202110233-TRIUMPH-STUDENT-WORKBOOK-MATHEMATICS-G08-PART1

EXERCISE 5.3 COMPOUND INTEREST 5.3.1 Key Concepts i. Simple interest is an increase on the principal. ii. Simple interest, I = PT R , where (P = Principal, T = Time, R = Rate of interest) 100 TR iii. Amount = Principal + Interest = P(1 + 100 ) . iv. Compound interest allows you to earn interest on interest. v. Amount at the end of ‘n’ years using compound interest = A = P 1+ R n . 100 vi. The time period after which interest is added to principal is called the conversion period. vii. When interest is compounded half yearly, then there are two conversion periods in a year, each after 6 months. In this case, half year rate will be half of the annual rate. 5.3.2 Additional Questions Objective Questions . 1. [AS1] The interest on Rs. 500 at 3% per annum for 3 years is (A) Rs. 503 (B) Rs. 1500 (C)Rs. 45 (D)Rs. 450 2. [AS2] The sum of money which when lent out at 9% per annum SI for 6 years gives Rs. 810 as interest is . (A) Rs. 1000 (B) Rs. 1500 (C)Rs. 1200 (D)None of these 3. [AS2] The annual installment that will discharge a debt of Rs. 4200 due in 5 years at 10% SI is . (A) Rs. 600 (B) Rs. 700 (C)Rs. 800 (D)Rs. 900 4. [AS2] The number of years in which a certain sum amounts to three times the principal at the rate of 16 2 % is . 3 (A) 12 years (B) 8 years (C)4 years (D)16 years EXERCISE 5.3. COMPOUND INTEREST 48

5. [AS2] The time in which Rs. 72 becomes Rs. 81 at 6 1 % p.a SI is . 4 (A) 112 years (B) 2 1 years 2 (C) 3 1 years (D)2 years 2 6. [AS1] The C.I on a certain sum for 2 years is Rs. 41 and S.I is Rs. 40. Then the rate per annum is . (A) 5% (B) 4% (D) 8% (C) 2 1 % 2 7. [AS1] The compound interest on Rs.1000 at 12% per annum for 1 1 years, compound annually 2 is . (A) Rs.187.20 (B) Rs.1720 (C) Rs.1910.16 (D) Rs.1782 8. [AS2] The rate percent per annum at which a sum of Rs. 7500 amounts to Rs. 8427 in 2 years, compounded annually is . (A) 4% (B) 5% (C) 6% (D) 8% 9. [AS3] The formula to find simple interest is . . (A) S .I = 100×P×R T (B) S.I = P×T ×R 100 (C) S .I = R×100 P×T (D)None of these 10. [AS3] The formula to find compound interest is n (A) C.I = P−P 1+ R 100 (B) C.I = P 1+ R n 100 (C)C.I = P+P 1+ R n 100 (D)C.I = P 1+ R n − P 100 EXERCISE 5.3. COMPOUND INTEREST 49

11. [AS4] Krishna deposits Rs. 30000 in a bank at 7% per annum. The compound interest for a certain time is Rs. 4347. The time for which Krishna deposited the money is . (A) 2 years (B) 2 1 years 2 (C)3 years (D)4 years 12. [AS4] Shilpa borrowed Rs. 2000 at 20% p.a. compounded half yearly. The amount of money she needs to discharge her debt after 1 1 years is . 2 (A) Rs. 2662 (B) Rs. 662 (C)Rs. 600 (D)Rs. 3662 Very Short Answer Type Questions 13 [AS1] Answer the following questions in one sentence. (i) Find the simple interest on Rs. 12000 at 8% p.a for 3 years. (ii) Find the compound interest when principal = Rs.1000, rate =10% per annum and time = 2 years. 14 [AS2] Answer the following questions in one sentence. (i) At what rate of simple interest on Rs. 4500 will become Rs. 5040 after 2 years? (ii) How many years will it take Rs. 8000 to earn a simple interest of Rs.1800 at 9% per annum? 15 [AS3] Answer the following questions in one sentence. (i) Let P be the principal and the rate of interest be R% per annum. If the interest is compounded annually, what is the formula to find the amount A? (ii) Write the formula to find rate ‘R’ if P is the principal, 'T' is the time period and S.I is the simple interest. (iii) Define conversion period. 16 [AS4] Answer the following questions in one sentence. (i) The present population of a town is 10000. The population of the town increases 5% annually. Find the population after 2 years. (ii) Madhu deposits Rs. 2000 in a bank at 10% per annum. Find the amount after 1 year if the interest is compounded annually. EXERCISE 5.3. COMPOUND INTEREST 50

Short Answer Type Questions 17(i) [AS1] The simple interest on a sum of money for a period of 3 years at 12% per annum is Rs. 6750. What will be the compound interest on the same sum at the same rate for the same period com- pounded annually? (ii) [AS1] Find the compound interest paid when a sum of Rs. 10, 000 is invested for 1 year and 3 months at 8 1 % per annum compounded annually. 2 18(i)[AS1] A sum taken for 1 1 years at 8% per annum is compounded half yearly. Find the number of 2 conversion times the interest is compounded and rate. (ii) [AS1] Find the compound interest on Rs. 15, 625 for 1 years at 8% per annum when compounded 12 half yearly. 19 [AS3] Define compound interest. Long Answer Type Questions 20 [AS1] A sum of money amounts to Rs. 2, 240 at 4% per annum simple interest in 3 years. Find the interest on the same sum for 6 months at 3 1 % per annum. 2 21 [AS1] Find the amount and the compound interest on Rs. 6,500 for 2 years, compounded annually, the rate of interest being 5% per annum during the first year and 6% per annum during the second year. 22 [AS1] Calculate compound interest on Rs. 1000 over a period of 1 year at 10% per annum, if interest is compounded quarterly. 23 [AS2] At what rate of compound interest will Rs. 20000 become Rs. 24200 after 2 years? EXERCISE 5.3. COMPOUND INTEREST 51

CHAPTER 6 SQUARE ROOTS AND CUBE ROOTS EXERCISE 6.1 PROPERTIES OF SQUARE NUMBERS 6.1.1 Key Concepts i. Square: The square of a number is the product of the number with the number itself. For a given number ‘x ’ the square of x is (x × x) , denoted by x2. e.g., 42 = 4 × 4 = 16 ii. Properties of perfect squares: a. A number ending in 2, 3, 7 or 8 is never a perfect square. b. A number ending in an odd number of zeroes is never a perfect square. c. The square of an even number is even. d. The square of an odd number is odd. e. The square of a proper fraction is always smaller than the fraction. e.g., (0.1)2 = 0.01 f. The sum of first ‘n’ odd natural numbers = n2. 6.1.2 Additional Questions Objective Questions 1. [AS3] The perfect square among the following is . (A) 425 (B) 1664 (C) 1200 (D) 1296 2. [AS3] The square of an odd number is . (A) Even (B) Odd (C) Prime (D) Composite 3. [AS2] The digit in the units place of the square of a number ending with 3 is . (A) 3 (B)6 (C) 9 (D) 2 EXERCISE 6.1. PROPERTIES OF SQUARE NUMBERS 52

4. [AS3] The square of the number 27 is . (A) 54 (B) 189 (C) 243 (D) 729 5. [AS2] The number whose square is 1225 is . (A) 35 (B) 625 (C) 125 (D) 25 Short Answer Type Questions 6(i) [AS2] a) The square of 33 is odd. Justify. b) The square of 806 is even. Justify. (ii) [AS2] Identify if the squares of the following numbers are odd or even: a) 342 b) 539 7(i) [AS3] a) Express 81 as the sum of 9 odd numbers. b) Express 100 as the sum of 10 odd numbers. (ii) [AS3] Write five numbers which are not perfect squares. Long Answer Type Questions 8 [AS2] (i) Find the perfect squares between 100 and 150. (ii) Check whether 243 is a perfect square or not. (iii) 225 is a perfect square. Justify. (iv) Show that 1225 is a perfect square. (v) Show that 450 is not a perfect square. 9 [AS2] (i) 1 = 12 1 + 3 = 22 1 + 3 + 5 = 32 What is the sum of the first 4 odd numbers? EXERCISE 6.1. PROPERTIES OF SQUARE NUMBERS 53

(ii) 12 = 1 Square of 11 = 121 Square of 111 = 12321 Square of 1111 = Complete the pattern. (iii) 1 + 2 + 1 = 22 1 + 2 + 3 + 2 + 1 = 32 = 42 1 + 2 + 3 + 4 + 5 + 4 + 3 + 2 + 1 = 52 Complete the pattern. (iv) 121 = 222 1 1 +2+ 12321 = 1 + 2 3332 2 + 1 +3+ = 1 + 2 + 44442 + 2 + 1 3+4+3 Complete the pattern. 10 [AS3] Express each of the following as the sum of two consecutive natural numbers. (i) 212 (ii) 132 (iii) 192 EXERCISE 6.1. PROPERTIES OF SQUARE NUMBERS 54

EXERCISE 6.2 PYTHAGOREAN TRIPLETS FINDING SQUARE ROOTS BY PRIME FACTORIZATION METHOD 6.2.1 Key Concepts i. If a2 + b2 = c2 then (a, b, c) is said to be a Pythagorean triplet. ii. If there are no common factors other than 1 among a, b, c then the triplet (a, b, c) is called primitive triplet. 6.2.2 Additional Questions Objective Questions 1. [AS3] A Pythagorean triplet among the following is . (A) (3, 4, 6) (B) (5, 12, 13) (C)(7, 14, 17) (D) None of these 2. [AS1] The square root of 51.84 is . (A) 7.8 (B) 6.8 (C) 7.2 (D) 8.2 3. [AS1] The square root of 3136 is . (A) 56 (B) 58 (C) 46 (D) 66 4. [AS1] The least number by which 1458 should be multiplied to get a perfect square is . (A) 4 (B) 3 (C) 1 (D) 2 5. [AS1] The least number by which 2645 should be divided to get a perfect square is . (A) 2 (B) 3 (C) 4 (D) 5 EXERCISE 6.2. PYTHAGOREAN TRIPLETS FINDING SQUARE ROOTS BY PR. . . 55

Very Short Answer Type Questions 6 [AS4] Answer the following questions in one sentence. A square coffee table has an area of 196 square metres. What is the length of one side of the coffee table? Short Answer Type Questions 7(i) [AS2] Check whether the triplet 6, 8, 10 is a Pythagorean triplet or not. (ii) [AS2] 9, 12, 15 is a Pythagorean triplet. Find whether its multiples also form a Pythagorean triplet. 8(i) [AS2] By repeated subtraction, find whether the number 55 is a perfect square or not. (ii) [AS2] By repeated subtraction, find whether the number 144 is a perfect square or not. Long Answer Type Questions 9 [AS1] (i) Find the square root of 7056 by prime factorization method. (ii) Find the factors of 9408 using prime factorization. By what number should it be divided tomake it a perfect square? (iii) Find the square root of 1764. 10 [AS1] Find the smallest number by which 35280 must be divided so that it becomes a perfect square. 11 [AS4] The area of a square field is 5184 m2 . A rectangular field, whose length is twice its breadth has its perimeter equal to the perimeter of the square field. Find the area of the rectangular field. EXERCISE 6.2. PYTHAGOREAN TRIPLETS FINDING SQUARE ROOTS BY PR. . . 56

EXERCISE 6.3 FINDING SQUARE ROOT BY DIVISION METHOD 6.3.1 Key Concepts i. For large numbers, prime factorisation method becomes lengthy and difficult so we use the division method. 6.3.2 Additional Questions Objective Questions 1. [AS1] The square root of 3481 is . (A) 59 (B) 61 (C) 51 (D) 69 2. [AS2] The square root of 350 lies between the numbers . (A) 15, 16 (B) 17, 18 (C)19, 20 (D) 18,19 √√ . 3. [AS1] If 9216 = 96 then 0.009216 = (B) 0.096 (D) 0.0096 (A) 0.96 (C) 9.6 EXERCISE 6.3. FINDING SQUARE ROOT BY DIVISION METHOD 57

4. [AS1] The square root of 368.64 is . (A) 1.92 (B) 0.19 (C) 19.2 (D) 19.8 √√ . 5. [AS1] 392 × 18 = (B) 18 (A) 84 (D)None of these (C) 54 6. [AS4] A square board has an area of 144 square units. The length of each side of the board is . (A) 11 units (B) 12 units (C)13 units (D)14 units 7. [AS4] The dimensions of rectangular field are 80 m and 18 m. The length of its diagonal is . (A) 28 cm (B) 32 m (C)82 m (D)82 cm 8. [AS4] 1024 plants are arranged so that number of plants in a row is the same as the number of rows. The number of plants in each row is . (A) 23 (B) 34 (C) 25 (D) 32 Very Short Answer Type Questions 9 [AS1] Answer the following questions in one sentence. The product of two numbers is 1296. If one number is 16 times the other, find the two numbers. 10 [AS2] Answer the following questions in one sentence. (i) Is 338 a perfect square? If not, find the smallest number by which it should be multiplied to get a perfect square. EXERCISE 6.3. FINDING SQUARE ROOT BY DIVISION METHOD 58

(ii) Show that 450 is not a perfect square. (iii) 360 is not perfect square. Given reason. 11 [AS4] Answer the following questions in one sentence. (i) 5929 students sit in an auditorium in such a manner that there are as many students in a row as there are rows in the auditorium. How many rows are there in the auditorium? (ii) A square table in a class room has an area of 1296 square metres. What is the length of one of its sides? (iii) 1521 trees were planted in a garden in such a way that there are as many trees in each row as there are rows in the garden. Find the number of rows and number of trees in each row. Short Answer Type Questions 12 [AS1] Find the square root of the number 7744 by division method. 13 [AS4] In an auditorium, the number of rows is equal to the number of chairs in each row. If the capacity of the auditorium is 2025, find the number of chairs in each row. Long Answer Type Questions 14 [AS1] Estimate the value of the following numbers to the nearest whole number. √ (i) √97 (ii) 250 √ (iii) 780 EXERCISE 6.3. FINDING SQUARE ROOT BY DIVISION METHOD 59

EXERCISE 6.4 CUBIC NUMBERS 6.4.1 Key Concepts i. Cube is a solid figure with six identical squares. ii. So, we require 1, 8, 27, 64. . . unit cubes to make cubic shapes. iii. These numbers are called cubic numbers or perfect cubes. 6.4.2 Additional Questions Objective Questions 1. [AS3] The perfect cube among the following is . (A) 1225 (B) 2744 (C) 780 (D) 255 2. [AS3] The cube of a number ending with 4, ends with the digit . (A) 2 (B) 8 (C) 4 (D) 6 3. [AS1] The least number by which 675 should be multiplied so that the product is a perfect cube is . (A) 3 (B) 5 (C) 2 (D) 7 4. [AS2] The least number by which 704 must be divided to obtain a perfect cube is . (A) 2 (B) 4 (C) 11 (D) 6 5. [AS3] The perfect cube of the following is . (A) 80000 (B) 8000 (C) 800 (D) 80 Very Short Answer Type Questions 6 [AS2] Answer the following questions in one sentence. (i) Show that 189 is not a perfect cube. EXERCISE 6.4. CUBIC NUMBERS 60

(ii) Prove that if a number is doubled, then its cube is eight times the cube of the given number. 7 [AS3] Answer the following questions in one sentence. (i) Define a perfect cube. (ii) Write cubes of first three multiples of 3. (iii) Write the ones digit of cube of 8. (iv) Express 28 as the product of prime factors. Short Answer Type Questions 8(i) [AS1] a) What is the smallest number by which 1323 is to be multiplied so that the product is a perfect cube? b) By which number 32 is to be multiplied by to make it a perfect cube? (ii) [AS1] What is the smallest number by which 675 should be multiplied so that the product is a perfect cube? 9(i) [AS1] What is the smallest number by which 1375 should be divided so that the quotient may be a perfect cube? (ii) [AS1] What is the smallest number by which 2916 should be divided so that the quotient is a perfect cube? 10(i) [AS2] a) How many cubic numbers are there between 50 and 250? b) What are the perfect cube numbers between 3000 and 5000? (ii) [AS2] How many perfect cubes are there between 500 and 1000? Long Answer Type Questions 11 [AS1] Find the digit in the units place of each of the following numbers: (i) 128 3 (ii) 212 3 (iii) 509 3 (iv) 647 3 (v) 783 3 12 [AS2] (i) Find whether 200 is a perfect cube or not. (ii) 1728 is a perfect cube. Justify. EXERCISE 6.4. CUBIC NUMBERS 61

EXERCISE 6.5 CUBE ROOTS 6.5.1 Key Concepts i. We require 8 unit cubes to form a cube of side 2 units. ii. Suppose a cube is formed with 64 unit cubes. Then the side of the cube is given by x , where x3 = 64. iii. Finding the number from its cube is called finding the cube root. It is the inverse operation of cubing. iv. The cube root of a number ‘x ’ is the number whose cube is x . It is denoted by √3x . √3 a v. For any two integers a and b, we have: 3√ √3 a × √3 b and 3 a = √3 b ab = b 6.5.2 Additional Questions Objective Questions . 1. [AS1] The cube root of 512 is (A) 2 (B) 4 (C) 6 (D) 8 2. [AS3] If the cube of a number ends with 8 then the number ends with . (A) 2 (B) 4 (C) 6 (D) 8 3. [AS3] If a number ends with 6 then its cube ends with . (A) 2 (B) 4 (C) 6 (D) 8 4. [AS3] The cube root of an even number is . (A) Odd (B) Prime (C) Co–prime (D) Even EXERCISE 6.5. CUBE ROOTS 62

5. [AS1] √3 13824 = (B) 24 (D)None of these (A) 14 (C) 34 Very Short Answer Type Questions 6 [AS1] Answer the following questions in one sentence. Find the cube root of the following numbers by prime factorization method: (i) 343 (ii) 2744 (iii) 1331 7 [AS4] Answer the following questions in one sentence. The volume of a cubical chalk piece box is 512 cubic metres. Find the length of its side. Short Answer Type Questions 8(i) [AS1] Estimate the cube root of: a) 21952 b) 2197 (ii) [AS1] Estimate the cube root of: a) 5832 b) 3375 Long Answer Type Questions 9 [AS2] (i) Which of the following is the cube of an odd natural number? (A) 2744 (B) 12167 (C) 4000 (D) 32768 (ii) A perfect cube may end with two zeros. (T/F) (iii) There is no perfect cube that ends with 8. (T/F) (iv) The cube of an even number is an even number. (T/F) (v) If a number ends with 5, what does its cube end with? EXERCISE 6.5. CUBE ROOTS 63

CHAPTER 7 FREQUENCY DISTRIBUTION TABLES AND GRAPHS EXERCISE 7.1 BASIC MEASURES OF CENTRAL TENDENCY 7.1.1 Key Concepts i. Information in the form of numerical figures are called observations. ii. Observations gathered initially are called raw data. iii. The difference between the highest and the lowest values of the observations in a given data is called its range. iv. The number of times a particular observation occurs is called its frequency. v. When the number of observations is large, the data is usually organized into groups called class intervals. vi. A table showing the frequencies of various class intervals is called a frequency distribution table. vii. The lower value of the class interval is called lower limit and upper value is called upper limit. viii. The difference between the upper limit and lower limit is called class size. ix. Arithmetic mean is simply the average. It is given by: x¯ = xi = S um o f observations N Number o f observations x. Arithmetic mean by deviation method: Arithmetic Mean = Estimated mean + Average of deviations = Estimated Mean + S um of deviations Number of observations x¯ = A + (xi−A) N xi. Median is simply the middle term of the distribution when it is arranged in either ascending or descending orders. xii. When ‘n’ is odd, n+1 th observation is the median. When ‘n’ is even, then the arithmetic mean 2 n th th of two middle observations i.e., 2 and n + 1 observations is the median of the data. 2 xiii. Mode is the most frequently occurring value in the data. EXERCISE 7.1. BASIC MEASURES OF CENTRAL TENDENCY 64

7.1.2 Additional Questions Objective Questions 1. [AS3] The most evidently used measure of central tendency is . (A) Mean (B) Median (C) Mode (D) None 2. [AS1] The mean of the data 20, 19, 13, 23, 17, 21, 24, 14, 18 and 22 is . (A) 191 (B) 1.91 (C) 19.1 (D) 0.191 3. [AS1] The median of the data 8, 9, 1, 6, 13, 10, 5 and 11 is . (A) 8 (B) 8.5 (C) 9 (D) 8.9 4. [AS1] The mode of the data 21.5, 34.5, 28, 31, 21.5, 30, 34.5, 33.5, 21.5, 34.5, 34.5, 28and 30 is . (A) 28 (B) 30 (C) 21.5 (D) 34.5 5. [AS3] The measure of central tendency which uses the mid values of classes in its calculation is . (A) Mean (B) Median (C) Mode (D) None 6. [AS4] The mean height of 8 students of a class of heights 142 cm, 145 cm, 150 cm, 148 cm, 152 cm, 138 cm, 141 cm and 144 cm is . (A) 140 cm (B) 145 cm (C)146 cm (D)148 cm 7. [AS4] The median wages of a person whose wages in a week from Sunday to Saturday are Rs. 8500, Rs. 8900, Rs. 8800, Rs. 8600, Rs. 9000, Rs. 9200 and Rs. 9100 is . (A) Rs. 8871 (B) Rs. 62100 (C)Rs. 8600 (D)Rs. 8900 EXERCISE 7.1. BASIC MEASURES OF CENTRAL TENDENCY 65

8. [AS1] The mean of 'n' observations is 3. Each observation is multiplied by 9 and one is added to it. The new mean is . (A) 28 (B) 27 (C) 13 (D)28 n 9. [AS1] The mean of 11 observations is 12. If one observation 17 is deleted then the new meanis . (A) 14.9 (B) 149 (C) 11.5 (D) 13.5 (A) 29 (B) 87 (D)None of these (C) 9 2 3 Very Short Answer Type Questions 11 [AS1] Answer the following questions in one sentence. (i) Find the arithmetic mean of the first five odd natural numbers. (ii) The sum of 25 observations in a data is 745. Find the mean of the data. (iii) Calculate the median of the data 38, 23, 51, 67, 46, 32, 59 and 21. (iv) The median of x , x, x , x , x , x , x is 17 . Find the value of x. 7 43652 2 (v) The mean of 24, 72, 48, 56, 68, 39, 48, x is 49. Find the value of x. 12 [AS1] Answer the following questions in one sentence. (i) If the median of the terms x − 3, x − 2, x + 4, x + 7 and x + 9 is 23, find the value of x. (ii) The mean of 12 observations is 18. If one observation 30 is deleted, find the new mean. Short Answer Type Questions 13 [AS1] Find the arithmetic mean of 5 observations 20, 21, 13, 8 and 22 taking the assumed mean as 7. 14(i) [AS1] The number of goals were scored by a team in a series of 10 matches are 2, 3, 4, 5, 0, 1, 3, 3, 4 and 3. Find the median number of goals scored by the team. EXERCISE 7.1. BASIC MEASURES OF CENTRAL TENDENCY 66

(ii) [AS1] The following observations are in ascending order. If the median of the data is 63, find the value of x. 20, 32, 48, 50, x, x + 2, 72, 78, 84, 95 15 [AS3] When will the estimated mean become the actual arithmetic mean? 16 [AS4] The number of runs scored by 10 batsmen in a one day cricket match are as given. 23, 54, 08, 60, 18, 29, 44, 05, 86. Find the average runs scored. 17 [AS4] The following are the marks obtained (out of 100 marks) by 30 students of Class VIII of a school. 10 20 36 92 95 40 50 56 60 70 92 88 80 70 72 70 36 40 36 40 92 40 50 50 56 60 70 60 60 88 Find the mean of the marks obtained by the 30 students. 18(i) [AS4] Find the mean from the following frequency distribution of marks in a competitive exam. Marks 5 10 15 20 25 30 35 40 45 50 Number of 10 40 70 71 75 30 18 72 8 6 students (ii) [AS4] The attendance in a school for five days is as follows: 400, 430, 425, 408, 410. Calculate the arithmetic mean by assuming a mean. Long Answer Type Questions 19 [AS1] If the median of x, x , x and x when x > 0, is 10, find the mean. 5 2 3 20 [AS4] Rajesh saw around 680 animals in a zoo that he visited in his holidays. After coming home he tabulated the number of all the different animals hehas seen in the zoo as following. EXERCISE 7.1. BASIC MEASURES OF CENTRAL TENDENCY 67

Beast Land Birds Water Reptiles Animals Animals 180 Animals 75 80 200 145 Answer the questions using the given table. (i) Which animals are the maximum in number? (ii) Which animals are the least in number in the zoo? (iii) Which animals are 10 less than half of the number of birds? (iv) How many water animals are there in the zoo? (v) How many reptiles has Rajesh seen in the zoo? 21 [AS4] The ages of 40 students in a sports club are given in the table. Age (in years) 11 12 13 14 15 Number of students 8 4 10 12 6 Find their mean age. 22 [AS4] The following table shows the weights of 50 persons in a group. Weight (in kg) 40 –44 44 –48 48 –52 52 –56 56 –60 Number of Persons 12 16 9 8 5 Find their mean weight. 23 [AS4] The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs.18. Find the value of the missing frequency ‘f’. Daily pocket 11 – 13 13–15 15–17 17–19 19–21 21–23 23–25 allowance (in Rs.) Number of children 7 69 13 f 54 EXERCISE 7.1. BASIC MEASURES OF CENTRAL TENDENCY 68

EXERCISE 7.2 ORGANISATION OF GROUPED DATA 7.2.1 Key Concepts i. Representation of classified distinct observations of the data with frequencies is called ‘Frequency Distribution’ or ‘Distribution Table’. ii. The difference between the upper and lower boundaries of a class is called the length of the class or class length denoted by ‘C’. iii. In a class, the initial value and end value of each class are called the lower limit and upper limit of that class respectively. iv. The average of upper limit of a class and the lower limit of the successive class is called the upper boundary of that class. v. The average of the lower limit of a class and the upper limit of the preceding class is called the lower boundary of that class. vi. The progressive total of frequencies from the last class of the table to the lower boundary of a particular class is called Greater than Cumulative Frequency (G.C.F.). vii. The progressive total of frequencies from the first class to the upper boundary of a particular class is called Less than Cumulative Frequency (L.C.F.). 7.2.2 Additional Questions Objective Questions 1. [AS3] The classes in which the upper limit of one class and the lower limit of the consecutive class are not equal are called . (A) Boundaries (B) True class limits (C)Inclusive classes (D)Exclusive classes 2. [AS3] The classes which are continuous i.e., the upper limit of one class is the same as the lower limit of immediate next class are called . (A) Boundaries (B) Class limits (C)Inclusive classes (D)Exclusive classes 3. [AS3] The total frequency from the beginning to the upper boundary of a particular class is called the cumulative frequency. (A) Greater than (B) Less than (C) Greater than or equal to (D) Less than or equal to EXERCISE 7.2. ORGANISATION OF GROUPED DATA 69

4. [AS3] If the Greater than Cumulative Frequency (G.C.F) of class 70 – 80 is 12 and that of 60 – 70 is 20 then the frequency of the class 60 – 70 is . (A) 5 (B) 240 3 (C) 32 (D) 8 5. [AS3] The difference between the upper and the lower boundaries of a class is called the . (A) Class length (B) Boundaries (C) Range (D)None of these 6. [AS1] In a given data, if the minimum value and range are 12 and 27 then the maximum value is . (A) 39 (B) 15 (C) 2.25 (D)None of these 7. [AS5] The graphs of the cumulative frequencies are called . (A) Ogive curves (B) Pictographs (C)Bar graphs (D) Histograms Very Short Answer Type Questions 8 [AS1] Answer the following questions in one sentence. (i) If the minimum value of a data is 27 and the range is 15, find the maximum value of the data. (ii) The minimum and maximum values of a data are 25 and 85 respectively. If there are 10 classes in the data, find the class length of each class. 9 [AS4] Answer the following questions in one sentence. (i) The following is a list of the names and heights of five boys in an eighth grade basketball team. Devendra –167 cm, Narendra –160 cm, Sanjay –165 cm, Piyush –175 cm and Girish –155 cm. Organise the data in the ascending order of heights. (ii) The marks scored by 10 students in test – 1 are Santosh – 77, Nagu – 40, Aman – 45, Amit – 81,Rakesh – 69, Sunil – 81, Vinod – 94, Anjali – 51, Aditi – 87 and Renu – 50. Arrange the data in alphabetical order. (iii) The marks scored by 20 students in a unit test out of 25 marks are as given. EXERCISE 7.2. ORGANISATION OF GROUPED DATA 70

12, 10, 08, 12, 04, 15, 18, 23, 18, 16, 16, 12, 23, 18, 12, 05, 16, 16, 12, 20. Find the number of students who scored less than 19. Short Answer Type Questions 10(i) [AS4] The heights (in cm) of 25 children are as given. 174, 168, 110, 142, 156, 199, 110, 101, 190, 102, 190, 111, 172, 140, 136, 174, 128, 124, 136,147, 168, 192, 101, 129, 114. Prepare a frequency distribution table taking the size of the class – interval as 20. (ii) [AS4] Consider the following marks (out of 50) scored in mathematics by 50 students of Class 8. 41, 31, 33, 32, 28, 31, 21, 10, 30, 22, 33, 37, 12, 05, 08, 15, 39, 26, 41, 46, 34, 22, 09, 11, 16, 22, 25, 29, 31, 39, 23, 31, 21, 45, 30, 22, 17, 36, 18, 20, 22, 44, 16, 24, 10, 28, 39, 28, 47, 17. Prepare a frequency distribution table. 11 [AS5] Rewrite the given table with true class limits. Classes 11 –15 16 –20 21 –25 26 –30 31 –35 36 –40 Frequency 4 37 5 7 2 12 [AS5] Given are the marks scored by 40 students in an examination. 38 32 24 26 28 25 20 18 36 30 32 38 26 24 20 16 36 16 18 22 24 26 28 10 34 30 20 24 26 22 32 28 38 20 24 22 28 30 28 10 Take class intervals 0 – 10, 11 – 20, 21 – 30 and construct a frequency distribution table for the given data. 13 [AS5] Create a cumulative frequency distribution for the frequency table given. Length 11 – 16 – 21 – 26 – 31 – 36 – 41 – (mm) 15 20 25 30 35 40 45 Frequency 2 4 8 14 6 4 2 EXERCISE 7.2. ORGANISATION OF GROUPED DATA 71

EXERCISE 7.3 GRAPHICAL REPRESENTATION OF DATA 7.3.1 Key Concepts i. Histogram is a graphical representation of frequency distribution of exclusive class intervals. ii. When the class intervals in a grouped frequency distribution are varying, we need to construct rectangles in histogram on the basis of frequency density. iii. Frequency density = Frequency of class. iv. Frequency polygon is a graphical representation of a frequency distribution (discrete/ continuous). v. In frequency polygon or frequency curve, class marks or mid values of the classes are taken on X–axis and the corresponding frequencies on the Y–axis. vi. Area of frequency polygon and histogram drawn for the same data are equal. vii. A graph representing the cumulative frequencies of a grouped frequency distribution against the corresponding lower/ upper boundaries of respective class intervals is called Cumulative Frequency Curve or “Ogive Curve”. 7.3.2 Additional Questions Objective Questions 1. [AS3] The lengths of the rectangles in a histogram are proportional to the . (A) Class interval (B) Frequency (C)Upper limits (D) Lower limits 2. [AS3] If the midpoints of the widths of rectangles on the top of the bars in a histogram are joined bystraight line segments then it is called a/ an . (A) Frequency curve (B) Ogive curve (C)Frequency polygon (D)None of these 3. [AS3] The point of intersection of less than and greater than ogive curves gives . (A) Mean (B) Median (C) Mode (D) Range EXERCISE 7.3. GRAPHICAL REPRESENTATION OF DATA 72

4. [AS3] To construct a histogram, the classes in the given data must be . (A) Raw data (B) Grouped data (C) Discrete (D) Continuous 5. [AS3] The curve constructed by taking the upper boundaries of classes on X–axis and the corresponding cumulative frequencies on Y–axis is known a curve. (A) Less than ogive (B) Greater than ogive (C) Frequency (D)None of these 6. [AS5] If a rectangle of height 7 cm represents 140 units then another rectangle of height 4.5 cm represents units. (A) 24.5 (B) 90 (C) 15.5 (D)None of these 7. [AS3] The mid points of the top ends of the bars in a histogram are joined by means of straight line segments to form a . (A) Frequency polygon (B) Frequency curve (C)Less than ogive curve (D)Greater than ogive curve 8. [AS3] The measure of central tendency which the common point of both the ogive curves gives is the . (A) Mean (B) Mode (C) Median (D)None of these 9. [AS5] In a pictograph 10 car pictures represent 5000 cars in real. In the same pictograph, 7 and a half car pictures represent cars in real. (A) 7000 (B) 375 (C) 37500 (D) 3750 10. [AS5] In a histogram every two consecutive rectangles have . (A)One common side (B) One common height (C)An equal area (D)None of these EXERCISE 7.3. GRAPHICAL REPRESENTATION OF DATA 73

Very Short Answer Type Questions 11 [AS5] Answer the following questions in one sentence. Read and understand the following vertical bar graph and answer the questions. (i) What was the production of wheat in the year 1990 – 91? (ii) In which year was the production minimum and in which year was it maximum? (iii) How many more lakhs of tonnes of wheat was produced in 1980 – 81 than in 1960 – 61? (iv) What is the total production of wheat in two years 1970 – 71 and 1980 – 81? (v) What is the increase in production of wheat in the year 1950 – 51 to the year 2000 – 2001? 12 [AS5] Answer the following questions in one sentence. In a bar graph, a rectangle of height 22 cm represents 27500000 people. Find the height of the rectangle which represents 43750000 people. Short Answer Type Questions 13(i) [AS5] The following table shows the expenditure per month of a family. Items Food Clothing Rent Education Miscellaneous 2000 1500 2500 1000 Expenditure 3500 (in Rs.) Draw a bar graph to represent the given data. (ii) [AS5] Draw a histogram for the given data. Salary (in 15 –20 20 –25 25 –30 30 –35 35 –40 thousand rupees) 35 30 45 40 10 No. of employees EXERCISE 7.3. GRAPHICAL REPRESENTATION OF DATA 74

14 [AS5] The following frequency polygon shows the marks scored by some students in a class test. Answer these questions based on the given frequency polygon. (i) What is the class size? (ii) How many students scored less than 20? (iii) How many students scored 25 or more? 15 [AS5] The table given shows the number of employees drawing different monthly salaries. Monthly 10000 – 15000 – 20000 – 25000 – 30000 – salary (in 15000 20000 25000 30000 35000 Rs.) 5 10 20 8 7 Number of employees Construct a histogram for the data. 16 [AS5] The table gives the age distribution of a group of students. Draw the cumulative frequency curve of less than type and hence obtain the median value. Age (in 4–5 5–6 6–7 7–8 8–9 9–10 10–11 11–12 12–13 13–14 14–15 15–16 years) Frequency 36 42 52 60 68 84 96 82 66 48 50 16 EXERCISE 7.3. GRAPHICAL REPRESENTATION OF DATA 75

17 [AS5] For the following frequency distribution, draw a cumulative frequency curve of more than type and hence obtain the median value. Class 0 –10 10–20 20–30 30–40 40–50 50–60 60–70 interval 5 15 20 23 17 11 9 Frequency 18 [AS5] During the medical checkup of 35 students of a class, their weights were recorded as follows: Weight 38–40 40–42 42–44 44–46 46–48 48–50 50–52 (in kg) 3245 14 4 3 No. of students Draw a less than type and a more than type ogive from the given data. Hence obtain the median weight from the graph. EXERCISE 7.3. GRAPHICAL REPRESENTATION OF DATA 76

CHAPTER 8 EXPLORING GEOMETRICAL FIGURES EXERCISE 8.1 CONGRUENCY OF SHAPES 8.1.1 Key Concepts i. Figures are said to be congruent if they have the same shape and size. ii. Figures are said to be similar if they have the same shape, but different size. iii. If we flip, slide or turn the congruent or similar shapes, their congruency or similarity remains the same. iv. The method of drawing enlarged or reduced similar figures is called dilation. v. We use ’ ’ symbol to represent congruency, and ’∼ ’ symbol for similarity. 8.1.2 Additional Questions Objective Questions 1. [AS3] If the shape and size of two figures are the same then they are called figures. (A) Similar (B) Congruent (C) Enlarged (D)None of these 2. [AS3] The method of drawing enlarged (or) reduced similar figures is called . (A) Congruency (B) Similarity (C) Dilation (D)None of these 3. [AS2] In ∆ABC, AB = AC = 3 cm and ∠A = 60 ◦, PQR is an equilateral triangle with side 3 cm then ABC and PQR are . (A) Congruent (B) Similar (C)Of different sizes (D)None of these EXERCISE 8.1. CONGRUENCY OF SHAPES 77

4. [AS3] The figures which are always similar are . (A) Rectangle (B) Trapezium (C) Parallelogram (D)All of these 5. [AS3] Two circles with radii r 1 and r2 are congruent, then . (A) r1 > r 2 (B) r1 < r2 (C)r1 = r2 (D)None of these 6. [AS1] ABC and LMN are congruent. If ∠B = 75◦ then the value of ∠M is . . (A) 65◦ (B) 55◦ (C) 75◦ (D) 100◦ DEF. If ∠P = ∠Q and ∠F = 50 ◦ , then the measure of ∠E is 7. [AS1] PQR (B) 65◦ (A) ◦ (D) 130◦ 50 (C) 105◦ Very Short Answer Type Questions 8 [AS1] Answer the following questions in one sentence. (i) In the adjacent figure, if PQR LMN, find LN. (ii) CAT GUN. If CA = 5 cm and LA = AT, find UN. (iii) Find the unknown value in the following similar figures. 9 [AS3] Answer the following questions in one sentence. (i) Identify the figures which are congruent. EXERCISE 8.1. CONGRUENCY OF SHAPES 78

(ii) Identify the figures which are congruent. 79 (iii) Identify the figures which are congruent. (iv) Identify the figures which are congruent. (v) Identify the figures which are congruent. 10 [AS3] Answer the following questions in one sentence. (i) Identify the similar figures among the following. (ii) Identify the similar figures among the following. EXERCISE 8.1. CONGRUENCY OF SHAPES

(iii) Identify the similar figures among the following. (iv) Identify the similar figures among the following. (v) Identify the similar figures among the following. 11 [AS3] Answer the following questions in one sentence. (i) Triangle ABC is congruent to triangle PQR. Name the congruent sides. (ii) ∆PQR ∆MNO Name the congruent angles. (iii) Right triangle ABC Right triangle MNO Name the congruent parts. (iv) If PQR YZX, write the name of the angle congruent to ∠Q. (v) If LMN DEF, ∠M = ∠N and ∠DEF = 55◦ then, what is the measure of ∠MLN? 12 [AS4] Answer the following questions in one sentence. Give any two real life examples for congruent shapes. EXERCISE 8.1. CONGRUENCY OF SHAPES 80

Short Answer Type Questions 13 (i) [AS1] Find the unknown value in the following similar figures. a) b) (ii) [AS1] Find the unknown value in the following similar triangles. a) b) c) EXERCISE 8.1. CONGRUENCY OF SHAPES 81

14(i) [AS2] In the figure, OA = OB and OD = OC. Show that a) AOD BOC and b) AD BC. (ii) [AS2] E and F are respectively the mid –points of equal sides AB and AC of ABC. Show that BF = CE. 15(i) [AS2] AB is a line segment and line l is its perpendicular bisector. If a point P lies on l, show that P is equidistant from A and B. EXERCISE 8.1. CONGRUENCY OF SHAPES 82

(ii) [AS2] In an isosceles triangle ABC with AB = AC, D and E are points on BC such that BE = CD. Show that AD = AE. 16 [AS3] What is the difference between congruent figures and similar figures? 17(i) [AS5] Draw dilation figures when the scale factor is given. Draw a triangle on a graph sheet and draw its dilation with scale factor 2. (ii) [AS5] Draw a square of side 3 cm and draw its dilation with a dilation factor 2. EXERCISE 8.1. CONGRUENCY OF SHAPES 83

EXERCISE 8.2 SYMMETRY 8.2.1 Key Concepts i. If a figure is folded or cut along a line such that each part exactly coincides with the other then the figure is said to be symmetrical and the line is called the line of symmetry or axis of symmetry. ii. Some figures may have more than one line of symmetry. iii. Symmetry is of three types, viz., line symmetry, rotational symmetry and point symmetry. iv. With rotational symmetry, the figure is rotated around a central point so that it appears two or more times the same as the original. The number of times for which it appears the same is called the order of rotation. v. Rotation means turning around a centre. vi. Flip is a transformation in which a plane figure is reflected across a line, creating a mirror image of the original figure. vii. The pattern formed by repeating figures to fill a plane without gaps or overlaps is called tessellation. 8.2.2 Additional Questions Objective Questions 1. [AS5] The number of lines of symmetry for a rhombus is . (A) 1 (B) 2 (C) 3 (D) 4 2. [AS5] The number of lines of symmetry for the figure is . (A) 4 (B) 3 (C) 2 (D) 1 3. [AS3] Every of a circle is an axis of symmetry for the circle. (A) Secant (B) Chord (C) Radius (D) Diameter EXERCISE 8.2. SYMMETRY 84

4. [AS3] The angle of rotational symmetry of a regular pentagon is . (A) 72 ◦ (B) 60◦ (C) 180◦ (D) 144◦ 5. [AS3] The angle of rotation of a figure is 45◦ . The order of its rotational symmetry is . (A) 2 (B) 4 (C) 6 (D) 8 6. [AS4] The letter of the English alphabet that has reflectional symmetry about a vertical mirror is . (A) A (B) B (C) C (D) D 7. [AS4] The letter of the English alphabet that has point symmetry is . (A) Q (B)R (C) P (D) S 8. [AS4] The letter that has a vertical line of symmetry is . (A) E (B) R (C) Q (D) V 9. [AS5] The sum of the number of dots on the opposite faces of a die is . (A) 6 (B) 8 (C) 7 (D) 9 EXERCISE 8.2. SYMMETRY 85

Very Short Answer Type Questions 10 [AS5] Answer the following questions in one sentence. (i) Find the axes of symmetry and the order of rotation for the following figure. (ii) Find the axes of symmetry and the order of rotation for the following figure. (iii) Find the axes of symmetry and the order of rotation for the following figure. 11 [AS5] Answer the following questions in one sentence. Draw all the lines of symmetry for the following figures. (i) (ii) EXERCISE 8.2. SYMMETRY 86

(iii) 12 [AS5] Answer the following questions in one sentence. (i) Draw the lines of symmetry for the following alphabet. (ii) Draw the lines of symmetry for the following alphabet. (iii) Draw the lines of symmetry for the following alphabet. Short Answer Type Questions 13 [AS3] What is the difference between line symmetry and point symmetry? 14 (i) [AS5] What are the basic shapes used in this tessellation? EXERCISE 8.2. SYMMETRY 87

(ii) [AS5] Draw a tessellation and name the basic shapes used in it. 15(i) [AS5] Which of the following have point symmetry? (ii) Draw two figures that have point symmetry. Long Answer Type Questions 16 [AS4] Complete the following table in which some drawing tools are given. Name of the tool Number of lines of symmetry i) The ruler ii) The divider iii) The compasses iv) The triangular piece with two equal sides v) The triangular piece with unequal sides 17 [AS5] Draw a tessellation containing hexagons, squares and triangles. EXERCISE 8.2. SYMMETRY 88

CHAPTER 9 AREA OF PLANE FIGURES EXERCISE 9.1 AREA OF TRAPEZIUM 9.1.1 Key Concepts i. Area of a trapezium = 1 (Sum of lengths of parallel sides) × (Distance between 2 1 them) = 2 h (a + b) ii. Area of a quadrilateral = 1 × Length of diagonal × Sum of the lengths of the perpendiculars 2 1 from the remaining two vertices on to the diagonal = 2 × d × (h1 + h2) iii. Area of a rhombus = Half of the product of diagonals = 1 × d1 × d2 2 iv. Area of a parallelogram = base × height = bh v. Area of a triangle = 1 × base × height = 1 × b × h 2 2 √ vi. Area of an equilateral triangle = 3 a2 4 9.1.2 Additional Questions Objective Questions 1. [AS1] The parallel sides of a trapezium are 9.7 cm and 6.3 cm, and the distance between them is 6.5 cm. The area of the trapezium is sq. cm. (A) 104 (B) 78 (C) 52 (D) 65 EXERCISE 9.1. AREA OF TRAPEZIUM 89

2. [AS1] ABCD is a trapezium in which AB = 40 m, BC = 15 m, CD = 28 m, AD = 9 m and CE ⊥AB. Then the area of trapezium ABCD is sq. m. (A) 306 (B) 316 (C) 296 (D) 284 3. [AS1] The length of a rectangular field is 12 m and the length of its diagonal is 15 m. The area of the field is sq. m. (A) 108 √ √ (B) 30 3 (D)None of these (C)12 15 4. [AS1] The area of a square field is 6050 sq. m. The length of its diagonal is m. (A) 110 (B) 112 (C) 120 (D) 135 5. [AS1] The area of the parallelogram one of whose sides measures 48 cm and the corresponding height measures 18.5 cm is sq. cm. (A) 444 (B) 888 (C) 1776 (D)None of these EXERCISE 9.1. AREA OF TRAPEZIUM 90

6. [AS1] The area of the trapezium given is . (A) 56 cm2 (B) 89 cm2 (C)410 cm2 (D)140 cm2 7. [AS3] A quadrilateral with two pairs of parallel opposite sides is called a . (A) Rhombus (B) Trapezium (C) Parallelogram (D) Kite 8. [AS3] The area of a rhombus is . (A) 1 d1d2 (B) d2 2 2 (C) bh (D) lb 9. [AS3] A quadrilateral with one pair of parallel opposite sides is called a . (A) Square (B) Rectangle (C)Trapezium (D) Rhombus 10. [AS4] A garden is in the form of a trapezium whose parallel sides are 40 m and 22 m. The perpendicular distance between them is 12 m. The area of the garden is . (A) 372 m2 (B) 462 m2 (C)732 m2 (D)550 m2 EXERCISE 9.1. AREA OF TRAPEZIUM 91

11. [AS4] The area of a hexagonal table top with each side 2 m is . (A) 11.392 m2 (B) 10.392 m2 (C)9.372 m2 (D)9.672 m2 Very Short Answer Type Questions 12 [AS1] Answer the following questions in one sentence. (i) The area of a rhombus is 50 cm2 and one of its diagonals is 15 cm. Find the other diagonal. (ii) Find the base of a parallelogram whose area is 128 cm2 and height 16 cm. (iii) Find the length of a rectangle, whose area is 154 cm2 and width 11 cm. (iv) Find the area of an equilateral triangle whose side is 6 cm. (v) Find the altitude of a trapezium, the sum of the lengths of whose bases is 8.5 cm and whose area is 34 cm2 . 13 [AS2] Answer the following questions in one sentence. (i) If the ratio of base of two triangles is x : y and that of their areas is a : b, then what is the ratio of their corresponding altitudes? 14 [AS3] Answer the following questions in one sentence. (i) Define a regular polygon. (ii) Write the formula to find the area of a quadrilateral. EXERCISE 9.1. AREA OF TRAPEZIUM 92

Short Answer Type Questions 15(i) [AS1] Find the area of the polygon given. CD BE AF⊥BE GD⊥BE CD = 14 cm BE = 18 cm AF = 6 cm DG = 8 cm. (ii) [AS1] The parallel sides of trapezium are 9 cm and 7 cm long and its area is 48 sq. cm. Find the distance between the parallel sides. 16(i) [AS4] Find the area of a rectangular park of length 200 m and breadth 150 m. (ii) [AS4] Find the area of the given field. 17(i) [AS4] The longer side of a rectangular hall is 24 m and the length of its diagonal is 26 m. Find the area of the hall. EXERCISE 9.1. AREA OF TRAPEZIUM 93

(ii) [AS4] In a four–sided field, the length of the longer diagonal is 128 m. The lengths of the perpendiculars from the opposite vertices upon this diagonal are 22.7 m and 17.3 m. Find the area of the field. 18 [AS1] Find the area of the following figure. 19 [AS1] Find the area of the shaded region in the following figure. 20 [AS1] Find the area of a trapezium whose parallel sides are AB = 16 cm, DC = 8 cm and non-parallel sides are BC = 10 cm and DA = 6 cm. 21 [AS1] Find the area of the field given. All the dimensions are given in metres. EXERCISE 9.1. AREA OF TRAPEZIUM 94

22 [AS1] Find the area of a quadrilateral whose sides measure 9 cm, 40 cm, 15 cm and 28 cm and the angle between the first two sides is a right angle. 23 [AS2] The length of a rectangle is increased by 30% and the width is decreased by the same percentage. What is the percentage change in its area? 24 [AS4] Find the length of a rectangular field whose breadth is 15 m and area is 180 m2. 25 [AS4] A rectangular lawn 75 m by 60 m, has two roads, each 4 m wide, running through the middleof the lawn, one parallel to length and the other parallel to breadth, as shown in the figure. Find the cost of gravelling the roads at Rs. 45 per sq. m. 26 [AS4] Find the breadth of the rectangular field given that its area is 256 cm.2 27 [AS4] A lawn is in the form of a rectangle whose sides are in the ratio 5 : 3. Its area is 3375 sq. m. Find the cost of fencing the lawn at Rs. 8.50 per metre. EXERCISE 9.1. AREA OF TRAPEZIUM 95

EXERCISE 9.2 AREA OF CIRCLE 9.2.1 Key Concepts i. Area of a circle = πr2, where r = radius. ii. Area of a ring = π R2 − r2 , where R = radius of outer circle and r = radius of inner circle. iii. Area of a sector = x × πr2, where x is the central angle and r is the radius. 360◦ 9.2.2 Additional Questions Objective Questions 1. [AS1] The area of a circle is 154 sq. m. Then its diameter is . (A) 7 m (B) 14 m (C) 3.5 m (D) 21 m 2. [AS2] The circumferences of two circles are in the ratio 2 : 3. The ratio between their areas is . (A) 2 : 3 (B) 3 : 2 (C)9 : 4 (D)4 : 9 3. [AS4] The circumference of a circular field is 242 m. Then its area is sq. m. (A) 9317 (B) 18634 (C) 4658.5 (D)None of these 4. [AS2] If the area of a circle is 49π sq. cm then its circumference is cm. (A) 14π (B) 21π (C) 28π (D) 7π 5. [AS2] On increasing the diameter of a circle by 40%, its area will be increased by . (A) 40% (B) 80% (C) 96% (D) 82% EXERCISE 9.2. AREA OF CIRCLE 96

6. [AS4] A circular garden has a circumference of 220 cm. There is a 7 cm wide border inside the garden along its boundary. The area of the border is . (A) 1368 m2 (B) 1683 m2 (C)1836 m2 (D)1386 m2 Short Answer Type Questions 7(i) [AS1] Find the length of the arc of the shaded part of the circle. (ii) [AS1] Find the length of the arc, which makes an angle 30° at the centre of the circle of radius 21cm. 8(i) [AS1] A chord of a circle of radius 14 cm makes a right angle at the centre. Find the area of the sector making the right angle. (ii) [AS1] The perimeter of a sector of a circle of radius 5.6 cm is 27.2 cm. Find its area. 9 [AS1] Find the area of the shaded region in the given figure. (Take π = 3.14 ) EXERCISE 9.2. AREA OF CIRCLE 97