[PDF] gemp101pdf - NCERT

950_6gemp101.pdf • Representation of integers on the number line and their addition and subtraction. • Properties of integers: -Integers are closed under addition, subtraction and multiplication. -Addition and multiplication are commutative for integers, i.e., a + b = b + a and a × b = b × a for any two integers a and b. -Addition and multiplication are associative for integers, i.e., (a + b) + c = a + (b + c) and ( a × b) × c = a × (b × c) for any three integers a, b and c. -Zero (0) is an additive identity for integers, i.e., a +

0 = 0 + a = a

for any integer a. -1 is multiplicative identity for integers, i.e., a × 1 = 1 × a = a for any integer a. -Integers show distributive property of multiplication over addition, i.e., a × (b + c) = a × b + a × c for any three integers a, b and c. • Product of a positive integer and a negative integer is a negative integer, i.e, a × (-b) = - ab, where a and b are positive integers. •Product of two negative integers is a positive integer, i.e., (- a ) × (-b) = ab, where a and b are positive integers. • Product of even number of negative integers is positive, where as the product of odd number of negative integers is negative, i.e., 2 (- ) (- ) ... (- ) ese1 1to e3 o ,hoer - fa aa = I- × × ... × fIand (2 1) (- ) (- ) ... (- ) uii 1to e3 o ,hoer - w a aa = -( -IEI IEImmmIEIw), where -pI pImmmpIfpIwIand o are positive integers. • When a positive integer is divided by a negative integer or vice-versa and the quotient obtained is an integer then it is a negative integer, i.e., - ÷ (- ) = (--)IcI = - a b , where - and are positive integers and - a b is an integer • When a negative integer is divided by another negative integer to give an integer then it gives a positive integer, i.e., ( l- ) ÷ (- ) = a b , where - and are positive integers and a b is also an integer. • For any integer -pI - + 1 = -IandI- b+b0Iis not defined. +÷ ↔Δ•?°≠=- →   = = • =  ° ÷-     ÷= -  =  = =  = •÷- =  ↔Δ•?°≠= →Madhre is standing in the middle of a bridge which is

20 m above the water level of a river. If a 35 m deep river

is flowing under the bridge (see Fig. 1.1), then the vertical distance between the foot of Madhre and bottom level of the river is: (a) 55 m(b)35 m (c) 20 m(d)15 m a +b

×c+b

Fig. 1.1

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Example

9:Find the odd one outl of the four optiolns given below: (a) (-3, -6)(b)(+1, -10)(c) (-2, -7)(d)(-4, -9)

Solution:Here -3 + (-6) = -9,

aaaaaaaaaaaaaaa +1 + (-10) = -9 and -2 + (-7) = -9 All the above pairs i.e. (-3, -6); (+1, -10); (-2, -7) give same answer on adding, whereas - 4 + (-9) = -13, gives a different answer. So, odd one out is (d).

Example

10:Match the integer in Column I to an integer in

Column II so that the sum is between -11 and -4

aaaa

Column I Column II

(a) -6(i)-11 (b)+1(ii)-5 (c) +7(iii)+1 (d)-2(iv)-13

Solution:(a)

↔ (iii) because -6 + (+1) = -5, which lies between -11 and -4. (b) a ↔(i) because +1 + (-11) = -10 which lies between -11 and -4. (c) ↔a(iv) because +7 + (-13) = -6 which lies between -11 and -4 (d) ↔a(ii) because -2 + (-5) = -7 which lies between -11 and -4.

Example

11:If a is an integer other than 1 and -1, match the following:

Column I Column II (a) a ÷ (-1)(i)a (b) 1 ÷ (a)(ii)1 (c) (-a) ÷ (-a)(iii)Not an integer (d) a ÷ (+1)(iv)-a

Solution:(a)

↔ (iv)(b)↔ (iii)(c)↔ (ii)(d)↔ (i)

Example

12:Write a pair of integers whose sum is zero (0) but difference

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Case 1

Correct responses = 22

Marks for 1 correct response = + 5

Marks for 22 correct respones = +110(22 × 5)

Marks scored = +110

Marks obtained for incorrect answer = 0

So, no incorrect response

And, therefore, 3 were unattempted

Case 2

Correct responses = 23

Marks from 23 correct responses = + 115(23 × 5)

Marks scored = + 110

Marks obtained for incorrect answers = 110 - (+115) = -5

Marks for 1 incorrect answer = -5

Number of incorrect responses = (-5) ÷ (-5) = 1

So, 23 correct, 1 incorrect and 1 unattempted.

Case 3

Correct responses = 24

Marks from 24 correct responses = + 120(24 × 5)

Marks scored = + 110

Marks obtained for incorrect answers = +110 - (+120) = -10 Number of incorrect responses = (-10) ÷ (-5) = 2 Thus the number of questions = 24 + 2 = 26. Whereas, total number of questions is 25. So, this case is not possible.

So, the possible ways are:

= 22 correct, 0 incorrect, 3 unattempted
= 23 correct, 1 incorrect, 1unattempted.
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Name of IonProton ChargeElectron ChargeIon Charge

Hydroxide ion +9--1

Sodium ion+11-+1

Aluminium ion+13-10-

Oxide ion +8-10-

Plan a Strategy •Some problems contain a lot of information. Read the entire problem carefully to be sure you understand all of the facts. You may need to read it over several times perhaps aloud so that you can hear yourself.

•Then decide which information is the most important(prioritise). Is there any information that is absolutelynecessary to solve the problem? This information is the mostimportant.

•Finally, put the information in order (sequence). Usecomparison words like before, after, longer, shorter, and so on

to help you. Write down the sequence before you try to solve the problem. Read the problem given below and then answer the questions that follow:

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"AFkk (a)Which location is closest to sea level? (b)Which location is farthest from sea level? (c) Arrange the locations from the least to the greatest elevation.

123.You are at an elevation 380 m above sea level as you start a motor

ride. During the ride, your elevation changes by the following metres:

540m, -268 m, 116 m, -152 m, 490 m, -844 m, 94m. What is your

elevation relative to the sea level at the end of the ride?

124.Evaluate the following, using distributive property.

(i)- 39 × 99(ii)(- 85) × 43 + 43 × ( - 15) (iii)53 × ( - 9) - ( - 109) × 53(iv)68 × (-17) + ( -68) × 3

125.If * is an operation such that for integers a and b we have

a * b = a × b + (a × a + b × b) then find (i) ( - 3) * (- 5) (ii)( - 6) * 2

126.If . is an operation such that for integers a and b we have

a . b = a × b - 2 × a × b + b × b (-a) × b + b × b then find (i) 4 . ( - 3) (ii) ( - 7) . ( - 1)

Also show that4 . ( - 3) 3 (- 3) . 4

and ( - 7) . ( - 1) 3 ( - 1) . (- 7)

127.Below u, v, w and x represent different integers, where u = -4 and

x

3 1. By using following equations, find each of the values:

u × v = u x × w = w u + x = w (a) v (b) w (c) x Explain your reasoning using the properties of integers.

128.Height of a place A is 1800 m above sea level. Another place B is

700 m below sea level. What is the difference between the levels of

these two places?

129.The given table shows the freezing points in

0

F of different gases at

sea level. Convert each of these into 0

C to the nearest integral value

using the relation and complete the table, 5

C (F 32)

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Mt. Everest which is 8,848m above sea level and the lowest point is challenger Deep at the bottom of Mariana Trench which is 10911m below sea level. What is the vertical distance between these two points?

Puzzle 1

Fill in the blank space of the following magic square so that the sum of the numbers in each row, each column and each of the diagonals is - 6. (i) -1 (ii)In this magic square, sum of the numbers in every row, column and each of the dialgonals is - 2. Fill lin the blank:

7 - 6

1

0 - 2

- 5 6 - 8

Puzzle 2

If a * b means a × b + 2 and

a # b means - a + b -(-3), then find the value of the following: (i) - 4 * 3(ii) (-3) * (-2) (iii) (-7) # (-3)(iv) 2 # (-4) (v) 7 * (-5)(vi) (-7 * 2) # 3 Next, match these answers with suitable letters by looking at the table below and arrange them in increasing order of integers to decode the name of the mathematician:

Integers-914-34-108-33 -21718

LettersP Y C T U I E G L D

Puzzle 3

'Equinoxes" are the two days of the year when the sun is directly above the earth"s equator, due to which the days and nights are of nearly equal length everywhere on the earth. Find the name of the month of autumn equinox using suitable properties of integers by solving the following questions. Match your answer with the letter given in the table and fill it in the box provided in each question. (a)(-1) × (-2) × (-3) × (-4) × (-5) (b)18946 × 99 - (-18946) (c) -1 + (-2) + (-3) + (-9) + (-8) (d)15 × (-99) (e)-143 + 600 - 257 + 400 (f)0 ÷ (-12) (g)-125 × 9 - 125 (h) (-1) × (-1) × ................... × (-1)

20 times

(i) -4 + 4 - 4 + 4 - ........ - 4

21 times

1 E -1485T -120S -30P -4R -1250B

1894600E

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