Balance of Payment refers to the statement of accounts recording all economic transactions of a given country, it receives payments from and makes payments to other countries. Balance of payments is a statement of accounts of these receipts and payments.

In other words Balance of Payment is overall records of all economic transactions of a country in a given period, with rest of the world.

Difference between stock and flow :-

Characteristics of Perfect competition :-

(1) Large Number of Buyers and sellers,

(2) Homogeneous product,

(3) Free Entry and Exit of Firms,

(4) Perfect Knowledge of the Market,

(5) Perfect Mobility of Factor,

(6) No Transportation cost.

a boy suffering from poor eye sight : Vitamin A

a person suffering from anaemia : Iron

Utility means – The capacity of a commodity to satisfy a particular want of consumer. Every commodity has some inherent want satisfying capacity. Characteristics of utility : 

(1) Utility is a psychological phenol menon. It depends upon the mental status of the consumer. 

(2) Utility is subjective which differs from person to person. Utility of a particular goods is not same for all consumers. 

(3) Utility is a relative concept because it changes from time and place. 

(4) Utility is related the expected satisfaction and not from actual realized satisfaction because expected satisfaction depends upon intensity of consumption.

Macroeconomics is the branch of economics that studies the behavior and performance of an economy as a whole. It focuses on the aggregate changes in the economy such as unemployment, growth rate, gross domestic product and inflation.

Micro Economics – It studies the economic activities of individual unit like individual form, individual household and individual industries. 

Macro Economics – It is associated with the study of groups at entire economy level, For example aggregate demand and aggregate supply of all goods and services and entire economy level, national income.national saving, national investment, total employment, total production.

Microeconomics is the study of economics at an individual, group, or company level. Whereas, macroeconomics is the study of a national economy as a whole. Microeconomics focuses on issues that affect individuals and companies. Macroeconomics focuses on issues that affect nations and the world economy.

Given linear programming problem is

7x + 7y - 7z = 2

-x + 11y + 7z = 1

11x + 5y + 7z = 0

It's matrix form is AX = B.

Where, A = \(\begin{bmatrix}7&7&-7\\-1&11&7\\11&5&7\end{bmatrix},\) x = \(\begin{bmatrix}x\\y\\z\end{bmatrix},\) B = \(\begin{bmatrix}2\\1\\0\end{bmatrix}\)

D = |A| = \(\begin{vmatrix}7&7&-7\\-1&11&7\\11&5&7\end{vmatrix}\) = \(7\begin{vmatrix}11&7\\5&7\end{vmatrix}-7\begin{vmatrix}-1&7\\11&7\end{vmatrix}\)\(-7\begin{vmatrix}-1&11\\11&5\end{vmatrix}\)

= 7(77 - 35) - 7(-7 - 77) - 7(-5 - 121)

 = 7 x 42 - 7x - 84 - 7x - 126

 = 294 + 588 + 882

 = 1764

D_x = \(\begin{vmatrix}2&7&-7\\1&11&7\\0&5&7\end{vmatrix}\)  = \(2\begin{vmatrix}11&7\\5&7\end{vmatrix}-7\begin{vmatrix}1&7\\0&7\end{vmatrix}\)\(-7\begin{vmatrix}1&11\\0&5\end{vmatrix}\)

 = 2(77 - 35) - 7(7 - 0) - 7(5 - 0)

 = 2 x 42 - 49 - 35

 = 84 - 84 = 0

D_y = \(\begin{vmatrix}7&2&-7\\-1&1&7\\11&0&7\end{vmatrix}\) =  \(7\begin{vmatrix}1&7\\0&7\end{vmatrix}-2\begin{vmatrix}-1&7\\11&7\end{vmatrix}\)\(-7\begin{vmatrix}-1&1\\11&0\end{vmatrix}\)

 = 7( 7 - 0) - 2(-7 - 77) - 7(0 - 11)

 = 49 - 2x - 84 + 77

 = 49 + 168 + 77

 = 294

D_z = \(\begin{vmatrix}7&7&2\\-1&11&1\\11&5&0\end{vmatrix}\) =  =  \(7\begin{vmatrix}11&1\\5&0\end{vmatrix}-7\begin{vmatrix}-1&1\\11&0\end{vmatrix}\)\(+7\begin{vmatrix}-1&11\\11&5\end{vmatrix}\)

 = 7(0 - 5) - 7(0 - 11) + 2(-5 + 121)

 = -35 + 77 - 252

 = -287 + 77

 = -210

x = \(\frac{D_x}{D}=\frac{0}{1764} = 0,\) 

y = \(\frac{D_y}{D}=\frac{294}{1784} =\frac16\) 

z = \(\frac{D_z}{D}=\frac{-210}{1764} =\frac{-10}{84}=\frac{-5}{42}\) 

Hence, solution of given linear programming is 

x = 0, y = 1/6 and z = -5/42

4x - 3y = 11;6x + 5y = 7

The matrix equation is \(\binom{4\,\,-3}{6\,\,-5}\) \(\binom{x}{y}\) = \(\binom{11}{7}\) 

                                       A      X  =   B

Now Δ =\( \begin{vmatrix} {4} & {-3} \\ {6}& {5} \\ \end{vmatrix}\) = 20 + 18 = 38 ≠ 0

∴ we can apply Cramer's rule

Δ_x = \(\begin{vmatrix} {11} & {-3} \\ {7}& {5} \\ \end{vmatrix}\) = 55 + 21 = 76

Δ_y =\(\begin{vmatrix} {4} & {11} \\ {6}& {7} \\ \end{vmatrix}\) = 28 - 66 = -38

x = \(\frac{Δx}{Δ}\) = \(\frac{76}{38}\) = 2, y = \(\frac{Δy}{Δ}\) = \(\frac{-38}{38}\) = -1

solution is (2,-1)

Given:

Let amount shares of A, B and C be Rs.7a, Rs.8a and Rs.12a respectively.

⇒ 12a = 8a  + 20

⇒ a = 5

Shares of A, B and C is Rs.35, Rs.40 and Rs.60 respectively.

∴ Required difference = 40 - 35 = Rs.5

Cube and cube root

Ramanujan, the great mathematician India ever had. He loved numbers experimented with numbers to a vast extent. Once G.H Hardy came to meet him in India. The taxi which he booked has a number 1729. When he met S. Ramanujan he said that this is a very dull number. Ramanujan replied quickly that this is not a dull number but its interesting number. Then also explained about cube and cube root.

He said that 1729 is the smallest number which can be expressed as the sum of two cubes in two different ways.
             1729 = 1728 + 1 = 12³ + 1³
             1729 = 1000 + 729 = 10³ + 9³

After that day 1729 is known as Hardy-Ramanujan number.

Perfect cubes or cube number:

When any number is obtained when a number is multiplied by itself three times is perfect cube or cube number.

or,
If in the prime factorisation of any number each factor appears three times, then the number is a perfect cube. Thus necessary for studying cube and cube root.
Example: 10 x 10 x 10 =1000, this number is perfect cube number.

Properties of perfect cube number:

1. Cube of even number is always even.
   Example: 2³ = 8, 4³ = 64 etc.
2. Cube of even number is always even.
   Example: 3³ = 27, 5³ = 125
3. We get a perfect cube by adding consecutive odd numbers in an amazing pattern.

           1 = 1 = 1³

           3 + 5 = 8 = 2³
           7 + 9 + 11 = 27 = 3³

          13 + 15 + 17 + 19 = 64 = 4³

           21 + 23 + 25 + 27 + 29 = 125 = 5³

The number of odd numbers in cube of any number is same as the number whose cube is obtained.

4. When prime factorization of a cube of a number then each prime factor appears 3 times.

Cube roots:

Cube root is simply the converse of cube of any number.
The symbol for representation of cube root is . it is special symbol as it has same symbol as square root only the redical attached with it.
Example: 8 = 2 x 2 x 2,  27 = 3 x 3 x 3 etc.

Example related to cube and cube root:

1. Find the value of (8)³.

       Solution:
            For getting cube of any number multiplies it by itself 3 times.

            Thus, (8)³ = 8 x 8 x 8 = 512

2. Find the value of (13)³.

      Solution:
         we can write 13 as 10 + 3
         thus, (13)³ = (10 + 3)³
         = (10)³ + (3)³ + 3 x 10 x 3(10 + 3)
         = 1000 + 9 + 90 x 13
         = 1009 +1170
         = 2179

3. Find the cube root of 1000.
   Solution:
   Directly we can assume the cube root as seeing it.
   In this case there are 3 zeroes on number than it is confirmed that the number must contain a zero.
   So we put that 10 x 10 x 10 = 1000, so cube root of 1000 is 10.

Given:

Sum of three numbers = 245

The ratio of first and second number = 2 : 3

The ratio of second and third number = 5 : 8

Calculation:

Let the three numbers be A, B, and C.

A : B = 2 : 3      ----(1)

B : C = 5 : 8      ----(2)

Multiply (1) with 5 and (2) with 3, to make the B ratio equal

⇒ A : B : C = 10 : 15 : 24

Let A = 10x, B = 15x, C = 24x

⇒ A + B + C = 245

⇒ 10x + 15x + 24x = 245

⇒ 49x = 245

⇒ x = (245/49)

⇒ x = 5

Second Number

⇒ 15 × 5

⇒ 75

∴ The second number is 75.

Square and square root:

 A natural number m is said to be square number or perfect square if it can be expressed in terms of n², where n is also a natural number. It is necessary to know about square and square root to solve number system related problems.
Example: 4 is a square number as 4 = 2²

Properties of perfect square:

1. All perfect square numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place.

2. None of these end with 2, 3, 7 or 8 at unit’s place.

3. If a number has 1 or 9 in the unit’s place, then it’s square ends in 1.

4. when a perfect square number ends in 6, the number whose square it is will have either 4 or 6 in unit’s place.

5. There are 2n non-perfect square numbers between n² and (n + 1)²
Example: between 2² and 3² the no. of non-perfect square no. are 2x2 =4 as (5, 6, 7, 8)

6. sum of first n odd natural numbers is n² also Remember it for square and square root related terms.
Example: 1+3+5 = 9 = 3²

7. Any perfect square number can be represented as sum of odd natural numbers starting from 1.

8. Also, Any odd perfect square number can be represented as sum of two consecutive natural number.
Example: 49 = 24+25, 121 = 60 + 61

9. For any natural number (n + 1) × (n – 1) = n² – 1.
Example: 6 x 4 = 5² – 1 = 24

10. Square numbers can only have even number of 0’s at the end.

11. If a perfect square is of n-digits, then its square root will have \(\frac{n}{2}\) digits if n is even, or \(\frac{n + 1}{2}\) if n is odd?

Pythagorean triplet:

For any natural number a > 1, we have (2a)²+(a² – 1)²=(a² + 1)²

So, 2a, a² – 1 and a²+ 1 forms a Pythagorean triplet.

By this rule, if any one of the members of Pythagorean triplet is given then you can find the rest. this also necessary for getting knowledge of square and square root.

Example:
1. 3² +  4² = 5² so, 3, 4, 5 are Pythagorean triplet.

2. If 5 is one of the members of the Pythagorean triplet then find the rest.

Solution:
Try 2a = 5 then a will not be an integer so take next
a² – 1 = 5 then a² = 6 again not an integer
try a² + 1 = 5 then a² = 4 thus, a = 2

2a = 4
a² – 1 = 3

Thus, Pythagorean triplet are 3, 4, 5.

Square root:

It is simply the inverse operation of square. Square and square root are in converse of each other.
There are many ways to find the square root of any number:

1. By just inversion of square.
Example: 3² = 9 and √9 = 3

2. By repeated subtraction of odd numbers.
Example: √9,
9 – 1 = 8, 8 – 3 = 5, 5 – 5 = 0, by 3 times consecutive subtraction we got zero thus √9 = 3.

3. By prime factorization.
√36 = 2x2x3x3 = 2² x 3² thus square root of 36 is 2x3 i.e. 6

4. By division method.
Try to solve examples based on square and square root. So that it will be easy for you to solve problems related to it.

2x - y = 3....(i)

5x + y = 4....(ii)

Adding (i) and (ii), we get

7x = 7

⇒ x = 7/7 = 1

y = 2x - 3 = 2 x 1 - 3 = -1

Hence, the solution of given equations are x = 1 and y - 1

Correct option is C) many times

Linear equation in one variable:

Earlier, we learnt about algebraic expression and algebraic equation. It is clear about the difference between algebraic expression and algebraic equation that algebraic equation involves equality and on the other hand there is no equality involved in algebraic expression.

Let’s clear it by seeing the examples:

Algebraic expressions are: 2x, 2x + 5, 8xy – 3, 7xy
Algebraic equations are: 4x = 7, 2x + 5 = y, 3x +2 = 8xy

Linear expression:

We can say linear expression as the algebraic expression which has only single degree on its variables.
As: 2x, 3y + 5 etc.

Linear Equation:

We denote linear equation the algebraic equation in which the highest power on the variables is equal to one.
Example: 2x + 3 = 0, 5x -2 = 0, 3x – y = z

Now it is clear that the linear equation can be of many types when number of variables is counted. In this section we will read only the linear equation where only one variable is involved.

Linear equation in one variable:

As we have seen that algebraic equation has equality involved in it. The same is in linear equation. There are two parts in these equations, the symbol of equality separates them. One part is LHS (left Hand Side) and the other side is RHS (Right Hand Side).

Any linear equation in one variable containing equations on both side, we solve it simply transposing the variables or literals on one side and the numeral on the other side. You can solve any linear equations involving a single variable by simply transposing like to like side.

In some cases, the algebraic equation is not a linear equation but after simplification, it converts into linear equation in one variable then it becomes easy to solve. We will have to identify for the given equation that if it can be simplified or not.

There are many applications of linear equation in one variable in everyday life. We can clearly see its uses in calculating numbers, ages, currency, perimeters and several other parameters.

Examples on linear equation in one variable:

1. Solve 2x + 4 = 8.
   Solution:
   As, 2x + 4 = 8
   when we subtract 4 from both side there will be no impact on its equality.
   2x + 4 – 4 = 8 – 4
   2x = 4
   now we divide each side by 2
   x = 2
2. What number should we add to twice the number 7 to get 30?
   Solution:
   Let us assume the number be x
   7 + x = 30
   let’s subtract 7 from both side
   7 + x – 7 = 30 – 7
   x = 23
   The required number is 23

Correct option is A) much harder

1. Everyone likes him for his honesty.

2. The police wanted a proof from Sathya for her explanation.

3. His speech is an attraction to all.

4. The punctuality and regularity of Suresh has earned him a good job.

5. The quick arrival of the policemen made us happy.

Correct option is C) but also

Correct option is A) us to move

Correct option is A) stand

Correct option is C) met

Correct option is C) good

Work is a path dependent quantity. It is known as a path function.

(4) Net force on particle in uniform circular motion is centripetal force (mv²/l)which provided by tension in string so the net force will be equal to tension i.e., T.

We are given

(i) BaCl₂ .2H₂O (s) + aq → BaCl₂ (aq), Δ_sol ΔH°₁ = 8.8 kJ mol^–1

(ii) BaCl₂ (s) + aq → BaCl₂ (aq), Δ_sol ΔH°₂ = – 20.6 kJ mol^–1 

We aim at

BaCl₂ (s) + 2H₂O → BaCl₂ .2H₂O (s), Δ_hyd H° = ? 

ΔH = ΔH₂ – ΔH₁ = – 29.4 kJ mol^–1 

Water is the solvent in which ionic compounds are generally soluble.

The four physical properties of non-metals are:

1. Non-metals are brittle, i.e. they cannot be beaten into sheets.

2. Non-metals are non-ductile, i.e., they cannot be drawn into wires.

3. They are bad conductors of heat and electricity except carbon (in the form of graphite).

4. They are dull [non-lusturous] [do not shine] except iodine and graphite.

5. They have comparitively low melting & boiling points.

6. They have low densities.

7. They may be solid, liquid or gas at room temperature.
Eg: bromine ⇒ liquid non metal.

8. Most of the solid non-metals are soft except carbon (in the form of diamond)

9. They are not strong i.e. they have low tensile strength.

Here, I = 5A, V = 220 V, t = 2h = 7200 sec
Power, P = VI
= 220 × 5 = 1100 W
Energy consumed = P × t
= 1100 W × 7200 sec
= 7920000 J

Ohm's law—“At constant temperature, current is directly proportional to its potential difference.” Mathematical expression on can be given as— 

V = IR

Where, V represents potential difference; I represents current and R = Resistance (constant).

These are the two tubes attached to the uterus one on either side.

(i) The fallopian tubes carry egg (ova) from ovary to the womb (uterus)

(ii) Fertilisation of the egg by the sperm may take place in the fallopian tube

Glucose is oxidised in the body during respiration.

Lungs are divided into very small sac like structures to increase the respiratory surface area. This facilitates efficient exchange of oxygen and CO₂ between the air and the blood.

The reaction between an acid and a base to give salt and water is called neutralization reaction.

Base + Acid ⇒ Salt + Water

Eg: HCl + NaOH → NaCl + H₂O

1. During day time suction pressure creates by transpiration (evaporation of water) helps in absorption and upward movement of water and minerals dissolved in it from roots to the leaves, us to the top of the plant.

2. At night, the effect of root pressure [created by movement of water from the soil to the xylem of roots] in transport of water is more important.

This is because of two reasons:

1. Alloys have higher resistivity than that of their constituent metals.

2. Alloys do not oxidise [or burn] readily at high temperature.

The need of using combination of resistances is to obtain a desired value of current in an electrical circuit, it can be done in three ways they are i) series combination, ii) parallel combination and iii) mixed combination.

Equivalent resistance: If a single resistance can replace the combination of resistances in such a manner that the current in the circuit remains unchanged, then that single resistance is called the equivalent resistance.

We are given :

(i) C (s) + O₂ (g)→ CO₂ (g), ΔH°₁ = – 393.5 kJ mol^–1

(ii) H₂(s) +1/2 O₂ (g) →H₂O(g) ΔH°₂ = – 241.8 kJ mol^–1

(iii) 2C (s) + 2H₂ (g) → C₂H₄ (g),ΔH°₃ = + 52.3 kJ mol^–1

We aim at : C₂H₄ (g) + 3O₂ (g) → 2CO₂ (g) + 2H₂O(g)

ΔH = – ΔH₃ + 2ΔH₁ + 2ΔH₂ 

= –1322.9 kJ mol^–1

We are given (i) C₂H₄ (g) + 3O₂ (g) → 2CO₂ (g) + 2H₂O

(l) ΔH = – 1410 kJ mol^–1

(ii) H₂ (g)+ 1/2 O₂(g)→ H₂O (l), ΔH = – 286.2 kJ mol^–1

(iii) C₂H₆ (g) + 31/2 O₂(g)→ 2CO₂ (g)+ 3H₂O (l), ΔH 

= – 1560.6 kJ mol^–1

We aim at: C₂H₄ + H₂ (g) → C₂H₆ (g), ΔH = ?

Equation (i) + Equation (ii) – Equation (iii) gives

C₂H₄ (g) + H₂ (g) → C₂H₆ (g),

ΔH = – 1410.0 + (–286.2) – (1560.6) = – 135.6 kJ mol^–1 

a. False. It does not influence the radio graphic contrast. 

b. True. 

c. True. 

d. True. 

e. True.

a. True. 

b. False. The line density of a grid (number of lead strips per cm) is usually 30–80 cm^–1 (typically 40). 

c. False. Usually aluminium, carbon fibre or plastic are used as inter space material. 

d. True. 

e. False. It is typically 8:1. High-ratio grids (12:1–16:1) are used with very large fields and high peak kV (kVp). 

a. False. It is usually made of molybdenum, in which the target annulus (made of tungsten alloy) is embedded. 

b. False. The opposite is true. Molybdenum has a low heat conductivity, which slows the conduction of heat towards the rotor bearings, protecting them from heat damage. 

c. True. They are usually lubricated with silver powder. 

d. False. Induction motors are used. These are brush less. 

e. False. The filament is offset towards the edge of the anode, in line with the target annulus.

a. False. It is conducted to the anode disc and transfers to the glass envelope by radiation. It cannot be transferred between the anode and glass by convection as the tube is evacuated. Some heat is transferred via conduction through the anode stem (this becomes the main process at lower temperatures as heat radiation becomes less significant). From the glass, heat is transferred to the insulating oil and the tube housing by convection. 

b. False. It is transferred by radiation and convection.

c. True. Blackening substantially increases the emissivity coefficient. 

d. True. 

e. True.

a. True. 

b. False. It is 6 mGy. 

c. True. (6 mGy versus 4 mGy). 

d. False. The reverse is true and is related to the amount of tissue that the X-ray beam needs to pass through. 

e. False. The skull has a greater sagittal length than coronal length.

Power factor = (15 x 745)/(0.83 x 1.732 x 440 x 22.36) = 0.79,  Lagging

φ = cos⁻¹ 0.79 = 37.8º I₁ = I_ph = 22.36/1.732 = 12.91 amp 

Active Current I_a = I₁ cos φ₁ = 12.91 × 0.79 = 10.2 amp

New Power-factor cos φ₂ = 0.95, φ₂ = 18.2º 

I₂ = 10.2/0.95 = 10.74 amp 

Capacitive current per phase = I₁ sin φ₁ − Ι₂sin φ₂ = 4.563 

Capacitive reactance per phase = 440/4.563 = 96.43 ohms 

Capacitance per phase = 33 μf 

These have to be delta-connected