots are real and equal
The complementary function (C.F) is (Ax + B) e^kx
The general solution is y = (Ax + B) e^kx

Question 5.
(D² – 2D – 15) y = 0
Solution:
The auxiliary equation is
m² – 2m + 15 = 0
m² + 3m – 5m – 15 = 0
m (m + 3) – 5 (m + 3) = 0
(m + 3) (m – 5) = 0
m = -3, 5
Roots are real and different
∴ The complementary function is
Ae^m₁x + Be^m₂x
C.F = Ae^-3x + Be^5x
∴ The general solution is
y = (Ae^-3x + Be^5x) ………… (1)
\(\frac { dy }{dx}\) = Ae^-3x (-3) + Be^5x (5)
\(\frac { dy }{dx}\) = -3Ae^-3x + 5Be^5x ………… (2)
\(\frac { d^2y }{dx^2}\) = 9Ae^-3x + 25Be^5x ……….. (3)
when x = 0; \(\frac { dy }{dx}\) = 0
-3 Ae° + 5Be° = 0
-3A + 5B = 0 ………. (4)
when x = 0; \(\frac { d^2y }{dx^2}\) = 2
Eqn (3) ⇒ 9Ae° + 25Be° = 2
9A + 25B = 2 ……… (5)
Solving equation (4) & (5)

Question 6.
(4D² + 4D – 3) y = e^2x
Solution:
The auxiliary equation is
4m² + 4m – 3 = 0
4m² + 6m – 2m – 3 = 0
2m (2m + 3) – 1 (2m + 3) = 0
(2m + 3) (2m – 1) = 0
2m = -3; 2m = 1
m = -3/2, 1/2
Roots are real and different
The complementary function is
Ae^m₁x + Be^m₂x

Question 7.
\(\frac { d^2y }{dx^2}\) + 16y = 0
Solution:
Given (D² + 16) y =0
The auxiliary equation is m² + 16 = 0
⇒ m² = -16
⇒ m = ± 4i
It is of the form α ± iβ, α = 0, β = 4
The complementary function (C.F) is e^0x [A cos 4x + B sin 4x]
The general solution is y = [A cos 4x + B sin 4x]

Question 8.
(D² – 3D + 2) y = e^3x which shall vanish for x = 0 and for x = log 2
Solution:
(D² – 3D + 2) y = e^3x
The auxiliary equation is
m² – 3m + 2 =0
(m – 1) (m – 2) = 0
m = 1, 2
Roots are real and different
The complementary function is
C.F = Ae^m₁x + Be^m₂x
C.F = A^x + Be^2x

when x = log 2; y = 0
Ae^{log 2} + Be^{2log 2} + \(\frac { e^{3xlog2} }{2}\) = 0
Ae^{log 2} + Be^{log (2)²} + \(\frac { e^{log2³} }{2}\) = 0
2A + 4B + \(\frac { 8 }{2}\) = 0
2A + 4B + 4 = 0
2A + 4B = -4 ……… (3)
Solving equation (2) & (3)
Eqn (2) × 2 ⇒ 2A + 2B = -1

Question 9.
(D² + D – 2) y = e^3x + e^-3x
Solution:
The auxiliary equation is
m² + m – 6 = 0
(m + 3) (m – 2) = 0
Roots are real and different
The complementary function is
C.F = Ae^m₁x + Be^m₂x
C.F = Ae^-3x + Be^2x

Question 10.
(D² – 10D + 25) y = 4e^5x + 5
Solution:
The auxiliary equation is
m² – 10m + 25 = 0
(m – 5) (m – 5) = 0
m = 5, 5
Roots are real and equal
C.F = (Ax + B) e^mx
C.F = (Ax + B) e^5x
P.I₍₁₎ = x. \(\frac { 4 }{2D-10}\) e^5x
Replace D by 5, 2D – 10 = 0 when D = 5

Question 11.
(4D² + 16D +15) y = 4e^{\(\frac { -3 }{2}\)}x
Solution:
The auxiliary equation is 4m² + 16m + 15 = 0
4m² + 16m + 10m + 15 = 0
2m (2m + 3) + 5 (2m + 3) = 0
(2m + 3) (2m + 5) = 0
2m = -3, -5
∴ m = -3/2, -5/2
Roots are real and different
C.F = (Ax + B) e^m₁x + Be^m₂x
C.F = Ae^{-3/2 x} + Be^{-5/2 x}

Question 12.
(3D² + D – 14) y – 13 e^2x
Solution:
The auxiliary equation is 3m² + m – 14 = 0
3m² – 6m + 7m – 14 = 0
3m (m – 2) + 7 (m – 2) = 0
(m – 2) (3m + 7) = 0
m = 2; 3m = -7
m = 2, -7/3
Roots are real and different
C.F = (Ax + B) e^m₁x + Be^m₂x
C.F = Ae^2x + Be^{-7/3 x}

Question 13.
Suppose that the quantity demanded Qd = 13 – 6p + 2\(\frac { dp }{dt}\) + \(\frac { d^2p }{dt^2}\) = and quantity supplied Qd = -3 + 2p where is the price. Find the equilibrium price for market clearence.
Solution:
For market clearance, the required condition is Qd = Qs

The auxiliary equation is
m² + 2m – 8 = 0
(m + 4) (m – 2) = 0
m = -4, 2
Roots are real and different

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Computer Science Guide Pdf Chapter 16 Data Visualization Using Pyplot: Line Chart, Pie Chart and Bar Chart Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Computer Science Solutions Chapter 16 Data Visualization Using Pyplot: Line Chart, Pie Chart and Bar Chart

12th Computer Science Guide Data Visualization Using Pyplot: Line Chart, Pie Chart and Bar Chart Text Book Questions and Answers

I. Choose the best answer (1 Mark)

Question 1.
Which is a python package used for 2D graphics?
a) matplotlib.pyplot
b) matplotlib.pip
c) matplotlib.numpy
d) matplotlib.plt
Answer:
a) matplotlib.pyplot

Question 2.
Identify the package manager for Python packages, or modules.
a) Matplotlib
b) PIP
c) plt.show()
d) python package
Answer:
b) PIP

Question 3.
Read the following code: Identify the purpose of this code and choose the right option from the following.
C:\Users\YourName\AppData\Local\Programs\Python\Python36-32\Scripts > pip – version
a) Check if PIP is Installed
b) Install PIP
c) Download a Package
d) Check PIP version
Answer:
d) Check PIP version

Question 4.
Read the following code: Identify the purpose of this code and choose the right option from the following.
C:\Users\Your Name\AppData\Local\Programs\Python\Python36-32\Scripts > pip list
a) List installed packages
b) list command
c) Install PIP
d) packages installed
Answer:
a) List installed packages

Question 5.
To install matplotlib, the following function will be typed in your command prompt. What does “-U”represents?
Python -m pip install -U pip
a) downloading pip to the latest version
b) upgrading pip to the latest version
c) removing pip
d) upgrading matplotlib to the latest version
Answer:
b) upgrading pip to the latest version

Question 6.
Observe the output figure. Identify the coding for obtaining this output.

a) import matplotlib.pyplot as pit
plt.plot([1,2,3],[4,5,1])
plt.show()

b) import matplotlib.pyplot as pit plt.
plot([1,2],[4,5])
plt.show()

c) import matplotlib.pyplot as pit
plt.plot([2,3],[5,1])
plt.showQ

d) import matplotlib.pyplot as pit pit .plot ([1,3], [4,1 ])
Answer:
a) import matplotlib.pyplot as pit
plt.plot([1,2,3],[4,5,1])
plt.show()

Question 7.
Read the code:
a. import matplotlib.pyplot as pit
b. plt.plot(3,2)
c. plt.show()
Identify the output for the above coding

Answer:
c

Question 8.
Which key is used to run the module?
a) F6
b) F4
c) F3
d) F5
Answer:
d) F5

Question 9.
Identify the right type of chart using the following hints.
Hint 1: This chart is often used to visualize a trend in data over intervals of time.
Hint 2: The line in this type of chart is often drawn chronologically.
a) Line chart
b) Bar chart
c) Pie chart
d) Scatter plot
Answer:
a) Line chart

Question 10.
Read the statements given below. Identify the right option from the following for pie chart. Statement A : To make a pie chart with Matplotlib, we can use the plt.pie() function.
Statement B : The autopct parameter allows us to display the percentage value using the Python string formatting.
a) Statement A is correct
b) Statement B is correct
c) Both the statements are correct
d) Both the statements are wrong
Answer:
b) Statement B is correct

II Answer the following questions (2 marks)

Question 1
Define Data Visualization.
Answer:
Data Visualization is the graphical representation of information and data. The objective of Data Visualization is to communicate information visually to users. For this, data visualization uses statistical graphics. Numerical data may be encoded using dots, lines, or bars, to visually communicate a quantitative message.

Question 2.
List the general types of data visualization.
Answer:

• Charts
• Tables
• Graphs
• Maps
• Infographics
• Dashboards

Question 3.
List the types of Visualizations in Matplotlib.
Answer:
There are many types of Visualizations under Matplotlib. Some of them are:

1. Line plot
2. Scatter plot
3. Histogram
4. Box plot
5. Bar chart and
6. Pie chart

Question 4.
How will you install Matplotlib?
Answer:

• Pip is a management software for installing python packages.
• We can install matplotlib using pip.
• First of all, we need to identify whether pip is installed in your PC. If so, upgrade the pip in your system.
• To check if pip is already installed in our system, navigate our command line to the location of Python’s script directory.
• You can install the latest version of pip from our command prompt using the following command:
  Python -m pip install -U pip
• To install matplotlib, type the following in our command prompt:
  Python -m pip install -U matplotlib

Question 5.
Write the difference between the following functions: plt. plot([1,2,3,4]), plt. plot ([1,2,3,4], [14, 9,16]).
Answer:

plt.plot([I,2,3,4])		plt.pIot([I,2,3,4],[1,4,9,16])
1	In plt.plot([1,2,3,4]) a single list or array is provided to the plot() command.	In plt.pIot([1,2,3,4],[l,4,9,16]) the first two parameters are ‘x’ and ‘y’ coordinates.
2	matplotlib assumes it is a sequence of y values, and automatically generates the x values.	This means, we have 4 co-ordinate according to these ‘list as (1,1), (2,4), (3,9) and (4,16).
3	Since python ranges start with 0, the default x vector has the same length as y but starts with 0. Hence the x data are [0,1,2,3].

III. Answer the following questions (3 marks)

Question 1.
Draw the output for the following data visualization plot.
import matplotlib.pyplot as plt
plt.bar([1,3,5,7,9],[5,2,7,8,2], label=”Example one”)
plt.bar([2,4,6,8,10],[8,6,2,5,6], label=:”Example two”, color=’g’)
plt.legend()
plt.xlabel(‘bar number’)
plt.ylabel(‘bar height’)
plt.title(‘Epic Graph\nAnother Line! Whoa’)
plt.show()
Output:

Question 2.
Write any three uses of data visualization.
Answer:

1. Data Visualization helps users to analyze and interpret the data easily.
2. It makes complex data understandable and usable.
3. Various Charts in Data Visualization helps to show the relationship in the data for one or more variables.

Question 3.
Write the coding for the following:
a) To check if PIP is Installed in your PC.
b) To Check the version of PIP installed in your PC.
c) To list the packages in matplotlib.
Answer:
a) To check if PIP is Installed in your PC.

1. In command prompt type pip – version.
2. If it is installed already, you will get version.
3. Command : Python – m pip install – U pip

b) To Check the version of PIP installed in your PC.
C:\ Users\ YourName\ AppData\ Local\ Programs\ Py thon\ Py thon36-32\ Scripts>
pip-version

c) To list the packages in matplotlib.
C:\ Users\ Y ou rName\ AppData\ Local\ Programs\ Py thon\ Py thon36-32\ Scripts>
pip list

Question 4.
Write the plot for the following pie chart output.
Coding:
import matplotlib. pyplot as pit
sizes = [54.2,8.3,8.3,29.2]
labels= [“playing”,”working”,”eating”,”sleeping”]
exp=(0,0,0.1,0)
pit.pie (sizes labels = labels,explode=exp,autopct=”%.If%%, shadow-True, counterclock=False,strartangle-90)
pit, axes (). set_aspect (“equal”)
pit. title (“Interesting Graph \nCheck it out”)
plt.show ()

IV. Answer the following questions (5 Marks)

Question 1.
Explain in detail the types of pyplots using Matplotlib.
Answer:
Matplotlib allows us to create different kinds of plots ranging from histograms and scatter plots to bar graphs and bar charts.

Line Chart:

• A Line Chart or Line Graph is a type of chart which displays information as a series of data points called ‘markers’ connected by straight line segments.
• A Line Chart is often used to visualize a trend in data over intervals of time – a time series – thus the line is often drawn chronologically.

Program for Line plot:
import matplotlib.pyplot as pit years = [2014, 2015, 2016, 2017, 2018] total_populations = [8939007, 8954518,, 8960387,8956741,8943721]
plt. plot (“years, total_populations)
plt. title (“Year vs Population in India”)
plt.xlabel (“Year”)
plt.ylabel (“Total Population”)
plt.showt

In this program,
Plt.titlet() →→7 specifies title to the graph
Plt.xlabelt) →→ specifies label for X-axis
Plt.ylabelO →→ specifies label for Y-axis
Output:

Bar Chart:

• A BarPlot (or Bar Chart) is one of. the most common type of plot. It shows the relationship between a numerical variable and a categorical variable.
• Bar chart represents categorical data with rectangular bars. Each bar has a height corresponds to the value it represents.
• The bars can be plotted vertically or horizontally.
• It is useful when we want to compare a given numeric value on different categories.
• To make a bar chart with Matplotlib, we can use the plt.bart) function.

Program:
import matplotlib.pyplot as pit
# Our data
labels = [“TAMIL”, “ENGLISH”, “MATHS”, “PHYSICS”, “CHEMISTRY”, “CS”]
usage = [79.8,67.3,77.8,68.4,70.2,88.5]
# Generating the y positions. Later, we’ll use them to replace them with labels. y_positions = range (len(labels))
# Creating our bar plot
plt.bar (y_positions, usage)
plt.xticks (y_positions, labels)
plt.ylabel (“RANGE”)
pit.title (“MARKS”)
plt.show()
Output:

Pie Chart:

• Pie Chart is probably one of the most common type of chart.
• It is a circular graphic which is divided into slices to illustrate numerical proportion.
• The point of a pie chart is to show the relationship of parts out of a whole.
• To make a Pie Chart with Matplotlib, we can use the plt.pie () function.
• The autopct parameter allows us to display the percentage value using the Python string formatting.

Program:
import matplotlib.pyplot as pit
sizes = [89, 80, 90,100, 75]
labels = [“Tamil”, “English”, “Maths”,
“Science”, “Social”]
plt.pie (sizes, labels = labels,
autopct = “%.2f”)
plt.axesfj.set aspect (“equal”)
plt.showt()
Output:

Question 2.
Explain the various buttons in a matplotlib window.
Answer:
Various buttons in a matplotlib window:

Home Button → The Home Button will help once you have begun navigating your chart. If you ever want to return back to the original view, you can click on this.
Forward/Back buttons → These buttons can be used like the Forward and Back buttons in your browser. You can click these to move back to the previous point you were at, or forward again.
Pan Axis → This cross-looking button allows you to click it, and then click and drag your graph around.
Zoom → The Zoom button lets you click on it, then click and drag a square that you would like to zoom into specifically. Zooming in will require a left click and drag. You can alternatively zoom out with a right click and drag.
Configure Subplots → This button allows you to configure various spacing options with your figure and plot.
Save Figure → This button will allow you to save your figure in various forms.

Question 3.
Explain the purpose of the following functions:
Answer:
a) plt.xlabel:
plt.xlabel Specifies label for X -axis
b) plt.ylabel:
plt.ylabel is used to specify label for y-axis
c) plt.title :
plt.title is used to specify title to the graph or assigns the plot title.
d) plt.legend():
plt.legend() is used to invoke the default legend with plt
e) plt.show():
plhshowQ is used to display the plot.

12th Computer Science Guide Data Visualization Using Pyplot: Line Chart, Pie Chart and Bar Chart Additional Questions and Answers

I. Choose the best answer (1 Mark)

Question 1.
………………. button is used to click and drag a graph around.
a) pan axis
b) home
c) zoom
d) drag
Answer:
a) pan axis

Question 2.
………………. charts display information as series of data points.
a) Bar
b) Pie
c) Line
d) Histogram
Answer:
c) Line

Question 3.
The representation of information in a graphic format is called …………………………
(a) chart
(b) graphics
(c) Infographics
(d) graphs
Answer:
c) Infographics

Question 4.
……………….refers to a graphical representation that displays data by way of bars to show the frequency of numerical data.
a) Bar chart
b) Line graph
c) Pie chart
d) Histogram
Answer:
d) Histogram

Question 5.
……………….represents the frequency distribution of continuous variables.
a) Bar chart
b) Line graph
c) Pie chart
d) Histogram
Answer:
d) Histogram

Question 6.
Find the Incorrect match from the following.
(a) Scatter plot – collection of points
(b) line charts – markers
(c) Box plot – Boxes
Answer:
(c) Box plot – Boxes

Question 7.
Which of the following plot we cannot rearrange the blocks from highest to lowest?
a) Line
b) Bar chart
c) Pie chart
d) Histogram
Answer:
d) Histogram

Question 8.
In ………………. graph, the width of the bars is always the same.
a) Line
b) Bar
c) Pie chart
d) Histogram
Answer:
b) Bar

Question 9.
The ………………. parameter allows us to display the percentage value using the Python string formatting in pie chart.
a) percent
b) autopct
c) pet
d) percentage
Answer:
b) autopct

Question 10.
Find the wrong statement.
If a single list is given to the plot( ) command, matplotlib assumes
(a) it is as a sequence of x values
(b) the sequence of y values
Answer:
(a) it is as a sequence of x values

Question 11.
………………. is the graphical representation of information and data.
a) Data visualization
b) Data Graphics
c) Data Dimension
d) Data Images
Answer:
a) Data visualization

Question 12.
………………. in data visualization helps to show the relationship in the data for more variables.
a) Tables
b) Graphics
c) Charts
d) Dashboards
Answer:
c) Charts

Question 13.
In a Scatter plot, the position of a point depends on its …………………………. value where each value is a position on either the horizontal or vertical dimension.
a) 2-Dimensional
b) 3-Dimensional
c) Single Dimensional
d) 4-Dimensional
Answer:
a) 2-Dimensional

Question 14.
……………………. plot shows the relationship between a numerical variable and a categorical variable.
(a) line
(b) Bar
(c) Scatter
(d) Box
Answer:
(b) Bar

Question 15.
…………………. is the representation of information in a graphic format.
a) Infographics
b) Graph
c) Symbol
d) Charts
Answer:
a) Infographics

Question 16.
…………………. is a collection of resources assembled to create a single unified visual display.
a) Infographics
b) Dashboard
c) Graph
d) Charts
Answer:
b) Dashboard

Question 17.
Matplotlib is a data visualization …………………. in Python.
a) control structure
b) dictionary
c) library
d) list
Answer:
c) library

Question 18.
Matplotlib allows us to create different kinds of …………………. ranging from histograms
a) Table
b) Charts
c) Maps
d) plots
Answer:
d) plots

Question 19.
Which function shows the percentage value in the pie chart?
(a) percent
(b) percentage
(c) slice
(d) auto pet
Answer:
(d) auto pet

Question 20.
…………………. command will take an arbitrary number of arguments.
a) show ()
b) plot ()
c) legend ()
d) title ()
Answer:
b) plot ()

Question 21.
The most popular data visualization library allows creating charts in few lines of code in python.
a) Molplotlib
b) Infographics
c) Data visualization
d) pip
Answer:
a) Molplotlib

II. Answer the following questions (2 and 3 Marks)

Question 1.
What is meant by Infographics?
Answer:
Infographics → An infographic (information graphic) is the representation of information in a graphic format.

Question 2.
Define Dashboard.
Answer:

• A dashboard is a collection of resources assembled to create a single unified visual display.
• Data visualizations and dashboards translate complex ideas and concepts into a simple visual format.
• Patterns and relationships that are undetectable in the text are detectable at a glance using the dashboard.

Question 3.
Write a note on matplotlib
Answer:

• Matplotlib is the most popular data visualization library in Python.
• It allows creating charts in few lines of code.

Question 4.
Write a note on the scatter plot.
Answer:

• A scatter plot is a type of plot that shows the data as a collection of points.
• The position of a point depends on its two dimensional value, where each value is a position on either the horizontal or vertical dimension.

Question 3.
What is Box Plot?
Answer:
Box plot: The box plot is a standardized way of displaying the distribution of data based on the five-number summary: minimum, first quartile, median, third quartile, and maximum.

III. Answer the following questions (5 Marks)

Question 1.
Write the key differences between Histogram and bar graph.
Answer:

Histogram		Bar graph
1	Histogram refers to a graphical representation; that displays data by way of bars to show the frequency of numerical data.	A bar graph is a pictorial representation of data that uses bars to compare different categories of data.
2	A histogram represents the frequency distribution of continuous variables.	A bar graph is a diagrammatic comparison of discrete variables.
3	The histogram presents numerical data	The bar graph shows categorical data
4	The histogram is drawn in such a way that there is no gap between the bars.	There is proper spacing between bars in a bar graph that indicates discontinuity.
5	Items of the histogram are numbers, which are categorised together, to represent ranges of data.	Items are considered as individual entities.
6	A histogram, rearranging the blocks, from highest to lowest cannot be done, as they are shown in the sequence of classes.	In the case of a bar graph, it is quite common to rearrange the blocks, from highest to lowest.
7	The width of rectangular blocks in a histogram may or may not be the same.	The width of the bars in a bar graph is always the same.

Question 2.
Explain the purpose of
i) plt.plot()
ii) pt.bar()
iii) plt.sticks()
iv) plt.pie
Answer:
i) plt.plot():
plt.plot() is used to make a line chart or graph with matplotlib.

ii) plt.bar():
plt.bar() is used to make a bar chart with matplotlib.

iii) plt.xticks():

• plt.xticks() display the tick marks along the x-axis at the values represented.
• Then specify the label for each tick mark.
• It is used bar chart.

iv) plt.pie ():
plt.pie () is used to make a pie chart with matplotlib.

Question 3.
Draw the output for the following python code:
import matplotlib.pyplot as pit
a = [1,2,3]
b = [5,7,4]
x = [1,2,3]
y = [10,14,12]
plt.plot(a,b, label=/Lable 1′)
plt.plot(x,y, label=’Lable 2′)
plt.xlabel(‘X-Axis’)
plt.ylabel(‘Y-Axis’)
plt. legend ()
plt. show ()
Output:

Question 4.
Draw the chart for the given Python snippet.
import matplotlib.pyplot as plt
plt.plot([l,2,3,4], [1,4,9,16])
plt.show()
Output:

Hands-on Practice

Question 1.
Create a plot. Set the title, the x and y labels for both axes,
import matplotlib.pyplot as pit
x=[1,2,3]
Y=[5,7,4]
plt.plot(x,y)
plt.xlable(‘X-AXIS’)
plt.ylabel(‘Y -AXIS’)
plt.title(‘LINE GRAPH)
plt.show()

Question 2.
Plot a pie chart for your marks in the recent examination.
import matplotlib.pyplot as pit
s=[60,85,90,83,95] l=[‘LANG’,’ENG’,’MAT ‘,’SCI’,’SS’]
plt.pie(s,labels=l)
plt.title(‘MARKS’)
plt.show()

Question 3.
Plot a line chart on the academic performance of Class 12 students in Computer Science for the past 10 years.
import matplotlib.pyplot as pit
x=[2009,2010,2011,2012,2013,2014,2015,20 16,2017,2018]
y=[56,68,97,88,92,96,98,99,100,100]
plt.plot(x,y) plt.xlable(‘YEAR’)
plt.ylabel(‘PASS % IN C.S’)
plt. show ()

Question 4.
Plot a bar chart for the number of computer science periods in a week, import matplotlib.pyplot as pit x=[“MON”,”TUE”,”WED”, “THUR”,”FRI”]
y=[6,5,2,1,7] plt.bar(x,y) pit. xlable (‘ DAY S’) plt.ylabel(‘PERIOD’) plt.showQ

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Business Maths Guide Pdf Chapter 4 Differential Equations Ex 4.6 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 4 Differential Equations Ex 4.6

Choose the most suitable answer from the given four alternatives:

Question 1.
The degree of the differential equation
\(\frac { d^2y }{dx^4}\) – (\(\frac { d^2y }{dx^2}\)) + \(\frac { dy }{dx}\) = 3
(a) 1
(b) 2
(c) 3
(d) 4
Solution:
(a) 1
Hint:
Since the power of \(\frac{d^{4} y}{d x^{4}}\) is 1

Question 2.
The order and degree of the differential equation \(\sqrt{\frac { d^2y }{dx^2}}\) = \(\sqrt{\frac { dy }{dx}+5}\) are respectively.
(a) 2 and 2
(b) 3 and 2
(c) 2 and 1
(d) 2 and 3
Solution:
(a) 1
Hint:
Squaring on both sides
\(\frac { d^2y }{dx^2}\) = \(\frac { dy }{dx}\) + 5
Highest order derivative is \(\frac { d^2y }{dx^2}\)
∴ order = 2
Power of the highest order derivative \(\frac { d^2y }{dx^2}\) = 1
∴ degree = 1

Question 3.
The order and degree of the differential equation
(\(\frac { d^2y }{dx^2}\))^3/2 – \(\sqrt{(\frac { dy }{dx})}\) – 4 = 0
(a) 2 and 6
(b) 3 and 6
(c) 1 and 4
(d) 2 and 4
Solution:
(a) 2 and 6
Hint:

Highest order derivative is \(\frac { d^2y }{dx^2}\)
∴ Order = 2
Power of the highest order derivative \(\frac { d^2y }{dx^2}\) is
∴ degree = 6

Question 4.
The differential equation (\(\frac { dx }{dy}\))³ + 2y^1/2 = x
(a) of order 2 and degree 1
(b) of order 1 and degree 3
(c) of order 1 and degree 6
(d) of order 1 and degree 2
Solution:
(b) of order 1 and degree 3
Hint:

Highest order derivative is \(\frac { dy }{dx}\)
∴ order = 1
Power of the highest order derivative \(\frac { dy }{dx}\) is 3
∴ degree = 3

Question 5.
The differential equation formed by eliminating a and b from y = ae^x + be^-x
(a) \(\frac { d^2y }{dx^2}\) – y = 0
(b) \(\frac { d^2y }{dx^2}\) – \(\frac { dy }{dx}\)y = 0
(c) \(\frac { d^2y }{dx^2}\) = 0
(d) \(\frac { d^2y }{dx^2}\) – x = 0
Solution:
(a) \(\frac { d^2y }{dx^2}\) – y = 0
Hint:

Question 6.
If y = ex + c – c³ then its differential equation is
(a) y = x\(\frac { dy }{dx}\) + \(\frac { dy }{dx}\) – (\(\frac { dy }{dx}\))³
(b) y + (\(\frac { dy }{dx}\))³ = x \(\frac { dy }{dx}\) – \(\frac { dy }{dx}\)
(c) \(\frac { dy }{dx}\) + (\(\frac { dy }{dx}\))³ – x\(\frac { dy }{dx}\)
(d) \(\frac { d^3y }{dx^3}\) = 0
Solution:
(a) y = x\(\frac { dy }{dx}\) + \(\frac { dy }{dx}\) – (\(\frac { dy }{dx}\))³
Hint:
y = cx + c – c³ ……… (1)
\(\frac { dy }{dx}\) = c
(1) ⇒ y = x\(\frac { dy }{dx}\) + \(\frac { dy }{dx}\) – (\(\frac { dy }{dx}\))³

Question 7.
The integrating factor of the differential equation \(\frac { dy }{dx}\) + Px = Q is
(a) e^∫pdx
(b) e^∫Pdx
(c) e^∫Pdy
(d) e^∫pdy
Solution:
(d) e^∫pdy

Question 8.
The complementary function of (D² + 4) y = e^2x is
(a) (Ax+B)e^2x
(b) (Ax+B)e^-2x
(c) A cos2x + B sin2x
(d) Ae^-2x + Be^2x
Solution:
(c) A cos2x + B sin2x
Hint:
A.E = m² + 4 = 0 ⇒ m = ±2i
C.F = e^0x (A cos 2x + B sin 2x)

Question 9.
The differential equation of y = mx + c is (m and c are arbitrary constants)
(a) \(\frac { d^2y }{dx^2}\) = 0
(b) y = x\(\frac { dy }{dx}\) + o
(c) xdy + ydx = 0
(c) ydx – xdy = 0
Solution:
(a) \(\frac { d^2y }{dx^2}\) = 0
Hint:
y = mx + c
\(\frac { dy }{dx}\) = m ⇒ \(\frac { d^2y }{dx^2}\) = 0

Question 10.
The particular intergral of the differential equation \(\frac { d^2y }{dx^2}\) – 8\(\frac { dy }{dx}\) + 16y = 2e^4x
(a) \(\frac { x^2e^{4x} }{2!}\)
(b) y = x\(\frac { e^{4x} }{2!}\)
(c) x²e^4x
(d) xe^4x
Solution:
(c) x²e^4x
Hint:

Question 11.
Solution of \(\frac { dx }{dy}\) + Px = 0
(a) x = ce^py
(b) x = ce^-py
(c) x = py + c
(d) x = cy
Solution:
(b) x = ce^-py

Question 12.
If sec^2x x isa na intergranting factor of the differential equation \(\frac { dx }{dy}\) + Px = Q then P =
(a) 2 tan x
(b) sec x
(c) cos 2 x
(d) tan 2 x
Solution:
(a) 2 tan x
Hint:
I.F = sec² x
e^∫pdx = sec²x
∫pdx = log sec² x
∫pdx = 2 log sec x
∫pdx = 2∫tan x dx ⇒ p = 2 tan x

Question 13.
The integrating factor of the differential equation is x \(\frac { dy }{dx}\) – y = x²
(a) \(\frac { -1 }{x}\)
(b) \(\frac { 1 }{x}\)
(c) log x
(c) x
Solution:
(b) \(\frac { 1 }{x}\)
Hint:

Question 14.
The solution of the differential equation where P and Q are the function of x is
(a) y = ∫Q e^∫pdx dx + c
(b) y = ∫Q e^-∫pdx dx + c
(c) ye^∫pdx = ∫Q e^∫pdx dx + c
(c) ye^∫pdx = ∫Q e^-∫pdx dx + c
Solution:
(c) ye^∫pdx = ∫Q e^∫pdx dx + c

Question 15.
The differential equation formed by eliminating A and B from y = e^-2x (A cos x + B sin x) is
(a) y₂ – 4y₁ + 5 = 0
(b) y₂ + 4y – 5 = 0
(c) y₂ – 4y₁ + 5 = 0
(d) y₂ + 4y₁ – 5 = 0
Solution:
(d) y₂ + 4y₁ – 5 = 0
Hint:
y = e^-2x (A cosx + B sinx)
y e^2x = A cosx + B sinx ………. (1)
y(e^2x) (2) + e^2x \(\frac { dy }{dx}\) = A(-sin x) + B cos x ………. (2)
Differentiating w.r.to x

Question 16.
The particular integral of the differential equation f (D) y = e^ax where f(D) = (D – a)²
(a) \(\frac { x^2 }{2}\) e^ax
(b) xe^ax
(c) \(\frac { x }{2}\) e^ax
(d) x² e^ax
Solution:
(a) \(\frac { x^2 }{2}\) e^ax

Question 17.
The differential equation of x² + y² = a²
(a) xdy + ydx = 0
(b) ydx – xdy = 0
(c) xdx – ydx = 0
(d) xdx + ydy = 0
Solution:
(d) xdx + ydy = 0
Hint:
x² + y² = a²
⇒ 2x + 2y \(\frac{d y}{d x}\) = 0
⇒ x dx + y dy = 0

Question 18.
The complementary function of \(\frac { d^y }{dx^2}\) – \(\frac { dy }{dx}\) = 0 is
(a) A + Be^x
(b) (A + B)e^x
(c) (Ax + B)e^x
(d) Ae^x + B
Solution:
(a) A + Be^x
Hint:
A.E is m² – m = 0
⇒ m(m – 1) = 0
⇒ m = 0, 1
CF is Ae^0x + Be^x = A + Be^x

Question 19.
The P.I of (3D² + D – 14) y = 13e^2x is
(a) \(\frac { 1 }{2}\) e^x
(b) xe^2x
(c) \(\frac { x^2 }{2}\) e^2x
(d) Ae^x + B
Solution:
(b) xe^2x
Hint:
(3D² + D – 14) y = 13e^2x

Question 20.
The general solution of the differential equation \(\frac { dy }{dx}\) = cos x is
(a) y = sinx + 1
(b) y = sinx – 2
(c) y = cosx + C, C is an arbitary constant
(d) y = sinx + C, C is an arbitary constant
Solution:
(d) y = sinx + C, C is an arbitary constant
Hint:
\(\frac { dy }{dx}\) = cos x
dy = cos x dx
∫dy = ∫cos x dx ⇒ y = sin x + c

Question 21.
A homogeneous differential equation of the form \(\frac { dy }{dx}\) = f(\(\frac { y }{x}\)) can be solved by making substitution.
(a) y = v x
(b) y = y x
(c) x = v y
(d) x = v
Solution:
(a) y = v x

Question 22.
A homogeneous differential equation of the form \(\frac { dy }{dx}\) = f(\(\frac { x }{y}\)) can be solved by making substitution.
(a) x = v y
(b) y = v x
(c) y = v
(d) x = v
Solution:
(a) y = v x

Question 23.
The variable separable form of \(\frac { dy }{dx}\) = \(\frac { y(x-y) }{x(x+y)}\) by taking y = v x and \(\frac { dy }{dx}\) = v + x \(\frac { dy }{dx}\)
(a) \(\frac { 2v^2 }{1+v}\) dv = \(\frac { dx }{x}\)
(b) \(\frac { 2v^2 }{1+v}\) dv = –\(\frac { dx }{x}\)
(c) \(\frac { 2v^2 }{1-v}\) dv = \(\frac { dx }{x}\)
(d) \(\frac { 1+v }{2v^2}\) dv = –\(\frac { dx }{x}\)
Solution:
(d) \(\frac { 1+v }{2v^2}\) dv = –\(\frac { dx }{x}\)
Hint:

Question 24.
Which of the following is the homogeneous differential equation?
(a) (3x – 5) dx = (4y – 1) dy
(b) xy dx – (x³ + y³) dy = 0
(c) y²dx + (x² – xy – y²) dy = 0
(d) (x² + y) dx (y² + x) dy
Solution:
(c) y²dx + (x² – xy – y²) dy = 0

Question 25.
The solution of the differential equation \(\frac { dy }{dx}\) = \(\frac { y }{x}\) + \(\frac { f(\frac { y }{x}) }{ f(\frac { y }{x}) }\) is
(a) f\(\frac { y }{x}\) = k x
(b) x f\(\frac { y }{x}\) = k
(c) f\(\frac { y }{x}\) = k y
(d) x f\(\frac { y }{x}\) = k
Solution:
(a) f\(\frac { y }{x}\) = k x

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Computer Science Guide Pdf Chapter 6 Control Structures Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Computer Science Solutions Chapter 6 Control Structures

12th Computer Science Guide Control Structures Text Book Questions and Answers

I. Choose the best answer (1 Mark)

Question 1.
How many important control structures are there in Python?
a) 3
b) 4
c) 5
d) 6
Answer:
a) 3

Question 2.
elif can be considered to be abbreviation of
a) nested if
b) if..else
c) else if
d) if..elif
Answer:
c)else if

Question 3.
What plays a vital role in Python programming?
a) Statements
b) Control
c) Structure
d) Indentation
Answer:
d) Indentation

Question 4.
Which statement is generally used as a placeholder?
a) continue
b) break
c) pass
d) goto
Answer:
c) pass

Question 5.
The condition in the if statement should be in the form of
a) Arithmetic or Relational expression
b) Arithmetic or Logical expression
c) Relational or Logical expression
d) Arithmetic
Answer:
c) Relational or Logical expression

Question 6.
Which is the most comfortable loop?
a) do..while
b) while
c) for
d) if..elif
Answer:
c) for

Question 7.
What is the output of the following snippet?
i=l
while True:
if i%3 ==0:
break
print(i/end=”)
i +=1
a) 12
b) 123
c) 1234
d) 124
Answer:
a) 12

Question 8.
What is the output of the following snippet?
T=1
while T:
print(True)
break
a) False
b) True
c) 0
d) no output
Answer:
b) True

Question 9.
Which amongst this is not a jump statement ?
a) for
b) goto
c) continue
d) break
Answer:
a) for

Question 10.
Which punctuation should be used in the blank?
if < condition >
statements-block 1
else:
statements-block 2
a) ;
b) :
c) ::
d) !
Answer:
b) :

II. Answer the following questions (2 Marks)

Question 1.
List the control structures in Python.
Answer:
There are three important control structures

1. Sequential
2. Alternative or Branching
3. Iterative or Looping

Question 2.
Write note on break statement.
Answer:

• The break statement terminates the loop containing it.
• Control of the program flows to the statement immediately after the body of the loop.
• When the break statement is executed, the control flow of the program comes out of the loop and starts executing the segment of code after the loop structure.
• If break statement is inside a nested loop (loop inside another loop), break will terminate the innermost loop.

Question 3.
Write is the syntax of if..else statement
Answer:
Syntax:
if:
statements – block 1
else:
statements – block 2

Question 4.
Define control structure.
Answer:
A program statement that causes a jump of control from one part of the program to another is called a control structure or control statement.

Question 5.
Write note on range () in loop
Answer:
Usually in Python, for loop uses the range() function in the sequence to specify the initial, final and increment values. range() generates a list of values starting from start till stop – 1.

III. Answer the following questions (3 Marks)

Question 1.
Write a program to display
Answer:
A
AB
ABC
ABCD
ABCDE
For i in range (1,6,1):
ch=65
for j in range (ch,ch+i,1):
a=chr(j)
print (a, end =’ ‘)
print ()

Question 2.
Write note on if..else structure.
Answer:
The if-else statement provides control to check the true block as well as the false block. Following is the syntax of ‘if-else’ statement.
Syntax:
if:
statements – block 1
else:
statements – block 2

Question 3.
Using if..else..elif statement write a suitable program to display largest of 3 numbers.
Answer:
a = int (input (“Enter number 1″)
b = int (input (” Enter number 2″)
c = int (input (” Enter number 3″)
if a > b and a > c:
put (” A is greatest”)
elif b > a and b > c:
print (“B is greatest”)
else:
print (“C is greatest”)

Question 4.
Write the syntax of while loop.
Answer:
The syntax of while loop in Python has the following syntax:
Syntax:
while:
statements block 1
[else:
statements block 2]

Question 5.
List the differences between break and continue statements.
Answer:

Break	Continue
Break statement terminates the loop containing it and control reaches after the body of the loop	Continue statement skips the remaining part of a loop and start with next iteration.

IV. Answer the following questions (5 Marks)

Question 1.
Write a detail note on for loop
Answer:
for loop:

• for loop is the most comfortable loop. It is also an entry check loop.
• The condition is checked in the beginning and the body of the loop
  (statements-block 1) is executed if it is only True otherwise the loop is not executed.

Syntax:
for counter_variable in
sequence:
statements – block 1
[else: # optional block statements – block 2]

• The counter, variable mentioned in the syntax is similar to the control variable that we used in the for loop of C++ and the sequence refers to the initial, final and increment value.
• Usually in Python, for loop uses the range () function in the sequence to specify the initial, final and increment values, range () generates a list of values starting from start till stop-1.

The syntax of range() follows:
range (start, stop, [step])
Where,
start – refers to the initial value
stop – refers to the final value
step – refers to increment value,
this is optional part.

Examples for range():
range (1,30,1) – will start the range of values from 1 and end at 29 range (2,30,2) – will start the range of values from 2 and end at 28 range (30,3,-3) – will start the range of values from 30 and end at 6E range (20) – will consider this value 20 as the end value ( or upper limit) and starts the range count from 0 to 19 (remember always range () will work till stop -1 value only)

Example-Program:
#Program to illustrate the use of for loop – to print single digit even number
for i in range (2,10,2):
print (i, end=”)

Output:
2468

Question 2.
Write a detail note on if..else..elif statement with suitable example.
Answer:

• When we need to construct a chain of if statement(s) then ‘elif’ clause can be
  used instead of ‘else’.
  Syntax:
  if < condition -1>:
  statements-block 1
  elif< condition -2>:
  statements-block 2
  else:
  statements-block n
• In the syntax of if..elif ..else mentioned above, condition -1 is tested if it is true then statements-block 1 is executed, otherwise, the control checks condition-Z, if it is true statements- block2 is executed and even if it fails statements-block n mentioned in else part is executed.
• ‘elif’ clause combines if..else- if ..else statements to one if..elif … else, “elif’ can be considered to be abbreviation of ‘else if’. In an’if’ statement there is no limit of ‘elif’ clause that can be used, but an clause if used should be placed at the end.

Example:

# Program to illustrate the use of nested if statement
Average – Grade
> =80 and above A
> =70 and above B
> =60 and <70 C
> =50 and <60 D
Otherwise E

Example-program
m1 = int (input(“Enter mark in first subject:”))
m2 = int (input(” Enter mark in second subject:”))
avg = (ml+ml)/2
if avg> =80:
print (“Grade: A”)
elif avg> =70 and avg< 80:
print (“Grade: B”)
elif avg> =60 and avg< 60:
print (“Grade: C”)
elif avg> =50 and avg< 60: .
print (“Grade: D”)
else:
print(“Grade: E”)

Output 1:
Enter mark in first
subject: 34
Enter mark in second
subject: 78
Grade: D

Question 3.
Write a program to display all 3 digit odd numbers.
Answer:
Odd Number (3 digits)
for a in range (100, 1000)
if a % 2 = = 1:
print b
Output:
101, 103, 105, 107, .. …… 997, 999

Question 4.
Write a program to display multiplication table for a given number.
Answer:
Coding:
num=int(input(“Display Multiplication Table of “))
for i in range(1,11):
print(i, x ,num, ‘=’, num*i)
Output:
Display Multiplication Table of 2
1 x 2 = 2
2 x 2 = 4
3 x 2 = 6
4 x 2 = 8
5 x 2 = 10
6 x 2 = 12
7 x 2 =14
8 x 2 = 16
9 x 2 =18
10 x 2 = 20
>>>

12th Computer Science Guide Control Structures Additional Questions and Answers

I. Choose the best answer ( I Mark)

Question 1.
Executing a set of statements multiple times are called…………………………..
(a) Iteration
(b) Looping
(c) Branching
(d) Both a and b
Answer:
(d) Both a and b

Question 2.
………… important control structures are available in python.
a) 2
b) 3
c) 4
d) many
Answer:
b) 3

Question 3.
Identify which is not a control structure?
(a) Sequential
(b) Alternative
(c) Iterative
(d) Break
Answer:
(d) Break

Question 4.
To construct a chain of if statement, else can be replaced by
a) while
b) ifel
c) else if
d) elif
Answer:
d) elif

Question 5.
Branching statements are otherwise called……………………………
(a) Alternative
(b) Iterative
(c) Loop
(d) Sequential
Answer:
(a) Alternative

Question 6.
Which statement is used to skip the remaining part of a loop and start with the next iteration?
a) continue
b) break
c) pass
d) condition
Answer:
a) continue

Question 7.
In the …………….. loop, the condition is any valid Boolean expression returning True or false.
a) if
b) else
c) elif
d) while
Answer:
d) while

Question 8.
How many blocks can be given in Nested if.. elif.. else statements?
(a) 1
(b) 2
(c) 3
(d) n
Answer:
(d) n

Question 9.
A loop placed within another loop is called as …………… loop structure.
a) entry check
b) exit check
c) nested
d) conditional
Answer:
c) nested

Question 10.
What types of Expressions can be given in the while loop?
(a) Arithmetic
(b) Logical
(c) Relational
(d) Boolean
Answer:
(d) Boolean

II. Answer the following questions (2 and 3 Marks)

Question 1.
Write a note on sequential statements?
Answer:
A sequential statement is composed of a sequence of statements which are executed one after another. A code to print your name, address, and phone number is an example of a sequential statement.

Question 2.
Write the syntax of for loop.
Answer:
Syntax:
for counter_variable in sequence:
statements-block 1
[else: # optional block
statement-block 2]

Question 3.
Define loops?
Answer:
Iteration or loop are used in a situation when the user needs to execute a block of code several times or till the condition is satisfied. A loop statement allows executing a statement or group of statements multiple times.

Question 4.
What is meant by Nested loop structure?
Answer:

• A loop placed within another loop is called a nested loop structure.
• A while; within another while; for within another for;
• For within while and while within for to construct nested loops.

Question 5.
Give the syntax of range O in for loop?
Answer:
The syntax of range ( ) is as follows:
range (start, stop, [step] )
Where,
start – refers to the initial value
stop – refers to the final value
step – refers to increment value, this is an optional part.

Question 6.
Write a note on the pass statement.
Answer:

• pass statement in Python programming is a null statement.
• pass statement when executed by the interpreter it is completely ignored.
• Nothing happens when the pass is executed, it results in no operation.

III. Answer the following questions (5 Marks)

Question 1.
Explain the types of alternative or branching statements provided by Python?
Answer:
The types of alternative or branching statements provided by Python are:

1. Simple if statement
2. if..else statement
3. if..elif statement

1) Simple if statement
Simple if is the simplest of all decision-making statements. The condition should be in the form of relational or logical expression.
Syntax:
if:
statements-block1
Example:
x=int (input(“Enter your age :”))
if x > =18:
print (“You are; eligible for voting”)
Output:
Enter your age :34
You are eligible for voting

2) if..else statement
The if.. else statement provides control to check the true block as well as the false block. Following is the syntax of ‘if..else statement.
Syntax:
if:
statements-block 1
else:
statements-block 2
Example:
a = int(input(” Enter any number :”))
if a%2==0:
print (a, ” is an even number”) else:
print (a, ” is an odd number”)
Output 1:
Enter any number:56
56 is an even number
Output 2:
Enter any number:67
67 is an odd number
Flowchart- if..else statement Execution

3) Nested if..elif…else statement:

• When we need to construct a chain of if statement(s) then ‘elif’ clause can be used instead of ‘else.
• ‘elif’ clause combines if..else-if.. elsestatements to one if ..elif… else. elif can be considered to be abbreviation of else if.
• In an ‘if statement there is no limit of ‘elif clause that can be used, but an ‘else clause if used should be placed at the end.

Syntax:
if<statements-block 1>:
elif :
statements-block 2
else:
statements-block n
Example:
a = int (input (“Enter number 1″)
b = int (input (” Enter number 2″)
c = int (input (” Enter number 3″)
if a > b and a > c:
put (” A is greatest”)
elif b > a and b > c:
print (“B is greatest”)
else:
print (“C is greatest”)

Question 2.
Explain while loop with example.
Answer:

• While loop belongs to entry check loop type, that is it is not executed even once
  if the condition is tested False in the beginning.
• In the while loop, the condition is any valid Boolean expression returning
  True or False.
• The else part of while is optional part of while. The statements blocki is kept
  executed till the condition is True.
• If the else part is written, it is executed when the condition is tested False.

Syntax:
while< condition >:
statements block 1
[else:
statements block 2]
Flowchart-while loop execution:

Example:

i=10	# intializing part of the control variable
while (i<=15):	# test condition
print (i,end=,\t/)	# statements – block1
i=i+1	# Updation of the control variable

Question 3.
Explain the Jump statement in python.
Answer:

• The jump statement in Python is used to unconditionally transfer the control from one part of the program to another.
• There are three keywords to achieve jump statements. in Python: break, continue, pass.

Flowchart -Use of break, continue statement in loop structure:

• break statement:
• The break statement terminates the loop containing it.
• Control of the program flows to the statement immediately after the body of the loop.
• A while or for loop will iterate till the condition is tested false, but one can even transfer the control out of the loop (terminate) with help of a break statement.
• When the break statement is executed, the control flow of the program comes out of the loop and starts executing the segment of code after the loop structure.
• If the break statement is inside a nested loop (loop inside another loop), the break will terminate the innermost loop. Syntax for break statement:
  break
  Example: for word in “Jump Statement”:
  ifword = = “e”:
  break print (word, end= “)
  Output: Jump Stat
  Flowchart- Working of break statement:

Working of break statement

continue statement: Continue statement unlike the break statement is used to skip the remaining part of a loop and start with the next iteration.

Syntax of continue statement:
continue Example:
for word in “Jump Statement”:
if word = = “e”:
continue print (word, end=”)
print (“\n End of the program”)

Output:
Jump Statement
End of the program

pass statement:

• pass statement is generally used as a placeholder.
• When we have a loop or function that is to be implemented in the future and not now, we cannot develop such functions or loops with empty body segments because the interpreter would raise an error.
• So, to avoid this we can use a pass statement to construct a body that does nothing.

Syntax of pass statement:
pass

Example:
forval in “Computer”:
pass
print (“End of the loop, loop structure will be built in future”)
Output: End of the loop, loop structure will be built in future

Question 4.
What kind of Nested ioop structure can be created?
Answer:

• A loop placed within another loop is called a nested loop structure.
• A while; within another while; for within another for;
• for within while and while within to construct nested loops.

HANDS-ON PRACTICE

Question 1.
Write a program to check whether the given character is a vowel or not.
Answer:
Coding:
ch=input (“Enter a character :”)
# to check if the letter is vowel
if ch in (‘a’, ‘A’, e , E , i , I , o ,O , u’, ‘U’):
print (ch/ is a vowel’)
Output:
Enter a character:e
e is a vowel

Question 2.
(i) Write a program to display all 3 digit even numbers.
(ii) Write the output for the following program.
Answer:
i=1
while (i<=6):
for j in range (1, i):
print(j, end=’\t’)
print (end=’ \ n’)
i+=1
i) Python Program:
for i in range(100,1000,2):
Print(i)
ii) Output: 1
1 2
1 2 3
1 2 3 4
1 2 3 4 5

Question 3.
Write a program to check if a number is Positive, Negative or zero.
Answer:
Coding:
num = float(input(” Enter a number: “))
if num > 0:
print(“Positive number”)
elifnum == 0:
print(“Zero”)
else:
print(“Negative number”)
Output:
Enter a number:5
Positive number

Question 4.
Write a program to display Fibonacci series 0112345 (up to n terms)
Answer:
Coding:
Number = int(input(“\n Please Enter the Range Number: “))
i = 0
First_ Value = 0
Second-Value = 1
while(i < Number):
if(i <= 1):
Next = i else:
Next = First-Value + Second_Value
First_Value = Second_Value
Second_Value = Next
print(Next)
i = i + 1
Output:
Please Enter the Range Number: 4
0
1
1
2
3

Question 5.
Write a program to display sum of natural numbers, up to n.
Answer:
Coding:
number = int(input(“Please Enter any Number:”))
total = 0
for value in range(l, number + 1):
total = total + value
print(“The Sum of Natural Numbers is total)
Output:
Please Enter any Number:5
The Sum of Natural Numbers is : 15

Question 6.
Write a program to check if the given number is a palindrome or not.
Answer:
Coding:
n=int(input(“Enter number:”))
temp=n
rev=0
while(n>0):
dig=n%10
rev=rev*10+dig
n=n//10
if(temp==rev):
print(“The number is a palindrome!”)
else:
print(“The number isn’t a palindrome!”)
Output:
isn’t a palindrome!

Question 7.
Write a program to print the following pattern
* * * * *
* * * *
* * *
* *
*
Answer:
Coding:
number = int(input(“Please Enter Pattern Number: “))
for i in range(number,0,-l):
for j in range(1,i+1,1):
print(“*”, end”)
print()
Output:
Please Enter Pattern Number:5
* * * * *
* * * *
* * *
* *
*

Question 8.
Write a program to check if the year is leap year or not.
Answer:
Coding:
def leap_year(y):
.’ if (y % 400 = = 0):
print(y, “is the leap year”)
elif(y%4 = = 0):
print(y, “is the leap year”)
else:
print(y, “is not a leap year”)
year = int(input(“Enter a year…”)
print(leap_year(year))
Output:
Enter a year… 2007
2007 is the leap year

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Economics Guide Pdf Chapter 3 Theories of  Employment and Income Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Economics Solutions Chapter 3 Theories of Employment and Income

12th Economics Guide Theories of Employment and Income Text Book Back Questions and Answers

PART – A

Multiple Choice questions

Question 1.
Every able bodied person who is willing to work at the prevailing wage rate is employed called as ……………..
a) Full employment
b) Under employment
c) Unemployment
d) Employment opportunity
Answer:
a) Full employment

Question 2.
Structural unemployment is a feature in a ……………
a) Static society
b) Socialist society
c) Dynamic society
d) Mixed economy
Answer:
c) Dynamic society

Question 3.
In disguised unemployment, the marginal productivity of labour is ……………..
a) Zero
b) One
c) Two
d) Positive
Answer:
a) Zero

Question 4.
The main concention of the classical Economic Theory is …………..
a) Under employment
b) Economy is always in the state of equilibrium
c) Demand creates its supply
d) Imperfect competition
Answer:
b) Economy is always in the state of equilibrium

Question 5.
J.B. say is a ……………
a) New classical Economist
b) Classical Economist
c) Modern Economist
d) New Economist
Answer:
b) Classical Economist

Question 6.
According to keynes, which type of unemployment prevails in capitalist economy?
a) Full employment
b) Voluntary unemployment
c) Involuntary unemployment
d) Under employment
Answer:
d) Under employment

Question 7.
The core of the classical theory of employment is
a) Law of Diminishing Return
b) Law of demand
c) Law of markets
d) Law of consumption
Answer:
c) Law of markets

Question 8.
Keynes attributes unemployment to ……………………..
a) A lack of effective supply
b) A lack of effective demand
c) A lack of both
d) None of the above
Answer:
b) A lack of effective demand

Question 9.
………………………….. Flexibility brings equality between saving and investment.
a) Demand
b) Supply
c) Capital
d) Interest
Answer:
d) Interest

Question 10.
…………………. theory is a turning point in the development of modern economic theory.
a) Keynes
b) Say’s
c) Classical
d) Employment
Answer:
a) Keynes

Question 11.
The basic concept used in keynes Theory of Employment and Income is ……………….
a) Aggregate demand
b) Aggregate supply
c) Effective demand
d) Marginal propensity to consume
Answer:
d) Marginal propensity to consume

Question 12.
The component of aggregate demand is …………………
a) Personal demand
b) Government expenditure
c) Only export
d) Only import
Answer:
b) Government expenditure

Question 13.
Aggregate supply is equal to ………….
a) ‘C + I + G
b) C+S+T + (x-m)
c) C+ S+ T+ (x – m)
d) C+ S+ T+ Rf
Answer:
d) C+ S+ T+ Rf

Question 14.
Keynes theory pursues to replace Laissez Faire by ………………….
a) No government intervention
b) Maximum intervention
c) State intervention in certain situation .
d) Private Sector Intervention
Answer:
c) State intervention in a certain situation

Question 15.
In Keynes theory of employment and income ……………………… is the basic cause of economic depression.
a) Less production
b) More demand
c) Inelastic supply
d) Less aggregate demand in relation to productive capacity
Answer:
d) Less aggregate demand in relation to productive capacity

Question 16.
Classical theory advocates ………………
a) Balanced budget
b) Unbalanced budget
c) Surplus budget
d) Deficit budget
Answer:
a) Balanced budget

Question 17.
Keynes theory emphasized on ………………….. equilibrium.
a) Very short run
b) Short-run
c) Very long run
d) Long run
Answer:
b) Short-run

Question 18.
According to classical theory, the rate of interest is a reward for …………….
a) Investment
b) Demand
c) Capital
d) Saving
Answer:
d) Saving

Question 19.
In Keynes theory, the demand for and supply of money are determined by …………………..
a) Rate of interest
b) Effective demand
c) Aggregate demand
d) Aggregate supply
Answer:
a) Rate of interest

Question 20.
Say’s law stressed the operation of ……………………. in the economy.
a) Induced price mechanism
b) Automatic price mechanism
c) Induced demand
d) Induced investment
Answer:
b) Automatic price mechanism

PART -B

Answer the following questions in one or two sentences.

Question 21.
Define full employment.
Answer:
Full employment refers to a situation in which every able-bodied person who is willing to work at the prevailing wage rate, is employed. In other words, full employment means that persons who are willing to work and able to work must have employment or a job.

Question 22.
What is the main feature of rural unemployment?
Answer:
The existence of disguised unemployment and seasonal unemployment is the main feature of rural unemployment.

Question 23.
Give a short note on frictional unemployment.
Answer:
Frictional Unemployment (Temporary Unemployment):

1. Frictional unemployment arises due to an imbalance between the supply of labour and demand for labour.
2. This is because of immobility of labour, lack of necessary skills, break down of machinery, shortage of raw materials etc.
3. The persons who lose jobs and in search of jobs are also included under frictional unemployment.

Question 24.
Give reasons for labour retrenchment at the present situation.
Answer:

• Capital intensive techniques.
• Invention and innovations
• Labour saving devices are reasons for retrenchment.

Question 25.
List out the assumptions of Say’s law.
Answer:
The Say’s Law of the market is based on the following assumptions:

1. No single buyer or seller of commodity or input can affect the price.
2. Full employment.
3. People are motivated by self-interest and self-interest determines economic decisions.
4. The laissez-faire policy is essential for an automatic and self-adjusting process of full employment equilibrium. Market forces determine everything right.
5. There will be perfect competition in labour and product market.
6. There is wage-price flexibility.
7. Money acts only as a medium of exchange.
8. Long-run analysis.
9. There is no possibility for overproduction or unemployment.

Question 26.
What is effective demand?
Answer:

• Effective demand denotes money actually spent by the people on products of industry.
• Effective demand equals national income.

Question 27.
What are the components of aggregate supply?
Answer:
Aggregate demand has the following four components:

1. Consumption demand
2. Investment demand
3. Government expenditure and
4. Net Export (export-import)

PART – C

Answer the following questions in a paragraph.
Question 28.
Explain the following in short
Answer:
(i) Seasonal unemployment
This type of unemployment occurs during certain seasons of the year.
(ii) Frictional unemployment
This type of unemployment arises due to an imbalance between supply and demand for labour.
iii) Educated unemployment
Sometimes educated people are underemployed or unemployed when qualification does not match the job.

Question 29.
Write a short note on the implications of Say’s law.
Answer:
Implications of Say’s Law:

1. There is no possibility for overproduction or unemployment.
2. If there exist unutilized resources in the economy, it is profitable to employ them up to the point of full employment. This is true under the condition that factors are willing to accept rewards on a par with their productivity.
3. As an automatic price mechanism operates in the economy, there is no need for government intervention. (However, J.M. Keynes emphasized the role of the State)
4. Interest flexibility brings about equality between saving and investment.
5. Money performs only the medium of exchange function in the economy, as people will not hold idle money.

Question 30.
Explain Keynes theory in the form of a flow chart.
Answer:

Question 31.
What do you mean by aggregate demand? Mention its components.
Answer:

1. The aggregate demand is the amount of money which entrepreneurs expect to get by selling the output produced by the number of labourers employed.
2. Therefore, it is the expected income or revenue from the sale of output at different levels of employment.
3. Aggregate demand has the following four components:
   • Consumption demand
   • InvestmenTdemand
   • Government expenditure and
   • Net Export (export-import)

Question 32.
Explain aggregate supply with the help of a diagram.
Answer:
In figure two aggregate supply curves are drawn with the assumption of fixed money wages and variable wages…

• Z curve is linear where money wages remain fixed; Z1.curve is non-linear since wage rate increases with employment.
• At full employment level (Nf) aggregate supply curve becomes inelastic.
• The slope of the aggregate supply curve depends on the relationship between employment and productivity.
• Based on this relation, the aggregate supply curve is expected to slope upwards. In reality, the aggregate supply curve will be like Z1.
• Therefore, the aggregate supply depends on the relationship between price and wages.

Question 33.
Write any five differences between classicism and Keynesianism.
Answer:

S.No	Keynesianism	Classicism
1.	Short-run equilibrium	Long-run equilibrium
2.	Saving is a vice	Saving is a social virtue
3.	State intervention is advocated	Laissez-faire policy
4.	Rate of interest is a flow	Rate of interest is a stock
5.	Demand creates its own supply	Supply creates its own demand

PART – D

Answer the following questions on the about page.

Question 34.
Describe the types of unemployment.
Answer:
1. Cyclical unemployment:
In a business cycle during the period of recession and depression, income and output fall leading to widespread unemployment. It can be cured by public investment or expansionary monetary policy.

2. Seasonal unemployment:
This type of unemployment occurs during certain seasons of the year.
Eg : Agriculture and Agro-based industries.

3. Frictional unemployment:
This type of unemployment arises due to an imbalance between supply and demand for labour.

4. Educated unemployment:
Sometimes educated people are underemployed or unemployed when qualification does not match the job.

5. Technical unemployment:
Modern technology being capital intensive requires less labourers and con-tributes to technological unemployment.

6. Structural unemployment:
Structural unemployment is due to a drastic change in the structure of society.

7. Disguised unemployment:
A person is said to be disguisedly unemployed if his contribution top out-put is less than what he can produce by working for normal hours per day. In this situation, the marginal productivity of labour is zero or less or negative.

Question 35.
Critically explain Say’s law of the market.
Answer:
Criticisms of Say’s Law: The following are the criticisms against Say’s law:

1. According to Keynes, supply does not create its demand. It is not applicable where demand does not increase as much as production increases.
2. The automatic adjustment process will not remove unemployment. Unemployment can be removed by an increase in the rate of investment.
3. Money is not neutral. Individuals hold money for unforeseen contingencies while businessmen keep a cash reserves for future activities.
4. Say’s law is based on the proposition that supply creates its own demand and there is no overproduction. Keynes said that overproduction is possible.
5. Keynes regards full employment as a special case because there is underemployment in capitalist economies.
6. The need for state intervention arises in the case of general overproduction and mass unemployment.

Question 36.
Narrate the equilibrium between ADF and ASF with a diagram.
Answer:

• Under the Keynes theory of employment, a simple two-sector economy is taken to understand the equilibrium between ADF and ASF.
• All the decisions concerning consumption expenditure are taken by the house-holds, while the business firms take decisions concerning investment.
• It is also assumed that the consumption function is linear and planned investment is autonomous.
  
• In the figure, the aggregate demand and aggregate supply reach equilibrium at point E. The employment level is No at that point.
• At ON1 employment, the aggregate supply is N₁R₁ But they are able to produce M₁N₁  M₁ R₁To the expected level of profit is M₁R₁ attain this level of profit, entrepreneurs will employ more labourers.
• The tendency to employ more labour will stop once they reach E. At all levels of employment beyond, ON_o, the AD curve is below the AS curve indicating loss to the producers.
• Hence they will never employ more than ON_o labor. Thus effective demand concept becomes a crucial point in determining the equilibrium level of output in the capitalist economy in the Keynesian system.
• The equilibrium level of employment need not be the full employment level (N)₁. The difference between (N_o –  N_f )is the level of unemployment.
• Thus Effective demand is significant to explain the under-employment equilibrium.

Question 37.
Explain the differences between classical theory and Keynes theory:
Answer:

Keynesianism	Classicism
1.Short- run equilibrium	Long-run equilibrium
2. Saving is a vice	Saving is a social virtue
3. The function of money as a medium of exchange and store of value	The function of money is to act as a medium of exchange
4. Macro approach to the national problem	Micro foundation to macro problems
5. State intervention is advocated	Champions of Laissez-faire policy
6. Applicable to both full employment and less than full employment level	Applicable only to the full employment situation.
7. Capitalism has inherent contradictions	Capitalism is well and good
8. Unbalanced budget	Balanced Budget
9. Rate of interest is a flow	Rate of interest is a stock
10. Demand creates its own supply	Supply creates its own demand.

12th Economics Guide Theories of Employment and Income Additional Important Questions and Answers

I. Match the following

Question 1.
a) J.B. Say – 1) Classical Economists
b) Adam Smith – 2) Law of Market
c) AC. Pigou – 3) Effective Demand
d) J.M.Keynes – 4) Wealth of Nations

Answer:
b) 2 4 1 3

Question 2.
a) Rural Areas – 1) Educated unemployment
b) Urban Areas – 2) Seasonal unemployment
c) Lack of employable skills – 3) Structural unemployment
d) Lack of demand – 4) Frictional unemployment

Answer:
a) 2 4 13

Question 3.
a) MPC – 1) C + I + G + (X – M)
b) MEC – 2) C + S + T + Rf
c) AD – 3) Marginal propensity to consume ’
d) AS – 4) Marginal Efficiency of capital

Answer :
d) 3 4 1 2.

II. Choose the correct pair

Question 1.
a) Long-run – Full employment
b) Market Economy – Government control
c) Keynes theory – Long run
d) Structural unemployment – Agriculture
Answer:
a) Long-run – Full employment

Question 2.
a) C – Private savings
b) S – Consumption Expenditure
c)T – Net tax payments ‘
d) Rf – Tax rate
Answer:
c) T – Net tax payments

Question 3.
a) Effective demand – Percapita Income
b) Equilibrium – Full employment
c) Aggregate supply function – Elastic
d) Say’s law – Underemployment
Answer:
b) Equilibrium – Full employment

III. Choose the incorrect pair

Question 1.
a) Price Mechanism- Lassiez faire Economy
b) J.B. Say – Classical theory
c) J.M. Keynes – Macro Economics
d) Law of the market – Underemployment
Answer:
d) Law of the market – Underemployment

Question 2.
a) Business cycle
b) Seasonal unemployment
c) New technology
d) Disguised unemployment
Answer:
b) Seasonal unemployment – Industrial sector

Question 3.
a) Aggregate supply price – Income
b) Consumption function – Straight line
c) Keynes theory – Induced Investment
d) Equilibrium – Underemployment
Answer:
d) Equilibrium – Underemployment

IV. Choose the correct statement

Question 1.
a) Equilibrium level of employment refers to full employment.
b) According to Say’s law demand creates its own supply.
c) Adam smith published his books “An enquiry into the nature and causes of the wealth of nations” in the year 1776.
d) Technical unemployment is due to drastic change in the structure of the society
Answer:
c) Adam smith published his books “An enquiry into the nature and causes of wealth of nations” in the year 1776.

Question 2.
Assumptions of Say’s law of markets :
a) Existence of underemployment.
b) Money is a prominent force in the economy.
c) Government interference is a must.
d) Long-run analysis.
Answer: Long run analysis.

Question 3.
a) Keynes supported the ideas of classical Economists.
b) An increase in the aggregate effective demand would increase the level of employment.
c) Effective Demand is less than National income.
d) Price refers to the cost of production.
Answer:
b) An increase in the aggregate effective demand would increase the level of employment.

V. Choose the incorrect statement

Question 1.
a) Keynes defines full employment as the absence of involuntary unemployment.
b) Full employment generally refers to the full employment of labour force of a country.
c) In developed countries the unemployment is purely structural.
d) Disguised unemployment occurs when more people are than what is actually required.
Answer:
c) In developed countries the unemployment is purely structural.

Question 2.
a) In Disguised unemployment and seasonal unemployment are seen in rural areas.
b) Due to Globalisation, a large number of people move from rural areas to urban areas.
c) Cyclical unemployment can be cured by public investment or expansionary monetary policy.
d) Seasonal unemployment occurs during certain seasons of the year.
Answer:
b) Due to Globalisation, a large number of people move from rural areas to urban areas.

Question 3.
a) In disguised unemployment marginal productivity of labour is positive.
b) Laissez-faire is a situation found in Market Economy.
c) According to J.B.Say “supply creates its own demand”.
d) Aggregate supply curve is inelastic.
Answer:
c) According to J.B.Say “supply creates its own demand”.

VI. Pick the odd one out:

Question 1.
The components of Aggregate supply are
a) Total Expenditure
b) Private savings
c) Net tax payments
d) Transfer payments to the foreigners
Answer:
a) Total Expenditure

Question 2.
The components of Aggregate Demand are
a) Consumption demand
b) Investment demand
c) Government expenditure
d) Net Import
Answer:
d) Net Import

VII. Analyse the reason:

Question 1.
Assertion (A): Aggregate supply is equal to the value of the national product.
Reason (R): Aggregate supply refers to the value of the total output of goods and services produced in an economy in a year.
Answer :
a) Assertion (A) and Reason (R) both are true and (R) is the correct explanation of (A)

Question 2.
Assertion (A): Supply creates its own demand.
Reason (R): Each product produced in the economy creates demand equal to its value in the market.
Answer:
a) Assertion (A) and Reason (R) both are true and (R) is the correct explanation of (A)

Question 3.
Assertion (A) : Seasonal unemployment occurs during certain seasons of the year.
Reason (R) : Seasonal unemployment exists during the downturn phase of the trade cycle in the economy.
Answer:
c) Assertion (A) is true, Reason (R) is false Options.

Question 4.
Assertion (A): Seasonal unemployment occurs during certain seasons of the year.
Reason (R): Seasonal unemployment exists during the downturn phase of the trade cycle in the economy.
Answer:
c) Assertion (A) is true, Reason (R) is false.
Options:
a)Assertion (A) and Reason (R) both are true and (R) is the correct explanation of (A).
b) Assertion (A) and Reason (R) both are true, but (R) is not the correct, explanation of (A).
c) Assertion (A) is true, Reason (R) is false.
d) Both (A) and (R) are false.

VIII. Choose the best Answer

Question 1.
Who is one of the greatest and most influential economists?
(a) J.M. Keynes
(b) Adam Smith
(c) Marshall
(d) Simon Kuznets
Answer:
(a) J.M. Keynes

Question 2.
Aggregate demand can be expressed as
a) AD = C + G + I + (M – X)
b) AD = I + G + C + (X – M)
c) AD = C + I + G + (X – M)
d) AD = C + I + G +(x – M)
Answer:
c) AD = C+I+G+(X-M)

Question 3.
The total stock of money circulating in an Economy is called –
(a) Money
(b) Capital
(c) Money Supply
(d) Finance
Answer:
(c) Money Supply

Question 4.
Capital intensive technology and innovations lead to ………………….. Unemployment.
a) Structural
b) Technical
c) Seasonal
d) Disguised
Answer:
b) Technical

Question 5.
How many components are there in the aggregate supply
a) 3
b) 4
c) 1
d) 2
Answer:
b) 4

Question 6.
…………………… is an increasing function of the level of employment –
(a) Aggregate supply function
(b) Aggregate demand function
(c) Aggregate consumption function
(d) Aggregate consumption expenditure
Answer:
(a) Aggregate supply function

Question 7.
Keynes defines full employment as the absence of ………………….. unemployment.
a) Structural
b) Involuntary
c) Seasonal
d) Voluntary
Answer:
b) Involuntary

Question 8.
…………………… Unemployment arises due to an imbalance between the supply of and demand for labour.
a) Frictional
b) Educated
c) Seasonal
d) Technical
Answer:
a) Frictional

Question 9.
Adam Smith published his book ‘An enquiry into the Nature and Causes of the Wealth of Nations” in ’ ……………………
a) 1767
b) 1776
c) 1676
d) 1176
Answer:
b) 1776

Question 10.
Frictional unemployment another name is called –
(a) Educated unemployment
(b) Seasonal unemployment
(c) Temporary unemployment
(d) Technical unemployment
Answer:
(c) Temporary unemployment

IX. Answer the following questions (2 Marks)

Question 1.
Write two approaches of the equilibrium level of Income in Keynesian theory?
Answer:
There are two approaches to the determination of the equilibrium level of income in Keynesian theory. These are:

1. Aggregate demand – Aggregate supply approach
2. Saving – An investment approach

Question 2.
What is marginal propensity to consume?
Answer:
Marginal propensity to consume is the additional consumption due to an additional unit of income.

Question 3.
Define “Marginal propensity to consume”?
Answer:
Marginal Propensity to Consume is the additional consumption due to an additional unit of income.

Question 4.
What is Underemployment?
Answer:
Underemployment is a situation where resources are not fully utilized in production.

Question 4.
Write the types of unemployment?
Answer:
Types of unemployment:

1. Cyclical Unemployment
2. Frictional Unemployment
3. Technical Unemployment
4. Disguised Unemployment
5. Seasonal Unemployment
6. Educated Unemployment
7. Structural Unemployment

Question 6.
State J.B.Say’s law of Market.
Answer:
J.B. Say enunciated the proposition that” Supply creates its own demand”.

X. Answer the following questions (3 Marks)

Question 1.
Mention the types of Unemployment.
Answer:
There are seven types of unemployment. They are

1. Cyclical Unemployment
2. Seasonal Unemployment
3. Frictional Unemployment
4. Educated Unemployment
5. Technical Unemployment
6. Structural Unemployment
7. Disguised Unemployment

Question 2.
Aggregate Supply Function meaning and components?
Answer:

1. Aggregate supply function is an increasing function of the level of employment.
2. Aggregate supply refers to the value of the total output of goods and services produced in an economy in a year.
3. In other words, aggregate supply is equal to the value of the national product, i.e., national income.

The components of aggregate supply are:

1. Aggregate (desired) consumption expenditure (C)
2. Aggregate (desired) private savings (S)
3. Net tax payments (T) (Total tax payment to be received by the government minus transfer payments, subsidy and interest payments to be incurred by the government)
4. Personal (desired) transfer payments to the foreigners (Rf) (e.g. Donations to international relief efforts)

Question 3.
What is the Marginal Efficiency of Capital?
Answer:

• Marginal Efficiency of Capital is the expected rate of return over costs of a new capital good.
• MEC depends on two factors namely prospective yield of a capital asset and sup-ply price of capital.

Question 4.
Name the Motives of Liquidity Preference.
Answer:
Liquidity preference is based on three motives namely

• Transaction Motive
• Precautionary Motive
• Speculative Motive

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Business Maths Guide Pdf Chapter 5 Numerical Methods Ex 5.3 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Ex 5.3

Choose the most suitable answer from the given four alternatives:

Question 1.
Δ²y₀ =
(a) y₂ – 2y₁ + y₀
(b) y₂ + 2y₁ – y₀
(c) y₂ + 2y₁ + y₀
(d) y₂ + y₁ + 2y₀
Solution:
(a) y₂ – 2y₁ + y₀
Hint:
Δ²y₀ = (E – 1)²y₀
= (E² – 2E + 1) y₀
= E²y₀ – 2Ey₀ + y₀
= y₂ – 2y₁ + y₀

Question 2.
Δf(x) =
(a) f (x + h)
(b) f (x) – f (x + h)
(c) f (x + h) – f(x)
(d) f(x) – f (x – h)
Solution:
(c) f (x + h) – f(x)
Hint:
Δf(x) = f (x + h) – f(x)

Question 3.
E≡
(a) 1 + Δ
(b) 1 – Δ
(c) 1 + ∇
(d) 1 – ∇
Solution:
(a) 1 + Δ

Question 4.
If h = 1, then Δ(x²) =
(a) 2x
(b) 2x – 1
(c) 2x + 1
(d) 1
Solution:
(c) 2x + 1
Hint:
Δ(x²) = (x + h)² – (x)²
= x² + 2xh + h² – x²
Δ(x²) = 2xh + h²
If h = 1 Δ (x²) = 2x + 1

Question 5.
If c is a constant then Δc =
(a) c
(b) Δ
(c) Δ²
(d) 0
Solution:
(d) 0

Question 6.
If m and n are positive integers then Δ^m Δⁿ f(x)=
(a) Δ^m+n f(x)
(b) Δ^m f(x)
(c) Δⁿ f(x)
(d) Δ^m-n f(x)
Solution:
(a) Δ^m+n f(x)

Question 7.
If ‘n’ is a positive integer Δⁿ[Δ^-n f(x)]
(a) f(2x)
(b) f(x + h)
(c) f(x)
(d) Δf(x)
Solution:
(c) f(x)
Hint:
Δⁿ [Δ^-n f(x)] = Δ^n-n f(x) = Δ⁰ f(x)
= f(x)

Question 8.
E f(x) =
(a) f(x – h)
(b) f(x)
(c) f(x + h)
(d) f(x + 2h)
Solution:
(c) f (x + h)

Question 9.
∇≡
(a) 1 + E
(b) 1 – E
(c) 1 – E^-1
(d) 1 + E^-1
Solution:
(c) 1 – E^-1

Question 10.
∇ f(a) =
(a) f(a) + f(a – h)
(b) f(a) – f(a + h)
(c) f(a) – f(a – h)
(d) f(a)
Solution:
(c) f(a) – f(a- h)

Question 11.
For the given points (x₀, y₀) and (x₁, y₁) the Lagrange’s formula is
(a) y(x) = \(\frac { x-x_1 }{x_0-x_1}\) y₀ + \(\frac { x-x_0 }{x_1-x_0}\) y₁
(b) y(x) = \(\frac { x_1-x}{x_0-x_1}\) y₀ + \(\frac { x-x_0 }{x_1-x_0}\) y₁
(c) y(x) = \(\frac { x-x_1 }{x_0-x_1}\) y₀ + \(\frac { x-x_0 }{x_1-x_0}\) y₀
(d) y(x) = \(\frac { x_1-x }{x_0-x_1}\) y₀ + \(\frac { x-x_0 }{x_1-x_0}\) y₀
Solution:
(a) y(x) = \(\frac { x-x_1 }{x_0-x_1}\) y₀ + \(\frac { x-x_0 }{x_1-x_0}\) y₁

Question 12.
Lagrange’s interpolation formula can be used for
(a) equal intervals only
(b) unequal intervals only
(c) both equal and unequal intervals
(d) none of these.
Solution:
(c) both equal and unequal intervals.

Question 13.
If f(x) = x² + 2x + 2 and the interval of differencing is unity then Δf(x)
(a) 2x – 3
(b) 2x + 3
(c) x + 3
(d) x – 3
Solution:
(b) 2x + 3
Hint:
Given:
f(x) = x² + 2x + 2
Δf(x) = f (x + h) – f (x)
since h = 1
Δf(x) = f (x – 1) – f (x)
= [(x + 1)² + 2(x + 1) + 2] – [x² + 2x + 2]
= [x² + 2x + 1 + 2x + 2 + 2] – [x² + 2x + 2]
= x² + 4x + 5 – x² – 2x – 2
= 2x + 3

Question 14.
For the given data find the value of Δ³y₀ is

(a) 1
(b) 0
(c) 2
(d) -1
Solution:
(b) 0
Hint:
From this data
y₀ = 12; y₁ = 13; y₂ = 15; y₃ = 18
Δ³y₀ = (E – 1)³ y₀
= (E³ – 3E² + 3E – 1) y₀
= E³y₀ – 3E²y₀ + 3Ey₀ – y₀
= y₃ – 3y₂ + 3y₁ – y₀
= 18 – 3 (15) + 3 (13) – 12
= 18 – 45 + 39 – 12
= 57 – 57 = 0

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Business Maths Guide Pdf Chapter 5 Numerical Methods Ex 5.2 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 5 Numerical Methods Ex 5.2

Question 1.
Using graphic method, find the value of y when x = 48 from the following data:

Solution:

The value of y when x = 48 is 6.8

Question 2.
The following data relates to indirect labour expenses and the level of output

Estimate the expenses at a level of output of 350 units, by using graphic method.
Solution:

Question 3.
Using Newton’s forward interpolation formula find the cubic polynomial.

Solution:
Since we use the Newton’s forward interpolation formula.
y_{(x= x0+nh)} = y₀ + \(\frac { n }{1!}\) Δy₀ + \(\frac { n(n-1) }{2!}\) Δ²y₀ + \(\frac { n(n-1)(n-2) }{3!}\) Δ³y₀ + ………
To find y at x
∴ x₀ + nh = x
0 + n(1) = x
∴ n = x

= 1 + x + (x² – x) (-1) + 2x (x² – 3x + 2)
y = 1 + x – x² + x + 2x³ – 6x² + 4x
y = 2x³ – 7x² + 6x + 1
∴ f(x) = 2x³ – 7x² + 6x + 1

Question 4.
The population of a city in a censes taken once in 10 years is given below. Estimate the population in the year 1955.

Solution:
Let the year be x and population be y. To find the population for the year 1955.
(ie) The value of y at x = 1955
Since the value of y is required near the beginning of the table, we use the Newton’s forward interpolation formula.
y_{(x= x0+nh)} = y₀ + \(\frac { n }{1!}\) Δy₀ + \(\frac { n(n-1) }{2!}\) Δ²y₀ + \(\frac { n(n-1)(n-2) }{3!}\) Δ³y₀ + ………
To find y at x = 1955
∴ x₀ + nh = 1955; x₀ = 1951, h = 10
⇒ 1951 + n(10) = 1955
10n = 1955 – 1951 ⇒ 10n = 4
n = \(\frac { 4 }{10}\) = 0.4

y = 35 + 2.8 – 1.08 + 0.064
= 37.864 – 1.08
y = 36.784
∴ Population in the year 1955 is 36.784 (lakhs)

Question 5.
In an examination the number of candidates who secured marks between certain interval were as follows:

Estimate the number of candidates whose marks are lessthan 70.
Solution:
Since the required mark is at the end of the table, we apply Backward interpolation formula. Let the marks be x and No. of candidates be y.

To find y at x = 70
x = x₀ + nh ⇒ 70 = 100 + n(20)
70 – 100 = 20n
20n = -30 ⇒ n = \(\frac { -30 }{20}\)
n = -1.5

= 235 – 25.5 – 12.375 – 1.125
= 235 – 39
= 196
∴ 196 candidates secured less than 70 marks

Question 6.
Find the value of f(x) when x = 32 from the following table

Solution:
Since the value of f(x) is required near the beginning of the table, we use the Newton’s forward interpolation formula.
y_{(x= x0+nh)} = y₀ + \(\frac { n }{1!}\) Δy₀ + \(\frac { n(n-1) }{2!}\) Δ²y₀ + \(\frac { n(n-1)(n-2) }{3!}\) Δ³y₀ + ………
To find y at x = 32
∴ x₀ + nh = 32;
30 + n(5) = 32
5n = 32 – 30 ⇒ 5n = 2
n = \(\frac { 2 }{5}\)
∴ n = 0.4

= 15.9 – 0.4 – 0.024 – 0.0128 – 0.00832
15.9 – 0.44512 = 15.45488
= 15.45
∴ when x = 32, f(x) = 15.45

Question 7.
The following data gives the melting point of a alloy of lead and zinc where ‘t’ is the temperature in degree c and p is the percentage of lead in the alloy

Find the melting point of the alloy containing 84 percent lead.
Solution:
Since the required value is at the end of the table, apply backward interpolation formula. To find T at p = 84
T_{(p= p0+nh)} = T_n + \(\frac { n }{1!}\) ∇T_n + \(\frac { n(n+1) }{2!}\) ∇²T₀ + \(\frac { n(n+1)(n+2) }{3!}\) Δ³T₀ + ………
To find T at P = 84
P_n + nh = 84
90 + n(10) = 84
10n = 84 – 90
10n = -6 ⇒ n = \(\frac { -6 }{10}\)
n = -0.6

= 304 – 16.8 – 0.24 – 0.091392
= 304 – 17.131392
= 286.86
Hence the melting point of the alloy is 286.86° c.

Question 8.
Find f(2.8) from the following

Solution:
Since the required value is at the end of the table, apply backward interpolation formula.

To find y at x = 2.8
∴ x₀ + nh = 2.8
∴ 3 + n(1) = 2.8
n = 2.8 – 3
n = -0.2

= 34 – 4.6 – 1.12 – 0.288
= 34 – 6.008
= 27.992
∴ f(2.8) = 27.992

Question 9.
Using interpolation estimate the output of a factory in 1986 from the following data

Solution:
Here the intervals are unequal. By Lagrange’s in-terpolation formula we have,
x₀ = 1974, x₁ = 1978, x₂ = 1982, x₃ = 1990
y₀ = 25, y₁ = 60, y₂ = 80, y₃ = 170, and x = 1986.

∴ output in 1986 is 108.75 (thousand tones)

Question 10.
Use lagrange’s formula and estimate from the following data the number of workers getting income not exceeding Rs. 26 per month.

Solution:
Here the intervals are unequal. By Lagrange’s In-terpolation formula we have,
x₀ = 15, x₁ = 25, x₂ = 30, x₃ = 35
y₀ = 36, y₁ = 40, y₂ = 45, y₃ = 48 and x = 26.

∴ Required No.of workers = 42 Persons (approximately)

Question 11.
Using interpolation estimate the business done in 1985 from the following data

Solution:
Here the intervals are unequal. By Lagrange’s formula we have,
x₀ = 1982, x₁ = 1983, x₂ = 1984, x₃ = 1986
y₀ = 150, y₁ = 235, y₂ = 365, y₃ = 525 and x = 1985.

∴ Business done in the year 1985 is 481.25 lakhs.

Question 12.
Using interpolation, find the value of f(x) when x = 15

Solution:
Here the intervals are unequal, By Lagrange’s in-terpolation formula we have,
x₀ = 3, x₁ = 7, x₂ = 11, x₃ = 19
y₀ = 42, y₁ = 43, y₂ = 47, y₃ = 60 and x = 15.

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Business Maths Guide Pdf Chapter 4 Differential Equations Ex 4.4 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 4 Differential Equations Ex 4.4

Question 1.
\(\frac { dy }{dx}\) – \(\frac { dy }{dx}\) = x
Solution:

The required solution is

Question 2.
\(\frac { dy }{dx}\) + y cos x = sin x cos x
Solution:
It is of the form \(\frac { dy }{dx}\) + Py = Q
Here P = cos x; Q = sin x cos x
∫Pdx = ∫cos x dx = sin x
I.F = e^∫pdx = e^sinx
The required solution is
Y(I.F) = ∫Q (IF) dx + c
Y(e^sinx) = ∫Q (I-F) dx + c
y (e^sinx) = ∫sin x cos x e^sinx dx + c

Question 3.
x\(\frac { dy }{dx}\) + 2y = x⁴
Solution:
The given equation can be reduced to
\(\frac { dy }{dx}\) + \(\frac { 2y }{x}\) = x³
It is of the form \(\frac { dy }{dx}\) + Py = Q
Here P = \(\frac { 2 }{x}\); Q = x³
∫pdx = ∫\(\frac { 2 }{x}\)dx = 2∫\(\frac { 1 }{x}\)dx = 2log x – log x²
I.F = e^∫Pdx = e^logx² = x²
The required solution is
y(I.F) = ∫Q (IF) dx + c
y(x²) = ∫x³ (x²) dx + c
x²y = ∫x⁵ dx + c
x²y = \(\frac { x^6 }{6}\) + c

Question 4.
\(\frac { dy }{dx}\) + \(\frac { 3x^2 }{1+x^3}\) = \(\frac { 1+x^2 }{1+x^3}\)
Solution:

Question 5.
\(\frac { dy }{dx}\) + \(\frac { y }{x}\) = xe^x
Solution:
\(\frac { dy }{dx}\) + py = Q
Here P = \(\frac { 1 }{x}\); Q = xe^x
∫Pdx = ∫\(\frac { 1 }{x}\) dx = log x
I.F = e∫Pdx = e^log = x
The required solution is
y (I.F) = ∫Q (I.F) dx + c

Question 6.
\(\frac { dy }{dx}\) + y tan x = cos³ x
Solution:
It is of the form \(\frac { dy }{dx}\) + Py = Q
Here P = tan x; Q = cos³ x
∫Pdx = ∫tan x dx = ∫\(\frac { sin x }{cos x}\) dx = -∫\(\frac { -sin x }{cos x}\) dx
= -log cos x = log sec x
I.F = e^∫Pdx = e^{log sec x} = sec x
The required solution is
y(I.F) = ∫Q(I.F) dx + c
y (sec x) = ∫cos³x (sec x) dx + c
y(sec x) = ∫cos³x \(\frac { 1 }{cos x}\) dx + c
y (sec x) = ∫cos²x dx + c
y (sec x)= ∫(\(\frac { 1+cos 2x }{2}\)) dx + c
y (sec x) = \(\frac { 1 }{2}\) ∫(1 + cos2x) dx + c
y (sec x) = \(\frac { 1 }{2}\) [x + \(\frac { sin2x }{2}\)] + c

Question 7.
If \(\frac { dy }{dx}\) + 2y tan x = sinx and if y = 0 when x = π/3 express y in terms of x
Solution:
\(\frac { dy }{dx}\) + 2y tan x = sinx
It is of the form \(\frac { dy }{dx}\) + Py = Q
Here P = 2tan x ; Q = sin x
∫Pdx = ∫2 tan x dx = 2∫tan xdx = 2 log sec x
log sec² x
I.F = e^∫Pdx = e^log(sec²x) = sec² x
The required solution is
y(I.F) = ∫Q(I.F) dx + c
y (sec² x) = ∫sin x (sec²x) dx + c
y(sec²x) = ∫sin x(\(\frac { 1 }{cos x}\)) sec x dx + c
y sec²x = ∫(\(\frac { sin x }{cos x}\)) sec x dx + c
y(sec²x) = ∫tan x sec x dx + c
⇒ y(sec²x) = sec x + c ………. (1)
If y = 0 when x = /3, then (1) ⇒
0(sec²(π/3)) = sec(π/3) + c
0 = 2 + c
⇒ c = -2
∴ Eqn (1) ⇒ y sec²x = sec x – 2

Question 8.
\(\frac { dy }{dx}\) + \(\frac { y }{x}\) = xe^x
Solution:
It is of the form \(\frac { dy }{dx}\) + Py = Q
Here P = \(\frac { 1 }{x}\); Q = xe^x
∫Pdx = ∫\(\frac { 1 }{x}\) dx = log x
I.F = e^∫Pdx = e^{log x} = x
The required solution is

Question 9.
A bank pays interest by contionous compounding, that is by treating the interest rate as the instantaneous rate of change of principal. A man invests Rs 1,00,000 in the bank deposit which accures interest, 8% over year compounded continuously. How much will he get after 10 years?
Solution :
Let P(t) denotes the amount of money in the account at time t. Then the differential equation govemning the growth of money is
\(\frac { dp }{dt}\) = \(\frac { 8 }{100}\)p = 0.08 p
⇒ \(\frac { dp }{p}\) = 0.08 dt
Integrating on both sides
∫\(\frac { dp }{p}\) = ∫0.08 dt
log_e P = 0.08 t + c
P = e^{0.08 t} + c
P = e^{0.08 t}. e^c
P = C₁ e^{0.08 t} ………. (1)
when t = 0, P = Rs 1,00,000
Eqn (1) ⇒ 1,00,000 = C₁ e°
C₁ = 1,00,000
∴ P = 100000 e^{0.08 t}
At t = 10
P= 1,00,000 . e^0.08(10)
= 1,00,000 e^0.8 {∵ e^0.8 = 2.2255}
= 100000 (2.2255)
p = Rs 2,25,550

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Business Maths Guide Pdf Chapter 4 Differential Equations Ex 4.1 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Business Maths Solutions Chapter 4 Differential Equations Ex 4.1

Question 1.
Find the order and degree of the following differential equations.
(i) \(\frac { dy }{dx}\) + 2y = x³
Solution:
Highest order derivative is \(\frac { dy }{dx}\)
∴ order = 1
Power of the highest order derivative \(\frac { dy }{dx}\) is 1
∴ degree = 1

(ii) \(\frac { d^3y }{dx^3}\) + 3(\(\frac { dy }{dx}\))³+ 2\(\frac { dy }{dx}\) = 0
Solution:
Highest order derivative is \(\frac { d^3y }{dx^3}\)
∴ order = 3
Power of the highest order derivative \(\frac { d^3y }{dx^3}\) is 1
∴ degree = 1

(iii) \(\frac { d^2y }{dx^2}\) = \(\sqrt{y – \frac { dy }{dx}}\)
Solution:
[ \(\frac { d^2y }{dx^2}\) ]² = y – \(\frac { dy }{dx}\)
Highest order derivative is \(\frac { d^2y }{dx^2}\)
∴ order = 2
Power of the highest order derivative \(\frac { d^2y }{dx^2}\) is 2
∴ degree = 2

(iv) \(\frac { d^3y }{dx^3}\) = 0
Solution:
Highest order derivative is \(\frac { d^3y }{dx^3}\)
∴ order = 3
Power of the highest order derivative \(\frac { d^3y }{dx^3}\) is 1
∴ degree = 1

(v) \(\frac { d^3y }{dx^3}\) + y + [ \(\frac { dy }{dx}\) – \(\frac { d^3y }{dx^3}\) ]^3/2 = 0
Solution:

Highest order derivative is \(\frac { d^3y }{dx^3}\)
∴ order = 3
Power of the highest order derivative \(\frac { d^3y }{dx^3}\) is 3
∴ degree = 3

(vi) (2 – y”)2 = y”² + 2y’
Solution:
(2)² – 2(2) (y”) + (y”)² = (y”)² + 2y’
4 – 4y” = 2y’
Highest order derivative is y”
∴ order = 2
Power of the highest order derivative y” is 2
∴ degree = 2

(vii) (\(\frac { dy }{dx}\))³ + y = x – \(\frac { dx }{dy}\)
Solution:

Highest order derivative is \(\frac { dy }{dx}\)
∴ order = 1
Power of the highest order derivative \(\frac { dy }{dx}\) is 4
∴ degree = 4

Question 2.
Find the differential equation of the following
(i) y = cx + c – c³
(ii) y = c (x – c)²
(iii) xy = c²
(iv) x² + y² = a²
Solution:
(i) y = cx + c – c³ ……. (1)
Here c is a constant which has to be eliminated
Differentiating w.r.t x, \(\frac{d y}{d x}\) = c …… (2)
Using (2) in (1) we get,
\(y=\left(\frac{d y}{d x}\right) x+\frac{d y}{d x}-\left(\frac{d y}{d x}\right)^{3}\) which is the required differential equation.

(ii) y = c (x – c)² ……… (1)
y = c (x² – 2cx + c²)
y = cx² – 2c²x + c³
Differentiating w.r. to x

Substituting this value of c and (x – c) in (1), we get

(iii) xy = c²
Differentiating w.r. to x
x(\(\frac { dy }{dx}\)) + y(1) = 0
∴ x(\(\frac { dy }{dx}\)) + y = 0

(iv) x² + y² = a²
Differentiating w.r. to x

Question 3.
Form the differential equation by eliminating α and ß from (x – α)² + (y – α)² = γ²
Solution:
(x – α)² + (y – α)² = γ² ……… (1)
where α and ß are parameters.
Since equation (1) contains two orbitary constants,
We differentiate it two times w.r.t. x
Differentiating (1) w.r.t. x, we get

Substituting the value of (x – α) and (y – ß) in (5) we get

This is the required differential equation.

Question 4.
Find the differential equation of the family of all straight lines passing through the origin.
Solution:
The general equation for a family of lines passing through the origin is
y = mx ……. (1)
Differentiating w.r.t x,
\(\frac{d y}{d x}\) = m ……. (2)
Using (2) in (1)
y = (\(\frac{d y}{d x}\)) x is the required differential equation

Question 5.
Form the differential equation that represents all parabolas each of which has a latus rectum 4a and whose axes are parallel to the x-axis.
Solution:
The equation of the family of the parabola is
(y – k)² = 4a (x – h) ……. (1)
where h and k are arbitrary constants,
[we have to differentiate the equation twice to eliminate h and k]
Differentiating equation (1) w.r.t. x

Question 6.
Find the differential equation of all circles passing through the origin and having their centers on the y axis, x² + (y – k)² = r²
Solution:
Equation of circle whose centre is (h, k)
(x – h)² + (y – k)² = r²
since the centre is on the y-axis (ie) (0, k) be the centre
(x – 0)² + (y – k)² = r²
x² + (y – k)² = r² ………. (1)
since the circle passing the origin (0, 0)
Eqn (1) becomes
0 + (0 – k)² = r²
k² = r² ⇒ r = k
Eqn (1) ⇒ x² + (y – k)² = k²
x² + y² – 2yk + k² = k²
x² + y² – 2yk = 0
x² + y² = 2yk ……….. (2)
Differentiating w.r.t. x

Question 7.
Find the differential equation of the family of parabola with foci at the origin and axis along the x axis, y² = 4a (x + a)
Solution:
Equation of parabola with foci at the origin and axis along the x-axis is
y² = 4a(x + a) ……… (1)
Differentiate w.r.t. x
2y \(\frac { dy }{dx}\) = 4a (1 + 0)
2y = \(\frac { dy }{dx}\) = 4a ⇒ a = \(\frac { y }{2}\), \(\frac { dy }{dx}\)
Substitute the value of a = \(\frac { y }{2}\) \(\frac { dy }{dx}\) in equ (1)