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The general form is 7 × 9 × a × d = 63ad

|(a², bc, ac+c²)(a²+ab,b², ac(ab, b²+bc, c²)| = 4a²b²c²

\(\Delta=\begin{vmatrix}a^2&bc&ac+c^2\\a^2+ab&b^2&ac\\ab&b^2+bc&c^2\end{vmatrix}\)

Taking a, b and c common from C₁, C₂ and C₃ respectively.

\(=abc\begin{vmatrix}a&c&a+c\\a+b&b&a\\b&b+c&a\end{vmatrix}\)

Using C₁ → C₁ + C₂ - C₃

\(=abc\begin{vmatrix}0&c&a+c\\2b&b&a\\b&b+c&c\end{vmatrix}\)

Using R₂ → R₂ - R₃

\(=abc\begin{vmatrix}0&c&a+c\\0&-c&a-c\\2b&b+c&c\end{vmatrix}\)

Expanding along C₁,

∆ = abc[0 - 0 + 2b(ca - c² + ca + c²)]

or ∆ = abc(4abc) or ∆ = 4a²b²c²

The Gauss divergence theorem states that the vector's outward flux through a closed surface is equal to the volume integral of the divergence over the area within the surface. The sum of all sources subtracted by the sum of every sink will result in the net flow of an area.

First we have to solve the homogeneous equation 

(3.1)  x²y" − 4xy' + 6y = 0. 

This is an Euler-Cauchy equation, so we look for solutions of the form y = x^m. The characteristic equation for m is 

m(m − 1) − 4m + 6 = 0. 

Note m(m − 1) − 4m + 6 = m² − 5m + 6 = (m − 2)(m − 3), so the roots are m = 2 and m = 3. Thus, we have basic solutions y₁ = x² and y₂ = x³ for the homogeneous equation (3.1). 

To solve the equation in the problem, we put it in standard form by dividing both sides by x². This gives us

y" -/xy' +  6/x2y = 1/x.

We look for a particular solution of this equation of the form y = u(x)y₁(x)+ v(x)y₂(x). The equations for u and v are 

u'x² + v'x³ = 0 

2u'x + 3v'x² = 1/x. 

If we multiply the top equation by −2 and the bottom equation by x, the system becomes 

−2u'x² − 2v'x³ = 0 

2u'x² + 3v'x³ = 1.

Adding these equations gives v'x³ = 1, so v' = 1/x³. Substituting this is the first equation gives u' = −1/x². Easy integrations then give

u = 1/x

v = − 1/2x² 

Plugging into the formula y = u(x)y¹(x) + v(x)y²(x) for our particular solution gives 

y = (1/x)x² − (1/2x²)x³ = 1/2x. 

Adding on the general solution of the homogeneous equation, we find 

y = 1/2x + C₁x² + C₂x³ 

for the general solution of the inhomogeneous equation.

Correct option is (d) Streamline

Tangent to streamline defines velocity vector

Correct option is (b) 360 kW 

d = 120 mm 

V = 40 m/s

Power = \(\cfrac12\) mv₁² = \(\cfrac12\) pav₁³

= \(\cfrac12\) x 1000 x \(\left(\cfrac{\pi}4\times0.12^2\right)\) x 40³

Power = 360 kW

Correct option is (a) 1, 2 and 3 only 

Let x² = y ⇒ 2x dx = dy

⇒ dx = dy/dx = dy/2√y

\(\therefore\int\limits_0^{\infty}e^{-x^2}dx\) = \(\int\limits_0^{\infty}\frac1{2\sqrt y}e^{-y}dy\)\(=\frac12\int\limits_0^{\infty}y^{\frac{-1}2}e^{-y}dy\) 

\(=\frac12\int\limits_0^{\infty}y^{\frac12-1}e^{-y}dy = \frac{\Gamma(\frac12)}2\) \((\because\int\limits_0^{\infty}x^{n-1}e^{-x}dx=\Gamma(x))\)

\(=\frac{\sqrt\pi}2(\because\Gamma(\frac12)=\sqrt\pi)^2\)

1. Reduced dependence on your local market

Your home market may be struggling due to economic pressures, but if you go global, you will have immediate access to a practically unlimited range of customers in areas where there is more money available to spend, and because different cultures have different wants and needs, you can diversify your product range to take advantage of these differences.

2. Increased chances of success

Unless you’ve got your pricing wrong, the higher the volume of products you sell, the more profit you make, and overseas trade is an obvious way to increase sales.  In support of this, UK Trade and Investment (UKTI) claim that companies who go global are 12% more likely to survive and excel than those who choose not to export.

3. Increased efficiency

Benefit from the economies of scale that the export of your goods can bring – go global and profitably use up any excess capacity in your business, smoothing the load and avoiding the seasonal peaks and troughs that are the bane of the production manager’s life. 

4. Increased productivity

Statistics from UK Trade and Investment (UKTI) state that companies involved in overseas trade can improve their productivity by 34% – imagine that, over a third more with no increase in plant. 

5. Economic advantage

Take advantage of currency fluctuations – export when the value of the pound sterling is low against other currencies, and reap the very real benefits.  Words of warning though; watch out for import tariffs in the country you are exporting to, and keep an eye on the value of sterling.  You don’t want to be caught out by any sudden upsurge in the value of the pound, or you could lose all the profit you have worked so hard to gain.

6. Innovation

Because you are exporting to a wider range of customers, you will also gain a wider range of feedback about your products, and this can lead to real benefits.  In fact, UKTI statistics show that businesses believe that exporting leads to innovation – increases in break-through product development to solve problems and meet the needs of the wider customer base.  53% of businesses they spoke to said that a new product or service has evolved because of their overseas trade.

7. Growth

The holy grail for any business, and something that has been lacking for a long time in our manufacturing industries – more overseas trade = increased growth opportunities, to benefit both your business and our economy as a whole.

1. on

2. on

3. behind

4. under

5. beside

6. behind

7. on

(i) Skylark’s voice / Cuckoo voice.

(ii) The speaker.

(iii) Cuckoo.

The King of Italy was a brother Monarch, and may be induced to do the thing cheaper. So the letter was written and  prompt reply was received .

Patient : Good morning, doctor. I have stomach ache. 

Doctor : Good morning. Since when? 

Patient : From yesterday, doctor. 

Doctor : Did you eat oily food from some platform stalls? 

Patient : Yes doctor. I took non-veg items from a platform shop. 

Doctor : The pain is due to bacterial infection. Take these tablets and tonic twice a day after food. 

Patient : O.K. doctor. How much is the bill amount? 

Doctor : It is Rs. 200. Take care of your health. 

Patient : Thank you doctor

Dona Laura said that she was so glad to be there and she was afraid that her seat would be .occupied. She wondered how beautiful the morning was. Petra said that the sun was hot. Dona Laura said that Petra was only twenty. She said further that she was feeling more tired than usual and told Petra if she so wished, to go and chat with the guard.

Women knew that the real value of forests was not the timber from a dead tree, but the springs and streams, food for their cattle, and fuel for their hearths. 

when in the blazing heat of Kudimianmalai’s stone quarries, you run into Manormani 22, training others, you know it’s all worth it. A quarry worker and Arivoli volunteer herself, she thinks it vital that her co-workers learn cycling. 

Customer: Excuse me. I’ve got a complaint to make.

Manager: Tell me what is the problem? 

Customer: I’ve been waiting for the past twenty minutes but no one seems to attend to this table.

Manager : Sorry I’ll send someone at once

Customer: Its ok. Thank you.

For culturing bacteria need culture medium which are of different types depending on species of bacteria.

Gifted Students: Gifted Students are those who have advanced mental potential or extraordinary ability in one or more areas such as creativity, intellectual, academic, sports etc . when compared to other student  of same background.

 Some Characteristics of Gifted Students :

• Advanced level of reasoning powers, logical thinking, and questioning attitude.
• Problem Solving Behavior
• May be asked to spend more time on questions that deal with creating as they are curious for knowledge.
• Advanced reading ability (rapid learner ).
• Thirst for knowledge.
• Unusual alertness.
• Independent thinkers.
• Interest in experimenting and doing things differently.
• Excellent Memory.
• Highly developed curiosity.
• Wide range of interest ( extreme focus in one area).
• Advanced level of vocabulary.

Given v² = \(\left(\cfrac{p_0-p}a\right)\)

p₀ - p = av²

p = p₀ - av² ......(i)

\(\because\) pv = nRT

p = \(\cfrac{nRT}v\)

\(\cfrac{nRT}v\) = p₀ - av²

T = \(\cfrac{p_0v}{nR}\) - \(\cfrac{av^3}{nR}\)....(ii)

\(\left(\cfrac{dT}{dv}\right)\) = 0

\(\left(\cfrac{dT}{dv}\right)\) = \(\cfrac{p_0}{nR}\) - \(\cfrac{3av^2}{nR}\) = 0

\(\cfrac{3av^2}{nR}\) = \(\cfrac{p_0}{nR}\)

v² = \(\cfrac{p_0}{3a}\)

v = \(\sqrt{\cfrac{p_0}{3a}}\)

there value put in equation (ii)

T_max  = \(\cfrac{p_0v}{nR}\) = \(\cfrac{av^3}{nR}\)

= \(\cfrac{p_0}{nR}\) \(\left(\sqrt{\cfrac{p_0}{3a}}\right)\) - \(\cfrac{a}{nR}\) \(\left({\cfrac{p_0}{3a}}\right)^{3/2}\)

T = \(\sqrt{\cfrac{p_0}{3a}}\) \(\left(\cfrac{p_0}{nR}-\cfrac{a}{nR}\times\cfrac{p_0}{3a}\right)\)

T = \(\sqrt{\cfrac{p_0}{3a}}\) \(\left(\cfrac{3p_0-p_0}{3nR}\right)\)

T = \(\cfrac{2p_0}{3nR}\) \(\sqrt{\cfrac{p_0}{3a}}\)

Give T = \(\cfrac{A\,p_0}{B\,nR}\) \(\sqrt{\cfrac{p_0}{Ba}}\)

then A = 2

B = 3

Then value at A + B equal to

= 2 + 3

= 5

Learning disabilities are disorders that affect brain functioning of understanding, reading, writing, spelling, reasoning or recalling ability. Learning disabilities can also cause problems in learning skills, language skills, mathematical operations and coordinating movements. Learning disabilities are continual and can cause continual life-long challenges. Children with learning abilities often have average or above-average intelligence.

 Diagnostic Test provides feedback on student's weaknesses and strengths in mathematics. This test may help teachers to find learning difficulties by identifying the ability, strengths, skills, and weaknesses of the students.

Steps of Diagnostic Testing:

• Identify students who are facing problems or need any academic assistance.
• Locating learning gaps and difficulties.
• Discovering causal factors of slow learning.
• Plan for specific efforts to overcome their problems and difficulties. Eg: Remedial Teaching. 

Diagnostic Assessment is used to identify areas of improvement for students and helps to guide them by knowing the gaps in their understanding. This test helps to recognize the issues that students have with the subject and helps to give proper instructions and feedback for better subject-related growth.

Learning Disabilities in mathematics can be assessed most appropriately by Diagnostic Test because it helps to know the difficulty in children's understanding.

 Aptitude test

• Aptitude tests are used for predicting success in future learning activities or occupations.
• Aptitude means the special ability or capacity besides the general intellectual ability which helps an individual to acquire a required degree of proficiency in a specific field.
• The aptitude tests are likely to test how an individual will perform in a specific area without proper guidance or training.
• These tests can be used to test the sensory and motor capabilities, aptitude in teaching, music, science, painting, mathematics, etc.
• These types of tests properly anticipate the future potentialities or capacities of an individual which helps us to select the best individual for a specialized activity.

​Achievement tests are administered on students to measure their learning outcomes. These tests are more prevalent in our schools. These tests show as to what has been learned by the students, rather than to predict future performance. 

A concept map is a visual representation and organization of the relationship between concepts.

• Concept maps are used to help students to organize and represent knowledge of the subject logically.
• Concept map begin  with the main idea, then broken down in to several topics.
• It is good for storing a large amount of information.
• It helps to displays connections between elements of the main idea of idea.
• Lines or Arrows may be used to represent how ideas are connected to one another.

Hence, Concept maps are most likely to increase ability to organize information logically.

(a) NNP at factor cost

These are also known as factor inputs. Factor inputs are mainly of four types – land, labour, capital and enterprise that help in the process of production.

Dividend is that part of profit of a corporate enterprise which is distributed among the shareholders.

Monotonic preferences- A consumer preferences are said to be monotonic if and only if, between any two bundles of the two goods, he prefers that bundle which has at least more quantity of one good but no less of other good.

Example - If bundle A (3,5) and bundle B (3,2) are available to the consumer, then consumer will prefer bundle A over bundle B as bundle A consists of more units of good 2 than bundle B.

Two Modes of Cashless Transactions are :

• Internet Bankiing 
•  Mobile App

The year 1921 is often referred to as the “Year of the Great Divide,” the last period in which India's population declined, due to the effects of famine.

Scarcity is an economic concept used to express the fact that resources are limited, but the demand for those resources is unlimited. Therefore, scarcity is the limitation of raw materials, goods, natural resources, services and so on, in a world of infinite human needs.

Price elasticity of demand

Answer is : (c) Tertiary

The problem of double counting is the problem of estimating the value of goods and services more than once. All enterprises treat their sales as final goods and services. This leads to the problem of double counting. This should be avoided because if the value of any commodity is counted more than once it will result in overestimation of the national income.

(i) Final goods and services method : We can avoid double counting by taking into account the value of final goods and services. 

(ii) Value added method : This problem can be avoided by taking into account net value added at each stage of production. The value added of each enterprise is added.

 (a) The statement is true. When the value of K = 1, MPC = 0

It is shown as K = 1/1–MPC

Given K = 1

K = 1/1– MPC

1 – MPC = 1

MPC = 1 – 1

= 0

Thus when investment multiplier is 1, the value of MPC is zero.

(b) The statement is false. The value of APS can be negative. It is shown as :

APC + APS = 1

APC = C/Y

if c = 1200 and Y = 1000

then APC = (1200/1000) = 1.2

1.2 + APS = 1

APS = – 0.2

The value of APS is generally less than 1.

(a) NDPFC = (ii) + (iv) + (ix) + (x)

= 200 + 700 + 600 + 400

= 3700

GDP_FC = NDP_FC + CFC

= (GFCF + S) – NDCF

= [1050 + (200 – 150)] – 1000

= 100

GDP_FC = 3700 + 100 = 3800 Crores.

(b) FIPA : Factor income paid to abroad.

NFIA = GNP_FC – GDP_FC

= 3750 – 3800

= (–) 50

NFIA = FIFA – FIPA

– 50 = 20 – FIPA

FIPA = 20 + 50 = 70 Crores.

A competitive firm in the short run can bear the maximum losses equivalent to its fixed cost.

Non-monetary transactions are quite evident in rural areas where payments for farm-labour are often made in kind rather than cash. But such transactions are not recorded, because they are outside the monetary system of exchange. To this extent GDP remains underestimated. Thus, non monetary exchanges make gross domestic product an inappropriate index of welfare.

(d) The sub-units are subordinate to the central government

Correct option is: (b) Secondary