202110219-TRIUMPH-STUDENT-WORKBOOK-MATHEMATICS-G07-PART1

5. [AS1] In a right angled triangle, the other two angles are in the ratio 2 : 3. Then the angles in order are . (A) 36◦, 54◦ (B) 54◦, 36◦ (C)45◦, 45◦ (D)None of these Short Answer Type Questions 6(i) [AS1] In ∆ABC, ∠A = 45◦ and ∠B = 30◦ find ∠C. (ii) [AS1] One angle of ∆ABC is 70◦ and the other two angles are equal. Find the measure of each of the equal angles. 7 [AS1] In a right angled triangle ABC, one of the angles other than the right angle is 40◦ . Show that the third angle is 50◦. 8 [AS2] Write the proof of the angle–sum property of a triangle. Long Answer Type Questions 9 [AS1] The angles of a triangle are in the ratio of 2 : 3 : 4. Find the angles. 10 [AS1] The angles of a triangle are in the ratio 3 : 5: 10. Find the measure of each angle. EXERCISE 5.3. PROPERTIES OF TRIANGLES 97

EXERCISE 5.4 EXTERIOR ANGLE OF A TRIANGLE 5.4.1 Key Concepts i. Exterior angle of triangle: When one side of a triangle is produced, the angle thus formed is called an exterior angle. In ∆COL , the side OL is produced to D. ∠CLD is an exterior angle. The exterior angle of a triangle is equal to the sum of two interior opposite angles. ∠COL + ∠OCL = ∠CLD 5.4.2 Additional Questions Objective Questions 1. [AS1] In a triangle two of the interior angles are 60◦and 45 ◦. Then the exterior angle at the third vertex is . (A) 75◦ (B) 105◦ (C) 285◦ (D) None of these 2. [AS1] In a triangle ABC, ∠A = ◦ and ∠B = 70◦. Then the exterior angle at C is . 50 (A) 60◦ (B) 120◦ (C) 70◦ (D) 50◦ EXERCISE 5.4. EXTERIOR ANGLE OF A TRIANGLE 98

3. [AS1] The exterior angle at one vertex of a triangle is 150 ◦ and one of the opposite interior angles is 87◦.Then the other angle is . (A) 150◦ (B) 87◦ (C) 237◦ (D) 63◦ 4. [AS1] The exterior angle at one vertex of a triangle is 127◦. Then the interior angle at that vertex is . (A) 127◦ (B) 307◦ (C) 53◦ (D) None of these 5. [AS1] If one of the interior angles of a triangle is 117◦, then the exterior angle at that vertex is . (A) 117◦ (B) 63◦ (C) 297◦ (D) None of these Short Answer Type Questions 6(i) [AS1] Find the value of y in the figure: (ii) [AS1] In the figure given, find the values of x and y. EXERCISE 5.4. EXTERIOR ANGLE OF A TRIANGLE 99

7 [AS1] One of the exterior angles of a triangle is 110◦and its interior opposite angles are equal to each other. Find the measures of the interior angles of the triangle. 8 [AS2] Prove that an exterior angle is equal to the sum of the interior opposite angles of a triangle. Long Answer Type Questions 9 [AS1] (i) One of the exterior angles of a triangle is 120◦and the interior opposite angles are in the ratio 2 : 3. Find the angles of the triangle. (ii) Find the value of angle 'x' in the figure. EXERCISE 5.4. EXTERIOR ANGLE OF A TRIANGLE 100

CHAPTER 6 RATIO –APPLICATIONS EXERCISE 6.1 RATIO 6.1.1 Key Concepts i. Ratio: A ratio is an ordered comparison of quantities of the same units. ii. We use the symbol ‘:’ to represent a ratio. The ratio of two quantities a and b is a : b and we read it as a is to b . iii. The two quantities ‘a’ and ‘b’ are called the terms of the ratio. The first quantity ‘a’ is called the first term or antecedent and the second quantity ‘b’ is called the second term or consequent. iv. We have to express both the quantities in same units before they are compared. 6.1.2 Additional Questions Objective Questions 1. [AS1] If a : b = 3 : 4 and b : c = 8 : 9, then a : c = . (A) 1 : 2 (B) 3 : 2 (C)1 : 3 (D) 2 : 3 2. [AS1] If A : B = 2 : 3 and B : C = 4 : 5, then C : A = . (A) 15 : 8 (B) 6 : 5 (C)8 : 5 (D)8 : 15 EXERCISE 6.1. RATIO 101

3. [AS1] If 2A = 3B and 4B = 5C, then A : C = . (A) 4 : 3 (B) 8 : 15 (C)3 : 4 (D) 15 : 8 4. [AS1] If A = 1 B and B = 1 C, then A : B : C = . 3 2 (B) 2 : 3 : 6 (D)3 : 1 : 2 (A) 1 : 3 : 6 (C)3 : 2 : 6 5. [AS1] If A = B = C5 , then A : B : C = . 3 2 (A) 3 : 2 : 5 (B) 4 : 3 : 5 (D)20 : 15 : 12 (C)5 : 4 : 3 Very Short Answer Type Questions 6 [AS4] Answer the following questions in one sentence. 30 students have planned for a picnic. The ratio of girls and boys is 3 : 7. How many boys are there in the picnic? Short Answer Type Questions 7 [AS1] Which of the following are equal ratios? (i) 3 : 4 and 15 : 20 (ii) 2 : 5 and 30 : 20 (iii) 3 : 4 and 6 : 8 8(i) [AS1] Express each phrase as a ratio in simplified form: a) The ratio of 8 to 12. b) The ratio of 3.2 to 16. c) The ratio of 80 g to 2 kg. d) The ratio of 2 feet to 2 yards. EXERCISE 6.1. RATIO 102

(ii) [AS1] Two numbers are in the ratio 4 : 7. Two-seventh of the larger number exceeds three-fourth of the smaller by 8. Find the numbers. 9 [AS4] A school has 42 sixth graders, 56 seventh graders, and 49 eighth graders. Write the ratio in the simplest form for the following: (i) The number of sixth graders to that of eighth graders. (ii) The number of seventh graders to the total students. EXERCISE 6.1. RATIO 103

EXERCISE 6.2 PROPORTION 6.2.1 Key Concepts i. Proportion: If two ratios are equal, then the two ratios are said to be in proportion. ii. We use the symbol ”: :” for proportion and read it as “is as”. iii. If two ratios a : b and c : d are equal, we write a : b : : c : d or a : b = c : d. Here ‘a’ and ‘d’ are called extremes and ‘b’ and ‘c’ are called means. iv. If a, b, c and d are in proportion, then we get a : b = c : d ⇒ a × d = b × c. i.e., Product o f extremes = Product o f means. 6.2.2 Additional Questions Objective Questions . 1. [AS1] If 7 : x :: 35 : 45 then x = (B) 15 (A) 11 (C) 9 (D) 5 2. [AS1] If 7 : x :: x : 28 then x = . (A) 14 (B) 28 (C) 7 (D)None of these 3. [AS1] In a proportion, 4 : 9 :: 16 : x, the value of x is . (A) 36 (B) 16 (C) 4 (D) 14 Very Short Answer Type Questions 4 [AS1] Answer the following questions in one sentence. (i) An art teacher bought several drawing kits that each contained 12 markers and 20 coloured pencils. She bought 84 markers in all. How many coloured pencils did she buy? (ii) A ship can travel 70 km in 4 hours. Find the distance (in km) that the ship could travel in 10 hours. EXERCISE 6.2. PROPORTION 104

(iii) Find the value of x in 2 : 7 :: 10 : x. (iv) A solution should contain 2 ounces of chemical for every 7 ounces of water. How much chemical should be added to 20 ounces of water to get the proper concentration? (v) If 9 tickets to a concert cost Rs. 1125, how much will 15 tickets cost? Short Answer Type Questions 5 [AS1] Solve the following proportions: (i) 15 : x = 25 : 40 (ii) 5 = x 13 2 (iii) 2x − 1 = 12.5 2 5 6 [AS2] Determine whether the following ratios are proportional or not. (i) 3 and 36 7 91 (ii) 6 and 54 11 99 EXERCISE 6.2. PROPORTION 105

EXERCISE 6.3 UNITARY METHOD 6.3.1 Key Concepts i. Unitary Method: The method in which we first find the value of one unit and then the value of the required number of units is known as the unitary method. ii. Direct proportion: If in two quantities, when one quantity increases, the other also increases in a way that their ratio is constant or vice-versa, then the two quantities are said to be in direct proportion. 6.3.2 Additional Questions Objective Questions 1. [AS1] In a map,1 cm represents 8 km. The actual distance represented by 80.5 cm in the map is . (A) 640 km (B) 642 km (C)644 km (D)648 km 2. [AS1] A car travels 120 km in 4 hours at a constant speed. The number of hours that the car takes to cover 210 km is . (A) 5 hours (B) 6 hours (C)7 hours (D)8 hours 3. [AS1] If 21 cows eat as much as 15 buffaloes, the number of cows that eat as much as 35 buffaloes is . (A) 45 (B) 49 (C) 56 (D) 54 EXERCISE 6.3. UNITARY METHOD 106

4. [AS1] The weight of 56 books is 7 kg. The weight of 128 such books is . (A) 14 kg (B) 15 kg (C)16 kg (D)18 kg 5. [AS1] 6 dozen eggs are bought for Rs. 108. The cost of 108 eggs is . (A) Rs. 171 (B) Rs. 162 (C)Rs. 153 (D)Rs. 180 Very Short Answer Type Questions 6 [AS2] State true or false. (i) If Shravya types 450 words in half an hour, then she can type 20 words in one minute. [] [AS2] Choose the correct answer. (ii) If 6 notebooks cost Rs. 45, then the cost of 8 note books is . (A) Rs. 60 (B) Rs. 75 (C)Rs. 50 (D)Rs. 80 [AS2] Answer the following questions in one sentence. (iii) John types 450 words in half an hour. How many words would he type in 7 minutes? (iv) A man is paid Rs. 770 for 7 days. If he works for 21 days, how much money will he get? (v) 8 persons can complete a task in 48 days. In how many days will 1 person complete the same task? Short Answer Type Questions 7(i) [AS1] The cost of 11 books is Rs.165. Find the cost of 7 such books. (ii) [AS1] In 3 hours, a train covers 183 km. Travelling at the same speed, what distance would the train cover in two and a half hours? 8(i) [AS2] The minute hand of a clock moves through an angle of 90◦ in 15 minutes. If the time passed is 22 minutes, by what angle does the minute hand move? EXERCISE 6.3. UNITARY METHOD 107

(ii) [AS2] At 12 noon, the angle made by the minute hand with the hour hand is 0◦. If the angle with respect to the hour hand increases to 72◦, what is the time? Long Answer Type Questions 9 The weight of 56 books is 7 kg. (i) What is the weight of 90 such books? (ii) How many such books weigh 7.5 kg? 10 [AS1] A recipe for 2 dozen corn muffins calls for 3 cups of flour. The number of muffins varies directly as the quantity of flour used. (i) Write a direct variation equation for the relationship between the number of cups of flour and the number of muffins. (ii) Estimate the number of cups of flour needed to make 6 dozen muffins. EXERCISE 6.3. UNITARY METHOD 108

EXERCISE 6.4 PERCENTAGES 6.4.1 Key Concepts i. Ratios can be expressed in the form of percentages. ii. The word percent means “per every hundred” or for a hundred. The symbol % is used to denote a percentage. iii.To convert a quantity into its equivalent percentage, express it as a fraction, multiply it by 100 and place a % symbol beside it. iv.If a quantity is increased by x% then the new quantity = Quantity × 100 + x 100 100 − x v. If a quantity is decreased by x% then the new quantity = Quantity × 100 6.4.2 Additional Questions Objective Questions 1. [AS1] 641 % expressed as a fraction is . (A) 1 (B) 1 8 16 (C) 4 (D) 1 25 25 2. [AS1] If x% of 75 = 12 then x = . (A) 8 (B) 10 (C) 12 (D) 16 3. [AS1] 5% of a number is 9. The number is . (A) 120 (B) 140 (C) 160 (D) 180 4. [AS1] A number increased by 20% gives 30. The number is . (A) 150 (B) 6 (C) 25 (D) 60 EXERCISE 6.4. PERCENTAGES 109

5. [AS1] 3 : 4 into percentage is . (A) 85% (C) 75% (B) 50% (D) 25% Short Answer Type Questions 6(i) [AS1] Find the percentage decrease in the price of a shirt when it is decreased from Rs. 80 to Rs. 60. (ii) [AS1] Express 45 cm as a percentage of 3 m. 7(i) [AS3] Convert 0.73 and 1.152 to percent. (ii) [AS3] Convert 1 and 3 to percent. 2 4 Long Answer Type Questions 8 [AS1] Find the mentioned percentage of each quantity. S. Percentage Quantity Working with Value No. 320 kg fractions 32 kg a) 10% 780 m b) 5% 10 × 320 kg c) 20% 100 d) 25% e) 50% Rs 800 f) 12.5% g) 2.5% Rs 560 722 g Rs 968 400 litres 9 [AS1] An alloy contains 32% copper, 40 % zinc and the rest nickel. Find in grams the quantity of each of the contents in 1 kg of alloy. 10 [AS1] A basket contains 350 eggs. If 12% of the eggs are rotten, how many eggs are good enough to be sold? EXERCISE 6.4. PERCENTAGES 110

11 [AS1] Mr. Narayana saves 20 % of his salary. If he recieves Rs. 14500 per month as his salary, find his monthly expenditure. 12 [AS1] In an examination, 96% of the candidates passed and 50 falied. How many candidates appeared for the exam? 13 [AS1] Find the percentage of pure gold in 22 – carat gold, if 24 – carat gold is 100% pure gold. 14 [AS2] (i) After a 25% reduction, a suit costs Rs. 450. What is its original price? (ii) 40% of the population of a town are males. If the total population of the town is 2,85,000, find the number of men in the town. 15 [AS3] Complete the following tables: (i) Fraction in the 2 7 simplest form 10 50 Fraction with 20 34 85 0.63 14 100 as 100 100 100 100 denominator Decimal 0.2 0.1 0.34 Percentage 20% 85% 60% (ii) Decimal 0.3 0.76 Fraction with 30 62 90 100 as 100 100 100 denominator Fraction in the 3 4 3 simplest form 10 5 25 Percentage 30% 36% 8% 16 [AS3] Look at the figures given and write the appropriate fractions, decimals and percentages representing the shaded parts in them. EXERCISE 6.4. PERCENTAGES 111

EXERCISE 6.4. PERCENTAGES 112

EXERCISE 6.5 PROFIT AND LOSS 6.5.1 Key Concepts i. If an object is sold at a price greater than that at which it was bought, there is profit. ii. If an object is sold at a price lower than that at which it was bought, there is a loss. iii. Profit and loss are calculated as the difference of the selling price and the cost price. iv. Profit = Selling price – Cost price v. Loss = Cost price – Selling price vi. Profit% or Loss % is calculated on the cost price. vii. Profit% = Pr o f it × 100% C .P L os s viii. Loss% = C .P × 100% ix. C.P =(100S.+P × 100 Pr o f it %) x. C.P = (10S0.P− × 100 loos %) 6.5.2 Additional Questions Objective Questions 1. [AS1] A man buys a book for Rs. 80 and sells it for Rs.100. His gain % is . (A) 20% (B) 25% (C) 120% (D) 125% 2. [AS1] A football is bought for Rs.120 and sold for Rs.105. The loss% is . (A) 12 1 % (B) 14 2 % 2 7 (C) 16 2 % (D) 13 1 % 3 3 3. [AS1] On selling a product bought for Rs.100, a man gains Rs. 20. His gain% is . (A) 20% (B) 25% (C) 18% (D) 22% EXERCISE 6.5. PROFIT AND LOSS 113

4. [AS1] If CP = Rs. 950, gain = 6% then SP = . (A) Rs. 1006 (B) Rs. 1008 (C)Rs. 1007 (D)Rs. 1009 5. [AS1] If SP = Rs. 8510, loss = 8% then CP = . (A) Rs. 9000 (B) Rs. 9250 (C)Rs. 9500 (D)Rs. 9750 Short Answer Type Questions 6(i) [AS1] A fruit seller bought some oranges for Rs. 2150 and sold them for Rs. 2500. Find his gain or loss. (ii) [AS1] By selling an article for Rs. 660, Pradeep loses Rs.60. At what price must he sell the article to gain 15%? 7 [AS4] The selling price of 10 pencils is equal to the cost price of 11 pencils. Find the gain percent. 8 [AS4] James purchased an article for Rs. 2200 and sold it for a profit of 20%. At what price did James sell the article? Long Answer Type Questions 9 [AS1] Complete the table: CP SP Net profit Profit % Net loss Loss% (in Rs.) (in Rs.) (in Rs.) (in Rs.) 800 1100 2300 500 30000 1000 35000 10 50000 12 EXERCISE 6.5. PROFIT AND LOSS 114

10 [AS1] If CP = Rs. 2500 and SP = Rs. 5200, find the gain percent. 11 [AS1] By selling a rickshaw for Rs. 9240, John loses 12%. For how much should he sell it to gain 12%? 12 [AS1] Find the SP of an article whose CP is Rs. 650 and gain % is 6%. 13 [AS1] A merchant marks his goods up by 60% and then offers a discount on the marked price. If the final selling price after the discount results in the merchant making no profit or loss, what was the percentage discount offered by the merchant? 14 [AS4] Sanjay buys packets of pens containing 12 pens in a packet. He repacks them into packets containing 10 pens in a packet. He sold these packets of 10 at the same rate as he bought the packets of 12. Did he make a profit? If so, what is the percentage of his profit? 15 [AS4] A shopkeeper bought 24 chairs at the rate of Rs. 450 per chair. He sold 16 of them at the rate of Rs. 600 per chair and the remaining at the rate of Rs. 400 per chair. Find his gain or loss percent. 16 [AS4] A tricycle was purchased for Rs.1120 and sold for Rs.1260. Find the gain and gain percent. 17 [AS4] An article was bought for Rs. 400 and sold for Rs. 336. Find the gain or loss percent. 18 [AS4] If the cost price of 6 chocolates is equal to the selling price of 5 chocolates, find the gain percent. 19 [AS4] Two cows were bought at the same cost each. One was sold at a profit of 5% and the other was sold at a loss of 7%. If the actual difference of their selling prices was Rs.144, what is the cost price of each cow? 20 [AS4] Sruthi bought a sweater at a sale for 30% off the original price and another 25% off the discounted price. If the original price of the sweater was Rs. 3000, what was its final price? 21 [AS4] Javed bought a shirt on sale for 25% off the original price and another 25 % off the discounted price. If the final price was Rs.1600, what was the price before the first discount? 22 [AS4] The cost of 15 pens is equal to the selling price of 20 pens. Find the loss percent. 23 [AS4] Find the SP of a table whose CP is Rs. 3300 and is sold at a loss of 10 %. EXERCISE 6.5. PROFIT AND LOSS 115

24 [AS4] An almirah was bought for Rs. 14360 and Rs. 240 was spent on its transportation. At what price should it be sold to gain 15%? 25 [AS4] A watch when sold at a profit of 6% yields Rs. 870 more than when it is sold at a loss of 6%. Find the cost price of the watch. 26 [AS4] If a merchant offers a discount of 30% on the list price, she makes a loss of 16%. What profit % or loss% will she make if she sells at a discount of 10% of the list price? 27 [AS4] A trader marks the price of a pair of shoes at 30% above the cost price, but allows a discount of 10%. If the cost price of the pair of shoes is Rs. 800, find the profit. 28 [AS4] Raman bought two cows for Rs. 4000 each and spent Rs.1000 on their transportation. If he sells each of them at Rs. 5500, find his gain. 29 [AS4] A shopkeeper bought a skirt for Rs. 480. A girl buys it from him at a discount of 1 % on the marked price which is Rs. 600. Find the profit of the shopkeeper. 122 30 [AS4] A trader marks a table at 20% above the cost price, but allows a discount of 10%. If the CP of the table is Rs.1500, find the profit obtained. 31 [AS4] Iqbal purchased a flat for Rs. 7,56,000 and spent Rs. 44,000 on its renovation. He sold it for Rs. 8,48,000. Find his profit or loss. EXERCISE 6.5. PROFIT AND LOSS 116

EXERCISE 6.6 SIMPLE INTEREST 6.6.1 Key Concepts i. When money is borrowed, interest is charged for the use of that money for a certain period of time. ii. When the money is paid back, the principal (amount of money that was borrowed) and the interest is paid back. iii. The amount of interest depends on the interest rate, the amount of money borrowed (principal) and the time for which the money is borrowed. iv. The formula for finding simple interest is: Interest = Princi pal × Rate × T ime . 100 SI = P ×R× T or PT R 100 100 6.6.2 Additional Questions Objective Questions . 1. [AS1] A sum amounts to Rs. 3626 in 219 days at 6% per annum simple interest. The sum is (A) Rs. 3000 (B) Rs. 3200 (C)Rs. 3500 (D)Rs. 3600 2. [AS1] The time in which Rs. 6000 amounts to Rs. 6360 at 8% per annum simple interest is . (A) 9 months (B) 8 months (C) 1 1 years (D) 1 1 years 4 2 3. [AS1] The simple interest on a sum for 5 years is 3 of the sum. The rate percent per 5 annum is . (A) 8% (B) 10% (C) 12% (D) 12 1 % 2 EXERCISE 6.6. SIMPLE INTEREST 117

4. [AS1] The simple interest at x % per annum for x years is Rs. x. Then the sum is . (A) Rs. x (B) Rs. 10x (C)Rs. 100x (D) R s. 100 x 5. [AS1] The rate percent per annum simple interest for which a sum doubles itself in 10 years is . (A) 8% (B) 10% (C) 12% (D) 12 1 % 2 Short Answer Type Questions 6(i) [AS1] Mr.Chawla took a loan of Rs. 64000 from a bank at the rate of 11% p.a. for three years at simple interest. How much interest and amount will he have to pay back after three years? (ii) [AS1] Gaurav borrowed Rs. 8500 from a bank. He paid 10% per annum and returned the amount after 3 years. How much interest did Gaurav pay in all? 7(i) [AS1] In how many years will a sum of money double itself at 10% interest per annum? (ii) [AS1] Suhail borrowed Rs. 12000 from his friend, who gave Rs. 2000 at 18% and the remaining amount at 15%. If he paid an interest of Rs. 5580, in how many years did he repay the loan? Long Answer Type Questions 8 [AS1] Calculate the interest and amount in the following cases: Principal in Rs. Rate p.a. Time Interest Amount (P) (R) (T) (I) (A) (a) 700 6% 2 years (b) 1000 8% 3 years (c) 10000 15% 2 years EXERCISE 6.6. SIMPLE INTEREST 118

9 [AS1] At what rate per cent per annum will Rs.7000 yield Rs.350 as simple interest in 2 years? 10 [AS1] What sum will amount to Rs. 1368 in three and a half years at the rate of 4% per annum? 11 [AS1] A sum of money lent out at a simple interest amount to Rs. 2200 in one year and Rs. 2800 in 4 years. Find the sum of money and the rate of interest. 12 [AS1] What sum will amount to Rs. 5525 at 10% per annum simple interest in 3 years? 13 [AS1] Find the simple interest on Rs. 4500 at 8% per annum for 73 days. Also find the amount. 14 [AS1] The simple interest on a certain sum is 16 of the sum. Find the rate percentage and 25 time, if both are numerically equal. 15 [AS1] The simple interest on a certain sum for 3 years at 8% per annum is Rs. 96 more than the simple interest on the same sum for 2 years at 9% per annum. Find the sum. 16 [AS4] At what rate percent per annum simple interest will a sum triple itself in 16 years? EXERCISE 6.6. SIMPLE INTEREST 119

CHAPTER 7 DATA HANDLING EXERCISE 7.1 ORGANISING DATA 7.1.1 Key Concepts i. Data: Information which is in the form of numbers or words and helps in taking decisions or drawing conclusions is called data. ii. Tables and graphs are the ways in which data is presented. iii. The numerical entries in the data are called ‘observations’. iv. Average or Arithmetic Mean (A.M) or Mean A.M. is equal to the sum of all the observations of a data set divided by the number of observations. It lies between the least and the highest values of the given data. 7.1.2 Additional Questions Objective Questions 1. [AS1] Each numerical figure in a data is called . (A) An array (B) A frequency (C) An observation (D) A statistics 2. [AS1] The mean of the first five natural numbers is . (A) 15 (B) 7.5 (C) 30 (D) 3 3. [AS1] The mean of the first five prime numbers is . (A) 2.5 (B) 5.6 (C) 17.5 (D) 28 EXERCISE 7.1. ORGANISING DATA 120

4. [AS1] The mean of the first six multiples of 5 is . (A) 2.5 (B) 12.5 (C) 17.5 (D) 17 5. [AS3] The subject that deals with the collection, presentation, analysis and interpretation of numerical data is . (A) Algebra (B) Geometry (C) Statistics (D) Arithmetic Very Short Answer Type Questions 6 [AS1] Answer the following questions in one sentence. (i) Find the arithmetic mean of 3, 0, –1, 7 and 11. (ii) The mean age of 5 children of a family is 12 years. If four of them are respectively 6, 1, 13 and 16 years of age, find the age of the fifth child. (iii) The time (in minutes) taken by various students in a test are 18, 20, 21, 23, 25, 25, 29, 31, 31, 37. Find the mean time (in minutes) taken by the students to complete the test. (iv) What is the mean of 20 and 30? (v) Find the mean of the first six prime numbers. 7 [AS1] Fill in the blanks. (i) The difference between the highest and the lowest values of the observations is known as . (ii) The mean of 6, 12, 15, 25, 30 and 35 is . (iii) If the mean of 6, 8, 5, x and 4 is 7, then the value of ’x’ is . (iv) 12 numbers have a mean 182. Their sum is . (v) The mean of 25, 30, 15, 22, 20, 12, 26, 20, 25, and x is 21. Then x = . EXERCISE 7.1. ORGANISING DATA 121

8 [AS1] Fill in the blanks. figures. (i) Data means information in the form of (ii) Data obtained in the form is called raw data. (iii) Numerical figures arranged in ascending or descending orders is called an . . (iv) The number of times a particular observation occurs is called its (v) Arranging the data in the form of a table is called of data. Short Answer Type Questions 9(i) [AS1] Find the mean of the first six odd natural numbers. (ii) [AS1] Find the mean: 7.6, 6.8, 8.5, 9.4, 5.9, 6.4, 9.1, 4.7 10(i) [AS4] Write the data given in ascending order. 7, 8, 7, 10, 6, 9, 7, 10, 5, 7, 6, 8, 5, 6, 7, 8, 9, 7, 6, 7, 8. (ii) [AS4] Given are the heights (in cm) of 16 students in a class. 154, 150, 152, 154, 154, 150, 148, 152, 152, 152, 154, 150, 152, 154, 152, 152 Arrange the data in ascending order and then prepare a frequency table. EXERCISE 7.1. ORGANISING DATA 122

EXERCISE 7.2 MODE 7.2.1 Key Concepts i. Mode: An observation of data that occurs most frequently is called the mode of the data. A data may have one or more modes and sometimes none. 7.2.2 Additional Questions Objective Questions . 1. [AS1] The mode of 3, 5, 1, 2, 4, 6, 0, 2, 2, 3 is (A) 1 (B) 2 (C) 4 (D) 6 2. [AS1] The mode of first five natural numbers is . (A) 0 (B) 1 (C) 5 (D) None of these 3. [AS1] The mode of the data 10, 8, 4, 7, 8, 11, 15, 8, 6, 8 is . (A) 10 (B) 8 (C) 4 (D) 7 4. [AS1] The mode of the data 27, 23, 39, 18, 27, 21, 27, 40, 36, 27 is . (A) 23 (B) 39 (C) 27 (D) 18 EXERCISE 7.2. MODE 123

5. [AS1] If the mode of the data 3, 4, p, 3, 8, 5 and 4 is 4 then 'p' = . (A) 3 (B) 4 (C) 8 (D) 0 Very Short Answer Type Questions 6 [AS1] Answer the following questions in one sentence. (i) The number of pairs of shoes of different sizes sold in a day in a shop is given in the table. Size of shoe 12345678 9 Number of pairs sold 1223453 7 2 What is the modal shoe –size? (ii) What is the observation which occurs most frequently in a data called? (iii) The marks obtained by 11 students of a class in a test are as given: 21, 15, 23, 21, 23, 18, 23, 18, 12, 20, 23 Find their modal marks. (iv) The weights (in kg) of 10 students of class 7 are 33, 28, 29, 26, 24, 32, 26, 32, 29 and 26. Find their modal weight. (v) The ages (in years ) of 11 cricket players are as given: 28, 34, 32, 41, 36, 32, 32, 38, 32, 40, 31 Find their modal age. Short Answer Type Questions 7(i) [AS1] a) Find the mode of the following years of experience of teachers in a school. 10, 12, 5, 4, 7, 6, 7, 4, 3, 2, 7, 1, 2, 4, 10, 1, 7, 6, 3, 4 b) Find the mode of the given set of numbers: 2, 3, 5, 3, 4, 3, 2, 2, 3, 2, 4, 2, 2 EXERCISE 7.2. MODE 124

(ii) [AS1] The following are the marks obtained by 80 students in a unit test, which is given for 10 marks. Find the mode of the data. Marks obtained 0 1 2 3 4 5 6 7 8 9 10 Total No .of 3 2 3 4 10 0 12 8 8 20 10 80 students EXERCISE 7.2. MODE 125

EXERCISE 7.3 MEDIAN 7.3.1 Key Concepts i. Median: Median is simply the middlemost observation, when all the observations are arranged in ascending or descending order. ii. In case of even number of observations, median is the average of the two middlemost observations. 7.3.2 Additional Questions Objective Questions 1. [AS3] After arranging the given data in ascending or descending order of magnitude, the value of the middle –most observation is called the of the data. (A) Median (B) Mean (C) Mode (D) Class 2. [AS3] If the number of observations ‘n’ is odd, then median is the term. (A) n (B) n − 1 2 2 (C) n + 1 (D) n2 2 2 3. [AS3] If the number of observations ‘n’ is even, then median is the mean of term and term. (A) n , n − 1 2 2 (B) n , n + 1 2 2 (C) n , n + 1 2 2 (D)None of these EXERCISE 7.3. MEDIAN 126

4. [AS1] The median of 3, 11, 7, 2, 5, 9, 9, 2, 10 is . (A) 9 (B) 7 (C) 10 (D) 11 5. [AS1] The median of 10, 32, 17, 19, 21, 22, 9, 35 is . (A) 40 (B) 20 (C) 30 (D) 10 Short Answer Type Questions 6(i) [AS1] The runs scored by 11 members of a cricket team are 25, 39, 53, 18, 65, 72, 0, 46, 31, 08 and 34. Find their median score. (ii) [AS1] The weight (in kg) of 10 students are 40, 52, 34, 47, 31, 35, 48, 41, 44 and 38. Find their median weight. 7(i) [AS2] How many distinct sets of three positive integers have a mean of 6, a median of 7 and no mode? (ii) [AS2] Write the differences among mean, median and mode. Long Answer Type Questions 8 [AS1] (i) What is the median weekly salary of workers in a farm whose salaries are Rs. 480, Rs. 600, Rs. 550, Rs. 500, Rs. 450, Rs. 480, Rs. 360, Rs. 650 and Rs. 700? (ii) Find the median salary of the following salaries of workers: Rs. 560, Rs. 850, Rs. 120, Rs. 360, Rs. 700, Rs. 840, Rs. 600. EXERCISE 7.3. MEDIAN 127

EXERCISE 7.4 PRESENTATION OF DATA 7.4.1 Key Concepts i. Pictographs represent data using pictures of objects. However, presenting data by a pictograph is often time consuming and difficult. ii. Bar graphs help in presenting data with much more ease. 7.4.2 Additional Questions Objective Questions 1. [AS3] A pictorial representation of numerical data in the form of rectangles is known as a . (A) Bar graph (B) Pictograph (C) Pie chart (D) Frequency table 2. [AS3] In a bar graph, the widths of all the rectangles are . (A) The same (B) Not the same (C) Varying (D) Cannot be said 3. [AS3] In a bar graph, the heights of the rectangles are . (A) The same (B) Varying (C)100 units (D) Cannot be said 4. [AS3] A graphical representation of data using bars of different heights is known as . (A) A histogram (B) A bar graph (C)A pie chart (D) None of these EXERCISE 7.4. PRESENTATION OF DATA 128

5. [AS3] In a pie chart, the angle around its centre is . (A) 90◦ (B) 0◦ (C) 180◦ (D) 360◦ Very Short Answer Type Questions 6 [AS4] Choose the correct answer. Read the following graph and answer the questions given. (i) The subject in which the student is the most proficient is . (A) English (B) Mathematics (C) Science (D) History (ii) The subject in which the student is the least proficient is . (A) English (B) Mathematics (C) Science (D) History (iii) The average marks obtained by the student is . (A) 57 (B) 63 (C) 80 (D) 48 EXERCISE 7.4. PRESENTATION OF DATA 129

(iv) The percentage obtained by the student is . (A) 80% (B) 63% (C) 57% (D) 90% (v) The ratio of the highest marks to the lowest marks obtained by the student is . (A) 2 : 11 (B) 9 : 2 (C)2 : 9 (D)11 : 2 7 [AS4] Choose the correct answer. Read the bar graph given and answer the following questions. (i) The state that is the largest producer of rice is . (A) U.P (B) W.B (C) M.P (D) Haryana (ii) The state that is the least producer of wheat is . (A) Maharashtra (B) W.B (C) M.P (D) Haryana (iii) The state that is the largest producer of wheat is . (A) M.P (B) Haryana (C) Maharashtra (D) U.P EXERCISE 7.4. PRESENTATION OF DATA 130

(iv) The state in which the total production of rice and wheat is the least is . (A) W.B (B) M.P (C) Maharashtra (D) Haryana (v) The state that is the least producer of rice is . (A) M.P (B) Haryana (C) W.B (D) U.P Short Answer Type Questions 8 [AS4] The following table shows the increase in population in a village between 2003 and 2013. Years 2003 2005 2007 2009 2011 2013 Population (in 86 78 82 85 94 100 thousands) (i) In the given data, are any two values equal? (ii) When was the population 85,000? (iii) What was the population in 2011? 9(i) [AS4] The graph given compares the heights and weights of various animals. EXERCISE 7.4. PRESENTATION OF DATA 131

a) The range of height is around cm. b) The weight of the ostrich is around kg. c) The ratio of the height to weight of a lion is about . (ii) [AS4] The following graph shows the points scored in five games in basketball. Study the graph and answer the questions that follow. a) How many games did the home team win? b) What is the highest score? c) What is the highest score of opponent’s team? 10 [AS5] The marks scored by Sunil in his SA2 are given in the graph. Study the graph and answer the following questions. EXERCISE 7.4. PRESENTATION OF DATA 132

(i) Sunil scored the highest marks in . (ii) The lowest mark is scored in . (iii) The second highest mark is scored in . Long Answer Type Questions 11 [AS4] In a survey, some students were asked about their favourite leisure activity. Their answers were used to draw this pie chart. Based on the pie–chart answer the following questions: 133 (i) Which is the most favourite leisure activity? (ii) Which is the least favourite leisure activity? EXERCISE 7.4. PRESENTATION OF DATA

12 [AS5] Consider this data collected from a survey of a colony regarding their involvement in different sports. Draw a double bar graph choosing an appropriate scale. Name of Cricket Basket Swimming Hockey Athletics the 1220 Ball 500 400 250 Sports 480 Number 610 320 250 100 of people 320 who just watch Number of people who participate 13 [AS5] The number of students in each grade of a test are A: 9, B: 12, C: 10 and D: 5. Draw a pie chart for this data. EXERCISE 7.4. PRESENTATION OF DATA 134

CHAPTER 8 CONGRUENCY OF TRIANGLES EXERCISE 8.1 CONGRUENCE OF LINE SEGMENTS 8.1.1 Key Concepts i. Two figures are said to be congruent if they are identical in shape and equal in size. ii. Two line segments are congruent if they have the same lengths. PQ RS AB CD iii. Two triangles are congruent if the corresponding angles and corresponding sides are equal. iv. We establish the congruency of the triangles by following some criteria. S.S.S. criterion: If three sides of a triangle are equal to the corresponding three sides of another triangle, then the triangles are congruent. FA = T I ; AN = IN ; FN = T N then ∆FAN △T IN 8.1.2 Additional Questions Objective Questions 1. [AS3] Two plane figures are said to be congruent, if they have the ________. (A) Same shape (B) Same size (C)Same shape and same size (D) Different size EXERCISE 8.1. CONGRUENCE OF LINE SEGMENTS 135

2. [AS3] Two congruent line segments are always______________. (A) Equal in length (B) Perpendicular to each other (C) Parallel to each other (D) Different in length 3. [AS3] Two congruent angles ________. (B) Are complementary (A) Have equal measure (D) Have different measures (C) Are supplementary 4. [AS3] Two circles are said to be congruent if they have _____________. (A) The same centre (B) The same radius (C) Different radii (D) Different diameters 5. [AS2] If △ABC △DEF, the true statement among the following is_______. (A) AC = DE (B) AB = DE (C)BC = EF (D)Both (B) and (C) Short Answer Type Questions 6(i) [AS1] In the given figure, triangles PQR and EDC are congruent to each other by the SSS criterion, write the measures of the corresponding congruent sides. EXERCISE 8.1. CONGRUENCE OF LINE SEGMENTS 136

(ii) [AS1] State the correspondence between the angles and sides in the following cases. a) △ABC △RPQ b) △ABC △PQR c) △ABC △QPR 7 [AS2] If △XYZ △DEF by SSS congruence condition and DE = 7.5 cm, then find the length of XY. 8(i) [AS2] PQR is isosceles with PQ = PR. S is the midpoint of QR. PQS PRS. State the condition of congruency satisfied by the two triangles. Also state why ∠Q = ∠R. (ii) [AS1] Given PR = QS, is PQ = RS? Why? 9(i) [AS1] In the figure, ABC and DBC are on the same base BC. Also, AB = DC and AC = DB. Is ABC DCB? Long Answer Type Questions 10 [AS2] ABCD is a rectangle and AC is one of its diagonals. Are the two triangles formed congruent? If yes, prove it. EXERCISE 8.1. CONGRUENCE OF LINE SEGMENTS 137

EXERCISE 8.2 SIDE-ANGLE-SIDE CONGRUENCY (SAS) 8.2.1 Key Concepts i. S.A.S. criterion: If two sides and an included angle of a triangle are equal to the corresponding twosides and the included angle of another triangle, then the two triangles are congruent. In △CAT and △PIG, CA = PI; ∠C = ∠P; CT = PG, then ∆CAT ∆PIG. 8.2.2 Additional Questions Objective Questions 1. [AS3] Two triangles are congruent if two sides and the included angle between them in one of the triangles are equal to corresponding sides and the angle included between them of the other triangle. This is the property of ________ congruence. (A) SAS (B) RHS (C) AAA (D) ASA 2. [AS2] If △DEF △BCA, then ___________. (A) ∠D = ∠A (B) ∠E = ∠A (C)∠F = ∠C (D)∠E = ∠C 3. [AS1] In △PQR, PQ = QR and ∠Q is twice that of ∠P. Then ∠Q = _________. (A) 90◦ (B) 36◦ (C) 144◦ (D) 108◦ EXERCISE 8.2. SIDE –ANGLE–SIDE CONGRUENCY (SAS) 138

4. [AS1] The value of ∠XYZ in the given figure is _________. (A) 30◦ (B) 120◦ (C) 90◦ (D) 60◦ 5. [AS2] ABC is an isoceles triangle with AB = AC and AD is the altitude. Then _________. (A) ∠B > ∠C (B) ∠B < ∠C (C)∠B = ∠C (D)∠B ∠C Short Answer Type Questions 6(i) [AS1] Study the figures given and name the pairs of congruent elements using the SAS congruence. EXERCISE 8.2. SIDE –ANGLE–SIDE CONGRUENCY (SAS) 139

(ii) [AS2] △ABC is isosceles where AB = AC and AD = AE. Prove that BE = CD. Long Answer Type Questions 7 [AS2] In the following figure, AB and CD bisect each other at P. Show that (i) △APC △BPD. (ii) AC BD. (iii) AC BD. EXERCISE 8.2. SIDE –ANGLE–SIDE CONGRUENCY (SAS) 140

EXERCISE 8.3 ANGLE-SIDE-ANGLE CONGRUENCY (ASA) 8.3.1 Key Concepts i. A.S.A. Criterion: If two angles and the included side of a triangle are equal to the corresponding two angles and the included side of another triangle then the triangles are congruent. In ∆MAT and ∆D I M, ∠A = ∠I; AT = I M; ∠T = ∠M. Hence, ∆MAT ∆D IM. 8.3.2 Additional Questions Objective Questions 1. [AS3] Two triangles are congruent if two angles and the side included between them in one of the triangles are equal to the corresponding angles and the side included between them of the other triangle. This is the property of ___________ congruence. (A) SAS (B) RHS (C) AAA (D) ASA 2. [AS3] Two triangles cannot be proved to be congruent by _______property. (A) SAS (B) SSS (C) ASA (D) AAA 3. [AS2] If ∠A = ∠R, ∠B = ∠P, AB = RP then the congruence condition by which, △ABC △RPQ is ___________. (A) SAS (B) ASA (C) SSS (D) RHS EXERCISE 8.3. ANGLE –SIDE –ANGLE CONGRUENCY (ASA) 141

4. [AS3] In PQR, the side included between ∠Q and ∠R is __________. (A) PQ (B) QR (C) PR (D)None of these 5. [AS3] In △PQR, the angle included between QR and RP is________. (A) ∠P (B) ∠Q (C) ∠R (D)None of these Short Answer Type Questions 6(i) [AS5] The following pairs of triangles are congruent to each other by the ASA congruency. Fill in the blanks. a) ∠PRS = 142 b) RS = EXERCISE 8.3. ANGLE –SIDE –ANGLE CONGRUENCY (ASA)

(ii) [AS5] Fill in the correct measures. a) ∠PMS = b) MS = c) ∠MS P = Long Answer Type Questions 7 [AS1] Show that △ABC △DCB if △EBC is an isosceles triangle in the following figure. EXERCISE 8.3. ANGLE –SIDE –ANGLE CONGRUENCY (ASA) 143

8 [AS5] Name the pairs of congruent parts using the ASA congruence in the following pairs of triangles. (i) (ii) EXERCISE 8.3. ANGLE –SIDE –ANGLE CONGRUENCY (ASA) 144

EXERCISE 8.4 RIGHT –ANGLE HYPOTENUSE SIDE CONGRUENCY (RHS) 8.4.1 Key Concepts i. R.H.S. criterion: In two right angled triangles, if the hypotenuse and a side of one triangle are correspondingly equal to the right angle and the corresponding side of another triangle, then the triangles are congruent. In ∆ABC and ∆DEF, ∠B = ∠E = 90◦ BC = EF (Hypotenuse) AC = DF (Side) Hence, ∆ABC ∆DEF. 8.4.2 Additional Questions Objective Questions 1. [AS2] In △ABC and △XYZ, ∠B = ∠X = 90◦ and BC = XZ. The additional information required to prove △ABC △YXZ by RHS congruence condition is ___________. (A) AB = XY (B) ∠A = ∠Y (C)∠C = ∠Z (D) AC = YZ 2. [AS3] In ABC, AD ⊥ BC, ∠B = ∠C. The property by which ADB ADC is ____________. (A) SAS (B) SSS (C) RHS (D) ASA EXERCISE 8.4. RIGHT-ANGLE HYPOTENUSE SIDE CONGRUENCY (RHS) 145

3. [AS3] Two right triangles are congruent if the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and one side of the other triangle This is the ___________ congruence condition. (A) SAS (B) RHS (C) SSS (D) ASA 4. [AS2] If BCD CBE by RHS congruence condition, then _____________ . (A) BC = AC (B) BD = CE (C)AB = CD (D)AE = AD 5. [AS3] Areas of two congruent squares . (A) Are equal (B) Are not equal (D)Cannot be determined (C) Are incomplete Short Answer Type Questions 6(i) [AS2] ABC is isosceles with AB = AC and AD ⊥ BC. Prove that ∠B = ∠C. EXERCISE 8.4. RIGHT –ANGLE HYPOTENUSE SIDE CONGRUENCY (RHS) 146