Real numbers Any number that we can think of in the real world is a real number . set of real number is denoted by R. real numbers include natural numbers (N) , whole numbers (W), Integers (Z/I), rational numbers (Q) and irrational numbers (R-Q) . For example:- 1,3.5,√7,3.¯2,22/7,π,1.234 etc are real numbers . In this chapter we also learn about LCM and HCF

Real numbers Any number that we can think of in the real world is a real number . set of real number is denoted by R. real numbers include natural numbers (N) , whole numbers (W), Integers (Z/I), rational numbers (Q) and irrational numbers (R-Q) . For example:- 1,3.5,√7,3.¯2,22/7,π,1.234 etc are real numbers . In this chapter we also learn about LCM and HCF

Real numbers Any number that we can think of in the real world is a real number . set of real number is denoted by R. real numbers include natural numbers (N) , whole numbers (W), Integers (Z/I), rational numbers (Q) and irrational numbers (R-Q) . For example:- 1,3.5,√7,3.¯2,22/7,π,1.234 etc are real numbers . In this chapter we also learn about LCM and HCF

LCM and HCF LEAST COMMON MULTIPLE (LCM)  COMMON MULTIPLE :- A common multiple of two numbers is a number which is exactly divisible by each of given numbers . e.g. 45 is a common multiple of 3,5,9,15  LEAST COMMON MULTIPLE :- The least common multiple of two or given numbers is the least number which is exactly divisible by each one of them . e.g.12 is common multiple of 2,3,4 & 24 is also common multiple of 2,3,4 but 12 is LCM of 2,3,4 HIGHEST COMMON FACTOR (HCF) Highest common factor is also known as greatest common divisor (GCD)  COMMON FACTOR :- A common factor of two or more numbers is a number which divides each of them exactly . e.g. 3 is a common factor of 3,12,18

SIMPLIFICATION Simplification is a process of reducing a complex expression into a simpler form. ‘VBODMAS’ Rule This rule gives the correct order in which various operations regarding simplification are to be performed, so as to find out the values of given expressions in simple ways. Let us see what these letters mean. Order of operations is as same as the order of letters in the ‘VBODMAS’ from the left to right as V B O D M A S Left to right Clearly, the order will be as follows First Vinculum bracket is solved, (Remember – 6 – 8 = - 14 but – 6- 8 = - (-2) = 2) Second Brackets are to be solved in order given below • First, small brackets (circular brackets ) ‘( )’ • Second , middle brackets (curly brackets) ‘{ }’ • Third , square brackets (big brackets) ‘[ ]’ Third Operation of ‘Of’ is performed. Fourth Operation of division is performed. Fifth Operation of multiplication is performed. Sixth Operation of addition is performed. Seventh Operation of subtraction is performed

SIMPLIFICATION Simplification is a process of reducing a complex expression into a simpler form. ‘VBODMAS’ Rule This rule gives the correct order in which various operations regarding simplification are to be performed, so as to find out the values of given expressions in simple ways. Let us see what these letters mean. Order of operations is as same as the order of letters in the ‘VBODMAS’ from the left to right as V B O D M A S Left to right Clearly, the order will be as follows First Vinculum bracket is solved, (Remember – 6 – 8 = - 14 but – 6- 8 = - (-2) = 2) Second Brackets are to be solved in order given below • First, small brackets (circular brackets ) ‘( )’ • Second , middle brackets (curly brackets) ‘{ }’ • Third , square brackets (big brackets) ‘[ ]’ Third Operation of ‘Of’ is performed. Fourth Operation of division is performed. Fifth Operation of multiplication is performed. Sixth Operation of addition is performed. Seventh Operation of subtraction is performed

SIMPLIFICATION Simplification is a process of reducing a complex expression into a simpler form. ‘VBODMAS’ Rule This rule gives the correct order in which various operations regarding simplification are to be performed, so as to find out the values of given expressions in simple ways. Let us see what these letters mean. Order of operations is as same as the order of letters in the ‘VBODMAS’ from the left to right as V B O D M A S Left to right Clearly, the order will be as follows First Vinculum bracket is solved, (Remember – 6 – 8 = - 14 but – 6- 8 = - (-2) = 2) Second Brackets are to be solved in order given below • First, small brackets (circular brackets ) ‘( )’ • Second , middle brackets (curly brackets) ‘{ }’ • Third , square brackets (big brackets) ‘[ ]’ Third Operation of ‘Of’ is performed. Fourth Operation of division is performed. Fifth Operation of multiplication is performed. Sixth Operation of addition is performed. Seventh Operation of subtraction is performed

Percentage Percent means per hundred. It is denoted by the symbol % .Here x% means x/100.Thus ,any percentage can be converted into an equivalent fraction by dividing it by 100. Short Tricks 1. When a value/number/quantity 'A' is increased or decreased by b%, then new value/number/quantity will be = (100 ±b)/100* A 2. If A is a% more than B, then b is less than a by [a/(100+a)* 100 ]% 3. If A is a % less than B, then b is more than a by [a/(100-a)* 100 ]% 4. When the value of an object is first changed (increased or decreased) by x% and then changed by y % then net effect is given as = [± x ±y + ((± x) (± y)))/100] % (+ ve sign indicates increase, - ve sign indicates decrease.) 5. If the price of a commodity increases or decreases by a%, then the decrease or increase in consumption so as not to increase or decrease the expenditure is equal to ( a/(100 ±a )) * 100%

Ratio If a and b are two quantities of same kind, then a/b is known as the ratio of a and b . It is written as a:b. The first of the ratio is called antecedent while the second term is called consequent. e.g., Ratio between 30kg and 50kg is 3: 5. Proportion The equality of two ratios is called proportion a, b, c, d are said to be in proportion if a : b = c : d or a : b : : c : d. Here, ‘a’ is the first second term ‘c’ is the third term and ‘d’ is the fourth term. Here, first and fourth terms are called extremes i.e., a and d while the second and third terms are called as means i.e., b and c. e.g., if 3 : 2 : : 135 : 90, then 3, 90 are the extremes while 2, 135 are called the means. In a proportion, we always have: Product of extremes = product of means a x d = b x c

Mixture To determine the mean value of the mixture when the prices of the individual items being mixed together and the proportion in which they are being mixed are given. Here, the value of the mixture is always higher than the lowest value and lower than highest value of the items being mixed. According to the Rule of Alligation, if two quantities are in a ratio, then (Quantity of cheaper)/(Quantity of dearer) = (Cost price of dearer-Mean price)/(Mean price-Cost price of cheaper) It can also be expressed as, Cost price of 1 unit Cost price of 1 unit quantity of cheaper (x) quantity of dearer (y) Mean price (m) (y-m) (m-x)

AVERAGE Average is the mean value of a set of numbers or values. Therefore average of a set of numbers is, Average= (X_1+ X_2+ X_3+ ……..〖+ X〗_n)/n or in other words, Average of some observations = (Sum of all observations)/(Number of all observations) Remarks:- If each observation of the given data is increased by same quantity, say x, then their average also increases by x. If each observation of the given data is decreased by same quantity, say x their then average also decreases by x. If each observation of the given data is multiplied same quantity, say x then their average also multiplied by x. If each observation of the given data is divided by same quantity, say, x then their average also divided by x.

In our daily life, we have to complete different kinds of work in different stipulated time and if we are not able to complete the work in definite time,we arrange some more persons for it.Different persons have different abilities to do work. Basic Rules 1. If time taken by a person to complete a work in x days,then work done in 1day is 1/x 2. If a person can do 1/x part in 1day , then he completes that work in x days

Clock A clock has two hands, the smaller one is called the hour hand or short hand while the larger one is called the minute hand or long hand. The face of a clock is a circular dial which subtends an angle of 360° at the centre. The circumference of the dial is divided into 12 equal parts to be called hour spaces. Every hour space is further divided into 5 equal parts to be called minute space. Hence, the total circumference is divided into 12 x 5= 60 equal minute space. Fundas In one hour, both the hands coincide once. In one hour, the hands are straight (point in opposite directions) once. In one hour, the hands are twice perpendicular to each other. In 60 min, the minute hand covers 360°. Thus, in 1 min the minute hand covers (360/60)° = 6° In 12 h, the hour hand covers 360°. Thus, in 1 min, the hour hand covers (360/(12*60))° = (1°)/2 Thus, in one minute, the minute hand gains ( 6 - 1/2) = 5 - 1/2, then the hour hand. When the two hands are at right angles, they are 15 min spaces apart When the two hands are in opposite directions, they are 30 min spaces apart In 60 min, the minute hand gains 55 min on the hour hand. The minute hand moves 12 times as fast as the hour hand.

CALENDAR ORDINARY YEAR -An ordinary year has 365 days LEAP YEAR-Leap year has 366 days (a)Any year (except a century) which is divisible by 4 is a leap year. (b)Century is a leap year if it is divisible by 400 Example: 1924, 1964, 1660, 1880, 1808 are all leap years. 400, 800, 1200, 1600, 2000 are all leap years. Total leap year /29 Feb in 100 years=24 Total leap year/29 Feb in 400 years=97 Total 29th Date in 100 years =1124 Total 29th Date in 400 years=4497 ODD DAYS - The number of days more than the complete weeks for a given period called odd days HOW TO CALCULATE ODD DAYS FOR "n" NUMBER OF DAYS: (a) Divide the number of days by 7. (b) The remainder so obtained is the number of odd days NUMBER OF ODD DAYS DAY OF THE WEEK 0 Sunday 1 Monday 2 Tuesday 3 Wednesday 4 Thursday 5 Friday 6 Saturday NOTE 1ordinary year =365 days =(52 weeks +1day) 1ordinary year =1odd day 1leap year =366days =(52 weeks+2days) 1leap year=2 odd days 100years =76 ordinary years +24 leap years =(76*1+24*2)odd days = 124 odd days =(17weeks+5days) = 5 odd days 200 years =( 5*2 =10 odd days) = 1week +3 days = 3 odd days 300 years =(5*3 = 15 odd days) = 2weeks +1 day = 1 odd day 400 years = (5*4 +1 = 21 odd days) = 3weeks +0 day = 0 odd day • Similarly, 800 year, 1200 year, 1600year, 2000year have 0 odd days • Last day of the century may be Sunday, Monday, Wednesday, Friday • Last day of the century cannot be Tuesday, Thursday, Saturday • The day on which ordinary year begins it ends on its same day Example: If 1st Jan 2001 was Monday then 31st Dec 2001 was also Monday REPETITION OF SAME CALENDAR IN YEARS IN CASE OF COMPLETE REPEAT (a) Divide the Given Year by 4 (b) You will get one of the following remainders 1, 2, 3, 0 (c) Add the following years in the remainders: Remainders Number of years to be added 1 6 2 11 3 11 0 28 IN CASE OF PARTIAL REPEAT (a) Year will match only from January 1st to February 28. (b) Ordinary year will match the Leap year and vice-versa. *Number of odd days must be multiple of 7.

• Binary system: In the binary system, only two symbols 0 and 1 are used. Since, in this system only two numbers are used, so its base or radix is 2. • Decimal system: in the decimal, we use 10 digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Since, 10 basic symbols are used in this system, so its base or radix 10. In this chapter we will do; 1. Decimal to binary conversion 2. Binary to decimal conversion

Profit and loss and discount Cost price (CP) the price at which an article is bought is called its cost price. All the overhead expenses in the transaction are added to the cost price . Selling price (SP) The price at which an article is sold is Called the selling price. Profit →if SP > CP then there is a profit Profit = SP-CP & P% = (SP-CP)/CP × 100 LOSS → If CP > SP then there is a loss. Loss = CP – SP & L% = (CP-SP)/CP × 100 MARKED PRICE OR LIST PRICE :- it is marked on the article Discount :- The Reduction allowed on the marked price of an article is called as discount . discount is always calculated on marked price . Discount = marked price – selling price D% = (MP-SP)/MP × 100

1. Principal: The money borrows or lend out for a certain period is called the principal or the sum. 2. Interest: Extra money paid for using other's money is called interest. 3. Simple Interest: If the interest on a sum borrowed for a certain period is reckoned uniformly, then it is called simple interest. Let principal = P, Rate = R% per annum (p.a.) and Time = T year. 4. Compound interest : Someti¬mes borrower & the lender agree to fix up a certain units of time, say yearly or halfly or quarterly to settle the previous account. After a specified period, the difference between the amount and money is borrowed called the compound interest (C.I.) for that period.

1. Principal: The money borrows or lend out for a certain period is called the principal or the sum. 2. Interest: Extra money paid for using other's money is called interest. 3. Simple Interest: If the interest on a sum borrowed for a certain period is reckoned uniformly, then it is called simple interest. Let principal = P, Rate = R% per annum (p.a.) and Time = T year. 4. Compound interest : Someti¬mes borrower & the lender agree to fix up a certain units of time, say yearly or halfly or quarterly to settle the previous account. After a specified period, the difference between the amount and money is borrowed called the compound interest (C.I.) for that period.

Mensuration :- Mensuration is a branch of mathematical science that is concerned with the measurement of areas and volumes of various geometric figures . It is of two types 2D and 3D mensuration 2D Mensuration applies to two dimensional figures like squares, rectangles, triangles, trapezium etc. 3D Mensuration applies to three dimensional figures like cube, cuboids, sphere, cone, cylinder etc. Area: Total space inside the boundary of a plane figure is called the area of that particular figure. Area is measured in square Unit. Perimeter:- It is the length of border around enclosed plane. Therefore, Sum of the sides of the a plane figure is the perimeter of that particular figure. Volume: volume is the space occupied by an object considering 3dimensional view.The formulas for calculating Volume vary with the shape of the object. The units of volume is cubic meters, cubic centimetre etc

Mensuration :- Mensuration is a branch of mathematical science that is concerned with the measurement of areas and volumes of various geometric figures . It is of two types 2D and 3D mensuration 2D Mensuration applies to two dimensional figures like squares, rectangles, triangles, trapezium etc. 3D Mensuration applies to three dimensional figures like cube, cuboids, sphere, cone, cylinder etc. Area: Total space inside the boundary of a plane figure is called the area of that particular figure. Area is measured in square Unit. Perimeter:- It is the length of border around enclosed plane. Therefore, Sum of the sides of the a plane figure is the perimeter of that particular figure. Volume: volume is the space occupied by an object considering 3dimensional view.The formulas for calculating Volume vary with the shape of the object. The units of volume is cubic meters, cubic centimetre etc

Mensuration :- Mensuration is a branch of mathematical science that is concerned with the measurement of areas and volumes of various geometric figures . It is of two types 2D and 3D mensuration 2D Mensuration applies to two dimensional figures like squares, rectangles, triangles, trapezium etc. 3D Mensuration applies to three dimensional figures like cube, cuboids, sphere, cone, cylinder etc. Area: Total space inside the boundary of a plane figure is called the area of that particular figure. Area is measured in square Unit. Perimeter:- It is the length of border around enclosed plane. Therefore, Sum of the sides of the a plane figure is the perimeter of that particular figure. Volume: volume is the space occupied by an object considering 3dimensional view.The formulas for calculating Volume vary with the shape of the object. The units of volume is cubic meters, cubic centimetre etc

Mensuration :- Mensuration is a branch of mathematical science that is concerned with the measurement of areas and volumes of various geometric figures . It is of two types 2D and 3D mensuration 2D Mensuration applies to two dimensional figures like squares, rectangles, triangles, trapezium etc. 3D Mensuration applies to three dimensional figures like cube, cuboids, sphere, cone, cylinder etc. Area: Total space inside the boundary of a plane figure is called the area of that particular figure. Area is measured in square Unit. Perimeter:- It is the length of border around enclosed plane. Therefore, Sum of the sides of the a plane figure is the perimeter of that particular figure. Volume: volume is the space occupied by an object considering 3dimensional view.The formulas for calculating Volume vary with the shape of the object. The units of volume is cubic meters, cubic centimetre etc

Mensuration :- Mensuration is a branch of mathematical science that is concerned with the measurement of areas and volumes of various geometric figures . It is of two types 2D and 3D mensuration 2D Mensuration applies to two dimensional figures like squares, rectangles, triangles, trapezium etc. 3D Mensuration applies to three dimensional figures like cube, cuboids, sphere, cone, cylinder etc. Area: Total space inside the boundary of a plane figure is called the area of that particular figure. Area is measured in square Unit. Perimeter:- It is the length of border around enclosed plane. Therefore, Sum of the sides of the a plane figure is the perimeter of that particular figure. Volume: volume is the space occupied by an object considering 3dimensional view.The formulas for calculating Volume vary with the shape of the object. The units of volume is cubic meters, cubic centimetre etc

Statistics is concerned with collection of data, presentation of data and analysis of data in the numerical form. Collection of data involves collection of primary as well as secondary data. Primary data is the data which is collected for the first time for statistical investigation. Secondary data refers to the data which is already collected and used for current statistical investigation. Measures of central tendency includes the measures of Mean, Median and Mode. Mean also known as average is the sum of all the items divided by number of items. Median refers to the middle number sorted ascending or descending. Mode refers to the mostly appearing figure in the given series.

Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. Probability is simply how likely something is to happen. Whenever we're unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. The analysis of events governed by probability is called statistics.

Time and distance Speed The distance travelled in a unit time is known as speed. Speed = (distance travelled)/(Time Taken) Units of speed are km/h, m/s etc. To convert speed of an object from km/h to m/s, multiply the speed by 5/18 . To convert speed of an object from m/s to km/h, multiply the speed by 18/5 . Average speed Average speed is the ratio of total distance covered to total time of journey. Average speed = (Total distance covered)/(Total time of journey) Rules for Solving Time and Distance Problems Rule 1 If a certain distance is covered with a speed of ‘x’ km/h and another equal distance with a speed of ‘y’ km/h then the average speed for the whole journey is the harmonic mean of the two speeds Average speed = (2xy/(x+y))km/h (This formula is applicable only when the distance is constant) Rule 2 If a certain distance is covered with a speeds of ‘x’ km/h and another distance with a speed of ‘y’ km/h but time interval for both journeys being same, then average speed for the whole journey is given by Average speed = ((x+y)/2)km/h Rule 3 If the ratio of speed A and B is x : y, then the ratio of time taken by them to cover the same distance is y : x.

Time and distance Speed The distance travelled in a unit time is known as speed. Speed = (distance travelled)/(Time Taken) Units of speed are km/h, m/s etc. To convert speed of an object from km/h to m/s, multiply the speed by 5/18 . To convert speed of an object from m/s to km/h, multiply the speed by 18/5 . Average speed Average speed is the ratio of total distance covered to total time of journey. Average speed = (Total distance covered)/(Total time of journey) Rules for Solving Time and Distance Problems Rule 1 If a certain distance is covered with a speed of ‘x’ km/h and another equal distance with a speed of ‘y’ km/h then the average speed for the whole journey is the harmonic mean of the two speeds Average speed = (2xy/(x+y))km/h (This formula is applicable only when the distance is constant) Rule 2 If a certain distance is covered with a speeds of ‘x’ km/h and another distance with a speed of ‘y’ km/h but time interval for both journeys being same, then average speed for the whole journey is given by Average speed = ((x+y)/2)km/h Rule 3 If the ratio of speed A and B is x : y, then the ratio of time taken by them to cover the same distance is y : x.

Time and distance Speed The distance travelled in a unit time is known as speed. Speed = (distance travelled)/(Time Taken) Units of speed are km/h, m/s etc. To convert speed of an object from km/h to m/s, multiply the speed by 5/18 . To convert speed of an object from m/s to km/h, multiply the speed by 18/5 . Average speed Average speed is the ratio of total distance covered to total time of journey. Average speed = (Total distance covered)/(Total time of journey) Rules for Solving Time and Distance Problems Rule 1 If a certain distance is covered with a speed of ‘x’ km/h and another equal distance with a speed of ‘y’ km/h then the average speed for the whole journey is the harmonic mean of the two speeds Average speed = (2xy/(x+y))km/h (This formula is applicable only when the distance is constant) Rule 2 If a certain distance is covered with a speeds of ‘x’ km/h and another distance with a speed of ‘y’ km/h but time interval for both journeys being same, then average speed for the whole journey is given by Average speed = ((x+y)/2)km/h Rule 3 If the ratio of speed A and B is x : y, then the ratio of time taken by them to cover the same distance is y : x.

Polynomial An expression in term of some variable(s) is called a polynomial. f(x)=2x-5 is a polynomial in variable x The expressions like 3x^2 –√(x,) 1/( x^2 7× +6) , 5x^3 - 4/x,etc., are not polynomials. Thus, a rational integral function of ‘x’ is said to be a polynomial, if the powers of ‘x’ in the terms of the polynomial are neither fractions nor negative. Thus, an expression of the form f (x) = a_n x^n + a_(n-1 ) x^(n-1) +……+ a_1x + a_o is called a polynomial in variable x where n be a positive integer anda_o, a_1, …….., a_n be constants ( real numbers). Degree of a polynomial The exponent of the highest degree term in a polynomial is known as its degree. e.g. , f(x) = 4x -3/2 is a polynomial in the variable x of degree 1. Linear polynomial A polynomial of degree one is called a linear polynomial. In general, f(x) = ax + b, where a ≠0 is a linear polynomial. Quadratic polynomial A polynomial of degree two is called a quadratic polynomial. In general, f (x) = 〖ax〗^(2 ) + bx + c, where a≠ 0 is a quadratic polynomial e.g. , f(x) = x^2 – 7x + 8 is a trinomial as it contains 3 terms. Cubic polynomial A polynomial of degree 3 is called a cubic polynomial in general f (x) = 〖ax〗^3 + 〖bx〗^2 + cx + d, a ≠ 0 is a cubic polynomial. f (x) = 〖2x〗^3 - x^(2 ) + 8x + 4 Biquadratic polynomial A fourth degree polynomial is called a biquadratic polynomial in general. f (x) =〖ax〗^4 + 〖bx〗^3 + 〖cx〗^2 + dx + e, a ≠ 0 is a biquadratic polynomial. Zero of a polynomial A real number ∝ is a zero (or root) of a polynomial f (x), if f (∝) = 0 e.g., if x = 1 is a root of the polynomial 〖3x〗^3 - 〖2x〗^(2 )+ x – 2, then f (1) = 0

Linear Equations when we equate two algebraic expressions using the signs of equality it forms an equation. Linear equations are first degree equations and may contain one or more variables. If the equation has only one variable, then the equation itself is sufficient to obtain the value of the variable. If the equation has two variables then two consistent equations are required to get the value of the variables. In general, an equation has n variables then n consistent equations are required to obtain all the value of the n variables. Linear Equation in One Variable These are first degree equations in one unknown. An equation of the form ax + b = 0 where a, b ∈ R and a ≠ 0 and x is the variable, is called a linear equation in one variable. We have, only one variable x whose value we have to find out. Linear Equation in two Variables These are first in two unknowns. An equation of the ax + by + c = 0, where a, b, c ∈ R and a ≠ 0, b ≠ 0 and x, y are variables is called linear equation in two variables have, two variables x and y whose values we have to find out. Any pair of values of x and y which satisfy the equations ax + by + c = 0, is called its solution. When two or more equations are satisfied by the same set values of the variables involved in them, then the termed as simultaneous equations.

Quadratic Equation These are second degree equations of the form 〖ax〗^(2 ) + bx + c = 0, where a, b, c ∈ R and a ≠ 0 are quadratic equations. Like a first degree equation in x has one value of x satisfying the equation, a quadratic equation in x will have two values of x that satisfy the equation. The value of x that satisfy the equation are called the roots of the equation. These roots may be real or imaginary. x^(2 ) + 5x + 6 = 0 has roots x = -2,- 3 Using Formula If the quadratic is 〖ax〗^(2 )+ bx + c = 0, then we can use the standard formula given below to find out the roots of the equation. If α and β are the roots of the quadratic equation, then ∝=(-b+√(b^2-4ac))/2a and β=(-b-√(b^2-4ac))/2a Sum and product of Roots of a Quadratic Equation If ∝ and β are the roots of the quadratic equation 〖ax〗^(2 ) + bx + c = 0 Sum of the roots = ∝ + β = - b/a Product of the roots = ∝β = c/a

Trigonometric Ratios The ratios between different sides of a right angled triangle w.r.t. its acute angles are called trigonometric ratios. Trigonometric ratios for right angled ∆ABC w.r.t angle A are given below. sin A = BC/AC = P/H cos A = AB/AC = B/H tan A = BC/AB = P/B cosec A = AC/BC = H/P sec A = AC/AB = H/B cot A =AB/BC =B/P Relation between Trigonometric Ratios sin A = 1/(cosec A) or cosec A = 1/sin⁡A cos A = 1/sec⁡A or sec A = 1/(cos A) tan A = sin⁡A/cos⁡〖A 〗 or cot A = cos⁡A/sin⁡A

Quadrants The axes X'OX and Y'OY divide the whole plane into four parts which are called quadrants. Here, OX and OX' are called the positive and negative directions respectively of x-axis and similarly OY and OY’ prime are the positive and negative directions, respectively of y-axis. In 1st quadrant, x > 0, y > 0 In 2nd quadrant, x < 0 y > 0 In 3rd quadrant, x < 0 , y < 0 In 4th quadrant, x > 0, y < 0 The coordinates of any point on the x-axis are of the form (x, 0) and on the y-axis are of the form (0, y).If the x-coordinate or abscissa of a point is zero, then it would be somewhere on the y-axis and if its y-coordinate or ordinate is zero, then it would be on x-axis. Distance Formulae The distance between any two points is the length of the line segment joining them. i.e. D = √(〖(x_2- x_1)〗^2+〖(y_2- y_1)〗^2 ) Or D = √(〖(Difference of bscissa)〗^2+ 〖(Difference of ordinates)〗^2 ) The distance between any two points is the length of the line segment joining them. Section formula Case I For external division x=(〖mx〗_2- 〖nx〗_1)/(m-n),y= (〖my〗_2- 〖ny〗_1)/(m-n) Case II For Internal division x=(〖mx〗_2+ 〖nx〗_1)/(m+ n),y= (〖my〗_2+ 〖ny〗_1)/(m+n) Area of Triangle Let A (x_1 y_1), B (x_2 y_2) and C (x_3 y_3) be the coordinates of the vertices of ∆ABC. Area of ∆ABC = 1/2 [x_1 (y_2- y_3 )+ x_2 (y_3- y_1 )+x_3 (y_1- y_2)]

Trigonometric Ratios The ratios between different sides of a right angled triangle w.r.t. its acute angles are called trigonometric ratios. Trigonometric ratios for right angled ∆ABC w.r.t angle A are given below. sin A = BC/AC = P/H cos A = AB/AC = B/H tan A = BC/AB = P/B cosec A = AC/BC = H/P sec A = AC/AB = H/B cot A =AB/BC =B/P Relation between Trigonometric Ratios sin A = 1/(cosec A) or cosec A = 1/sin⁡A cos A = 1/sec⁡A or sec A = 1/(cos A) tan A = sin⁡A/cos⁡〖A 〗 or cot A = cos⁡A/sin⁡A Trigonometric Ratios The ratios between different sides of a right angled triangle w.r.t. its acute angles are called trigonometric ratios. Trigonometric ratios for right angled ∆ABC w.r.t angle A are given below. sin A = BC/AC = P/H cos A = AB/AC = B/H tan A = BC/AB = P/B cosec A = AC/BC = H/P sec A = AC/AB = H/B cot A =AB/BC =B/P Relation between Trigonometric Ratios sin A = 1/(cosec A) or cosec A = 1/sin⁡A cos A = 1/sec⁡A or sec A = 1/(cos A) tan A = sin⁡A/cos⁡〖A 〗 or cot A = cos⁡A/sin⁡A

GEOMETRY Geometry is the most important topic of maths section in PSTET. Geometry can be defined as the study of shapes. Plane Geometry is related to the properties and relation of plane figures, such as angles, triangles, other polygons and circles. Line is defined by its length but has no breadth. A line contains infinite points. Through a given point , there passes infinite lines. Line Segment is the part of the line that contains two points and all points between them. The two points are called end points. Ray is a line segment when extended infinitely in one direction Parallel lines Two lines in the same plane are said to be parallel, if they never meet. Transversal is a line which cuts a pair of parallel is called a transversal.

GEOMETRY Geometry is the most important topic of maths section in PSTET. Geometry can be defined as the study of shapes. Plane Geometry is related to the properties and relation of plane figures, such as angles, triangles, other polygons and circles. Line is defined by its length but has no breadth. A line contains infinite points. Through a given point , there passes infinite lines. Line Segment is the part of the line that contains two points and all points between them. The two points are called end points. Ray is a line segment when extended infinitely in one direction Parallel lines Two lines in the same plane are said to be parallel, if they never meet. Transversal is a line which cuts a pair of parallel is called a transversal.

GEOMETRY Geometry is the most important topic of maths section in PSTET. Geometry can be defined as the study of shapes. Plane Geometry is related to the properties and relation of plane figures, such as angles, triangles, other polygons and circles. Line is defined by its length but has no breadth. A line contains infinite points. Through a given point , there passes infinite lines. Line Segment is the part of the line that contains two points and all points between them. The two points are called end points. Ray is a line segment when extended infinitely in one direction Parallel lines Two lines in the same plane are said to be parallel, if they never meet. Transversal is a line which cuts a pair of parallel is called a transversal.

GEOMETRY Geometry is the most important topic of maths section in PSTET. Geometry can be defined as the study of shapes. Plane Geometry is related to the properties and relation of plane figures, such as angles, triangles, other polygons and circles. Line is defined by its length but has no breadth. A line contains infinite points. Through a given point , there passes infinite lines. Line Segment is the part of the line that contains two points and all points between them. The two points are called end points. Ray is a line segment when extended infinitely in one direction Parallel lines Two lines in the same plane are said to be parallel, if they never meet. Transversal is a line which cuts a pair of parallel is called a transversal.

GEOMETRY Geometry is the most important topic of maths section in PSTET. Geometry can be defined as the study of shapes. Plane Geometry is related to the properties and relation of plane figures, such as angles, triangles, other polygons and circles. Line is defined by its length but has no breadth. A line contains infinite points. Through a given point , there passes infinite lines. Line Segment is the part of the line that contains two points and all points between them. The two points are called end points. Ray is a line segment when extended infinitely in one direction Parallel lines Two lines in the same plane are said to be parallel, if they never meet. Transversal is a line which cuts a pair of parallel is called a transversal.

TRAINGLES Summary In this chapter you have studied the following points : 1. Two figures having the same shape but not necessarily the same size are called similar figures. 2. All the congruent figures are similar but the converse is not true. 3. Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (i.e., proportion). 4. If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. 5. If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. 6. If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar (AAA similarity criterion). 7. If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar (AA similarity criterion). 8. If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar (SSS similarity criterion). 9. If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar (SAS similarity criterion). 10. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. 11. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other. 12. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras Theorem). 13. If in a triangle, square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

TRAINGLES Summary In this chapter you have studied the following points : 1. Two figures having the same shape but not necessarily the same size are called similar figures. 2. All the congruent figures are similar but the converse is not true. 3. Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (i.e., proportion). 4. If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. 5. If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. 6. If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar (AAA similarity criterion). 7. If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar (AA similarity criterion). 8. If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar (SSS similarity criterion). 9. If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar (SAS similarity criterion). 10. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. 11. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other. 12. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras Theorem). 13. If in a triangle, square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

TRAINGLES Summary In this chapter you have studied the following points : 1. Two figures having the same shape but not necessarily the same size are called similar figures. 2. All the congruent figures are similar but the converse is not true. 3. Two polygons of the same number of sides are similar, if (i) their corresponding angles are equal and (ii) their corresponding sides are in the same ratio (i.e., proportion). 4. If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio. 5. If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side. 6. If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio and hence the two triangles are similar (AAA similarity criterion). 7. If in two triangles, two angles of one triangle are respectively equal to the two angles of the other triangle, then the two triangles are similar (AA similarity criterion). 8. If in two triangles, corresponding sides are in the same ratio, then their corresponding angles are equal and hence the triangles are similar (SSS similarity criterion). 9. If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio (proportional), then the triangles are similar (SAS similarity criterion). 10. The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. 11. If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then the triangles on both sides of the perpendicular are similar to the whole triangle and also to each other. 12. In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (Pythagoras Theorem). 13. If in a triangle, square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Arithmetic Progression An arithmetic progression is a sequence in which terms increase or decrease by a constant number called the common difference. E.g., The sequence 2, 6, 10, 14, 18, 22, .… Is an arithmetic progression, whose first term is 2 and common difference is 4. An arithmetic progression is represented by a, (a + d), (a + 2d), (a + 3d), ….., a + (n – 1) d. Here a = first term d = common difference n = number of terms in the progression The general term of an arithmetic progression is given by Tn = a + (n – 1) d The sum of n terms of an arithmetic progression is given by S_n = n/2 [2a + (n – 1) d] 0r S_n = n/2[a+l] where l is the last term of arithmetic progression. If a, b, c are in arithmetic progression, then b = (a+c)/2 where b is the arithmetic mean. Fundas If the same quantity is added or multiplied to each term of an AP, then the resulting series is also an AP. If three terms are in AP, then they can be taken as (a – d), a, (a + d). If four terms are in AP, then they can be taken as (a – 3d), (a – d), (a + d), (a +3d). If five terms are in AP, then they can be taken as (a – 2d), (a – d), a, (a + d), (a + 2d).

Mensuration

The universe is everything. It includes all of space, and all the matter and energy that space contains. It even includes time itself and, of course, it includes you. Earth and the Moon are part of the universe, as are the other planets and their many dozens of moons.

The solar system comprises the sun, 8 planets their moon and other non stellar objects . The sun is at the centre of the solar system and all the planets rolled around it in elliptical orbit . The sun is the nearest star to the earth .The plants are classified in 2 types inner planets and outer plants .Mercury, venus ,earth ,mars are the inner planets .

The ocean (also the sea or the world ocean) is the body of salt water that covers approximately 70.8% of the surface of Earth and contains 97% of Earth's water.[1] Another definition is "any of the large bodies of water into which the great ocean is divided".[2] Separate names are used to identify five different areas of the ocean.

Cyclones In meteorology, a cyclone is a large scale air mass that rotates around a strong center of low atmospheric pressure, counterclockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere .

In geography we learn about origin of universe ,mountains ,lakes ,rivers ,glaciers In this chapter we learnt about so many geographical terms like apogee perigee aphilian perihelion So here are some mcq related this chapter

Dams of india Description - In this lecture we will learn The concept of the dams of india with tricks . The location of dam is very important topic So with the tricks we will learn the river and the state of a dam

Winds Wind is the movement of air, caused by the uneven heating of the Earth by the sun and the Earth's own rotation. There are 3 types of winds- trade winds,Westerly winds, easterlies winds.

Geography of India Longitude and latitude of india /Tropic of cancer states/Physical features of india/ Himalaya/Plaines of india /Peninsular plateau of india /Coastal plains of india /location of India geographical facts of India/ map of india with sources/islands of India/Thar desert/Geographical regions of india /Size of india /Neighbouring countries of india

Volcanism is the eruption of molten rock (magma) onto the surface of a planet. A volcano is the vent through which magma and gases are discharged. Magma that reaches the surface is called “lava.” Volcanos are named for Vulcan — the Roman god of fire!

The layers of atmosphere The composition and characteristics of the atmosphere Of the earth vary with height from the surface . Temperature pressure and intensity of the most important characteristics of the atmosphere that vary with altitude .On the basis of the thermal and other characteristics the atmosphere can be divided into a number of almost concentric layers .

International boundaries Description - In this lecture we will learn international boundaries with very interesting tricks .It national boundaries are very important topic .With the help of tricks we can learn easily these international boundaries .

An eclipse occurs when one heavenly body such as a moon or planet moves into the shadow of another heavenly body. Let's learn about the two types of eclipses on Earth. What Is a Lunar Eclipse? The Moon moves in an orbit around Earth. At the same time, Earth orbits the Sun.

An earthquake Description -An earthquake (also known as a quake, tremor or temblor) is the shaking of the surface of the Earth resulting from a sudden release of energy in the Earth's lithosphere that creates seismic waves.

The structure of the earth is divided into four major components: the crust, the mantle, the outer core, and the inner core. Each layer has a unique chemical composition, physical state, and can impact life on Earth's surface.

Lakes of india- In this lecture we are going to learn the lakes of india in very entertaining manner .With these tricks you can easily learn all the lakes of india .In this lecture we learn the lakes with their states .

There are three kinds of rock: igneous, sedimentary, and metamorphic. Igneous rocks form when molten rock (magma or lava) cools and solidifies. Sedimentary rocks originate when particles settle out of water or air, or by precipitation of minerals from water. They accumulate in layers.

Ocean currents are the continuous, predictable, directional movement of seawater driven by gravity, wind (Coriolis Effect), and water density. Ocean water moves in two directions: horizontally and vertically. Horizontal movements are referred to as currents, while vertical changes are called upwellings or downwellings.

French Revolution, also called Revolution of 1789, revolutionary movement that shook France between 1787 and 1799 and reached its first climax there in 1789.The feudal regime had been weakened step-by-step and had already disappeared in parts of Europe.

The French Revolution opened up the possibility of creating a dramatic change in the way in which society was structured. Not everyone in Europe, however, wanted a complete transformation. Some were ‘conservatives’, while others were ‘liberals’ or ‘radicals’.

The French Revolution opened up the possibility of creating a dramatic change in the way in which society was structured. Not everyone in Europe, however, wanted a complete transformation. Some were ‘conservatives’, while others were ‘liberals’ or ‘radicals’.

The Nazi Party then began to eliminate all political opposition and consolidate its power. Hindenburg died on 2 August 1934 and Hitler became dictator of Germany by merging the offices and powers of the chancellery and presidency.

The problem of deforestation has become more complicated for some reasons. For a huge increase in population, the food demand increased and the cultivation area was extended by clearing forests.

Pastoralism is a way of keeping animals such as cattle, sheep, that involves moving from one place to another to find water and food. Nomads are people who do not live in one place but move from one area to another to earn their living.Mainly pastoral communities are found in mountainous regions..

Peasants cultivated open fields which were strips of land near their villages. These strips were of varying quality. This was a measure to ensure that everyone had a mix of good and bad land.

National awakening also grew out of an intellectual reaction to the Enlightenment that emphasized national identity and developed an authentic view of cultural self-expression through nationhood.

The Indo-China region of modern-day Vietnam, Cambodia and Laos struggled with colonisation from the Europeans just like India. They too fought back to gain their independence and this where their nationalist movement started. Vietnam actually gained formal independence even before India

The making of the global world has a long history of trade, migration of people in search of work, the movement of capital, etc.Globalisation’ is often referred to as an economic system that has emerged Since last 50 years and so on .

The Age of Industrialisation begins by explaining the scenario before the Industrial Revolution and how it changed over time in terms of labour, setting up of factories, etc.In Britain, the most dynamic industries were cotton and metals.

In India and as in many other colonies, the growth of modern nationalism is intimately connected to the anti-colonial movement. People began discovering their unity in the process of their struggle with colonialism. The sense of being oppressed under colonialism provided a shared bond that tied many different groups together. The Congress under Mahatma Gandhi tried to forge these groups together within one movement. But the unity did not emerge without conflict.

The modern city worldwide has developed over the last 200 years. Three historical processes have shaped modern cities in decisive ways. The rise of capitalism. The establishment of colonial rule over large parts of the world.

In this lecture you will learn about Development in Indian Economy. By watching this video, you will learn about: Concept of Development. Income and other criteria’s for measuring the development of the economy. Different development goals for the different category of persons. Concept of Infant Mortality ratio, Literacy ratio, Body mass Index, Per Capita Income and National Income etc. Human Development Index prepared by UNDP. World Bank Report and HDI. Concept of Sustainable Development, SDGs of India. Objective Questions on the development topic.

In this lecture you will learn about the different sectors of Indian Economy. By watching this video, you will learn about: Sector of Indian Economy: Primary sector, secondary sector and tertiary sector. Sectors on the basis of operation and ownership. MGNREGA scheme of generating employment, GDP (Gross Domestic Product) and Disguised unemployment.

In this lecture you will learn about the concept of Money and Credit. By watching this video, you will learn about: Concept of Money. Barter System and concept of Double Coincidence of Wants. Evolution of Money. Modern forms of money: Currency, Deposits in Banks and Cheques. Loan activities of banks, Two different credit situations and the terms of credit. Formal Sector Credit in India: Formal and Informal Sector. Self Help Groups for people. Objective Questions on the topic of Money and Credit.

In this lecture you will learn about the concept of Globalization and its Impact on Indian Economy. By watching this video, you will learn about: Production across countries and interlinking the production across countries. Foreign trade and integration of foreign markets. Concept of Globalization; Factors enabling Globalization: Technology, Trade liberalization, Foreign Investment policy. World Trade Organization, Structure of WTO, World Trade Group. Impact of Globalization on Indian Economy.

Consumer Rights are referred to a set of laws that represent the right to be informed about the quantity, quality, purity, potency, price of goods and their standards so that the consumer is protected against all sorts of unfair trade practices. The Consumer Bill of Rights upholds the right to safety of every citizen.

Geography MCQ's - Lecture 19

Geography MCQ's - Lecture 20

Geography MCQ's - Lecture 21

Geography MCQ's - Lecture 22

Geography MCQ's - Lecture 23

Geography MCQ's - Lecture 24

Introductory (Reasoning) - Lecture 1

Number Series Number series is a sequential arrangement of numbers following a certain defined pattern. In this section, we deal with questions in which a series of numbers, (which are generally called the terms of the series) is given. These numbers/terms follow a certain pattern throughout the series. Candidates are asked either to find a missing term or to find the wrong term of the series.

Blood relation is an important topic for all sorts of competitive exams like Banking , SSC, RBI grade B , PSPCL, PPSC , Punjab Sub Inspector etc. It almost carries a weightage of 3 to 5 marks and may vary exam to exam. So, basically questions Blood relations involve analysis of certain blood relations and then inferring on the basis of the given information. In the questions , a chain of relationships is given in the form of information and on the basis of which, relation between any two members of the chain is asked. Following are some of the basic tricks which helps to solve blood relations questions easily and efficiently . After understanding the basic concept relation , Solve blood relations question in an MCQ format which helps you in exams.

Coding –decoding is one of the most important topic reasoning. It is repeatedly asked in competitive exams and also in AFPI. It carries a weightage of 2-3 marks. Generally, there are three different types of coding – decoding. • Pattern based • Direct based • Elimination based While solving questions of coding –decoding one should memorise alphabetical positions of alphabets from A to Z (1-26) and opposite pairs like B is opposite to Y.

Direction is one of the most important topic of reasoning. Direction has its relevance in all types of competitive exam like banking exams, SSC, UPSC, PPSC, AFPI etc. The position towards which someone or something moves or faces is known as direction. Basically there are four main directions are: 1. North 2. South 3. East 4. West CARDINAL DIRECTION : A direction between two consecutive main directions is called sub or cardinal direction. 1. North-east 2. North-west 3. South-east 4. South-west Here, angle formed between two consecutive main directions is 90 degree and angle formed between cardinal and main direction is 45 degree.

Order and ranking There are two types of ranking test in analytical reasoning: 1. Sequential Order of arrangement: In this type of questions, persons or objects are arranged based on the comparison of parameers such as age, marks, salary, weight etc. 2. Ranking: If a certain person sitting in a row, let L be the rank from left end and R be the rank from right end. Total = L+R-1

Missing number series is one such topic which has its importance in MRSAFPI Exam. It carries weight age of 3 marks. So, Below are basic tricks which can help you to solve questions based on number series. • Before solving questions always observe the flow of series whether it is increasing order or in decreasing order. • If there is gradual increase in series , then it has highly chances of addition. • If there is gradual decrease in series, then it has highly chances of subtraction • If there is huge increase in series then probably there would be multiplication or along with it addition and subtraction or square and cube. • If there is huge decrease in series then probably there would be division or along with it addition and subtraction or square and cube • Memorise square from 1 to 50 and cubes from 1 to 13. • One should good with tables as they can help a lot while solving questions.

Letter series is series of alphabets which arranged in certain pattern . It is one of the easiest topic in reasoning which helps to score more in exam. To solve letter series one can take help of numeric postions of alphabets from A(1) TO Z(26). There are 10 MCQS given below on letter series for your practice.

Analogy means similarity shared by two things that are compared. Questions on verbal analogy judge your ability to understand the diverse relationship between various elements, things etc. Analogy topic is an important topic of reasoning and it has its relevance in MRAFPI. Questions asked on analogy carries a weightage of 2-3 marks. There are three types of analogy : • Number based analogy questions • Alphabet based analogy questions • Vocab based analogy questions

Odd one out / classification in these type of questions , a group of words , letters or numbers is given. On behalf of alphabetical values and their positions letters form a group same as numbers follow mathematical operations/ rules, hence form a group. Candidates are required to select the option which does not belong to that same group.

Symbol and Notation is one of the most scoring part of reasoning. In these type of questions, a mathematical equation is given with arthimetical notification. Candidates are required to solve these questions by using basic concept of BODMAS rule.

Logical venn-diagram is one of the important topic in non-verbal reasoning. It has important role in competitive exam like MRSAFPI and other sort of competitive exams. A venn diagram shows all possible relations among the given groups of elements in a single figure. The most common type of questions that we encounter in venn – diagram are based on circular venn diagrams.

Counting figure is an interesting topic of non-verbal reasoning. In this topic, one has to count number of triangles or square or rectangles in the given figure. If one knows certain basic tricks to count the figures then this topic can be scoring. For instance, counting the numbers of squares in figure one can easily take sum of squares of number of columns if number of rows and column are equal .

Cube and Dice is one of the most important topic of non-verbal reasoning. Generally, questions asked on this topic are based on opposite and related side of a dice or cube. Candidates are required to relate the given dice and cube and transform the relation between more than once dice or cube. • Ordinary dice : A dice in which the total of two adjacent side equals to 7 , although the total opposite sides does not equal to 7. • Standard Dice : A dice in which total of two opposite sides equals to 7 although the total of two adjacent does not equal to 7.

Mirror and Water image is an important topic of non- verbal reasoning. This is the most scoring topic also. It is based on basic concept of visualisation .