CHAPTER -2 LINEAR EQUATIONS IN ONE VARIABLE

Cube and Cube Root( as per PSEB )

Cube and Cube Root

Points to Remember:

1.   If a number is multiplied by itself thrice then the product obtained is cube of the number. Let a number be ‘a’ then cube of a = a x a x a which is written as a³ and read as cube of a.

2.   1, 8, 27 ….. 1000 are perfect cubes.

3.   Cube of an odd number is odd.

4.   Cube of an even number is even.

5.   Cube of positive number is positive.

6.   Cube of negative number is negative.

7.   If unit’s place digits of a number is 1, 4, 5, 6, 9 and 0 then unit’s place digit of its cube is 1, 4, 5, 6, 9 and 0 respectively.

8.   If unit’s place digit of a number is 2 then unit’s place digit of its cube is 8.

9.   If unit’s place digit of a number is 8 then unit’s place digit of its cube is 2.

10. If unit’s place digit of a number is 3 then unit’s place digit of its 

      cube is 7.

11. If unit’s place digit of a number is 7 then unit’s place digit of its 

      cube is 3.

Points to Remember:

1.   If the cube of a number m = m3 = n then cube root of n is m and we will write 3Ön = m.

2.   To find the cube of a number, we add 1+ (n) (n – 1) x 3 in the previous cube ; when n = 1, 2, 3 …… etc.

e.g.  for n = 1, 1 + (1) (1 – 1) x 3 =1

        for n = 2, 1 + (2) (2 – 1) x 3 =7

        for n = 3, 1 + (3) (3 – 1) x 3 =19

        for n = 4, 1 + (4) (4 – 1) x 3 =37

       ( ………………………………….)

        For n  =  9, 1 + (9) (9 - 1) x 3 = 217

Thus by adding 1, 7, 19, 37, 61 ………… 217 etc, to the previous cube, the next cube is obtained.

3.   If by subtracting above numbers i.e. 1, 7, 19, 37, 61, 91, 169, 217,

      271 etc. in order from the given number, the remainder is zero, then  

      the number is perfect cube otherwise it is not a perfect cube.

Points to Remember:

1.         The numbers having unit’s place digit

(i)  0, 1, 4, 5, 6 and 9 then their cubes have unit’s place digit as 0,      

      1, 4, 5, 6 and respectively.

(ii)  2, then unit’s place digit of its cube is 8

(iii) 8, then unit’s place digit of its cube is 2

(iv) 3, then unit’s place digit of its cube is 7

(v)  7, then unit’s place digit of its cube is 3

          Therefore, the unit’s place digit of cube root of a number is known by the unit’s place digit of the number.

2.      (i)  Cube of one digit number has maximum three digits.

          (ii)  Cube of two digit number has maximum six digits.

Points to Remember:

1.      Cube of a negative number is negative and cube root of a negative number is also negative 3Ö-m=-3Öm

2.      For a rational number p, ³Öp =³Öp

                                              q     q    Öq   

3.      If there are two numbers a and b then  ³Ö a x b = ³Ö a x ³Öb   

Square and Square Root ( as per PSEB )

Square and Square Root

Points to Remember:

1.   When a number is multiplied be itself then the product is the square of the number.

2.   Square of an odd number is odd.

3.   Square of an even number is even.

4.   Perfect square has only 0, 1, 4, 5, 6 or 9 digits at its unit’s place.

5.   If a number has its unit’s place digits 2, 3, 7 or 8 then it is not a perfect square.

6.   If a number has its unit’s place digits 1 0r 9 then its square has 1 as unit’s place digit.

7.    If unit’s place digit of a number is 2 or 8 then its square has 4 as unit’s place digit.

8.   If unit’s place digit of a number is 3 or 7 then its square has 9 as unit’s place digit.

9.   If unit’s place digit of a number is 4 or 6 then its square has unit’s place digit 6.

10. If unit’s place digit of a number is 5 then its square has unit’s place digit 5 and ten’s place digit 2 i.e. the square ends with 25.

11. If unit’s place digit of a number is 0 (zero) then its square has  

 unit’s and ten’s place digit as zero.

12. When a perfect square is divided by 3 remainder is 0 or 1.

Square and Square Root

Points to Remember:

1.   By multiplying a number by itself we get the square of the number.

2.   Square of positive and negative numbers is always a positive number and no negative number is a perfect square.

3.   By using the identities (a+b)² = a²+ 2ab + b² and (a – b) ² = a² – 2ab + b² we can find the square of numbers.

Points to Remember:

1.   If n= m2 then n is square root of m and we write Ön = m.

2.   If we go on subtracting 1, 3, 5, …… in order, at one stage the remainder will be zero. If the number of subtractions is n (say n times) then he square root of this number is n.

Points to Remember:

1.   Square root by prime-factorization method:

(i) Prime factors of number are made.

(ii) Pairs of equal prime factors of number are made.

(iii) One factor out of a pair of equal factors is taken.

(iv) By multiplying the one-one factor taken from pair of equal 

      factors square root of the number is obtained.

    2. If the prime factors are not in pairs then number is not a perfect   

        square because perfect square is formed by multiplying two equal

        numbers.

          We know that if equal prime factors are not in pairs then number is not a perfect square but this can be made another number by multiplying or dividing it by some prime factors which is a perfect square.

          To find the number of digits in the square root of a number, starting from unit place put dot (·) on the alternate digits of the number. The number of digits in square root is the same as the number of dots.

Points to Remember:

1. Rational number: A number which can be put in the form of p/q   

    Where p, q are integers, q ≠ 0 (i.e. q is not zero), p and q having  

    no common factor is called rational number.

2. For taking square root of rational number we write Öp = Öp

                                                                                    q    Öq         

Points to Remember:

1.   To find the square root of decimal numbers

(a) When these numbers are perfect squares:

 (i)  Starting from decimal on right and left side make the pairs.

(ii)  Start doing square root as usual.

(iii)  Whenever turn of decimals comes, put decimal in quotient.

(iv)  Proceed till remainder is zero. The obtained quotient is square root of the number.

        (b) When numbers are not perfect squares: In such questions to find out square root upto certain decimal places we put zero after the decimal point to complete the pairs because by putting zero after the decimal the value of number does not change.

area

Chap 11 MENSURATION

Chap 11

MENSURATION

SOME IMPORTANT FACTS

1.       PERIMETER : The sum of lengths of all sides of a plane figure is called the perimeter of the figure.

2.       Area : The measurement of the region enclosed by plane figure is called the area of the plane figure.

3.       Area of Rectangle            :               Length  X Breadth

Length   =  Area  / Breadth   ;       Breadth   = Area / length

4.       Area of four walls  :  = [ 2 ( length + Breadth )  x height )

5.       Area of Square  =  (side )²    =  (diagonal )²  /2

6.       Side of Square  =              √Area

7.       Area of Parallelogram  =  Base  x Height

Base   =   Area / Height  ;  Height   =  Area  / Base

8.       Area of Quadrilateral     =             Diogonal  x ( Sum of perpendicular from opposite vertices on it )

9.       Area of Rhombus             =             product of Diagonals/2

10.   Area of Triangle                               =             ( Base  x Height ) / 2

11.   Area of Trapezium          =             ( Sum of parallel sides ) X ( distance between them )

2

12.                        The Circumference /Perimeter of a circle   =       2 πr

13.   Area of a circle of radius r            =             πr²  

14.   Cuboid  ;              ii)            Volume of a Cuboid =    Length x Breadth x Height  =  ( lbh ) Cu. Units

i)             Total Surface Area of a Cuboid  =  2 ( lb + bh + lh ) sq. units

iii)           Lateral Surface Area of a Cuboid =  [ 2 ( l + b ) x h ] Sq. units

15.   Cube   ;                 i)             Volume of a Cube =    (Edge )³ =  a³  Cu. Units

i)             Total Surface Area of a Cube  =  6 a² sq. units

iii)           Lateral Surface Area of a Cube =  4a²   Sq. units

16.   Cylinder ; suppose a cylinder of Height  ‘h’ and radius ‘ r’

a.       Volume                                                =             πr²h Cu. Units

b.     Lateral Surface Area       =             2 πrh sq. Units

c.       Total surface Area           =             (2 πrh + 2πr² ) sq. Units

17.         i)                       1 cm³        =             1 ml

     ii)                      1 liter      =             1000 cm³

i)                                    1 m³       =             1000000 cm³  =  1000  liters 

Chap 11 COMPARING QUANTITIES

Chap 11

COMPARING QUANTITIES

SOME IMPORTANT FACTS

RATIO

1.       The ratio of a number ‘a’ to another number ‘b’ ( b ≠ 0 ) is a fraction a / b and is written as   a : b .

2.       In the ratio a : b , the first term  a is called antecedent and second term b is called consequent.

3.       The ratio is said to be in simplest form if its two terms have no common factors other than 1.

4.       The ratio of two numbers is usually expressed in its simplest form .

5.       A ratio with its second term 100 is also called a percent.

PERCENTAGE

6.     To convert a ratio into a per cent , we write it as a fraction and multiply it by 100 and put per cent sign (%)

7.     Increase %    =          ( increase x 100 / original value ) %

Decrease % =          ( Decrease x 100 / original value ) %

PROFIT AND LOSS

8.     The money paid by the shopkeeper to buy the goods from a manufacturer is called the Cost Price ( CP) of the shopkeeper.

9.     The price at which a shopkeeper sells the goods called the selling price (SP) of the shopkeeper.

10.                        Effective Price = Cost Price + Overhead  Charges

11.                        If  SP< CP, then there is a loss ; Loss = CP – SP

12.                        If SP > CP, then there is profit ; Profit = SP - CP

13.                         Profit or Loss is calculated on the Cost price of good.

14.                        Gain %  = ( Gain  X 100) / CP   and Loss %  =   ( Loss  x 100 ) / CP

15.                         SP  = ( 100 + Profit %)CP / 100 ;  SP    = ( 100 - Loss% )CP /100

16.                        CP  = (100 x SP) / (100 + Profit %),  CP = ( 100x SP ) / (100 – Loss %)

17.                        Discount : Sometimes shopkeeper offer certain percent of rebate on the marked price for cash payments . This rebate is known as discount.

Discount = Marked price x Rate of Discount

SP = Marked Price  - Discount

18.                        SALES TAX : The Government levies it at a specified rate on the sale price of the items and it differs from item to item and from state to state .

Sales Tax  = Selling Price  X Rate of sales Tax

SIMPLE INTEREST

19.  The money borrowed from a lender is called the principal.

20.  The additional money paid by the borrower to the lender at the end of the specified period is called the INTEREST.

21.  The total money which the borrower pays back to the lender at the end of the specified period is called the AMOUNT.

22.  Interest is said to be Simple if it is calculated on the original Principal throughout the loan period.

23.  If P = Principal, R = Rate of Interest per annum and T = Time, then the simple interest is given by           SI  =  P x R x T

100

COMPOUND INTEREST

24.  If the interest is added ( compounded ) with the Principal after a specified period of time to form a new Principal and the interest for the subsequent period is calculated on this new   Principal, then the interest thus obtain is called the Compound Interest .

25.    If P = Principal, R = Rate of Interest per annum and T = Time, then

a)      Amount after n years ( Compounded annually ) =  P(1 +  R/100)ⁿ  

b)      Amount after n years ( compounded Half yearly) =  P(1 +  R/200)ⁿ  

c)      Amount after n years ( compounded quarterly ) =  P(1 +  R/400)ⁿ  

d)      If the interest is compounded annually but time being a fraction, says 3⅔ years , then

Amount  = P(1 +  R/100)ⁿ   x  (1 +  2R/300)  

e)      If the rates be R₁ % for first year R₂ % for second year, R₃ % for third year, than Amount after 3 years

P(1 +  R₁ /100)x (1 +  R₂/100) x (1 +  R₃/100)   

GROWTH AND DEPRECIATION

26.  let P be the population at the beginning of a certain year.

a)      If the constant rate of growth is R% p.a., then                                                             Population after n years =  P x (1 +  R/100)ⁿ 

b)      If the rate of growth is R₁ % for first year R₂ % for second year, R₃ % for third year, than population after third years

P(1 +  R₁ /100)x (1 +  R₂/100) x (1 +  R₃/100)   

c)      If the Population decrease at R% p.a then  

Population after n years =  P x (1 -  R/100)ⁿ 

27.  If V is the value of a machine at a certain time and R% p.a is the rate of depreciation , then

Value of machine after n years = V x (1 -  R/100)ⁿ