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Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 11 Integral Calculus Ex 11.6 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.6

Question 1.
\(\frac{x}{\sqrt{1+x^{2}}}\)
Answer:
Put 1 + x² = u
2x dx = du
x dx = \(\frac{1}{2}\) du

Question 2.
\(\frac{x^{2}}{1+x^{6}}\)
Answer:

Put x³ = u
3x² dx = du
x² dx = \(\frac{1}{3}\) du

Question 3.
\(\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}\)
Answer:

Question 4.

Answer:

Question 5.

Answer:

Question 6.
\(\frac{\cot x}{\log (\sin x)}\)
Answer:

Question 7.

Answer:

Question 8.

Answer:

Put a² + b² sin²x = u
(0 + b² × 2 sin x cos x) dx = du
b² sin 2x dx = du
sin 2x dx = \(\frac{1}{b^{2}}\) du

Question 9.
\(\frac{\sin ^{-1} x}{\sqrt{1-x^{2}}}\)
Answer:

Question 10.
\(\frac{\sqrt{x}}{1+\sqrt{x}}\)
Answer:

Question 11.

Answer:

Question 12.
α β x^α-1 e^{-β xα}
Answer:
∫α β x^α-1 e^{-β xα}
Put β x^α = u
α β x^α-1 dx = du

Question 13.
\(\tan x \sqrt{\sec x}\)
Answer:

put cos x = u
– sin x dx = du
sin x dx = – du

Question 14.
x (1 – x)¹⁷
Answer:
∫x (1 – x)¹⁷ dx
put 1 – x = u
-dx = du

Question 15.
sin⁵ x cos³ x
Answer:
∫ sin⁵ x cos³ x dx = ∫ sin⁵ x cos² x . cos x dx
= ∫ sin⁵ x (1 – sin² x) . cos x dx
= ∫ (sin⁵ x – sin⁷ x) . cos x dx
= ∫ sin⁵ x cos x dx – ∫ sin⁷ x cos x dx
put u = sin x
du = cos x dx

Question 16.

Answer:

Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 11 Integral Calculus Ex 11.11 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.11

Integrate the following with respect to x:

Question 1.
(i) \(\frac{2 x-3}{x^{2}+4 x-12}\)
Answer:
Let 2x – 3 = A \(\frac{\mathrm{d}}{\mathrm{d} x}\) (x² + 4x – 12) + B
2x – 3 = A (2x + 4) + B
2x – 3 = 2Ax + 4A + B
2A = 2 ⇒ A = 1
4A + B = – 3
4 × 1 + B – 3 ⇒ B = – 3 – 4 = – 7
2x – 3 = 1 (2x + 4) – 7

(ii) \(\frac{5 x-2}{2+2 x+x^{2}}\)
Answer:
Let 5x – 2 = A \(\frac{\mathrm{d}}{\mathrm{d} x}\) (x² + 2x + 2) + B
5x – 2 = A (2x + 2) + B
5x – 2 = 2Ax + 2A + B
2A = 5 ⇒ A = \(\frac{5}{2}\)
2A + B = -2
2 × \(\frac{5}{2}\) + B = -2 ⇒ B = -2 – 5 = -7
5x – 2 = \(\frac{5}{2}\) (2x + 2) – 7

Put x² + 2x + 12 = t
(2x + 2) dx = dt

Put x + 1 = u
dx = du

(iii) \(\frac{3 x+1}{2 x^{2}-2 x+3}\)
Answer:
Let 3x + 1 = A \(\frac{\mathrm{d}}{\mathrm{d} x}\) (2x² – 2x + 3) + B
3x + 1 = A (4x – 2) + B
3x + 1 = 4Ax – 2A + B

Question 2.
(i) \(\frac{2 x+1}{\sqrt{9+4 x-x^{2}}}\)
Answer:
Let 2x + 1 = A \(\frac{\mathrm{d}}{\mathrm{d} x}\) (9 + 4x – x²) + B
2x + 1 = A(4 – 2x) + B
2x + 1 = 4A – 2Ax + B
-2A = 2 ⇒ A = -1
4A + B = 1 ⇒ 4 (-1) + B = 1
B = 1 + 4 = 5
B = 5
2x + 1 = – 1 (4 – 2x) + 5

Put 9 + 4x – x² = t
(4x – 2) dx = dt

Put x – 2 = u
dx = du

(ii) \(\frac{x+2}{\sqrt{x^{2}-1}}\)
Answer:
Let x + 2 = A \(\frac{\mathrm{d}}{\mathrm{d} x}\) (x² – 1) + B
x + 2 = A (2x) + B
2A = 1 ⇒ A = \(\frac{1}{2}\)
B = 2

(iii) \(\frac{2 x+3}{\sqrt{x^{2}+4 x+1}}\)
Answer:
Let 2x + 3 = A \(\frac{\mathrm{d}}{\mathrm{d} x}\) (x² + 4x + 1) + B
2x + 3 = A (2x + 3) + B
2A = 2 ⇒ A = 1
4A + B = 3
4 × 1 + B = 3 ⇒ B = 3 – 4
B = -1
(2x + 3) = (2x + 4) – 1

Put x² + 4x + 1 = t
(2x + 4) dx = dt

Put x + 2 = u
dx = dt

Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 11 Integral Calculus Ex 11.10 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.10

Find the integrals of the following:

Question 1.
(i) \(\frac{1}{4-x^{2}}\)
Answer:

(ii) \(\frac{1}{25-4 x^{2}}\)
Answer:

(iii) \(\frac{1}{9 x^{2}-4}\)
Answer:

Question 2.
(i) \(\frac{1}{6 x-7-x^{2}}\)
Answer:

(ii) \(\frac{1}{(x+1)^{2}-25}\)
Answer:

(iii) \(\frac{1}{\sqrt{x^{2}+4 x+2}}\)
Answer:

Question 3.
(i) \(\frac{1}{\sqrt{(2+x)^{2}-1}}\)
Answer:

(ii) \(\frac{1}{\sqrt{x^{2}-4 x+5}}\)
Answer:

(iii) \(\frac{1}{\sqrt{9+8 x-x^{2}}}\)
Answer:

Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 11 Integral Calculus Ex 11.9 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.9

Integrate the following with respect to x:

Question 1.
e^x (tan x + log sec x)
Answer:
Let I = ∫e^x (tan x + log sec x) dx
Take f(x) = log sec x
f'(x) = \(\frac{1}{\sec x}\) sec x tan x
f'(x) = tan x
[∫e^x [f (x) + f (x)] dx = e^x f(x) + c]
∴ I = e^x log |sec x| + c

Question 2.

Answer:

[∫ e^x [f (x) + f (x)] dx = e^x f(x) + c]

Question 3.
e^x sec x (1 + tan x)
Answer:
Let I = \(\mathrm{I}=\int e^{x}(\sec x+\sec x \tan x) d x\)
Take f(x) = sec x
f ‘ (x) = sec x tan x
This is of the form of \(\int e^{x}\left[f(x)+f^{\prime}(x)\right] d x\) = e^x f(x) + c
= e^x sec x + c

Question 4.

Answer:

Take f(x) = tan x
f'(x) = sec² x
[∫ e^x [f (x) + f (x)] dx = e^x f(x) + c]
I = e^x tan x + c

Question 5.

Answer:

Put tan^-1 x = t
\(\frac{1}{1+x^{2}}\)dx = dt
x = tan t
∴ I = ∫e^t [1 + tan t + tan²t] dt
= ∫e^t [1 + tan² t + tan t] dt
= ∫e^t [sec²t + tan t] dt
= ∫e^t [tan t + sec²t] dt
f(x) = tan t
f'(x) = sec²t
[∫ e^x [f (x) + f (x)] dx = e^x f(x) + c]
∴ I = e^t tan t + c
I = e^tan-1(x) . x + c
I = x e^tan-1(x) + c

Question 6.

Answer:

Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 11 Integral Calculus Ex 11.8 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.8

Integrate the following with respect to x:

Question 1.
(i) e^ax cos bx
Answer:

(ii) e^2x sin x
Answer:

(iii) e^-x cos 2x
Answer:

Question 2.
(i) e^-3x sin 2x
Answer:

(ii) e^-4x sin 2x
Answer:

(iii) e^-3x cos x
Answer:

Tamilnadu State Board New Syllabus Samacheer Kalvi 11th Maths Guide Pdf Chapter 11 Integral Calculus Ex 11.7 Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 11th Maths Solutions Chapter 11 Integral Calculus Ex 11.7

Integrate the following with respect to x:

Question 1.
(i) 9x e^3x
Answer:

(ii) x sin 3x
Answer:

(iii) 25 x e^-5x
Answer:

(iv) x sec x tan x
Answer:
∫ x sec x tan x dx
u = x, u’ = 1 , u” = 0
dv = sec x tan x dx , v = ∫ sec x tan x dx = sec x
v₁ = ∫ v dx = ∫ sec x dx = log |sec x + tan x|
v₂ = ∫ v₁ dx = ∫ log |sec x + tan x| dx
∫ u dv = uv – u’ v₁ + u” v₂ – u”’ v₃ + ………….
∫x sec x tan x dx = x sec x – 1 × log |sec x + tan x| + 0 × ∫ log |x sec x tan x| + c
∫x sec x tan x dx = x sec x – log |sec x + tan x| + c

Question 2.
(i) x log x
Answer:

(ii) 27 x² e^3x
Answer:

(iii) x² cos x
Answer:

(iv) x³ sin x
Answer:
∫ x³ sin x
u = x³, u’ = 3x², u” = 6x , u”’ = 6 , u’^v = 0
dv = sin x dx ⇒ v = ∫ sin x dx = – cosx
v₁ = ∫v dx = ∫- cos x dx = – ∫ cos x dx = – sin x
v₂ = ∫ v₁ dx = ∫ – sin x dx = – ∫ sin x dx = – (- cos x) = cos x
v₃ = ∫ v₂ dx = ∫ cos x dx = sin x
v₄ = ∫ v₃ dx = ∫ sin x dx = – cos x
∫ u dv = uv – u’ v₁ + u” v₂ – u”’ v₃ + u’^vv₄ – ………..
∫x³ sin x dx = x³ (- cos x) – 3x² (- sin x) + 6x (cos x) – 6 sin x + 0 (- cos x) + c
= -x³ cos x + 3x² sin x + 6x cos x – 6 sin x + c

Question 3.
(i)
Answer:

(ii) x⁵ e^x2
Answer:

(iii)
Answer:

(iv)
Answer:

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 9 Applications of Integration Ex 9.2 Textbook Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 9 Applications of Integration Ex 9.2

Question 1.
Evaluate the following integrals as the limits of sums:
(i) \(\int_{0}^{1}\) (5x + 4)dx
Solution:
(i) Here f(x) = 5x + 4, a = 0, b = 1
Hence we get

(ii) \(\int_{1}^{2}\) (4x² – 1)dx
Solution:
Here f(x) = 4x² – 1, a = 1, b = 2
We use the formula

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 9 Applications of Integration Ex 9.1 Textbook Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 9 Applications of Integration Ex 9.1

Question 1.
Find an approximate value of \(\int_{1}^{1.5}\) x dx by applying the left – end rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}
Solution:
Here a = 1, b = 1.5, n = 5, f(x) = x
So, the width of each subinterval is

The left hand rule for Riemann sum,
S = [f(x₀) + f(x₁)) + f(x₂) + f(x₃) + f(x₄)] ∆x
= [f(1) + f(1.1) + f(1.2) + f(1.3) + f(1.4)] (0.1)
= [1 + 1.1 + 1.2 + 1.3 + 1.4] (0.1)
= [6] (0.1)
= 0.6.

Question 2.
Find an approximate value of \(\int_{1}^{1.5}\) x² dx by applying the right – end rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}
Solution:
Here a = 1, b = 1.5, n = 5, f(x) = x²
So, the width of each subinterval is
h = Δx = \(\frac{b-a}{n}\) = \(\frac{1.5- 1}{5}\) = 0.1
The partition of the interval is given by
x₁ = 1.1
x₂ = 1.2
x₃ = 1.3
x₄ = 1.4
x₅ = 1.5
The right – end rule for Riemann sum with equal width Δx is
S = [f(x₁) + f(x₂) + ….. + f(x_n)] Δx
∴ S = [f(1.1) + f(1.2) + f(1.3) + f(1.4) + f(1.5)](0.1)
= (1.21 + 1.44 + 1.69 + 1.96 + 2.25) × 0.1
= 8.55 × 0.1
= 0.855

Question 3.
Find an approximate value of \(\int_{1}^{1.5}\) (2 – x) dx by applying the mid – point rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}
Solution:
Here a = 1, b = 1.5, n = 5, f(x) = 2 – x
So, the width of each subinterval is
h = Δx = \(\frac{b-a}{n}\) = \(\frac{1.5- 1}{5}\) = 0.1
The partition of the interval is given by
The partition of the interval is given b.y
x₀ = 1, x₁ = 1.1, x₂=1.2, x₃ = 1.3, x₄ = 1.4, x₅ = 1.5
The mid – point rule for Riemann sum with equal width Δx is

∴ S = (0.95 + 0.85 + 0.75 + 0.65 + 0.55) × 0.1
= 3.75 × 0.1
= 0.375

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Accountancy Guide Pdf Chapter 10 Computerised Accounting System-Tally Text Book Back Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Accountancy Solutions Chapter 10 Computerised Accounting System-Tally

12th Accountancy Guide Computerised Accounting System-Tally Text Book Back Questions and Answers

I Multiple Choice Questions

Choose the correct answer
Question 1.
Accounting report prepared according to the requirements of the user is
(a) Routine accounting report
(b) Special purpose report
(c) Trial balance
(d) Balance sheet
Answer:
(b) Special purpose report

Question 2.
Function key F 11 is used for
(a) Company Features
(b) Accounting vouchers
(c) Company Configuration
(d) None of these
Answer:
(a) Company Features

Question 3.
Which sub menu displays groups, ledgers and voucher type in tally?
(a) Inventory vouchers
(b) Accounting vouchers
(c) Company Info
(d) Account Info
Answer:
(d) Account Info

Question 4.
What are the predefined Ledger(s) in Tally?
(i) Cash
(ii) Profit & Loss A/c
(iii) Capital A/c
(a) Only (i)
(b) Only (ii)
(c) Both (i) and (ii)
(d) Both (ii) and (iii)
Answer:
(c) Both (i) and (ii)

Question 5.
Contra voucher is used for
(a) Master entry
(b) Withdrawal of cash from bank for office use
(c) Reports
(d) Credit purchase of assets
Answer:
(b) Withdrawal of cash from bank for office use

Question 6.
Which is not the default group in Tally
(a) Suspense account
(b) Outstanding expense
(c) Sales account
(d) Investments
Answer:
(b) Outstanding expense

Question 7.
Salary account comes under which of the following head?
(a) Direct Incomes
(b) Direct Expenses
(c) Indirect Incomes
(d) Indirect Expenses
Answer:
(d) Indirect Expenses

Question 8.
₹ 25,000 withdrawn from bank for office use. In which voucher type, this transaction will be recorded
(a) Suspense account
(b) Outstanding expense
(c) Sales account
(d) Investment
Answer:
(a) Suspense account

Question 9.
In which vouchers type credit purchase of furniture is recorded in Tally
(a) Receipt voucher
(b) Journal voucher
(c) Purchase voucher
(d) Payment voucher
Answer:
(b) Journal voucher

Question 10.
Which of the following options is used to view Trial Balance from Gateway of Tally?
(a) Gateway of Tally → Reports → Trial Balance
(b) Gateway of Tally → Trial Balance
(c) Gateway of Tally → Reports → Display → Trial Balance
(d) None of these
Answer:
(c) Gateway of Tally → Reports → Display → Trial Balance

II Very Short Answer Questions

Question 1.
What is automated accounting system?
Answer:
Automated accounting is an approach to maintain up-to-date accounting records with the aid of accounting software.

Question 2.
What are accounting reports?
Answer:
Accounting report is a compilation of accounting information that are derived from the accounting records of a business concern. Accounting reports may be classified as routine reports and special purpose reports.

Question 3.
State any five accounting reports.
Answer:

1. Daybooks/Joumal
2. Ledger
3. Trial Balance
4. Income statement
5. Balance sheet

Question 4.
What is Accounting Information System (AIS)?
Answer:
Accounting Information System (AIS) collects financial data, processes them, and provides information to the various users. To provide information AIS requires data from another information system that is manufacturing, marketing, and human resources. Similarly, other information systems require data from Ais in order to provide information.

Question 5.
What is a group in Tally.ERP 9?
Answer:
A group is a collection of ledgers of the same nature. There are predefined groups of accounts which are widely used in accounts of any organization. Groups are categorized as primary groups and subgroups.

III Short Answer Questions

Question 1.
Write a brief note on accounting vouchers.
Answer:
A voucher is a document which contains details of transactions. Transactions are to be recorded through voucher entries. Tally has a set of predefined vouchers such as Purchase, Sales, Payment, Receipt and Contra.

To view the list of voucher types:
Gateway of Tally → Masters Accounts Info → Voucher Type →Display
As per the requirements of users, additional voucher type can be created.

Following are some of the major accounting vouchers used in an organisation:

• Receipt Voucher
• Payment Voucher
• Contra Voucher
• Purchase Voucher
• Sales Voucher
• Journal Voucher

Question 2.
What are the pre-defined ledgers available in Tally.ERP 9?
Answer:
Tally has two predefined ledgers, cash and profit and loss A/c. The user has to create various other ledgers based on their requirements. Predefined group/ledger cannot be deleted.

Question 3.
Mention the commonly used voucher types in Tally.ERP 9?
Answer:
Following are some of the major accounting vouchers used in an organization:

1. Receipt Voucher
2. Payment Voucher
3. Contra Voucher
4. Purchase Voucher
5. Sales Voucher
6. Journal Voucher

Question 4.
Explain how to view profit and loss statement in Tally ERP 9?
Answer:
F10: A/c Reports > Profit & Loss A/c > AltF1 (detailed)
(or)
Gateway of Tally > Reports > Profit & Loss A/c > AltF1 (detailed)

Question 5.
Explain any five applications of computerized accounting system.
Answer:
Application of accounting software – Tally

(1) Starting Tally:
Tally can be started in either of the ways given below:
Click on Start > All programs . Tally.ERP 9 > Tally.ERP 9 icon
(Or)
Click Tally.ERP. 9 icon (shortcut) on the desktop

(2) Creation of a company:
It is essential to enter the particulars of the company for which accounts are to be maintained in Tally. In order to create a company, the following steps are to be followed:
Company Info>Create Company

(3) Gateway of Tally:
After creation of a company and whenever Tally is started a screen Gateway of Tally appears. It shows the company selected under List of Selected Companies on the left pane.

Gateway of Tally is shown on the right pane which contains menu options such as Transactions and Reports. The shortcut key for each menu option is a letter from the option’s name, which will be highlighted in red.

Vertical button bar is placed on the extreme right of screen which contains buttons such as F1 and F2 for quick interaction with Tally.ERP 9.

Bottom horizontal information panel displays details of product, version, licence, configuration and calculator.

(4) Closing Tally
The following are the two ways for closing Tally:
Press Esc Key > Press Enter or Y or Click on yes
(or)
Press Ctrl + Q

(5) Select a company
If multiple companies are created, to choose a particular company, after opening Tally, click F1 (Select Cmp) on the vertical button bar. Tally.ERP 9 displays the Select Company screen, with a list of companies that are already created. Select the company for which accounting has to be done.

IV Exercises

Question 1.
Record the following transactions in Tally.
(a) Devi commenced a business with a capital of ₹ 4,00,000
(b) An account was opened with Indian Bank and deposited ₹ 60,000
(c) Purchased furniture by paying cash ₹ 15,000
(d) Goods purchased on credit from Sumathy for ₹ 50,000
(e) Cash sales made for ₹ 10,000
(f) Goods purchased from Raja for ₹ 5,000 and paid by cheque
(g) Goods sold to Arun on credit for ₹ 70,000
(h) Money is withdrawn from bank for office use ₹ 25,000
(i) Part payment of ₹ 30,000 made to Sumathy by cheque
(j) Arun made part payment of ₹ 10,000 by cash
(k) Salaries paid to staff through ECS ₹ 36,000
(l) Carriage on purchases of ₹ 6,000 paid by cash
(m) Purchased computer from Muthu Ltd. on credit ₹ 44000
Solution

Question 2.
The following balance sheet has been prepared from the books of pearl on 1 – 4- 2018,

During the year the following transactions took place.

(a) Wages paid by cash ₹ 4,000
(b) Salaries paid by cheque ₹ 10,000
(c) Cash purchases made for ₹ 4,000
(d) Good purchased on credit from Yazhini ₹ 30,000
(e) Goods sold on credit to Jothi ₹ 40,000
(f) Payment made to Yazhini through NEFT ₹ 6,000
(g) Cash received from Peter ₹ 10,000
(h) Cash sales made for ₹ 4,000
(i) Depreciate buildings at 20%
(j) Closing stock on 31.03.2013 ₹ 9,000

You are required to prepare trading and profit and loss account for the year ended 31-03-2019 and balance sheet as on that using Tally.
Solution

12th Accountancy Guide Computerised Accounting System-Tally Additional Important Questions and Answers

III Short Answer Questions

Question 1.
What do you mean by a Computerised Accounting system?
Answer:
Computerized accounting system refers to the system of maintaining accounts using f computers. It involves the processing of accounting transactions through the use of a computer in order to maintain and produce accounting records and reports. Computerized accounting system takes accounting transactions as input that are processed through accounting software to generate various reports.

Question 2.
What are the applications of the Computerised Accounting system?
Answer:
The application of CAS are as follows:
(1) Maintaining accounting records: In CAS, accounting records can be maintained easily and efficiently for the low time period. It does not require a large amount of physical space. It facilitates fast and accurate retrieval of data and information.

(2) Inventory management: CAS facilitates efficient management of inventory. Fast-moving, slow-moving, and obsolete inventory can be identified. Updated information about the availability of inventory, level of inventory, etc., can be obtained instantly.

(3) Payroll preparation: Payroll involves the calculation of the amount due to an employee, pay of an employee may be calculated based on hours/days worked or units produced. CAS records the attendance of employees, computes the amount of salary, makes deductions such as provident fund, income ta, etc.

(4) Report generation: CAS helps to generate various routine and special purpose reports.

(5) Data import/export: Accounting data and information can be imported from or exported to other users within the organization as well as outside the organization.

(6) Taxation: CAS helps to compute various taxes and to deduct these and deposit the same to the Government account.

Question 3.
What are the steps involved in designing accounting reports?
Answer:
Following are the steps involved in designing accounting reports:

1. Define the objective of generating the report.
2. Specify the structure of the report
3. Creating database queries to interact with the database to retrieve, modify, add, or delete data from the records.
4. Write notes on the following
   • Contra Voucher
   • Purchase Voucher
   • Journal Voucher

(a) Contra Voucher:
A transaction involving both cash account and bank account is recorded using contra voucher. The transaction may be for deposit of cash into bank account or withdrawal of cash from bank account To record Contra:
Gateway of Tally > Transactions > Accounting Voucher > F4:Contra

(b) Purchase Voucher:
Purchase voucher are used for recording both cash and credit purchases of goods. To record Purchases:
Gateway of Tally > Transactions > Accounting Vouchers > F9:Purchase

(c) Journal Voucher:
Journal voucher are used for recording transactions involving other than cash, bank, purchases and sales such as depreciation, provision for bad bebts.
To record Journal:
Gateway of Tally > Transactions > Accounting vouchers > F7:Journal

Tamilnadu State Board New Syllabus Samacheer Kalvi 12th Maths Guide Pdf Chapter 8 Differentials and Partial Derivatives Ex 8.8 Textbook Questions and Answers, Notes.

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 8 Differentials and Partial Derivatives Ex 8.8

Choose the most suitable answer from the given four alternatives:

Question 1.
A circular template has a radius of 10 cm. The measurement of the radius has an approximate error of 0.02 an. Then the percentage error in the calculating the area of this template is
(a) 0.2 %
(b) 0.4 %
(c) 0.04 %
(d) 0.08 %
Solution:
(b) 0.4 %
Hint:
Radius of a circular plate = 10 cm
error in radius dr = 0.02
A = πr²
dA = 2πrdr

Question 2.
The percentage error of the fifth root of 31 is approximately how many times the percentage error in 31?
(a) \(\frac{1}{31}\)
(b) \(\frac{1}{5}\)
(c) 5
(d) 31
Solution:
(b) \(\frac{1}{5}\)
Hint:
(i.e.) the percentage error in the nth root of a number is approximately \(\frac{1}{n}\) times the percentage error in the number.

Question 3.
If u (x, y) = e^{x2 + y2} then \(\frac{\partial u}{\partial x}\) is equal to
(a) e^{x2 + y2}
(b) 2xu
(c) x²u
(d) y²u
Solution:
(b) 2xu
Hint:
u = e^{x2 + y2}
\(\frac{\partial u}{\partial x}\) = e^{x2 + y2} (2x) = 2xu

Question 4.
If v (x, y) = log (e^x + e^y), then \(\frac{\partial v}{\partial x}\) + \(\frac{\partial u}{\partial y}\) is equal to
(a) (e^x + e^y)
(b) \(\frac{1}{e^x+e^y}\)
(c) 2
(d) 1
Solution:
(d) 1
Hint:
v = log (e^x + e^y)

= 1

Question 5.
If w (x, y) = x^y, x > 0, then \(\frac{\partial w}{\partial x}\) is equal to
(a) x^y log x
(b) y log x
(c) y x^y-1
(d) x log y
Solution:
(c) y x^y-1
Hint:
w(x, y) = x^y
\(\frac{\partial w}{\partial x}\) = y x^y-1

Question 6.
If f (x, y) = e^xy, then \(\frac{\partial^2 f}{\partial x \partial y}\) is equal to
(a) xy e^xy
(b) (1 + xy)e^xy
(c) (1 + y) e^xy
(d) (1 + x)e^xy
Solution:
(b) (1 + xy)e^xy
Hint:
f(x, y) = e^xy
\(\frac{\partial f}{\partial y}\) = x e^xy (x) = xe^xy
\(\frac{\partial^2 f}{\partial x \partial y}\) = x e^xy (y) + e^xy = e^xy(1 + xy)

Question 7.
If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in our calculation of the volume is
(a) 0.4 cu.cm
(b) 0.45 cu.cm
(c) 2 cu.cm
(d) 4.8 cu.cm
Solution:
(d) 4.8 cu.cm
Hint:
Side of the cube = 4 cm
Volume V = x³
dV = 3x²dx
= 3 × 16 × 0.1
= 4.8 cu. cm

Question 8.
The change in the surface area S = 6x² of a cube when the edge length varies from x₀ to x₀ + dx is
(a) 12 x₀ + dx
(b) 12 x₀ dx
(c) 6 x₀ dx
(d) 6 x₀ + dx
Solution:
(b) 12 x₀ dx
Hint:
Surface Area S = 6x²
dS = 12 × dx
dS = 12 x₀ dx

Question 9.
The approximate change in volume V of a cube of side x meters caused by increasing the side by 1% is
(a) 0.3 x m³
(b) 0.03 x m³
(c) 0.03 x² m³
(d) 0.03 x³ m³
Solution:
(c) 0.03 x² m³
Hint:
Volume of the cube V = x³
dV = 3x² dx
= 3x² × 0.01
= 0.03 x² m³

Question 10.
If g (x, y) = 3x² – 5y + 2y², x(t) = e^t and y(t) = cos t then \(\frac{dg}{dt}\) is equal to
(a) 6 e^2t + 5 sin t – 4 cos t sin t
(b) 6 e^2t – 5 sin t – 4 cos t sin t
(c) 3 e^2tt + 5 sin t + 4 cos t sin t
(d) 3 e^2t – 5 sint + 4 cos t sin t
Solution:
(a) 6 e^2t + 5 sin t – 4 cos t sin t
Hint:
x = e^t, y = cos t

= 6 x e^t + (-5 + 4y) – sin t
= 6 e^t e^t + 5 sin t – 4 cos t sin t
= 6e^2t + 5 sin t – 4 cos t sin t

Question 11.
If f(x) = \(\frac{x}{x+1}\), then its differential is given by
(a) –\(\frac{x}{(x+1)^2}\) dx
(b) \(\frac{x}{(x+1)^2}\) dx
(c) \(\frac{x}{x+1}\) dx
(d) –\(\frac{x}{x+1}\) dx
Solution:
(b) \(\frac{x}{(x+1)^2}\) dx
Hint:

Question 12.
If u (x, y) = x² + 3xy + y – 2019, then \(\frac{\partial u}{\partial x}\)| _{(4, -5)} is equal to
(a) -4
(b) -3
(c) -7
(d) 13
Solution:
(c) -7
Hint:
u (x, y) = x² + 3xy + y – 2019
\(\frac{\partial u}{\partial x}\) = 2x + 3y
\(\frac{\partial u}{\partial x}\)| _{(4, -5)} = 8 – 15 = -7

Question 13.
Linear approximation for g(x) = cos x at x = \(\frac{π}{2}\) is
(a) x + \(\frac{π}{2}\)
(b) -x + \(\frac{π}{2}\)
(c) x – \(\frac{π}{2}\)
(d) -x – \(\frac{π}{2}\)
Solution:
(b) -x + \(\frac{π}{2}\)
Hint:
linear approximation

Question 14.
If w (x, y, z) = x² (y – z) + y² ( z – x)+ z²(x – y) then \(\frac{\partial w}{\partial x}\) + \(\frac{\partial w}{\partial y}\) + \(\frac{\partial w}{\partial z}\) is
(a) xy + yz + zx
(b) x (y + z)
(c) y (z + x)
(d) 0
Solution:
(d) 0
Hint:
w (x, y, z) = x² (y – z) + y² ( z – x)+ z²(x – y)

= 2xy – 2xz – y² + z² + x² + 2yz – 2yx – z² – x² + y² + 2zx – 2zy
= 0

Question 15.
If f (x, y, z) = xy + yz + zx, then f_x – f_z is equal to
(a) z – x
(b) y – z
(c) x – z
(d) y – x
Solution:
(a) z – x
Hint:
f (x, y, z) = xy + yz + zx
\(\frac{\partial f}{\partial x}\) = y + z
\(\frac{\partial f}{\partial z}\) = y + x
f_x – f_z = y + z – y – x = z – x