202110721-PERFORM-STUDENT-WORKBOOK-MATHEMATICS-G08-FY_Optimized

PRACTICE SHEET - 4 (PS-4) I. Choose the correct option. 1. Which of the following gives the number of times a particular value occurs in a data? (A) Frequency (B) Class (C) Class mark (D) Raw data 2. In a class 10 – 20, what is the name of the value 10? (A)Lower boundary (B)Upper boundary (C)Upper limit (D)Lower limit 3. What is a graph without gaps between the bars called? (A) Pie chart (B) Histogram (C) Bar graph (D) Pictograph 4. What does a circle graph show? (A) The maximum frequency of the data (B) The minimum frequency of the data (C) The range of the data (D)The relationship between the whole and its parts 5. The following table shows the percentages of students who like different flavours of ice-cream. Which of the following pie – charts represent the data in the table? (A) (B) (C) (D) All of these 6. Study the following pie – chart showing the expenditure of a person in a month. Identify the item on which he spent the least amount. (A) House rent (B) Transport (C) Food (D) Clothes 7. The following pie – chart shows the expenditure of a person in a month. On which item did he spend 10% of his income? (D) Both (A) and (B) (A) Clothes (B) House rent (C) Savings 37

PRACTICE SHEET - 4 (PS-4) 8. Study the following pie chart which shows the distribution of marks secured by Giri in different subjects in his half-yearly examination. If he scored 216 marks in Mathematics, how much did he score in Hindi? (A) 168 (B) 140 (C) 120 (D) 240 9. The outcomes of tossing a coin are __________. (A) Always heads (B) Always tails (C) Either heads or tails (D)Both (A) or (B) 10. The experiment whose outcome(s) cannot be predicted in advance is called __________. (A) a random experiment (B) an impossible event (C) a possible event (D) probability II. Short Answer Questions. 11. A fair die is thrown once. What is the probability that an even number appears? 12. Study the given frequency table. Which subject(s) has/have the tally marks incorrectly marked in the table? 13. The number of hours in a week that different number of students watch TV is represented in the follow- ing histogram. Study the given histogram and answer the following questions. (i) How many students watched the TV for 3 to 5 hours in the week? (ii) How many students watched the TV for the least number of hours in the week? 38

PRACTICE SHEET - 4 (PS-4) III. Long Answer Questions. 14. The marks obtained by 50 students in a mathematics exam are as follows: 52, 59, 64, 77, 15, 21, 51, 54, 72, 56, 68, 36, 65, 52, 60, 27, 23, 50, 38, 42, 63, 47, 75, 12, 33, 26, 34, 48, 58, 59, 62, 51, 48, 50, 41, 55, 57, 65, 54, 43, 56, 44, 30, 46, 67, 53, 39, 35, 47, 43. Construct a grouped frequency distribution table for the given data. 15. The number of hours spent by Farah for various activities is given in the following table. Activities No. of hours Study 5 Watch TV 2 Play 3 Sleep 8 Others 6 Represent this data as a bar graph. 39

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. A frequency distribution table is prepared for 4. The vehicles passing a toll booth is marked as marks obtained in a class of 30 students. One follows: Car (C), BUS(B), VAN(V)  (3 Marks) of the class interval is 50-60. What is the upper The following data was collected during the limit?  (1 Mark) peak hour. CVCVBVBCVBVVVBCCCBVCBBVCVCV CCVVCCCCVVBVBVB Make a frequency distribution table. Draw a bar graph to illustrate it. 2. From a well shuffled deck of 52 cards, find the probability of getting  (2 Marks) a) Queen card b) Red colour king card 5. Two dice are rolled, find the probability that the sum is  (3 Marks) a) equal to 1 b) equal to 5 3. Observe the given histogram of marks of students from class 8. Answer the following question  (2 Marks) a) What is the total number of students in the class? b) How many students have scored above 45 6. Draw a pie chart for following activities of and how many students have scored below school going child.  (4 Marks) 10? Sleep – 9hrs, c) How many students passed the test, passing School – 7hrs, marks being 30? Home work – 2 hrs, d) What group of marks are scored by Play – 2 hrs, maximum of students? Others – 3 hrs, TV- 1 hr 40

6. Squares and Square Roots Learning Outcome By the end of this chapter, you will be: • Understand the method to find the smallest • Understand and identify perfect squares. number to be divided or multiplied to make the • Understand the properties and patterns in squares. number a perfect square. • Understand the method to find square root by • Understand the method to find the square root by prime factorization method. long division method. • To find the smallest number to be subtracted or added to make the number a perfect square. Concept Map Square and Square Root Square of a Number Patterns Pythagorean Triplets Square Root Square Root by Square Root by Square Root by Long division Method Repeated Subtraction Prime factorization Smallest number to be Smallest number to be added to make a perfect multiplied to make a square perfect square Smallest number to be Smallest number to be subtracted to make a divided to make a perfect square perfect square (a+b)2 = a2 + 2ab + b2 Key Points • Pythagorean triplets For any natural number m > 1, we have (2m)2 + • If a natural number m can be expressed as n2, where n is also a natural number, then m is a square (m2−1)2 = (m2 + 1)2 number So 2m, m2−1 and m2 + 1 forms a Pythagorean • The numbers 1, 4, 9, 16 ... are square numbers. triplet. These numbers are also called perfect squares. • Square root of a number is the number whose • All square numbers ends with 0, 1, 4, 5, 6 or 9 at square is given number m = n2 unit’s place Square root of m, m = n Square root is denoted by expression • If a number ends with 1 or 9 in the units place its • Finding square root through repeated subtraction square ends with 1 We know sum of the first n odd natural numbers is • If a number ends with 2 or 8 in the units place its n2. So in this method we subtract the odd number, square ends with 4 starting from 1 until we get the reminder as zero. The count of odd number will be the square root. • If a number ends with 3 or 7 in the units place its Example: 36 square ends with 9 Then, (i) 36−1 = 35 • If a number ends with 4 or 6 in the units place its (ii) 35−3 = 32 square ends with 6 41 • If a number ends with 5 in the units place its square ends with 5 • If a number ends with 0 in the units place its square ends with 0 • Square of a number can be found using identity,

6. Squares and Square Roots (iii) 32−5 = 27 digit as 5. Get the remainder. (iv) 27−7 = 20 o Step 6: Since the remainder is 0 and no digits are (v) 20−9 = 11 (vi) 11−11 = 0 left in the given number, therefore the number So 6 odd numbers, Square root is 6 on the top is square root Example: • Finding square root through prime factorization In this method, we find the prime factorization of   25 2 6 25 the number. We will get same prime number occurring in pair –4 45   2 25 for perfect square number. Square root will be giv- en by multiplication of prime factor occurring in – 2 25 pair.    0 Example: 81 81 = (3 × 3) × (3 × 3) 42 81 = 3 × 3 = 9 • Smallest number to be multiplied or divided to form a perfect square Find prime factorization of given number, the number which does not occur as pair is the smallest number to be multiplied or divided to make the given number a perfect square. Example: 72 = 2 x 2 x 2 x 3 x 3 Smallest number to be multiplied or divided is 2. • Finding square root by division method This can be well explained with the example: o Step 1: Place a bar over every pair of digits starting from the digit at one’s place. If the number of digits in it is odd, then the left-most single digit too will have a bar.   In the below example 6 and 25 will have separate bar. o Step 2: Find the largest number whose square is less than or equal to the number under the extreme left bar. Take this number as the divisor and the quotient with the number under the extreme left bar as the dividend. Divide and get the remainder. In the below example 4 < 6, So taking 2 as divisor and quotient and dividing, we get 2 as reminder. o Step 3: Bring down the number under the next bar to the right of the remainder.   In the below example we bring 25 down with the reminder, so the number is 225. o Step 4: Double the quotient and enter it with a blank on its right.   In the below example, it will be 4. o Step 5: Guess a largest possible digit to fill the blank which will also become the new digit in the quotient, such that when the new divisor is multiplied to the new quotient the product is less than or equal to the dividend.   In this case 45 × 5 = 225 so we choose the new

6. Squares and Square Roots • Smallest number to be subtracted to make a given number a perfect square Find the square root of the given number using long division method, the number left as remainder is the smallest number to be subtracted from the given number to make it a perfect square Example: Find the smallest number which must be subtracted from 3250 to obtain a perfect square number. Also, find the square root of that number. Answer: Using division method, we have,   5  7 5 3250 – 25 107   7  50 – 7  49      1 Here, remainder is 1. Thus, we must subtract 1 from 3250 to get perfect square number. Therefore, required square number = 3250−1 = 3249.And, 3249 = 57 • Smallest number to be added to make a given number as perfect square Find the square root of given number by direct method, find the square of successor of the quotient. The given number subtracted from the square of the successor of the quotient is the smallest number to be added. Example: Find the smallest number which must be added to 1825 to obtain a perfect square number. Also, find the square root of that number. Using division method we have Here, remainder is 61 which means square of 42 is less than 1825. The next number after 42 is 43 whose square is 1849. So, number to be added to 1825 = 432−1825 = 1849−1825 = 24. Thus, we must add 24 to 1825 to get perfect square number. Therefore, required square number = 1825 + 24= 1849. And, 1849 = 43 43

6. Squares and Square Roots Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Prime factors • Prime numbers, Prime factorization PS – 1 Squares • Introducing the concept of squares and explaining the term perfect square PS – 2 • Properties of square number PS – 3 • The digits 1, 4,5,6,9 with which a perfect square will end. • Patterns in square number • Finding the square of a number without actual multiplication. Square root • Finding square root of a number by repeated subtraction PS – 4 • Finding square root of a perfect square by prime factorization method. • Finding the smallest number to be multiplied or divided to make a number a perfect square. • Finding the square by long division method. • Finding the smallest number to be added or subtracted to make a number a perfect square. • Square root of decimal numbers • Square root by successive subtraction Worksheet for “Square and Square Roots” PS – 5 Evaluation with Self Check or ---- Self Evaluation Sheet Peer Check* 44

PRACTICE SHEET - 1 (PS-1) 1. Find the prime factors of the following numbers: i) 128  ii) 305  iii) 158  iv) 625  v) 309 vi) 175  vii) 1463  viii) 468  ix) 816  x) 1001 PRACTICE SHEET - 2 (PS-2) 1. Write a Pythagorean triplet whose one member is 24. 2. What will be the unit’s digit of the squares of following numbers (i) 23  (ii) 42  (iii) 91  (iv) 155  (v)2036 3. The squares of which of the following numbers could be odd. (i) 289  (ii) 225  (iii) 464  (iv) 361  (v) 676 4. If 132 = 168   1032 = 10609   10032 = 1006009   1000032 = ? 5. Why is 4000 not a perfect square? 6. Find the square of following numbers: (i)64  (ii) 105  (iii) 218  (iv) 92  (v) 109 7. Find the square of 72 without actual multiplication. 8. Which of the following numbers cannot be the square of a number? Give reasons. (i) 363 (ii) 567 (iii) 7056 (iv) 168 (v) 4356 PRACTICE SHEET - 3 (PS-3) 1. Find the square root of the following numbers by repeated subtraction: (i) 81  (ii) 144 2. Find the square root by prime factorisation method: (i) 2916  (ii) 1764  (iii) 1156  (iv) 324  (v) 1521 (vi) 2304  (vii) 1764  (viii) 2025  (ix) 4225  (x) 4624 3. Find the smallest number to be multiplied to make the following numbers a perfect square, find the square root of the resulting number: (i) 2420  (ii) 4032 4. Find the smallest number to be divided to make the following numbers a perfect square, find the square root of the resulting number: (i) 1805  (ii) 4046  (iii) 7623  (iv) 7350  (v) 6358 45

PRACTICE SHEET - 4 (PS-4) 1. Find the square root of following numbers by long division method: (i) 2209  (ii) 2809  (iii) 9409  (iv) 15129  (v) 27225 (vi) 339889  (vii) 432964  (viii) 528529  (ix) 962361  (x) 11881 2. Find the square root of the following numbers to two places of decimal. (i)17  (ii) 10.5  (iii) 65  (iv) 5.18  (v) 108 3. Find the square root of the following number to three places of decimal: (i) 67  (ii) 168  (iii) 35  (iv) 96.25  (v) 258 4. Find the smallest number to be subtracted to make the following numbers a perfect square: (i) 690  (ii) 3277  (iii) 13091  (iv) 83625  (v) 966293 5. Find the smallest number to be added to make the following numbers a perfect square: (i) 41995  (ii) 369519  (iii) 969920  (iv) 431541 (v) 8825 6. Find the greatest and least 4 digit number which is a perfect square. Also find the square root of the number so obtained. PRACTICE SHEET - 5 (PS-5) I. Choose the correct option. 1. Which of these is not a perfect square? (A) 145 (B) 156 (C) 169 (D) 225 2. Identify the square of 18. (A) 575 (B) 196 (C) 529 (D) 324 3. The ones digit of the square of which of the following numbers is 1? (A) 32 (B) 49 (C) 72 (D) 93 4. Which of the following numbers has 6 in its ones place? (A) 81² (B) 60² (C) 24² (D) 53² 5. The natural number N can be expressed as the sum of odd natural numbers starting with 1. What type of number is N? (A) A perfect square number (B) Not a perfect square number (C) A perfect cube number (D) Both (B) and (C) 6. A Pythagorean triplet is _____________. (A) 3, 5, 7 (B) 8, 15, 17 (C) 11, 13, 15 (D) 8, 10, 12 7. The prime factorisation of 256 is __________. (A) 2 × 2 × 2 × 2 × 2 × 2 (B) 2 × 2 × 2 × 2 × 2 (C) 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 (D) 2 × 2 × 2 × 2 × 2 × 2 × 2 8. What is the number equal to (2 × 2 × 2)²? (A) 81 (B) 36 (C) 16 (D) 64 9. The prime number by which 3200 is to be multiplied by to make it a perfect square is ____________. (A) 7 (B) 5 (C) 2 (D) 3 10. The square root of 4096 is __________. (A) 64 (B) 24 (C) 56 (D) 36 46

PRACTICE SHEET - 5 (PS-5) II. Short Answer Questions. 11. Find the square root of 2304 by prime factorisation method. 12. Between which two numbers does the square root of 450 lie? III. Long Answer Questions. 13. The area of a square field is 48400 sq. m. Find its perimeter. 14. Using long division method, find the square root of the decimal number 98.01. 15. Find the least number that must be added to 6225 to make it a perfect square. 47

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. Why is 243 not a perfect square? (1 Mark) 6. Find the square root of 10609 by long division method.(2 Marks) 2. Find the unit’s digit of the square of 342. (1 Mark) 7. There are 2312 marbles to be arranged such that there are same numbers of marbles in each row as the numbers of rows are. Find the number of marbles left out. (3 Marks) 3. Find the square of 53. (1 Mark) 4. Find 4624 by prime factorization method. (2 Marks) 8. What least number must be added to 5607 to make the sum a perfect square? Find this perfect square 5. Find the smallest number to be multiplied to (3 Marks) 9072 to make it a perfect square. (2 Marks) and its square root. 48

7. Cubes and Cube Roots • Find cube root of a perfect cube by estimation • Find the cube root of a cube by prime factorization Learning Outcome method By the end of this chapter, you will be: • Define a perfect cube • Find the smallest number to be multiplied or divided to form a perfect cube Concept Map Key Points Example 2: Smallest number to be divided to make 108 a perfect cube. • Cubes of numbers 108 = 3 × 3 × 3 × 2 × 2 • Perfect cubes Factor 3 occurs thrice and 2 occurs twice, smallest number to be divided is 2 × 2 = 4 Every number can be expressed as the product of • Cube root by estimation method power of it’s prime factors. If the powers of all the - Make groups of 3 digits from unit (one’s) place. prime factors are multiples of 3, then the number is said to be a Perfect cube. This 1st group. The remaining number makes 1331=11x11x11 is a perfect cube the 2nd group. • Smallest number to be multiplied to make a - The unit’s digit of the 1st group will decide the number a perfect cube unit digit of the cube root. (If the unit digit of the Example 1: Find the smallest number to be cube is 6 then the unit digit of cube root will also multiplied to make 72 a perfect cube. 6 as 6 × 6 × 6 = 216). 72 =2 × 2 × 2 × 3 × 3 - Find the cube of numbers between which the The factor 2 occurs thrice, 3 occurs twice, smallest 2nd groups lie number to be multiplied is 3. - Take the smaller number as its ten’s digit. Example 2: Smallest number to be multiplied to Example 1: Find the cube root of 238328. 750 to make a perfect cube. 238328 is an even number so its cube root has to be 750 = 5 × 5 × 5 × 3 × 2 an even number. The factor 5 occurs thrice, factor 3 and 2 occurs Make two groups 238 328 once. The smallest numbers to be multiplied is 3 × First group 328 - unit digit is 8 so unit digit of cube 3 × 2 × 2 = 36 root will be 2. • Smallest number to be divided to make a number Second group 238 - it lies between 63 = 216 and 73 a perfect cube = 343. Example 1: Smallest number to be divided to make So take the smaller number 6 as ten’s place. 24 a perfect cube. 3 238328 = 62 24 = 2 × 2 × 2 × 3 • Cube root of perfect cubes by prime factorization Factor 2 occurs thrice and 3 occurs once, smallest method number to be divided is 3. 49

7. Cubes and Cube Roots Prime factorize the given number and group the factors in sets of three. From each set of three numbers one number is taken out of the cube root. The product is called cube root of the given number. Denoted by 3 . 3 1000 = 3 2u2u2u 5u5u5 = (2) × (5) = 10 Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Pre-requisite knowledge: • Square and Square root Square and Square root • Concept of perfect square and smallest number to be multiplied or divided to form a perfect square Cubes • Explaining the concept of perfect cubes • Prime factorization method to find the smallest number PS – 1 to be multiplied to form a perfect cube • Prime factorization method to find the smallest number to be divided to form a perfect cube Cube Root • Finding cube root of a perfect cube by prime factorization method Worksheet for “Cubes and Cube Roots” PS – 2 Evaluation with Self Check or ---- Self Evaluation Sheet Peer Check* 50

PRACTICE SHEET - 1 (PS-1) 1. Find the one’s digit of the cube of each of the following: (i) 243  (ii) 391  (iii) 1568  (iv) 2497  (v) 1089 2. Which of the following numbers are perfect cubes? (i) 1152  (ii) 42875  (iii) 2916  (iv) 2744  (v) 10125 3. Find the smallest number to be multiplied to make the following numbers a perfect cube: (i) 3456  (ii) 52728  (iii) 196608  (iv)  300125  (v) 78608 4. Find the smallest number to be divided to make the following numbers a perfect cube (i) 93312  (ii) 24565  (iii) 209952  (iv) 109774  (v) 677376 5. Find the cube root of following numbers: (i)10648  (ii) 1728  (iii)54872  (iv) 91125  (v) 17576  (vi) 35937  (vii) 12167  (viii) 140608  (ix) 24389  (x) 132651 6. Three numbers are in the ratio 1:2:3 and sum of their cubes is 36000. Find the numbers. PRACTICE SHEET - 2 (PS-2) I. Choose the correct option. 1. Which of these is obtained by multiplying a number by itself three times? (A) The square of the number (B) The cube of the number (C) The square root of the number (D) The cube root of the number 2. Identify the cube root of 729. (A) 9 (B) 19 (C) 29 (D) 33 3. The ones digit of the cube of which of the following numbers is 5? (A) 36 (B) 44 (C) 73 (D) 95 4. Which of the following numbers has 6 in its ones place? (A) 81³ (B) 60³ (C) 26³ (D) 53³ 5. The number of perfect cubes between 1 and 100, both inclusive is _______. (A) 1 (B) 0 (C) 2 (D) 4 6. The cube of an even number is _____________. (A) never even (B) always odd (C) always even (D) either odd or even 7. The prime factorisation of the cube of 15 is __________. (A) 3³ × 5³ (B) 3³ × 2³ (C) 3³ × 5³ (D) 3³ × 7³ 8. What is the number equal to (3 × 3 × 3)³? (A) 19681 (B) 19683 (C) 19861 (D) 18694 9. The number by which 400 is to be multiplied by to make it a perfect cube is ____________. (A) 10 (B) 50 (C) 20 (D) 30 10. The cube root of 4096 is __________. (A) 24 (B) 12 (C) 8 (D) 16 II. Short Answer Questions. 11. Find the cube root of 216 by prime factorisation method. 12. What is the cube root of 1000? 13. Estimate the cube root of 614125? III. Long Answer Questions. 14. Is 49000 a perfect cube? If yes, find its cube root. If not, find the smallest number by which 49000 is to be multiplied to make it a perfect cube, and find its cube root. 15. Find the least number by which 6125 has to be divided to make it a perfect cube. Find its cube root. 51

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. (i) Check if 200 is a perfect cube. 4. Three numbers are in the ratio 4:5:7 and sum of (ii) Find the cube of 32. their cubes is 4256. Find the numbers. (3 Marks) (iii) F ind the units digit of the cube of 354 without actual calculation. (3 Marks) 2. (i) Find the smallest number to be divided to 5. Evaluate : 3 27 + 3 0.008  (2 Marks) make 596288 a perfect cube. (ii) Find the smallest number to be multiplied to make 137781 a perfect cube.  (4 Marks) 3. Find cube root of 2500470.  (3 Marks) 52

8. Comparing Quantities Learning Outcomes At the end of this chapter, you will be able to: • Identify and solve problems based on cost price, • Find the ratio between two given quantities. selling price, profit and loss, discount and discount • Understand the concept of selling price, cost percent. price, profit and loss. • Differentiate between simple interest and compound interest. Concept Map Key Points like amount spent on repairs, labour charges, transportation, etc. • Ratio: Comparing by division is called ratio. • If SP > CP then Profit = SP - CP and Quantities written in ratio have the same unit. Profit % = Profit x100 Ratio has no unit. Equality of two ratios is called proportion. Cost Price • If CP > SP then Loss = CP - SP and • Product of extremes = Product of means Loss % = Loss x100 • Percentage: Percentage means for every hundred. Cost Price The result of any division in which the divisor is • Discount is a reduction given on marked price.  100 is a percentage. The divisor is denoted by a special symbol %, read as percent. Discount = Marked Price – Sale Price. • Profit and Loss: • Discount can be calculated when discount (i)  Cost Price (CP): The amount for which an article is bought. percentage is given.  Discount = Discount % of (ii)  Selling Price (SP): The amount for which an Marked Price article is sold. • Additional expenses made after buying an article • Additional expenses made after buying an article are included in the cost price and are known as are included in the cost price and are known as overhead expenses. These may include expenses 53

8. Comparing Quantities overhead expenses. CP = Buying price + Overhead A P ©¨§1  R ·n  P is principal, R is rate of interest, expenses 100 ¹¸ • Sales tax is charged on the sale of an item by the government and is added to the Bill Amount. Sales n is time period tax = Tax% of Bill Amount (ii) Amount when interest is compounded half • Simple Interest: If the principal remains the same for the entire loan period, then the interest paid is ©¨§1 R ·n/ 2 100 ¸¹ called simple interest. SI P u R uT yearly A P    100 • Compound interest is the interest calculated on the previous year’s amount (A = P + I) (i) Amount when interest is compounded annually  Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Ratio and proportion • Introduction to ratio • Converting ratio to percentage • Converting percentage to ratio PS – 1 • Proportion Profit, loss and discount • Cost price, selling price, profit, loss • Profit % and Loss % • Marked Price, Discount PS – 2 • Discount % Simple interest and Compund Interest • Simple interest • Compound interest, Compounded PS – 3 half yearly, Compounded annually • Depreciation Worksheet for “Comparing Quantities” PS – 4 Evaluation with Self Check or Peer ---- Self-Evaluation Sheet Check* 54

PRACTICE SHEET - 1 (PS-1) 1. Find the ratio of the following: (i) 50m to 2 km (ii) 3 kg to 600 gms (iii) Rs 2 to 75 paise 2. Convert the following ratio into percentage: (i) 3:5 (ii)  5:4 3. 60% of 40 students are good in playing football, and the rest are good in cricket. How many are good in cricket? 4. Arun spends 80% of his monthly income. If his monthly salary is Rs 16000 . How much does he save? 5. In an election candidate A got 45% votes, candidate B got 30% votes and candidate C got the rest of the votes. If total votes casted in the election were 30000. Find the votes got by each candidate. 6. Ashok got a 20% increase in his earnings. If he was earning Rs 2, 40, 000 , Find his new earning. 7. Deepti went to school for 216 days in a full year. If her attendance is 90% , find total working days of the school. 8. A garden has 2000 trees. 12% of these are mango trees, 18% are lemon and the rest are orange trees. Find the number of orange trees. 9. Rohit deposits 12% of his income in a bank. He deposited Rs. 1440 in the bank during 2017. What was his total income for the year 2018? 10. A certain school has 300 students, 142 of whom are boys. It has 30 teachers, 12 of which are men. What percent of total number of students and teachers in the school are female? 55

PRACTICE SHEET - 2 (PS-2) 1. A student buys a pen for Rs 150 and sells for Rs 175 . Find his profit Percentage. 2. A shopkeeper bought 5 kg oranges at Rs 15 per Kg and sold it at Rs 17 per kg. Find the total profit in the overall transaction. 3. Akash bought a refrigerator for Rs 15000 and spent Rs 1500 on transportation and sold it for Rs 20000 . Find his profit percentage. 4. A fruit seller bought 30 kgs of apples for Rs 52 per kg, out of which 4 kg apples were rotten. He sold the rest of the apples at Rs 58 per kg. Find overall gain or loss percent. 5. Rakesh bought two televisions for Rs 9000 each. He sold one at a profit of 20% and one at a loss of 15% . Find profit% or loss % on the whole transaction. 6. A shopkeeper bought 90 articles for Rs 2700 and sold them at a profit of 20 % on whole. Find the selling price of each article. 7. A vendor sold two air-conditioners for Rs 15000 each. On one he made profit of 10% and on the other he had a loss of 5 %. Find his overall gain% or loss%. 8. Rudra purchased a car for Rs 73500. He spent Rs 10300 on repairs and paid Rs 2600 for insurance. Then he sold it for Rs 84240. What was his loss or gain %. 9. The cost price of an article is 90 % of its selling price. What is the profit or loss percent? 10. A man sold a radio set for Rs 250 and gained one ninth of its cost price. Find its cost price and gain percent. 11. A shopkeeper buys pens at 5 for Rs 28 and sells at 25% profit. Find the amount to be paid by a boy to buy 3 pens. 12. By selling 8 pens, Shyam loses equal to cost price of 2 pens. Find his loss percent. 13. A shop keeper bought rice for Rs 4500 . He sold one third at 10% profit.What price must he sell the rest of the rice so as to gain 12% on overall transcation? 14. Ram sold a watch at 5% profit, had he sold for Rs 24 more, he would have gained 11% . Find the cost price of the watch. 15. An article is marked at Rs 1300 and sold at Rs 1144 . Find the discount percent. 16. The marked price of a television set is Rs 23600 and is available at a discount of 8%. Find the selling price of television set. 17. An article is marked at Rs 2250. By selling it a discount of 12 %, the dealer makes a profit of 10 %. Find the selling price of the article and cost price of the article for the dealer. 18. A shopkeeper marks his goods at 30 % above the cost price and then gives a discount of 10 %. Find his gain %. 19. The cost price of an article is 25% below the marked price. If the article is available at 15% and its cost price is Rs 2400 . Find its marked price, selling price and profit percentage. 56

PRACTICE SHEET - 3 (PS-3) 1. Find the interest and amount on Rs 1500 for 2 years at 8 % rate of simple interest. 2. Find the interest and amount on Rs 2600 in 2� years and 3 months at 10% per annum simple interest. 3. Find the principal which will amount to Rs 4000 in 4 years at 6.25% per annum. 4. In how many years will Rs 950 produce Rs 399 as simple interest 7% ? 5. Raj borrows Rs 8000 , out of which Rs 4500 at 5% simple interest and remaining at 6% simple interest. Find the total interest paid by him after 4 years. 6. A sum of money lent out in simple interest doubles itself in 8 years. Find the rate of interest. 7. Rs 4000 will amount to Rs 5000 in 8 years. In how many years will Rs 2100 amount to Rs 2800 at same rate of interest? 8. A sum of Rs 8000 is invested for 2 years at 10% per annum compound interest. Find the interest for first year and the amount payable at the end of 2 years. 9. Raman invested Rs 15000 at an interest of 12% p.a compounded half yearly. Find the amount he gets after 1 1 years. 2 10. Find the C.I when a sum of Rs 20, 000 is invested for 1 year and 5 months at 10% per annum compounded annually. 11. Ram borrows Rs 8000 at 8% per annum simple interest for 3 years, Rahul borrows same amount for 10% per annum compound interest for same time. Find the difference in the interest. 12. A motorcycle was bought for Rs 45000 . Its value depreciated by 10% per annum. Find the value of the motorcycle after 2 years. 57

PRACTICE SHEET - 4 (PS-4) I. Choose the correct option. 1. Identify the correct factor used to convert speed in km/h to m/s. 7 5 18 -5 (A) (B) 18 (C) 18 (D) 5 18 2. The equation used to find the selling price (S.P) of an article is _______. (A) S.P = Cost Price + Profit (B) S.P = Cost Price – Profit (C) S.P = Cost Price – Loss (D) Both (A) and (C) 3. Which of these equations is correct? (A) Discount = Marked Price – Selling price (B) Discount = Marked Price – Discount % of Marked Price (C) Marked Price = Selling price – Discount (D) Selling Price = Marked price + Discount 4. Mr. Kiran bought a pen for Rs. 30 and sold it for Rs 50. What is his profit percent? 1 (B) 2 1 (D) 20% (A) 66 3 % 66 % (C) 33 3 % 3 5. A discount of 5 % is offered on a chair costing Rs. 250. How much should a person who buys the chair pay for it? (A) Rs. 235.50 (B) Rs. 225 (C) Rs. 230 (D) Rs. 237.50 6. A sales tax of 10% is charged on an item that costs Rs. 400. What is the bill amount? (A) Rs. 440 (B) Rs. 360 (C) Rs. 410 (D) Rs. 340 7. The number of trees in a locality is 20000. Every year, there is an increase of 5% in planting more trees. How many trees will be there in the locality after 2 years? (A) 20020 (B) 22050 (C) 22000 (D) 25000 8. Somesh borrowed Rs 1000 for 2 years at the rate of 5% p.a. simple interest. How much interest does Somesh get? (A) Rs. 110 (B) Rs. 150 (C) Rs. 100 (D) Rs. 120 9. In computing the amount at compound interest, with interest compounded half – yearly, the rate R is given by ____________. 2R R (A) 2R (B) 5 (C) R (D) 2 10. The loss incurred by selling an article costing Rs. N at Rs. N/2 is __________. (A) 100% (B) 50% (C) 25% (D) 10% II. Short Answer Questions. 11. Hameed paid Rs. 6000 for a cell phone, which included a GST of 20%. What is the actual cost of his cell phone? 12. A set of permanent markers is sold for Rs. 805 including a GST of 15%. What is the actual cost of the set of markers? 13. A grocer sold 2 bags of rice for Rs. 5460 each. He made a profit of 5% on one and a loss of 5% on the other. Did he gain or lose overall? Find the value of loss or gain. III. Long Answer Questions. 14. Hari purchased a book for Rs. 450, including a 10% VAT. What is the actual cost of the book (Round- ed to the nearest 10s)? 15. Rupa borrowed Rs. 10000 from a bank for 2 years at the rate of 10% p. a. simple interest and Rs. 10000 from her friend for the same period at the rate of 10 % p.a. compounded annually. On which sum did she pay a greater interest? How much is it? 58

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. The total number of students in a class is 40. If there are 12 boys and the rest girls. Find the ratio between boys and girls and the percentage of girls in the class. (2 Marks) 2. A shopkeeper bought 20 pens for Rs 5 each. At what price must he sell each pen so to gain 20% on whole? (2 Marks) 3. Find the amount at the end of 2 years that will be received on a deposit of Rs 20, 000 at 15% per annum simple interest. (2 Marks) 4. A man sold a watch at a loss of 5% , had he sold it for Rs 104 more, he would have gained 8% . Find the selling price of the watch. (3 Marks) 5. A man bought two bags for Rs 750 each. He sells one at 6% gain and other at 4% loss. Find the total gain or loss on the overall transaction. (3 Marks) 6. On what sum will the compound interest at 5% per annum for 2 years compounded annually be Rs 164? (3 Marks) 59

9. Algebraic Expressions and Identities Learning Outcome • Multiply a monomial by monomial and monomial by a polynomial By the end of this chapter, you will be: • Identify the terms and coefficients in an algebraic • Multiply a polynomial by polynomial • Understand and solve algebraic expressions expression • Differentiate between monomial, binomial and using standard identities polynomials • Add and subtract algebraic expressions Concept Map Key Points Algebraic expressions having two terms is called binomials. • Algebraic expression is the expression having Example: x + xy, 2p + 11q constants and variable. o Trinomials Example: 12x, 3x  7, 3xy  6x  3, 5 y  4 Algebraic expressions having three terms is called trinomials • Terms, coefficient and factors Example: x + xy + 1, 2p + 11q + pq Example: 15x + 4 o Polynomials Terms are 15x and 4. An expression containing, one or more terms with non–zero coefficient (with variables having non • Terms are added to form algebraic expressions. negative exponents) is called a polynomial. • Terms themselves can be formed as the product of Example: x + y + z+ 2, 3pq, 11xyz−10x • Like and Unlike Terms factors. When the variable part of the terms is same, they The term 15x is the product of its factors 15 and x. are called like terms. The term 4 is made up of just one factor, i.e., 4. Example: 2x, 3x are like term • The numerical factor of a term is called its numerical coefficient or simply coefficient. The 5x2 and −9x2 are like terms. coefficient in the term 15x is 15 and the coefficient When the variable part of the terms is not same, in the constant term 4 is 4. they are called unlike terms. • Monomials, Binomials, Trinomials Example: 2x, 3y are unlike terms, o Monomials Algebraic expressions having one term is called 5x2 and −9zx2 are unlike terms monomials. • Addition and Subtraction of Algebraic Expressions Example: 6xy, –2z, 10y o Binomials o Addition 60

9. Algebraic Expressions and Identities Add the like terms. 2. (−10pq33) × (6p35) = −60p36q33 Write the final algebraic expression o Multiply the binomials Example: Add the following expression x−y + xy, 1. (2a + 6b) and (4a−3b) y−z + yz, z−x + zx 2. (x−1) and (3x−2) Solution: = (x−y + xy) + (y−z + yz) + (z−x + zx) Ans: Let us first understand how to multiply the Arranging the like term together terms (a + b) and (c + d). Multiplication can be done by distributive law = x−x−y + y−z + z + xy + yz + zx (a + b) (c + d) = a (c + d) + b (c + d) = xy + yz + zx  (As x−x = y−y = z−z = 0) = (a × c) + (a × d) + (b × c) + (b × d) o Subtraction We will use the same concept in all the question Subtract the expression together on the same below line, change the sign of all the term which is to be 1. (2a + 6b) and (4a−3b) subtracted and then arrange the like term together. = 2a × 4a−2a × 3b + 6b × 4a−6b × 3b Add the like terms. = 8a²−6ab + 24ab –18b2 Write the final algebraic expression = 8a² + 18ab−18b2 Example: Subtract 4x−7xy + 3y + 12 from 12x−9xy 2. (x−1) and (3x−2) + 5y−3 = x × 3x−2x−1 × 3x + 2 Solution: = (12x−9xy + 5y−3)−(4x−7xy + 3y + 12) = 3x2 – 2x−3x + 2 While subtracting, we need to remember signs are = 3x2 – 5x + 2 reversed after ‘–’ sign once bracket is opened, i.e. + o Multiply Binomial by Trinomial becomes−and−becomes + 1. (3x + 1) (4x2−7x + 1) = 12x−9xy + 5y−3−4x + 7xy−3y−12 Ans: By distributive law Arranging the like term together = 3x (4x2−7x + 1) + 1 (4x2−7x + 1) = 8x−2xy + 2y−15 = 12x3−21x2 + 3x + 4x2−7x + 1 • Multiplication of Algebraic expression = 12x3−17x2−4x + 1 General steps for Multiplication o Multiply Trinomial by Trinomial 1) We have to use distributive law and distribute (a + b + c) (a + b−c) each term of the first polynomial to every term of Answer: By distributive law the second polynomial.    = a (a + b−c) + b (a + b−c) + c (a + b−c) 2) When you multiply two terms together you must    = a2 + ab−ac + ab + b2−bc + ac + bc−c2 multiply the coefficient (numbers) and add the    = a2 + b2−c2 + 2ab exponents • Identity 3) Also as we already know An identity is an equality, which is true for all values      (+) × (+) = (+) of the variables in the equality.      (+) × (−) = (−) (a+b)2 = a2 + 2ab + b2      (−) × (−) = (−) (a−b)2 = a2−2ab + b2      (−) × (−) = (+) (a+b) (a−b) = a2−b2 4) Group like terms (x+a) (x+b) = x2 + (a+b) x + ab It is true for all the values of a and b. Types of Multiplication An equation is true only for certain values of its o Multiplication of monomial to monomial variables. An equation is not an identity. o Multiplication of monomial to binomial, 61 trinomial or more terms polynomials o Multiplication of binomial, trinomial or more terms polynomials to monomial o Multiplication of binomial to binomial, trinomial or more terms polynomials o Multiplication of trinomial to trinomial or more terms polynomials • Multiplication of Monomials 1. a2 × (2b22) × (4a26) 2. (−10pq33) × (6p35) Ans: Multiplying the constant and using the exponent property given above we get 1. a2× (2b22) × (4a26) = 8a28b22

9. Algebraic Expressions and Identities Example 1. (y + 1) (y + 1) 2. (2x + 1) (2x−1) 3. (2z−3) (2z−3) Answer: 1. (y + 1) (y + 1)    = (y + 1)2 (a + b)2 = a2 + 2ab + b2   = y2 + 2y + 1 2. (2x + 1) (2x−1) (a + b) (a−b) = a2−b2   =4x2−1 3. (2z−3) (2z−3)    = (2z−3)2 (a−b)2=a2−2ab + b2   =4z2−12z + 9 Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET PS – 1 Algebraic expressions • Variable, constant, algebraic expression, Integers transposition, PS – 2 • BODMAS PS – 3 • Like term, unlike terms, numerical coefficients • Addition and subtraction of like terms PS – 4 PS – 5 Algebraic Expressions: • Revising definition of terms, variables, coefficients Self Evaluation Sheet Addition and subtraction • Classification of monomial, binomial, polynomial • Addition of algebraic expressions • Subtraction of algebraic expressions Algebraic Expressions: • Multiplication of Multiplication of algebraic Monomial by monomial expression Monomial by binomial Monomial by polynomial Polynomial by polynomial Standard Identity • (a+b)2 = a2 + 2ab + b2 • (a−b)2 = a2−2ab + b2 • (a+b) (a−b) = a2−b2 • (x+a) (x+b) = x2 + (a+b) x + ab Worksheet for “Algebraic Expressions and Identities” Evaluation with Self Check –––– or Peer Check* 62

PRACTICE SHEET - 1 (PS-1) 1. Write the coefficient of x in the following expressions. (i) 2x−3y (ii) 3−x + 4y (iii) 3 + 5x + 7a (iv) 2−8x + 3y 2. Classify as monomial, binomial trinomial (i) 3x + 4y (ii) 2xy (iii) 3x−9y + 3z (iv) 2x−3 (v) 15x−8y + 17z 3. Identify the like terms –2xy, 4yx, –5x2y, 20xy2, –13yx2, –5yx2, –12x2y 4. Simplify: (i) 3a + 5b−7a + 8b (ii) 4x2 + 6y2−5x2 + 9y2 + z2 (iii) 2 (a + 3b)−(2b + c)−4(a + b) (iv) 5 (a−5b + 3c)−5 (2a + c) (v) 3xy−x2−5yx + y2−3x2−5y2 (vi) (a2−5ab + 3b2)−(2a2−8ab + 3b2) (vii) a (b−7) + b (a + 7)−6ab (viii) (4pq−3q2) + (7pq−9p2) (ix) 4a + 2b + 3ab−9ab + 4b (x) 2xy + 3xy2 + 7x2y−9x2y−8xy PRACTICE SHEET - 2 (PS-2) 1. Add the following (i) ab−8ac, 5ab + 9bc, 8ab−6ac (ii) 2a + 5b−3ab, 5a + 3b−5ab, 4ab−2a + 9b (iii) 2x2y + 7xy2, 5x2y−6xy2, 6xy2−7x2y (iv) 9ab + 2a2−b2, 3a2 + 6b2, 6ab−2a2 + 3b2 (v) 8xy−y2 + 6x2, 7x2 + 8y2−9xy 2. Subtract 3a + 4b−3c + 5 from 9a + 6b−3c + 9. 3. Subtract 4x2−6y2 + 6xy from 9xy + 8y2−6x2. 4. Add x2−7y2 + 6z2 and 5x2−6z2, subtract the sum from 5y2−9z2. 5. From the sum of 3ab + 5bc + 8ca and 6bc−9ab + 7ca, subtract 5ab + 9bc−3ca. 63

PRACTICE SHEET - 3 (PS-3) 1. Find the following products: (i) 3, 4x  (ii) –5a, 6a (iii) –8x, 6xy  (iv) –5xy, –8xy  (v) 8ab, 6ba  (x) –5x2, 6x3, 8x4y (vi) 5x, 6x2, 7x3   (vii) 3a, 4bc, 5ac  (viii) x2y3z, –3yz2, 5x3y2  (ix) 3, 5y, –5x, –10xy  2. Simplify (i) 4x (3x + 6y) (ii) –3a (5a−7c) (iii) (x2–y2) 2xy (iv) 1 xy (3x + 6y) (v) −6 ab ( 3a2b−6ab2) 3 15 3. Multiply (i) (2x + 5) (3x−8  (ii) (2x2−3y2) (4x2 + 5y2) (iii) (a + 3b) (2x−7y)  (iv) (4−2x) (3−5x) (v) (2x−3y) (2x + 5y)  (vi) (2xy−6y) (4xy−3y)  (vii) (a + 2b) (x−2y) 4. Sides of a rectangle are 3x + 4y and 2x−5y. Find the area of rectangle and its value when x=4, y=1. 5. Simplify (i) (x2−4) (x + 4) + 16 (ii) (a−b) (2a + b) + (a−2b) (a + b) (iii) (2x−3y) (2x + 3y + 4)−8x PRACTICE SHEET - 4 (PS-4) 1. Use a suitable identity to find the following products (i) (x + 6) (x + 6)  (ii) (3x−5) (3x−5)  (iii) (2x + 5) (2x−5)  (iv) (−7a + 3b) (–7a + 3b)   (v) (5a−7b) (−7b + 5a) 2. Solve using identity (i) (3x + 2) (3x−3)  (ii) (5x + 2) (5x + 3)  (iii) (2a + 3b) (2a−7b)  (iv) (3x2 + 5) (3x2 + 2  (v) (pqr−4) (pqr + 3) 3. Simplify (i) (3x + 4)2−(3x−4)  (ii) (3x−5y)2 + (3x + 5y)2   (iii) (xy + 2yz)2−4xy2z  (iv) (3x−5y)2−(5x−3y)2   (v) (5m + 4n)2−(5m−4n)2−80mn 4. Using suitable identity , find the value of (i) (52)2  (ii) (995)2  (iii) 10.1 × 10.2  (iv) 9.8 × 10.2  (v) (132)2−(68)2  (vi) 101 × 103  (vii) 105 × 95  (viii) 52 × 53 5. The cost of one pen is (2x + y) and Rohit bought (2x + y) pens. Find the total cost in terms of x and y. If x = 5 and y = 2, find the total amount paid by him. 64

PRACTICE SHEET - 5 (PS-5) I. Choose the correct option. 1. Choose the numerical coefficient in the expression(-102t²p³). (A)-102t² (B)-102p³ (C)-10² (D) t2p³ 2. The value of the expression p(p – 3) for p = (-1) is _______. (A)2 (B)-2 (C)-4 (D) 4 3. Which identity is used to evaluate 16m² – 25n²? (A)x² – y² = (x + y) ( x – y) (B) (x + y)² = x² + 2xy + y² (C)(x – y)² = x² - 2xy + y² (D)Both (B) and (C) 4. 9082 = ______________. (A)(900 + 8) (900 + 8) (B)(900 + 8) (900 – 8) (C)(908 + 8) (908 – 8) (D) (800 + 9) (800 – 8) 5. The product (5t + 7r)(5t + 7r) is _______. (A)25t² +70tr – 49r² (B) 25t² – 70tr +49r² (C)25t² – 70tr – 49r² (D) 25t² +70tr +49r² 6. The value of 252 – 162is _____________. (C)31 × 9 (A)41 × 9 (B) 41 × 19 (D) 40 × 9 7. The expansion of (4x – 2) (4x – 2) is __________ . (A) 16x² + 16x + 4 (B) 16x² - 16x + 4 (C) 16x² - 16x - 4 (D) 16x² + 16x + 4 8. Identify the expansion of (x + a) (x + b). (A) x² + (a + b)x – ab (B) x² - (a + b)x + ab (C) x² + (a + b)x + ab (D) x² - (a + b)x - ab 9. The expression that can be evaluated using the identity a² – b² = (a + b) (a – b)is ____________. (A)45² – 36² (B) 54 × 46 (C) (12 – 7) (12 + 7) (D) All of these 10. The product (m + n3) (m – n3) has __________ terms. (A) 3 (B) 2 (C) 4 (D) 1 II. Short Answer Questions. 11. Add the following. xy – yz, yz – zx, zx – xy, 125 12. Find the difference of 25p – 10 and 46 – 20p. 13. Simplify the expression (3x³ + x² + 4x – 2) (5x – 7) and evaluate it for x = (-3). III.Long Answer Questions. § 2 5 · 2 ©¨ 5 2 ¸¹ 14. Find the value of x - y + 4xy 15. Show that (x – y)( x + y) + (y – z) (y + z) + (z – x) (z + x) = 0 65

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. Add xy−2yz, 4xy + 3zx, 5yz−8zx.  (1 Mark) 6. Solve ( –4x + 3y) (3y−4x) using suitable identity. (3 Marks) 2. Find the product of (6−3x) (4 + x).  (1 Mark) 7. Show that (2x + 3y)2−24xy = (2x−3y)2  (2 Marks) 3. Find (102)2 using a suitable identity.  (2 Marks) 8. Subtract x (y + z) + y (2z−x) from y (z−2x)−x (3z−y).(2 Marks) 4. Subtract 6x2−4xy + 5y2 from 8y2 + 6xy−3x2. (2 Marks) 5. Simplify 9842−162.  (2 Marks) 66

10. Visualising Solid Shapes • Identify front, side and top view of various objects. Learning Outcomes • Map spaces around him/her. • Tabulate the number of faces, edges and vertices At the end of this chapter, you will be able to: • Identify 2D and 3D objects. for polyhedrons. • Identify different shapes in nested objects. Concept Map Key Points • Polyhedrons are solids made up of polygonal regions which are called its faces; these faces • 3D objects have different views from different meet at edges which are line segments; and positions. the edges meet at vertices which are points. • A map is different from a picture. • For any polyhedron, • A map depicts the location of a particular F+V–E=2 object/place in relation to other objects/ where ‘F’ stands for number of faces, V stands places. for number of vertices and E stands for • Symbols are used to depict the different number of edges. This relationship is called objects/places. Euler’s formula. • There is no reference or perspective in a map. • Maps involve a scale which is fixed for a particular map. 67

10. Visualising Solid Shapes Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Views of 3D shapes • Recognising 2D and 3D objects. PS – 1 • Recognising different shapes in PS – 2 nested objects. PS – 3 • Views of 3D objects from different PS – 4 Self-evaluation Sheet positions Mapping space around us • Drawing maps of spaces around Faces, edges and vertices • Tabulate the number of faces, edges and vertices for polyhedrons • Verify Eulers formula Worksheet for “Visualising Solid Shapes” Evaluation with self- check or Peer check* 68

PRACTICE SHEET - 1 (PS-1) 1. Match the following. 2. Identify if the following figures are 2D or 3D. (i) (ii) (iii) (iv) (v) (vi) (vii) 69

PRACTICE SHEET - 1 (PS-1) 3. Draw the front view, top view and side view for the cubes shown below. i. ii. iii. iv. 4. Draw the front view, side view and top view of the given objects. 70

PRACTICE SHEET - 2 (PS-2) 1. Vijay is going home from school. He moved 2 km north direction, 3 km east direction, 5 km in south direction and at last he moved 3 km in west direction. Draw the road map using a proper scale and find the actual distance from his home to school. 2. Draw a map of your living room using proper scale and symbols for different objects. 3. Draw a map giving instructions to your friend so that she reaches your house without any difficulty. PRACTICE SHEET - 3 (PS-3) 1. Tabulate the number of faces, edges and vertices for the following polyhedrons. i. ii. iii. 2. Verify Euler’s formula for the following. 71

PRACTICE SHEET - 4 (PS-4) I. Choose the correct option. 1. Which of these shapes is 2 dimensional? 2. Identify a 3 - dimensional shape. (A) (B) (C) (D) All of these 3. Solids made up of polygonal regions are called _____________. (D) Both (B) and (C) (A) Polyhedrons (B) Cuboids (C) Prisms 4. Choose the top view of the solid shape given. (A) (B) (C) (D) 5. The number of edges in a cube is _______. (C) 10 (D) 12 (A) 6 (B) 8 6. What shape will the front view of a mud brick look like? (A) Rectangle (B) Square (C) Cuboidal (D) Triangular 7. A polyhedron with congruent polygons for its top and base is called a __________. (A) cube (B) prism (C) pyramid (D) All of these 8. The lateral faces of a pyramid are _____. (A) squares (B) rectangles (C) triangles (D) circles 9. A polyhedron with a 6-sided polygonal base is called a ____________. (A) pentagonal prism (B) hexagonal pyramid (C) triangular pyramid (D) hexagonal prism 10. A prism with all rectangular faces is called a ___________ . (A) cube (B) cuboid (C) cone (D) cylinder II. Short Answer Questions. 11. Mention the similarities of a cylinder with a prism. 12. Verify Euler’s formula for the given solid. 13. Given here are the front, side and top views of a solid. 72

PRACTICE SHEET - 4 (PS-4) III. Long Answer Questions. 14. Complete the given table using the Euler’s formula for solids. Faces 8 B 12 Edges A 30 30 Vertices 6 12 C 15. Verify Euler’s formula for the given octahedron. 73

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. Tabulate the number of faces, edges and 6. In the given polyhedrons, determine how many vertices for the following polyhedrons.(4 Marks) faces meet at the apex. (2 Marks) i. i. ii. ii. iii. iv. 2. Draw a map of your school compound using proper scale and symbols for various features like playground main building, garden etc. (3 Marks) 3. Front view of a cylindrical pen is _______. (1 Mark) 4. Write the Euler’s formula for any polyhedron.  (2 Marks) 5. Draw the front view, top view and side view for the building shown below. (3 Marks) Building 74

11. Mensuration Learning Outcomes At the end of this chapter, you will be able to: • Understand the concepts of solid shapes, volume • Understand the difference between area and and surface area. perimeter. • Understand the unit conversions of volume and • Area of quadrilateral and special types of surface area. quadrilaterals and parallelograms. Concept Map Key Points • Area o The surface covered by the border line of the figure is the area of the plain shape. o Unit of the area is square of the unit of length. • Perimeter o The perimeter is the length of the boundary of the plane shape. o The unit of the perimeter is same as the unit of length. 75

11. Mensuration • Area and Perimeter of Some 2D Shapes Shape Image Area Perimeter (Side)2 4 × Side Square Rectangle Length × Breadth 2(Length +Breadth) Triangle 1 × Base × height Sum of three sides 2 Parallelogram Base × Height 2( Sum of adjacent sides) Circle πr2 2πr • Area of Trapezium • A trapezium is a quadrilateral whose two sides are parallel. And if its non-parallel sides are equal then it is said to be an isosceles trapezium. • Area of trapezium is half of the product of the summation of the parallel sides and the perpendicular distance between them. 76

11. Mensuration COVERAGE DETAILS PRACTICE SHEETS • Area of rectangle, square, circle PS – 1 Work Plan • Area of trapezium • Area of Polygon CONCEPT COVERAGE Mensuration • Surface area and volume of cube, Cuboid PS – 2 • Surface area of cylinder PS – 3 Worksheet for \"Mensuration\" ---- Self-Evaluation Sheet Evaluation with Self Check or Peer Check* 77

PRACTICE SHEET - 1 (PS-1) 1. A square field with side 70m has same perimeter as a rectangle of breadth 35m. Find which of them has greater area. 2. Mr Rakesh has a rectangular plot 100 m long and 80 m wide. He wants to construct a house of length 70 m and width 55 m in the middle and make a garden around the house. Find the cost of laying grass in the garden at the rate of Rs 7.50 per sq m. 3. A flooring tile is in the shape of a parallelogram of side 35 cm and corresponding height 10cm. Find the number of tiles required to cover an area of 1050 m2. 4. Find the area of a trapezium whose parallel sides are 57 m and 39 m and the distance between them is 28 cm. 5. The area of a trapezium is 352 cm2. The distance between the parallel sides is 16 cm. If one of the parallel sides is 25 cm, find the other side. 6. Find the area of rhombus whose lengths of diagonals are 36 cm and 22.5 cm. 7. One side of a rhombus is 24cm and its corresponding altitude is 10cm. If one of its diagonals is 12cm, find the length of the other diagonal. 8. If one of the parallel sides of the trapezium is 3 times the other side and the distance between their parallel sides is 100 m and its area is 16000 m2. Find the length of its sides. 9. Floor of a building consists of 1500 tiles each in the shape of a rhombus each of whose diagonals are 55 cm and 40 cm. Find the cost of polishing the floor at the rate of Rs 15 per m2. 10. Find the perimeter of a semicircle of radius 3.5 cm including its diameter. 11. If the circumference of the circular sheet is 176 cm, Find its area. 12. Find the perimeter and area of a protractor of diameter 28 cm. 78

PRACTICE SHEET - 2 (PS-2) 1. Find the volume of a cube whose surface area is 2400 cm2. 2. A box of dimensions 90 cm ×75 cm ×50 cm is to be covered with cloth of width 300 cm. Find the length of the cloth required to cover 50 such boxes. 3. A painter is painting the walls and ceiling of a building 15 m high, 9 m long and 7.5 m wide. Find the number of cans of paint required if each Can can cover an area of 75 m2. 4. A cuboid measure 75 m × 60 m ×50 m and a cube measure 60 m. Find the difference in surface area and volume. 5. Three cubes each of side 2 cm is placed next to each other. Find the surface area of the cuboid hence formed. 6. The lateral surface area of a hollow cylinder is 4224 cm2. It is cut along its length to form a rectangular sheet. If the width of the sheet is 33 m, find its perimeter. 7. The perimeter of the floor of a room is 104 m. If the height of the room is 9 m. Find the cost of painting the wall at the rate of Rs 10 per m2. 8. How many times does the volume of a cube change if its sides are tripled? 9. How many bricks will be required for a wall which is 8� m long, 6 m high and 22.5 cm thick, if each brick measures 25 cm by 11.25 cm by 6 cm? 10. Find the height of a cuboid whose volume is 490 cm3 and base area is 35 cm2. 11. A godown is in the form of cuboid measuring 60 m × 40 m × 20 m. How many cuboidal boxes of volume 0.8 m3 can be placed in it? 12. Find the volume of cylinder whose base radius is 14 cm and height is 35 cm. 13. A cylindrical tank has a capacity of 5632 cm3. If the diameter of the base is 16 cm, find its depth. 14. Find the curved surface area of cylinder whose radius is 8 cm and height is 14 cm. 15. How much area will a garden roller of diameter 1.4 m and 2 m long cover in 5 revolutions? 16. A rectangular paper of width 28 m is rolled along its length to form a cylinder of radius 15m. Find the volume of the cylinder. 17. Find the height of the cylinder of volume 3.08m3 and diameter of base is 70 cm. PRACTICE SHEET - 3 (PS-3) I. Choose the correct option. 1. The measure of the distance around a closed plane figure is called its _______. (A) area (B) length (C) breadth (D) perimeter 2. The region of a plane covered by a closed plane figure is called its __________. (A) perimeter (B) area (C) surface area (D) volume 3. The area of a parallelogram with base ‘b’ units and height ‘h’ units is given by _____________ sq. units. bh (A) 2 (B) b²h (C) bh (D) π bh 79

PRACTICE SHEET - 3 (PS-3) 4. The perimeter of the given rectangle is _______________. (A) 40 cm (B) 40 cm² (C) 20 cm (D) 20 cm² 5. The perimeter of the given semicircle is _______. (A) 38.47 cm² (B) 38 cm² (C) 12.25 cm (D) 38.47 cm 6. Identify the surface area of a cube of edge 4 cm. (A) 64 cm (B) 96 cm (C) 96 cm² (D) 64 cm² 7. The area of the base of a cuboid is 6 cm² and its height is 2 cm. The volume of the cuboid is __________. (A) 12 cm² (B) 12 cm³ (C) 24 cm³ (D) 72 cm³ 8. A cylinder has a base area of 154 cm² and height 6 cm. Its volume is _____. (A) 924 cm³ (B) 942 cm³ (C) 924 cm² (D) 924 cm 9. Identify the correct statement among the following. (A) 1 m3 = 100000 cm³ (B) 100 ml = 1 cm³ (C) 1 ml = 1 cm³ (D) 1 m³ = 100l 10. The volume of a cuboid is 48 cm3. If its height and length are 2 cm and 6 cm, its width is ___________ . (A) 4 cm (B) 6 cm (C) 2 cm (D) 8 cm II. Short Answer Questions. 11. A cylinder has a circular base of radius 4 cm and a height of 3 cm. What is its curved surface area? 12. Find the edge of a cube whose total surface area is 864 cm². 13. Study the solids in the given figures. Explain which solid has greater capacity. III. Long Answer Questions. 14. A rectangular tin sheet of width 20 cm is folded to a make a cylinder of radius 14 cm. Find the volume of the cylinder so formed. If the tin sheet costs Rs. 4.50 per sq. cm, find the amount to be spent to buy the sheet needed to make the cylinder. 15. Compute the radius of a cylinder of height 15 cm and total surface area 11000 cm². 80

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. A square of side 16 cm has the same area as a rectangle of length 8 cm. Find the breadth of the rectangle.(2 Marks) 2. Find the area of rhombus whose diagonals are 14.5 cm and 20 cm. (2 Marks) 3. Perimeter of base of a cylinder is 77 cm. Find the volume of cylinder if its height is 30 cm. (2 Marks) 4. A rectangular plot has length 70 m and breadth 50 m, a path 5 m wide runs along the inner boundary and a garden in the middle, find the cost of putting grass in the garden at the rate of Rs 10 per m². (3 Marks) 5. A quadrilateral PQRS has a diagonal PR of length 24 cm. perpendiculars drawn from Q and S on PR are 12 cm and 14 cm respectively. Find the area of quadrilateral PQRS. (3 Marks) 6. A cylinder has a radius of 21 cm and height of 15 cm. Find the cost of milk stored in it if the cost of 1litre of milk is Rs 20 . (3 Marks) 81

12. Exponents and Powers • Identify and use the laws of exponents. • Learning the representation of number in Learning Outcomes standard form and usual form. At the end of this chapter, you will be able to: • Understand the power notation form of representation of a number. • Identify positive and negative exponents. Concept Map Key Points 1215 =1000 + 200 + 10 + 5 = 1 × 103 + 2 × 102 + 2 × 101 + 5 × 100 • What is exponent and base? So these are all positive exponents. 23 = 2 × 2 × 2  Base = 2, Exponent = 3 Decimal number can be expressed as exponents 32 = 3 × 3  Base = 3, Exponent = 2 also Negative exponents 1215.15 = 1000 + 2-00 + 10 + 5 +.10 +.05=  1 × 103 + Here we will be looking how to interact with these 2 × 102 + 2 × 101 + 5 × 100 + 1/10 + 5/100 type of expression =1 × 103 + 2 × 102 + 2 × 101 + 5 × 100 + 10-1 + 5 × 10-2 2-10 So this has both the negative and positive 11-1 exponents. Here we can see that exponents are negative. • Laws of Exponents Now negative exponents can be converted into Here are the laws of exponents when a and b are positive exponents like this non-zero integers and m, n are any integers. 2-10 = 1/210 a-m = 1/am 11-1 = 1/11 am / an = am-n we can say that for any non-zero integer a, (am)n  = amn a-m = 1/am am x bm  = (ab)m where m is a positive integer. am    is the am / bm   = (a/b)m multiplicative inverse of a-m. a0 =1 (a/b)-m = (b/a)m • How to express the decimal number in exponent (1)n = 1 for infinitely many n. form? (-1)p = 1  for any even integer p We already know that any non -decimal number (-1)n = -1 for any odd integer n can be expressed in exponent form like below 82

12. Exponents and Powers Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEETS PS – 1 Laws of exponents • Multiplicative inverse • Expanded form PS – 2 • Laws of exponents PS – 3 • Missing values Self-Evaluation Sheet Power notation • Standard form • Usual form Worksheet for “Exponents and Powers” Evaluation with Self Check or Peer ---- Check* 83

PRACTICE SHEET - 1 (PS-1) 1. Find the multiplicative inverse of: (i) 3−4 (ii) 15−6 (iii) 7−3 2. Evaluate: (i) 4−2 (ii) −53 (iii) 1 3 (iv) 2−3 (v) −6−2 2 3. Express as power notation with positive exponent. (i) 33 y 37 (ii) 1 (iii) ( 25 y 23 ) u 23 (iv) ( 43 u 46 ) y43 23 (v) ( 33 u 73 ) 4. Expand the following numbers using exponents (i) 112.53 (ii) 1.5625 (iii) 2056.8 5. Find the value of:  (ii) 41 u 51 y 52 (i) ( 41  60 ) u 42 (iii) ( 1 )1  ( 1 )1  ( 1 )1 (iv) {( 3 )2 u § 4 ·2 } 254 5 ¨© 5 ¹¸ 6. (i) ( 5 )2 u ( 3 )3 u ( 3 )0 (ii) § 3 ·4 u ( 2 )2 (iii){( 3 )−2 }2 955 ©¨ 5 ¸¹ 5 2 7. Find the value of x for which ( 5 )4 u ( 5 )5 § 5 ·3x 33 ¨© 3 ¸¹ 8. By what number should 61 multiplied so that the product becomes 91 ? 9. If 53x1 y 25 125 , find x. 10. Find the multiplicative inverse of (7)2 y (90)1 PRACTICE SHEET - 2 (PS-2) 1. Express the following in standard form. (i) 0.00000000000056 (ii) 0.0000000000000000829 (iii) 0.00000000000000000963 (iv) 315200000000000000 (v) 152700000000000000000 (vi) 980000000000000 2. Express the following in usual form. (i) 4.05 u109 (ii) 5.9 u107 (iii) 4.756 u105 (iv) 3.165×107 (v)  4.978 ×109 (vi) 3.005 ×1014 3. Some birds travels 15000 km to migrate to suitable conditions. Write the distance in m in scientific notation. 4. Special balances can measure upto 0.000000001 grams, express in standard form. 5. Express 2 years in seconds in standard form. 84

PRACTICE SHEET - 3 (PS-3) I. Choose the correct option. 1. a–m = _______ (A)am (B) − 1 (C) - am (D) 1 2. a0 = _______ am am (A) 0 (B) 1 (C) - 1 (D) a 3. 131= _____________ (A) 0 (B) 1 (C) 13 (D) 13 + 1 4. The multiplicative inverse of 6-4 is _______________. (A) 6⁴ (B) 4-6 (C) 4⁶ (D) All of these 5. The formula used to simplify (-22)³ × (-22)⁶ is _______. (A) aᵐ × a-n = aᵐ+n (B) am ÷ aⁿ = am+n (C) am ÷ an = am – n (D) am × an = am+n 6. Identify the surface area of a cube of edge 4 cm. (A) 64 cm (B) 96 cm (C) 96 cm² (D) 64 cm² 7. Which of these can be expressed as a negative exponent? (A) 8z (B) 1 (C) 83 (D) 8 8m 8. The simplified form of 24 × 25 is _____. [ ] (A) 2⁹ (B) 2¹ (C) 2-1 (D) 220 9. Identify the number that has the same value as 130. 1 13 (B) 1 (C) 13 (D) − 13 (A) − 13 13 13 10. § x ·p can be simplified as ___________. ¨ y ¸ © ¹ (A) x p (B) (xy)p (C) xp – yp (D) xp + yp yp II. Short Answer Questions. 11. Simplify 43 × 4-8 and express your answer as a positive power. 12. Express 153 as a power of 3. 13. Express the given numbers as directed: a) In standard form: (i) 25400000 (ii) 0.0000019 b) In usual form: (i) 14.23 × 104 (ii) 9 × 10-6 III. Long Answer Questions. °­§ 1 ·1  § 1 ·1 ½°1 ®¯°¨© 2 ¹¸ ¨© 3 ¹¸ ¾ 14. Evaluate: ¿° 3432 u 35 u 95 15. Simplify: 275 u 495 85

SELF-EVALUATION SHEET Marks: 15 Time: 30 Mins 1. Expand number 9078.21 using exponents. 6. Express the following in standard form (1 Mark) (3 Marks) (i) 0.000063 (ii) 9060000 (iii) 0.000000001094 2. Simplify and express the result with positive exponent:  (2 Marks)    7. Simplify (a) 93 u 32 y 40 (b) 54 y 34 x 32 x 5−2  ( 36 y 310 ) x 3−5 (3 Marks) 3. Find the value of x when 42x6 y16 256  (2 Marks) 4. Find the multiplicative inverse of a) 1 10100 b) 1 (2 Marks) 7−2 5. Evaluate § § 1 ·2  § 1 ·1 ·3 (2 Marks) ©¨¨ ©¨ 4 ¹¸ ©¨ 4 ¸¹ ¹¸¸ 86