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Page 3 : MATHEMATICS / X / 2021 – 22, , Preface, This Study Material is an in-house academic exercise undertaken by, the Maths teachers of KVS Ernakulam Region under the supervision of, a subject expert, Shri K P Sudhakaran, Principal, KV Peringome, to, provide the students a comprehensive, yet concise, support tool for, consolidation of learning., It consists of 7-chapters in capsule form with the gist of the lesson and, questions in VSA, SA and LA forms. This material is developed, keeping in mind the latest CBSE curriculum and pattern of the question, paper. It will definitely provide the students a valuable window on, precise information and it covers all essential components that are, required for effective revision of the subject., Hoping this material will prove to be a helpful tool for quick revision, and will serve the purpose of enhancing students’ confidence level to, help them perform better., Best of Luck., , 3

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Page 6 : MATHEMATICS / X / 2021 – 22, , CONTENT INDEX, S No, , CONTENT, , 1, , RATIONALISED CURRICULUM, , 2, , UNIT 1 - ALGEBRA, , 3, , 4, , QUADRATIC EQUATIONS, , 8, , , , ARITHMETIC PROGRESSIONS, , 15, , UNIT 2 - GEOMETRY, , , CIRCLES, , 26, , , , CONSTRUCTIONS, , 41, , UNIT 3 - TRIGONOMETRY, SOME APPLICATIONS OF TRIGONOMETRY, , 43, , UNIT 4 - MENSURATION, , , 6, , 7, , , , , 5, , PAGE NO, , SURFACEAREA AND VOLUME, , 53, , UNIT 5 - STATISTICS AND PROBABILITY, , , STATISTICS, , 63, , 7, , CBSE SAMPLE QUESTION PAPER- BASIC, , 73, , 8, , MARKING SCHEME- BASIC, , 77, , 9, , CBSE SAMPLE PAPER - STANDARD, , 83, , 10, , MARKING SCHEME - STANDARD, , 87, , 11, , PRACTICE PAPER 1 - BASIC, , 91, , 12, , MARKING SCHEME, , 94, , 13, , PRACTICE PAPERS 2 - BASIC, , 98, , 14, , MARKING SCHEME, , 101, , 15, , PRACTICE PAPERS 1 - STANDARD, , 108, , 16, , MARKING SCHEME, , 114, , 19, , PRACTICE PAPER 2- STANDARD, , 124, , 20, , MARKING SCHEME, , 128, , 6

Page 7 : MATHEMATICS / X / 2021 – 22, , MATHEMATICS (CODE NO. 041), RATIONALISED CURRICULUM (2021-22), UNIT, 1, , 2, , TERM 1, ALGEBRA, QUADRATIC EQUATIONS, , , , ARITHMETIC PROGRESSIONS, , GEOMETRY, CIRCLES, CONSTRUCTIONS, , 7, , SOME APPLICATIONS OF, TRIGONOMETRY, , MENSURATION, , , 8, , 9, , TRIGONOMETRY, , , 4, , 10, , , , , , 3, , WEIGHTAGE, , 6, , SURFACE AREA AND VOLUME, , STATISTICS & PROBABILITY, , , 8, , STATISTICS, , TOTAL, , 40, , INTERNAL ASSESSMENT, , 10, , TOTAL, , 50, , INTERNAL ASSESSMENT, INTERNAL ASSESSMENT, , TERM 1, , PERIODIC TESTS, , 3, , MULTIPLE ASSESSMENT, , 2, , PORTFOLIO, , 2, , ENRICHMENT ACTIVITIES, , 3, , TOTAL MARKS, , 10 MARKS FOR THE TERM, , 7

Page 8 : MATHEMATICS / X / 2021 – 22, , ALGEBRA, QUADRATIC EQUATIONS, IMPORTANT FORMULAS & CONCEPTS, Quadratic Polynomial, A polynomial of the form ax² + bx + c, where a, b and c are real numbers and a≠0 is called a, quadratic polynomial., The standard form of a Quadratic Equation, The standard form of a quadratic equation is ax² + bx + c = 0, where a, b and c are real numbers, and a≠0.‘a’ is the coefficient of x² . ‘b’ is the coefficient of x and ‘c’ is the constant term., Roots of a Quadratic equation, The values of x for which a quadratic equation is satisfied are called the roots of the quadratic, equation., If α is a root of the quadratic equation ax² + bx + c = 0, then aα² +bα+c=0., A quadratic equation can have two distinct real roots, two equal roots or real roots may not, exist., Methods of solving a Quadratic Equation, 1., , Factorization method, In this method, factorisation can be done using splitting the middle term, , 2. Using Quadratic Formula, Quadratic Formula is used to obtain the roots of a quadratic equation directly from the, standard form of the equation., Quadratic formula: The roots of a quadratic equation ax² + bx + c = 0 are given, provided b² – 4ac ≥ 0., Here, the value b² – 4ac is known as the discriminant and is generally denoted by D. The value, of discriminant helps us to determine the nature of roots for a given quadratic equation. The, rules are:, 1. If D = 0 ⇒ The roots are Real and Equal., 2. If D > 0 ⇒ The two roots are Real and distinct., 3. If D < 0 ⇒ No Real roots exist., 8

Page 9 : MATHEMATICS / X / 2021 – 22, , SHORT ANSWER TYPE QUESTIONS, SECTION - A (2 MARK QUESTIONS), Q1., , For what value of p for equation 2 x² – 4x + p = 0 will have real roots?, , Q2., , One year ago, a man was 8 times as old as his son. Now his age is equal to the square, of his son’s age. Then find their present ages, , Q3., , Find the sum of the roots of the quadratic equation 3 x² – 9x + 5 = 0?, , Q4., , If, , Q5., , A natural number, when increased by 12, equals 160 times its reciprocal. Find the, number?, , Q6., , If the one root of the equation 4x2 – 2x + p – 4 = 0 be the reciprocal of other, find the, value of p, , Q7., , is a root of the equation x² + kx – = 0 then find the value of k?, , Find the roots of the following quadratic equations by factorisation:, , x 2-3x-10=0, , Q8., , Find two numbers whose sum is 27 and product is 182., , Q9., , The sum of the squares of two consecutive natural numbers is 313, then find the, numbers, , Q10., , Write the quadratic equation whose one root is 3 + √2, , Q11., , If -5 is a root of the quadratic equation 2x2 + px -15 = 0 and the quadratic equation, p (x2 + x) + k = 0 has equal roots, find the value of k., , Q12., , For what value of p, the equation px2 + 6x + 4p= 0 has product of roots equal to sum, of roots?, , SHORT ANSWER TYPE QUESTIONS, Section B- 3 Mark questions, Q1., , Find two consecutive positive integers, the sum of whose squares is 365., , Q2., , If 2 is a root of the equation x2 + bx +12 =0, find the value of ‘b’ and find the other, root., , Q3., , Find the nature of roots of equation 9x2 + 12x + 4 = 0, , Q4., , Determine the discriminant of the equation: 2x2 – 7x + 3 = 0, , Q5., , Find two numbers whose sum is 27 and product is 182., , Q6., , Solve: x +, , Q7., , Solve by factorization: 9x2 – 3x – 20 = 0, , Q8., , Find k, if 2kx2 + 6x + 5 = 0 has equal roots, , = 3 (x ≠ 0), , 9

Page 10 : MATHEMATICS / X / 2021 – 22, , 3x2 - 2√6 x + 2 = 0, , Q9., , Find the roots of the quadratic equation:, , Q10., , The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm,, find the other two sides., , Q11., , Find the value of p, for which one root of the quadratic equation px 2 – 14x + 8 = 0 is 6, times the other, , Q12., , Find the value of k, for which the quadratic equation (k- 12) x 2 + 2 (k-12) x + 2 =, 0 has equal roots, , Q13., , The sum of a number and its reciprocal is, , . Find the number., , Q14., , Find the discriminant of the equation 3x2– 2x + = 0 and hence write the nature of its, roots. Find them, if they are real., , Q15., , Three consecutive natural numbers are such that the square of the middle number, exceeds the difference of the squares of the other two by 60. Find the numbers, , LONG ANSWER TYPE QUESTIONS), SECTION - C (4 MARKS QUESTIONS), Q1., , If -5 is a root of the quadratic equation 2x2 + px – 15 = 0 and the quadratic equation, p (x2 + x) + k = 0 has equal roots, find the value of k., , Q2., , Solve the following quadratic equation for x: 4x2 + 4bx – (a2 – b2) = 0, , Q3., , The sum of the areas of two squares is 468 m2. If the difference of their perimeters is, 24 m, find the sides of the two squares., , Q4., , A train travels 180 km at a uniform speed. If the speed had been 9 km/ hour more, it, would have taken 1 hour less for the same journey. Find the speed of the train., , Q5., , In a flight of 600 km, an aircraft was slowed due to bad weather. Its average speed, for the trip was reduced by 200 km/hr and the time of flight increased by 30 minutes., Find the original duration of the flight., , Q6., , The difference of squares of two numbers is 180. The square of the smaller number, is 8 times the larger number. Find the two numbers., , Q7., , The sum of reciprocals of Rehman’s ages (in years) 3 years ago and 5 years from, now is, , Q8., , . Find his present age., , A motor boat whose speed is 24 km/h in still water takes 1 hour more to go 32 km, upstream than to return downstream to the same spot. Find the speed of the stream., , 10

Page 11 : MATHEMATICS / X / 2021 – 22, , Q9., , The diagonal of a rectangular field is 16 metres more than the shorter side. If the, longer side is 14 metres more than the shorter side, then find the lengths of the sides, of the field., , Q10., , Solve the following quadratic equation for x: 9x2 – 6b2 x – (a4 – b4) = 0, , CASE STUDY QUESTIONS, CASE STUDY 1:, Nidhi and Ria are very close friends. Nidhi’s parents own a Maruti Alto. Ria’s parents own, a Toyota Liva. Both the families decide to go for a picnic to Somnath temple in Gujrat by their, own cars. Nidhi’s car travels x km/hr while Ria’s car travels 5 km/hr more than Nidhi’s car., Nidhi’s car took 4 hours more than Ria’s car in covering 400km., , 1. What will be the distance covered by Ria’s car in two hours? What is the speed of, Nidhi’s car?, 2. What is the speed of Ria’s car? How much time Ria took to travel 400 km, CASE STUDY 2, An Auditorium was booked for School Annual Day Celebrations and the, seats are arranged in a particular manner. The number of rows are equal to the number of seats, in each row. When the number of rows was doubled and the number of seats in each row was, reduced by 10, the total number of seats increased by 300., , 11

Page 12 : MATHEMATICS / X / 2021 – 22, , Based on the above information answer the following questions, 1., , If x is taken as number of row in original arrangement which quadratic equation, describe the situation? How many number of rows are there in the original, arrangement?, , 2. How many seats are there in the auditorium in original arrangement? How many seats, are there in the auditorium after re-arrangement?, CASE STUDY 3, Some students planned a picnic to Wayanad as a part of their Scout and guide activities. The, total budget for picnic was Rs.2000 for each student. But 5 students failed to attend the picnic, and thus the contribution for each student is increased by Rs.20.The details of other, expenditures are given in the table below, , Article, , Cost per, student, , Entry ticket, , Coffee, , Food, , 5, , 10, , 25, , Travelling, Cost, , 50, , Ice Cream, , 15, , 1. If x is the number of students planned for picnic, write the correct quadratic equation that describe, the situation? What is the number of students planned for picnic?, 2. What is the number of students who attended the picnic? What is the total expense for this picnic?, , 12

Page 13 : MATHEMATICS / X / 2021 – 22, , Answer Key, , Short answer (2 Marks), SECTION A, Question, , Answer, , Question, , Answer, , 1, , 𝑃≤2, , 7, , x = 5 and 8, , 7 years, 49 years, , 8, , Numbers are 14 and, 13, , 3, , 3, , 9, , 12 & 13, , 4, , K=2, , 10, , 𝑥 − 6𝑥 + 7 = 0, , 8, , 11, , P=8, , 12, , 2, , 5, 6, , 𝑘=, , 7, 4, , 𝑝=, , −3, 2, , Section – B Short Answer (3 Marks), Question, , Answer, , Question, , 13 and 14, , 9, , 1, , Answer, , 𝑥=, , 2, ,𝑥 =, 3, , 2, 3, , b=-8, other root 6 10, , Other two sides are, 5cm and 12cm, , 3, , Two equal roots, , 11, , p=3, , 4, , 25, , 12, , k = 14, , 5, , 13 and 14, , 13, , The number is 4, , 2, , 6, , 3 + √5 3 − √5, ,, 2, 2, , 14, , D = 0, Equal roots 𝑇ℎ𝑒𝑦 𝑎𝑟𝑒 ,, , 7, , 𝑥=, , 5, 4, 𝑎𝑛𝑑 −, 3, 3, , 15, , The numbers 9, 10, 11, , 8, , 9, 10, , 13

Page 14 : MATHEMATICS / X / 2021 – 22, , SECTION C Long answer Type Questions, 1, , P = 7 and k =, , 6, , 18 & 12 OR 18 & -12, , 2, , −𝑏 + 𝑎 −𝑏 − 𝑎, ,, 2, 2, , 7, , 7 years, , 3, , 18m and 12m, , 8, , 8 km/hr, , 4, , 36 km/ hr, , 9, , 10 m and 24 m, , 1 hour, , 10, , 𝑏 +𝑎 𝑏 −𝑎, ,, 3, 3, , 5, , CASE STUDY BASED QUESTIONS, CASE STUDY 1, , CASE STUDY 2, , CASE STUDY 3, , 1. a) 2(x+5)km, , 1. a) 𝑎𝑥 − 20𝑥 − 300 = 0, , 1. a) 𝑎𝑥 − 5𝑥 − 500 = 0, , b) 20km/hr, 2. a) 25 km/hr, b) 16 hours, , b) 30, 2. a) 900, b) 1200, , b) 25, 2. a) 20, b) Rs 2100, , ******************************************************, , 14

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Page 16 : MATHEMATICS / X / 2021 – 22, , SHORT ANSWER TYPE QUESTIONS, SECTION A (2 Mark Questions), Q1. How many two digits numbers are divisible by 3?, Q2. In an AP, the common difference is -4, the seventh term is 4, and then find the first, term?, Q3. Which term of AP 8,14,20,26.....will be 72 more than its 41st term?, Q4. Write the nth term of AP, , ,, , ,, , , ..............?, , Q5. Find the middle term of 6, 13, 20........ 216?, Q6. The 8th term of an AP is zero. Then find its 38th term?, Q7. Find the sum of all two-digit positive odd numbers?, Q8. Three numbers are in AP and their sum is 21, find the middle number?, Q9., , If 7 times the 7th term of an AP is equal to 11times its 11th term. Then find its 18th, term., , Q10. The consecutive terms of an AP are 2, x, 26, find the value of x?, Q11. For what value of p is 2p+1, 13, 5p-3, are 3 consecutive terms of an AP?, Q12. Which term of the AP: 3, 8, 13, 18 ... is 78?, Q13. Write the 5th term from the end of the AP 3,5,7,9 ....... 201?, Q14. From the given AP: 8, 10, 12... Find the sum of its last 10 terms if it has 60 terms?, Q15. Find the number of terms of an AP 5, 9, 13 ...185?, Q16. If an AP has 8 as the first term, -5 as the common difference and its first 3 terms are 8,, A, B, then find (A+B)?, Q17. Find the 21st term of an AP whose first two terms are -3 and 4?, Q18. If an =5-11n, then find its common difference?, Q19. How many terms of AP 18, 16, 4, ...? should be taken, so that their sum is 0?, Q20. In an AP, if a=3.5, d=0, n=101, then find the value of an?, Q21. Which term of the following AP 27, 24, 21............is zero?, Q22. Find the 10th term of the sequence√2, √8 , √18, … … … . ?, Q23. Find the common difference of the AP ,, , ,, , Q24. Which term of the AP 12, 7, 2, -3 is -98?, Q25. Find the value of 𝑥 𝑓𝑜𝑟 𝑤ℎ𝑐ℎ 𝑥 + 2 , 2𝑥 ,2𝑥 + 3 are three consecutive terms of an AP?, , 16

Page 17 : MATHEMATICS / X / 2021 – 22, , SHORT ANSWER QUESTIONS, SECTION B (3 Mark Questions), Q1. If the 3rd and 9th term of an AP are 4 and -8 respectively, then which term of this AP is zero?, Q2. Find the 25th term of an AP. -5, -5/2,0,5/2, …., Q3. The first three terms of an A.P are 3y-1, 3y +5 and 5y +1 respectively then find y., Q4. The fifth term of an A. P is 20 and the sum of its seventh and eleventh terms is 64., Find the common difference., Q5. Find whether 100 is a term of the A P 20,28,36 …, Q6. How many two-digit numbers are divisible by 7?, Q7. If the ratio of the sums of first n terms of two A. P’s is (7n+1): (4n+27), find the ratio, of their 𝑚 𝑡𝑒𝑟𝑚𝑠., Q8., , Find the sum of all odd numbers between 0 and 50., , Q9. If 𝑚, , term of an A.P is and 𝑛 term is , find the sum of first mn terms., , Q10. In an A.P if sum of its first n terms is 3𝑛 + 5𝑛 𝑎𝑛𝑑 𝑖𝑡𝑠 𝑘, , 𝑡𝑒𝑟𝑚 𝑖𝑠 164,find the value of 𝑘., , Q11. Find the common difference of an AP, whose first term is ½ and the 8th term is 17/6.Also ,find the, ratio of 4th term and 50th term., , Q12. How many terms of the AP 24, 21, 18, . . . must be taken so that their sum is 78?, Q13. Determine the A.P. whose 4th term is 18 and the difference of 9th term from the 15th term is 30., Q14. The sum of the first 9 terms of an A.P. is 171 and the sum of its first 24 terms is 996. Find the first, term and common difference of the A.P., , Q15. Find the number of natural numbers between 101 and 999 which are divisible by both 2 and, 5, , LONG ANSWER QUESTION, SECTION C (4 Mark Questions), Q1. In an A. P. if 𝑆 + 𝑆 =167 and 𝑆 = 235, then find the A.P., where 𝑆 denotes the sum, of the first n terms., Q2. The first and the last term of A.P. are 5 and 45 respectively. If the sum of all its terms, is 400, find its common difference., Q3. The sum of 3rd and 7th terms of an A.P. is 6 and their product is 8. Find the sum of the, first 20 terms of the A.P, Q4. If 1 + 4 + 7 + 10 ……. + x = 287 find the value of x., , 17

Page 18 : MATHEMATICS / X / 2021 – 22, , Q5. The ratio of the sums of first m and first n terms of an A.P.is 𝑚 ∶ 𝑛 .Show that the, ratio of its 𝑚, , and the 𝑛 terms is (2m-1): (2n - 1)., , Q6. If the ratio of the 11th term of an A.P. to its 18th term is 2 :3, find the ratio of the sum, of the first five term to the sum of the first 10 terms, Q7. If the 𝑝, , term of an A.P. is 1/q and the 𝑞, , terms of the A.P. is, , term is 1/p, prove that sum of the first pq, , ., , Q8. The sum of first n terms of an A.P, are 𝑆 , 𝑆 , 𝑆 𝑟𝑒𝑠𝑝𝑒𝑐𝑡𝑖𝑣𝑒𝑙𝑦. 𝑇ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚 𝑜𝑓 each A.P is 1 and common, differences are 1,2 and 3 respectively .Prove that 𝑆 + 𝑆 = 2𝑆, Q9. If pth, qth and rth terms of an A.P. are a, b, c respectively, then show that (a – b)r + (b –, c)p+ (c – a)q = 0., Q10. The first and the last terms of an AP are 8 and 350 respectively. If its common, difference is 9, how many terms are there and what is their sum?, , CASE STUDY QUESTIONS, CASE STUDY QUESTION I, Birthdays are important for each one of us. Smriti is celebrating her birthday. She, invited her friends for a party. She arranged a number card game. In this game, number cards, are distributed among her friends such that they are following an Arithmetic progression., Smriti made sure that each of her friends who stood in a row gets a card. The first three cards, marked 2x, x+10 and 3x+2 is given to Rahul, Sonu and Sanjay respectively, , 18

Page 19 : MATHEMATICS / X / 2021 – 22, , Based on the above information answer the questions, 1., , Sonu is a curious child. She wants to find the sum of the number cards obtained by, her, Rahul, and Sanjay. From the above given information, help her to do so., , 2., , Smriti has called only a few of her close friends as the Covid pandemic is spreading., Ratan is her friend who gets the last card with the number 56. If so, find how many of, Smriti’s friends are attending the birthday party., , CASE STUDY QUESTION II, As a part of this one-week long festival, students of Durgapura Higher Secondary, School thought of planting trees in and around their school to reduce air pollution. It, was decided that each section of each class would plant twice as many plants as class, which they belong to. There were 4 sections of each standard from 1 to 12. So, if, thereare four sections in class 1 say 1A, 1B , 1C and I D, then each section would plant 2, trees. Similarly, each section of class 2 would plant 4 trees and so on. Thus, the number, of trees planted by classes 1 to 12 formed an AP given by 8, 16, 24, ,.... Ratan ,who is a, student of Class 10 B decided to frame a set of questions and answers based on the, above information . Help him to do so., , 1., , Find the total number of trees planted by class 10 students of all the sections together. Also, find the total number of trees planted by students of Ratan’s class alone., , 2. The members of the Nature Club of the School decided to find the total number of trees, planted by the students of the school altogether. Help them to do so., , CASE STUDY QUESTION III, Accumulation of plastics in the environment creates problems for wildlife and, their habitats as well as for human. Plastics are a threat to the environment. The children of, 19

Page 20 : MATHEMATICS / X / 2021 – 22, , Avantipur decided that they would contribute their service to put an end to the usage of, plastics in their village. They fixed posters and hoisted placards which depicted the ill effects, of plastics on human health and environment. They started their work on June 𝟏𝟓𝒕𝒉 They, started collecting the thrown off plastic bottles in their locality and started counting them., To their astonishment, they found that the number of plastic bottles that they collected each, day were in Arithmetic Progression which went like this: 417 ,404 ,391, ……, , 1. How many bottles did they collect on June 25, , ?, , 2. The children of Avantipur wanted to make their village a plastic free zone. Identify the, day on which they got 1 bottle which was their dream day, , ANSWER KEY, SECTION A, Qn no, , ANS, , Qn.no, , ANS, , Qn no, , ANS, , Qnno, , ANS, , Qn.no, , ANS, , 1, , 𝑛 = 30, , 6, , 30𝑑, , 11, , 𝑝=4, , 16, , 1, , 21, , 10th, , 2, , 𝑎 = 28, , 7, , 2475, , 12, , 𝑛 = 16, , 17, , 137, , 22, , √200, , 3, , 𝑛 = 53, , 8, , 𝑎=7, , 13, , 193, , 18, , −11, , 23, , -1, , 4, , 1 + (𝑛 − 1)𝑚, 𝑚, , 9, , zero, , 14, , 1170, , 19, , 𝑛 = 19, , 24, , 23rd, , 5, , a16=111, , 10, , 𝑥 = 14, , 15, , 𝑛 = 46, , 20, , 3.5, , 25, , 5, , SECTION B, 1), , 𝑎 + 2𝑑 = 4 𝑎𝑛𝑑 𝑎 + 8𝑑 = −8. 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑖𝑛𝑔 𝑡ℎ𝑒𝑚 𝑤𝑒 𝑔𝑒𝑡 𝑑 = −2. 𝐴𝑙𝑠𝑜 𝑤𝑒 𝑔𝑒𝑡 𝑎 = 8, Let 𝑎, , = 0 ; 𝑎 + (𝑛 − 1)𝑑 = 0 ; 8 + (𝑛 − 1)(−2) = −8 ;, , 𝑛 = 5, 𝐻𝑒𝑛𝑐𝑒 5𝑡ℎ 𝑡𝑒𝑟𝑚 𝑜𝑓 𝑡ℎ𝑒 𝐴𝑃 𝑖𝑠 𝑧𝑒𝑟𝑜 ;, 2), , 55, use 𝑎, , 𝑎 + (𝑛 − 1)𝑑, where a=-5, d= 5/2, 20

Page 24 : MATHEMATICS / X / 2021 – 22, , qth term = A + (q – 1)D = b …(ii), rth term = A + (r – 1)D = c … (iii), L.H.S. = (a – b)r + (b – c)p + (c – a)q, = [A + (p – 1)D – (A + (q – 1)D)]r + [A + (q – 1)D – (A + (r – 1)D)]p + [A + (r – 1)D – (A, + (p – 1)D)]q, = [(p – 1 – q + 1)D]r + [(q – 1 – r + 1)D]p + [(r – 1 – p + 1)D]q, = D[(p – q)r + (q – r)p + (r – p)q], = D[pr – qr + qp – rp + rq – pq], = D[0] = 0 = R.H.S., 10), , 𝑎 = 350, 𝑎 + (𝑛 − 1)𝑑 = 350, 𝑛 = 39, 𝑆 = (𝑎 + 𝑎, , ), , = 6981, , CASE STUDY QUESTIONS, , Q NO:, CASE STUDY :1, , HINTS/SOLUTION, , MARKS, , (i)To find the sum of the number on the cards,first find x., As the terms are in AP,, (x+10)-2x = (3x+2)-(x+10) ⇒ x = 6, ∴ Sum of the number cards of Rahul, Sonu and Sanjay is 2x +, x+10 + 3x+2=6x +12 =48, , 1, 1, , (ii) To find the number card ,find the first term(a) and, common difference(d) ⇒ a = 12 and d= 4, Let Ratan occupy the 𝑛, , position., , ⇒ 𝑎 = a +(n-1)d ⇒ 56 = 12 +(n-1)4 ⇒n =12. Hence ,, Smriti’s 12 friends attended the party., CASE STUDY :2, , (i), , 1, , 1, , 8,16,24,…., , To find the total number of trees planted by Class 10 students, of all the 4 sections together, find 𝑎, 𝑎, , = a +9d ⇒ 𝑎, , .Here a= 8 , d= 8, , = 8 +(9 x 8) = 80 trees, , The total number of trees planted by students of Ratan’s, , 1, , class(X B ) =80 /4 =20 trees, 1, 24

Page 25 : MATHEMATICS / X / 2021 – 22, , (ii) Sum of trees planted by the students of the school, altogether= 𝑆 =, , [2a +(n-1)d ], =, , 1, 1, , [(2 x8 )+ (11 x 8)], , =624 trees, CASE STUDY :3, , (i) 417 ,404 ,391 ,……, As the children started collecting plastics on on June 15, ,June 25, ⇒𝑎, , falls on the 11, , 1, , day ⇒n = 11, 1, , = a +(11-1)d =287 bottles, , (ii)The AP is 417 ,404 ,391 ,……, a=417 ,d =(-13), Let the day on which they got 1 bottle be the 𝑛, , day ⇒ 𝑎, , = 1 ⇒ a +(n-1)d =1, ⇒ 417 +(n-1)(-13) =1 ⇒ n = 33, Their dream day was on the 33, , 1, day starting from June, , 1, , 15 . Hence the day falls on July 17, , *****************************************************, , 25

Page 26 : MATHEMATICS / X / 2021 – 22, , Circles, Important Concepts, Tangent to a circle, A tangent to a circle is a line that intersects the circle at only one point, , O, P, *, , There is only one tangent at a point on a circle, , *, , There are exactly two tangents to a circle through a point lying outside the circle., , *, , The tangent at any point of a circle is perpendicular to the radius through the point of, contact., , *, , The length of tangents drawn from an external point to a circle are equal., , Short Answer Questions, SECTION A (2 MARK QUESTIONS), Q1. Prove that the line segments joining the points of contact of two parallel tangents is a, diameter of the circle., Q2. O is the centre of the circle and BCD is a tangent to it at C., Prove that < 𝐵𝐴𝐶+< 𝐴𝐶𝐷 = 90, , Q3. In the figure quadrilateral ABCD is drawn to circumscribe a circle., Prove that AD + BC = AB + CD, , 26

Page 27 : MATHEMATICS / X / 2021 – 22, , Q4. Prove that the tangents drawn at the end- points of the diameter of a circle are parallel., Q5. Two concentric circles have centre O, OP= 4cm, OB = 5cm. AB is a chord of the outer, circle and tangent to the inner circle at P. Find the length of AB., Q6. Two tangents PA and PB are drawn to a circle with centre O such that < 𝐴𝑃𝐵 = 1200., Prove that OP=2AP, Q7. In the isosceles triangle ABC in fig. AB = AC, show that BF = FC, , Q8. In the fig. a circle is inscribed in a ∆𝐴𝐵𝐶 with sides AB = 12cm, BC = 8 cm and, AC = 10cm. Find the lengths of AD, BE and CF, , Q9. In fig. circle is inscribed in a quadrilateral ABCD in which < 𝐵= 900. If AD = 23cm,, AB = 29cm, and DS = 5cm, find the radius ‘r’ of the circle, , Q10., , In fig. two circles touch each other externally at C. Prove that the common tangent at, , C bisects the other two tangents, , 27

Page 28 : MATHEMATICS / X / 2021 – 22, , Q11., , In fig. circle touches the side BC of a triangle ABC at the point P and AB and AC, , produced at Q and R. Show that AQ = (𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 ∆𝐴𝐵𝐶), , Q12., , Find the actual length of sides of ∆𝑂𝑇𝑃, , Q13., , In fig. all three sides of the triangle touch the circle. Find the value of x., , Q14., , Two tangents PR and PQ are drawn from external point P to a circle with centre O., Prove that PROQ is a cyclic quadrilateral., , Q15., , Prove that tangents drawn at the ends of a chord make equal angles with the chord, , 28

Page 29 : MATHEMATICS / X / 2021 – 22, , SHORT ANSWER QUESTIONS, SECTION B (3 MARK QUESTIONS), Q1. If an angle between two tangents drawn from a point P to a circle of radius ‘a’ and centre, O is 60°, then prove that AP = a√3., Q2. In the figure common tangents AB and CD to two circles with centre O and ‘O I intersects, at E. Prove that AB = CD., , Q3. If all the sides of a parallelogram touch a circle, then prove that the parallelogram is a, rhombus., Q4. XY and XIYI are two parallel tangents to a circle with centre O and another tangent AB, with point of contact C, intersecting XY at A and XIYI at B, is drawn. Prove that, ∠𝐴𝑂𝐵 = 90°., Q5., , In the figure, a circle is inscribed in a quadrilateral ABCD in which ∠𝐵 = 90°. If AD =, , 23 cm , AB = 29 cm and DS = 5 cm, find the radius of the circle., , Q6. In figure tangent segments PS and PT are drawn to a circle with centre O such that ∠𝑆𝑃𝑇, = 120°. Prove that OP = 2PS., , 29

Page 30 : MATHEMATICS / X / 2021 – 22, , Q7. In fig. 3, PQ and PR are tangents to the circle with centre O and S is a point on the circle, such that SQR 50 and SRM  60 . Find QSR., , Q8. Two tangents TP and TQ are drawn to a circle with centre O from an external point T., Prove that PTQ=2OPQ., Q9. In fig, two circles with centres A and B touch each other externally at K. find the length, of segment PQ. (Given PA=13 cm , BQ=5 cm , PS=12 cm AND QT=3 cm), , Q10., , PA and PB are the two tangents to a circle with centre O in which OP is equal to the, diameter of the circle. Prove that APB is an equilateral triangle., , Q11., , 11. In fig. Chords AB and CD intersect at P. If AB = 5 cm, PB = 3 cm and PD = 4 cm., Find the length of CD., , Q12., , The tangent at a point C of a circle and a diameter AB when extended intersect at P., If ∠PCA = 110°, find ∠CBA., , 30

Page 31 : MATHEMATICS / X / 2021 – 22, , Q13., , Prove that the angle between the two tangents drawn from an external point to a circle, is supplementary to the angle subtended by the line-segment joining the points of, contact at the centre, , LONG ANSWER QUESTIONS, SECTION C (4 MARK QUESTIONS), Q1. In fig PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents drawn at P, and Q intersect at T. Find the length of TP., , Q2. Prove that the lengths of tangents drawn from an external point to a circle are equal., Q3. In fig, two equal circles with centres O and OI, touch each other at X. OOI produced meet, the circle with centre OI at A. AC is tangent to the circle with centre O, at the point C., OID is perpendicular to AC. Find the value of, , ., , Q4. The radius of the in-circle of a triangle is 4 cm and the segments into which one side is, divided by the point of contact are 6 cm and 8 cm. Determine the other two sides of the, triangle., Q5. In fig, tangents PQ and PR are drawn from an external point P to a circle with centre O,, such that ∠RPQ= 30°. A chord RS is drawn parallel to the tangent PQ. Find ∠RQS., , Q6. Prove that opposites sides of a quadrilateral circumscribing a circle subtend, supplementary angles at the centre of the circle., , 31

Page 32 : MATHEMATICS / X / 2021 – 22, , Q7. In fig AB is diameter of a circle with centre O and QC is a tangent to the circle at C. If, ∠CAB=30°, find ∠CQA and ∠CBA., , Q8. In fig, O is the centre of a circle of radius 5 cm. T is a point such that OT = 13cm and OT, intersect circle at E. If AB is a tangent to the circle at E, find the length of AB, where TP, and TQ are two tangents to the circle., Q9. In ∆ABC, AB= 8cm , BC=6cm and CA= 4 cm. With the vertices of triangle as centre, three, circles are described, each touching the other two externally, find the radii of each, circle., Q10., , In a right triangle ABC in which, ∠B = 90°, a circle is drawn with AB as diameter, , intersecting the hypotenuse AC at P. Prove that the tangent to the circle at P bisect BC., Q11. A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments, BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6, cm respectively. Find the sides AB and AC., , CASE STUDY QUESTIONS, Case Study 1, An international school in Hyderabad organised an Interschool Throwball, Tournament for girls just after the pre-board exam. The throw ball team was very, excited. The team captain Anjali directed the team to assemble in the ground for, practices. Only three girls Priyanshi, Swetha and Aditi showed up. The rest did not, come on the pretext of preparing for pre-board exam. Anjali drew a circle of radius, 5 m on the ground. The centre A was the position of Priyanshi. Anjali marked a point, N, 13 m away from centre A as her own position. From the point N, she drew two, tangential lines NS and NR and gave positions S and R to Swetha and Aditi. Anjali, 32

Page 33 : MATHEMATICS / X / 2021 – 22, , throws the ball to Priyanshi, Priyanshi throws it to Swetha, Swetha throws it to, Anjali, Anjali throws it to Aditi, Aditi throws it to Priyanshi, Priyanshi throws it to, Swetha and so on., , 1., 2., , Find the distance between Swetha and Anjali. Which theorem is used and why is it, used?, If ∠ SNR = θ, find ∠ NAS. Write the reason for your answer., , Case Study – 2, , Varun has been selected by his School to design logo for Sports Day T-shirts for, students and staff . The logo design is as given in the figure and he is working on the, fonts and different colours according to the theme. In given figure, a circle with centre O, is inscribed in a ΔABC, such that it touches the sides AB, BC and CA at points D, E and, F respectively. The lengths of sides AB, BC and CA are 12 cm, 8 cm and 10 cm, respectively., , 33

Page 34 : MATHEMATICS / X / 2021 – 22, , 1. Find the length of AD, 3. If radius of the circle is 4cm, find area of ∆ABC., Answer Key, SECTION A (Short answer Questions), Q1. Consider the circle with centre at O, PQ & RS are two parallel tangents to it touching at A and B respectively., Join OA and OB, Now OA perpendicular to PQ (∴ radius is perpendicular to tangent), and OB perpendicular to RS, ∴OA∥OB, But OA and OB pass through O, ∴AB is straight line through centre, ∴AB is a diameter, <OCD= 900, , Q2., , (∵ radius is perpendicular to tangent at the point of contact), , <OCA +<ACD = 900, <OAC + < ACD = 900 (∵ OC = OA , <OCA=<OAC), <BAC + < ACD = 900, Q3. AS = AP ,,,,,,,,,,,,,,,,,,,,,(i) (Length of tangents drawn from an external point to a circle are, equal), DS=DR……………..(ii), CQ=CR…………….. (iii), BQ=BP………………(iv), Adding (i), (ii) ,(iii) and (iv) we get, AS + DS+ CQ + BQ = AP + DR + CR + BP, AD +BC = AB + CD, Q4., , Let AB be a diameter of the circle. Two tangents PQ and RS are drawn at points A and, B respectively., 34

Page 35 : MATHEMATICS / X / 2021 – 22, , Radius drawn to these tangents will be perpendicular to the tangents., Thus, OA ⊥ PQ and OB ⊥ RS, ∠OAP = 90º, ∠OAQ = 90º, ∠OBR = 90º, ∠OBS = 90º, It can be observed that, ∠OAP = ∠OBS (Alternate interior angles), ∠OAQ = ∠OBR (Alternate interior angles), Since alternate interior angles are equal, lines PQ and RS will be parallel., Q5., , OP = 4 cm, OB = 5 cm, We know that the radius is perpendicular to the tangent at the point of contact., In right triangle OPB,, OB2 = OP2 + PB2, (5)2 = (4)2 + PB2, PB2 = 25 - 16 = 9, PB = 3 cm, We know that perpendicular from the centre to the chord bisect the chord., AB = 2PB = 6 cm, , 35

Page 36 : MATHEMATICS / X / 2021 – 22, , Q6., , In ΔOAP and ΔOBP,, OP = OP, , (Common), , ∠OAP = ∠OBP (90°) (Radius is perpendicular to the tangent at the point of contact), OA = OB (Radius of the circle), ∴ ΔOAP is congruent to ΔOBP (RHS criterion), ∠OPA = ∠OPB = 120°/2 = 60° (CPCT), In ΔOAP,, Cos ∠OPA = cos 60° = AP/OP, Therefore, 1/2 =AP/OP, Thus, OP = 2AP, Hence, proved., Q7. AB= AC (given), ie AE +BE = AG + GC, BE = GC (Length of tangents drawn from an external point to a circle are equal), BF = CF, , ( ∵ BE = BF and GC = CF), , Q8. Let AD= x cm, BD = 12 – x, BE = 12 – x, CE = 8 – (12 – x), CE = x – 4 …….. (i), AF = x, CF = 10 – x --------(ii), From (i) and (ii) , we get, 36

Page 38 : MATHEMATICS / X / 2021 – 22, , x-8 = 6cm, there fore x = 14cm, , Q14., Given : Tangents PR and PQ from an external point P to a circle with centre O., To prove : Quadrilateral QORP is cyclic., Proof : RO and RP are the radius and tangent respectively at contact point R., ∴∠PRO=90∘, Similarly ∠PQO=90∘, In quadrilateral OQPR, we have, ∠P+∠R+∠O+∠Q=3600, ⇒∠P+∠900+∠O+∠900=3600, ⇒∠P+∠O=3600−1800=1800, These are opposite angles of quadrilateral QORP and are supplementary., ∴ Quadrilateral QORP is cyclic, hence, proved., 15., , Given: - A circle with centre O, PA and PB are tangents drawn at ends A and B on chord AB., To prove: - ∠PAB=∠PBA, Construction: - Join OA and OB, Proof: - In △AOB, we have, OA=OB, , (Radii of the same circle), , 38

Page 41 : MATHEMATICS / X / 2021 – 22, , Q11. AB = 15 cm, , AC = 13 cm, , Case Study, CASE STUDY 1, , CASE STUDY 2, , 1), , 1) 7, , 12m ., , 2) 90° – (θ/2), , AD=AF=x cm, BD=BE=y cm, CF=CE= z cm, AB = x + y= 12cm, BC = y + z = 8 cm, CA= z + x = 10 cm, AB+BC+CA= 30 cm, x +y +y +z +z +x = 30, x + y + z = 15, AD= 7cm, 60cm2, , Sum of the four angles of a, quadrilateral is 3600., , Area of ∆ABC = Area of ∆OAB + Area of ∆OBC +Area, of ∆OCA, , Pythagoras Theorem because Δ NSA, is a right-angled triangle (NS ⊥ SA), , Also ∠ NAS = ∠ NAR, , Constructions, Key Points, 1. Division of a line segment in a given ratio., 2. Construction of tangents to a circle, , Short Answer Questions, Section A (2 Mark Questions), Q1. Draw a line segment of length 8.4 cm and divide it in the ratio 7:5, Q2. Draw a circle of radius 4cm. From a point 8cm away from its centre, construct pair, of tangents to the circle., 41

Page 42 : MATHEMATICS / X / 2021 – 22, , Short Answer Questions, SECTION B (3 MARK QUESTIONS), Q1. Draw a line segment AB of length 7 cm. Taking A as centre, draw a circle of, radius 3 cm and taking B as centre, draw another circle of radius 2 cm., Construct tangents to each circle from the centre of the other circle ., Q2. Construct a pair of tangents to a circle of radius 4 cm from a point which is, at a distance of 6 cm from its centre., Q3. Draw a line segment of length 8 cm and divide it internally in the ratio 4:5., Q4. Draw two concentric circles of radii 3 cm and 5 cm. Construct a tangent to, smaller circle from a point on the larger circle. Also measure its length., Q5. Draw a circle of radius 3 cm. From a point P, 7 cm away from its centre, draw two tangents to the circle. Measure the length of each tangent., Q6. Construct two tangents PT and PQ to a circle of radius 4 cm and centre O such that, ∠TOQ=120°., Q7. To a circle of radius 5 cm, draw two tangents which are inclined to each, other at an angle of 60°., Q8. Draw a circle of radius 3·5 cm. Draw two tangents to the circle which are, perpendicular to each other., Q9. Draw a line segment of 6 cm and divide it in the ratio 3 : 2., Q10. Draw a line sgment AB of length 7 cm. Using a ruler and compasses, find a point P on, AB such that, , = ., , Q11. Draw a circle of radius 3 cm. Take two points P and Q on one of its extended, diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from, these two points P and Q., Answer Key, Short Answer Questions, 4. The length of the tangent = 4 cm., 5. The length of the tangent = 6.3 cm, , 42

Page 43 : MATHEMATICS / X / 2021 – 22, , SOME APPLICATION OF TRIGONOMETRY, HEIGHTS AND DISTANCES: Trigonometry is used for finding the heights and distances, of various objects, without measuring them., Line of sight is the line drawn from the eye of the observer to the point on the object viewed, by the observer., Horizontal level is the horizontal line through the eye of the observer., , ANGLE OF ELEVATION, The angle of elevation is relevant for objects above horizontal level., It is the angle formed by the line of sight with the horizontal level., , ANGLE OF DEPRESSION, The angle of depression is relevant for objects below horizontal level., It is the angle formed by the line of sight with the horizontal level., , 43

Page 45 : MATHEMATICS / X / 2021 – 22, TRIGONOMETRIC RATIOS OF SOME SPECIFIC ANGLES, , MIND, MINDMAP, MAP, , 45

Page 46 : MATHEMATICS / X / 2021 – 22, , VERY SHORT ANSWER, SECTION A (2 MARK QUESTIONS), Q1. An airplane at an altitude of 200m observes the angles of depression of opposite points, on the two banks of a river are to be 450 and 600.Find the width of the river. (Take, √3=1.73), Q2. A tree 12m high is broken by the wind in such a way that its top touches the ground and, makes an angle 600 with the ground. At what height from the bottom, the tree is broken, by the wind. (Take √3=1.73), Q3. At some time of the day the length of the shadow of a tower is equal to its height. Find, the sun’s altitude at that time., Q4. A ladder 15m long makes an angle of 600 with the wall. Find the height of the point, where the ladder touches the wall., Q5. A vertical pole 20m long casts a shadow 20 √3m long. Find the sun’s altitude. At the, same time a tower casts a shadow 90m long. Determine the height of the tower., Q6. The tops of two towers of heights x and y standing on level ground, making angles 30 0, and 600 respectively at the Centre of the line joining their feet. Find x: y., Q7. From a balloon vertically above a straight road, the angles of depression of two cars at, an instant are found to be 450 and 600.If the cars are 100m apart, find the height of the, balloon., Q8. The angle of elevation of the top of the first storey of a building is 30 0 at a point on the, ground distant 15m from its foot. How high its second storey will be if the angle of, elevation of the top of the second storey at the same point is 450., Q9. From a bridge, 25m high, the angle of depression of a boat is 45 0. Find the horizontal, distance of the boat from the bridge., Q10. A 1.8m tall girl stands at a distance of 4.6m from a lamp post and casts a shadow of, 5.4m on the ground. Find the height of the lamp post., Q11. Two poles are 25m and 15m high and the line joining their tops make an angle of 45 0, with the horizontal. Find the distance between these poles, Q12. If two towers of height h1 and h2 subtend angles of 600 and 300 respectively at the, midpoint of the line joining their feet, then find the value of h 1: h2, Q13. If the height of a vertical pole is √3 times the length of its shadow on the ground, then, what is the angle of elevation of the sun at that time, , SHORT ANSWER, SECTION B (3 MARK QUESTIONS), Q1. From a point on the ground, the angles of elevation of the bottom and top of a water, tank kept on the top of the 30 m high building are 30 0 and 450 respectively. Find the, height of the water tank?, Q2. From the top of a multi-storeyed building, 90m high, the angles of depression of the top, and the bottom of a tower are observed to be 300 and 600 respectively. Find the height, of the tower?, 46

Page 47 : MATHEMATICS / X / 2021 – 22, , Q3. Two ships are there in the sea on either side of a lighthouse in such a way that the ships, and the base of the lighthouse are in the same straight line. The angles of depression of, two ships as observed from the top of the lighthouse are 60 0 and 450. If the height of the, lighthouse is 200m, find the distance between the two ships., , (Use √ 3 = 1.73), , Q4. From the top of a 300 metre high light-house, the angles of depression of two ships,, which are due south of the observer and in a straight line with its base, are 60 0 and, 300.Find their distance apart?, Q5. A Statue, 1.6 m tall, stands on the top of a pedestal. From a point on the ground, the, angle of elevation of the top of the statue is 600 and from the same point, the angle of, elevation of the top of the pedestal is 450. Find the height of the pedestal?, , (Use, , √ 3 = 1.73), Q6., , A peacock is sitting on the top of a tree. It observes a serpent on the ground making an, angle of depression of 300. The peacock with the speed of 300 metre/ minute catches, the serpent in 12 seconds. What is the height of the tree?, , Q7. An aero plane, at an altitude of 1200 m, finds that two ships are sailing towards it in the, same direction. The angles of depression of the ships as observed from the aeroplane, are 600 and 300 respectively. Find the distance between the two ships?, Q8. If the angles of elevation of the tops of two statues of heights m1 and m2 are 600 and 300, respectively from the mid-point of the line segment joining their feet, then find the ratio, m1: m2?, Q9. From the top of a 7m high building, the angle of elevation of the top of a cable tower is, 600 and the angle of depression of its foot is 450. Determine the height of the tower?, Q10. The angle of elevation of the top of a hill from the foot of a tower is 60 0 and the angle, of elevation of the top of the tower from the foot of the hill is 300. If the tower is 50 m, high, find the height of the hill?, Q11. The shadow of a tower standing on level ground is found to be 40 m longer when the, Sun’s altitude is 30° than when it is 60°. Find the height of the tower., Q12. Two pillars of equal heights are on either side of a road, which is hundred metres wide., The angles of elevation of the tops of the pillars are 60 0 and 300 at a point on the road, between the pillars. Find the position of the point between the pillars?, Q13. An observer 1.5 m tall is 20.5 m away from a tower 22 m high. Determine the angle of, elevation of the top of the tower from the eye of the observer?, , 47

Page 48 : MATHEMATICS / X / 2021 – 22, , LONG ANSWER, SECTION C ( 4 MARK QUESTIONS), Q1. The angle of elevation of a cloud from a point 100 metre above the surface of a lake is, 300 and angle of depression of the reflection of cloud in the lake is 600. Find the, height of the cloud., Q2. From the top of a tower 60m high, the angles of depression of the top and bottom of a, vertical lamppost are observed to be 300 and 600 respectively. Find:, Q1. The horizontal distance between the tower and the lamppost., Q2. The height of the lamp post., Q3. From a point on a cricket ground, the angle of elevation of the top of a tower is found, to be 300 at a distance of 225 m from the tower. On walking 150 m towards the tower,, again the angle of elevation is found. Find the new angle of elevation and the height, of the tower?, Q4. From the top of a tower, the angle of depression of an object on the horizontal ground, is found to be 600. On descending 20 m vertically downwards from the top of the, tower, the angle of depression of the object is found to be 300. Find the height of the, tower., Q5. From a window 15metres high above the ground in a street, the angles of elevation and, depression of the top and foot of another house on the opposite side of the street, , are, , 30° and 45° respectively. Show that the height of the opposite house is 23.65 m., (Use √ 3 = 1.73), Q6. The angle of elevation of an aeroplane from a point on the ground is 60°. After a, flight of 30 seconds, the angle of elevation changes to 30°. If the plane is flying at a, constant height of 3600 √ 3 m, find the speed of the plane in km per hour., Q7. An aeroplane is flying at a height of 300 m above the ground. Flying at this height the, angle of depression from the aeroplane of two points on the banks of a river in, opposite directions are 45° and 30° respectively. Find the width of the river. (Use √ 3, = 1.732), Q8. As observed from the top of 100m high lighthouse from the sea level, the angles of, depression of two ships are 30° and 45°. If one should be exactly behind the other on, the same side of the Lighthouse, find the distance between the two ships. (Use √ 3 =, 1.732), , 48

Page 49 : MATHEMATICS / X / 2021 – 22, , Q9. From a point P on the ground the angle of elevation of top of the tower is 30° and that, of a flag staff fixed on the top of a tower is, 60°. If the length of a flagstaff is 5m, find, the height of the tower., Q10. The angle of elevation of the top of a vertical Tower from a point on the ground is 60°., From another point 10 m vertically above the first, its angle of elevation is 30°. Find, the height of the tower., Q11. The angle of elevation of the top of a hill at the foot of a tower is 60° and the angle of, depression of the top of the tower of the foot of the hill is 30°. If the tower is 50 m, high find the height of the hill?, , CASE STUDY QUESTIONS, CASE STUDY 1, , A group of students of class x visited India Gate on an education trip. The teacher and, students had interest in History as well. The teacher narrated that India Gate, official name, Delhi Memorial, originally called All-India War Memorial, monumental sandstone arch in, New Delhi, dedicated to the troops of British India who died in wars fought between 1914, and 1919.The teacher also said that India Gate, which is located at the eastern end of the, Rajpath (formerly called the Kingsway), is about 138 feet (42 meters) in height., Based on the above information answer the following questions, 1. They want to see the tower at an angle of 600.So they want to know the distance, where they should stand and hence find the distance., 2. If the altitude of the sun is at 600, then what is the height of the vertical tower that will, cast a shadow of length 20m?, , 49

Page 50 : MATHEMATICS / X / 2021 – 22, , CASE STUDY 2:, LIGHT HOUSE, A boy is standing on the top of light house. He observed that boat P and boat Q are, approaching to light house from opposite directions. He finds that angle of depression of boat, P is 450 and angle of depression of boat Q is 300. He also knows that height of the light house, is 100m., , Based on the above information, answer the following questions., (i), , What is the length of CD?, , (ii), , What is the length of BD?, , CASE STUDY 3:, , A boy 4 m tall spots a pigeon sitting on the top of a pole of height 54m from the ground. The, angle of elevation of the pigeon from the eyes of boy at any instant is 60 0. The pigeon flies, , 50

Page 51 : MATHEMATICS / X / 2021 – 22, , away horizontally in such a way that it remained at a constant height from the ground. After 8, seconds, the angle of evaluation of the pigeon from the same point is 45 0 (take √3 = 1.73 ) ., Based on the above information, answer the following questions., 1. Find the distance of first position of the pigeon from the eyes of the boy., 2. How much distance the pigeon covers in 8 second, , ANSWER KEY, VERY SHORT ANSWER, Q No., , Answer, , Q No., , Answer, , 1, , x + y = 315.33m, , 8, , (15-5√3)m, , 2, , 9, 10, , 25m, , 3, , 5.567, 450, , 4, , 7.5m, , 11, , 10m, , 5, , 30√3m, , 12, , 3:1, , 6, , 1:3, , 13, , 600, , 7, , 50(3+√3)m, , Short Answer Type, Q No., Option, , m, , 1, , 30(√3-1), , Long Answer, Q, Answer, No., 1, 200m, , 2, , 60 metres, , 2, , a) 20√3m, b) 40m, , 3, , 315.33 m, , 3, , Angle of Elevation= 600, Height = 75√3m, , 4, 6, , 30m, 864 km/hr, , 4, 5, , 200√3 m, 2.2 m (approx…), , 6, 7, , 30m, 800√3 m, , 7, , Width of river= 819.6m, , 8, , Distance between two, ships= 73.2 m, , 8, , 3:1, , 9, , Height of the tower = 2.5m, , 51

Page 52 : MATHEMATICS / X / 2021 – 22, , 9, , 7(√3+1) m, , 10, , Height of the tower = 15 m, , 10, , 150m, , 11, , Height of the hill = 150m, , 11, , 20√3 m, , 12, , 25m, , 13, , 45 ˚, , CASE STUDY QUESTIONS, CASE STUDY 1, , CASE STUDY 2, , 1) 14√3, , 1) 100 m, , 2) 20 √3 m, , 2) 100√3 m, , CASE STUDY 3, 1), , √, , m, , 2) 21.09m, , 52

Page 53 : MATHEMATICS / X / 2021 – 22, , UNIT IV - MENSURATION, SURFACE AREAS AND VOLUMES, IMPORTANT FORMULAE AND CONCEPTS, , 53

Page 54 : MATHEMATICS / X / 2021 – 22, ,  Surface areas and volumes of combinations of solids, Surface areas and volumes of combinations of solids of any two of the, following:, cubes, cuboids, spheres, hemispheres and right circular cylinders/cones., , , Conversion of Solid from One Shape to Another, Problems involving converting one type of metallic solid into another and other, mixed problems.(Problems with combination of not more than two different, solids)., , * Deleted Topics, Frustum of a cone (Total surface area and volume of Frustum of a cone, , SHORT ANSWER QUESTIONS, SECTION A (2 MARK QUESTIONS), Q1. A sphere of diameter 18 cm is dropped into a cylindrical vessel of diameter 36 cm, partly, filled with water. If the sphere is completely submerged, then, , calculate the rise, , of water level (in cm)., Q2. Find the number of solid spheres, each of diameter 6 cm that can be made by, melting a solid metal cylinder of height 45 cm and diameter 4 cm., Q3. Volume and surface area of a solid hemisphere are numerically equal. What is the, diameter of hemisphere?, Q4. If the total surface area of a solid hemisphere is 462 cm2, find its volume. (π = 3.14), Q5. Two cubes, each of side 4 cm are joined end to end. Find the surface area of the, resulting cuboid., Q6. A vessel is in the form of a hemispherical bowl surmounted by a hollow cylinder of, same diameter. The diameter of the hemispherical bowl is 14 cm and the total height, of the vessel is 13 cm. Find the total (inner) suface area of the vessel. (Use π = 22/7), Q7. The largest possible sphere is carved out of a wooden solid cube of side 7 cm., Find the volume of the wood left., Q8. A cone of height 20 cm and radius of base 5 cm is made up of modelling clay. A, child reshapes it in the form of a sphere. Find the diameter of the sphere., Q9. A solid sphere of radius 10.5 cm is melted and recast into smaller solid cones,, each of radius 3.5 cm and height 3 cm. Find the number of cones so formed. (Use π, = 22/7), 54

Page 55 : MATHEMATICS / X / 2021 – 22, , Q10. What is the capacity of a cylindrical vessel with a hemispherical portion raised, upward at the bottom?, Q11. A solid piece of iron in the form of a cuboid of dimension 49 cm × 33 cm × 24 cm, is melted to form a solid sphere. Find the radius of sphere., Q12. A vessel is in the form of a hemispherical bowl surmounted by a hollow cylinder of, same diameter. The diameter of the hemispherical bowl is 14 cm and the total height, of the vessel is 13 cm. Find the total (inner) suface area of the vessel. (Use π = 22/7), Q13. In Figure, is a decorative block, made up of two solids-a cube and a hemisphere. The, base of the block is a cube of side 6 cm and the hemisphere fixed on the top has a, diameter of 3.5 cm. Find the total surface area of the block. (Use π = 22/7), , Q14. A conical vessel, with base radius 5 cm and height 24 cm, is full of water. This water, is emptied into a cylindrical vessel of base radius 10 cm. Find the height to which the, water will rise in the cylindrical vessel. (Use π = 22/7), Q15. A 21 m deep well with diameter 6 m is dug and the earth from digging is evenly, spread to form a platform 27 m x 11 m. Find the height of the platform., , SHORT ANSWER QUESTIONS, SECTION B (3 MARK QUESTIONS), Q16. 12 Solid spheres of the same size are made by melting a solid metallic cone of, base, , radius 1cm and height of 48 cm. Find the radius of each sphere., , Q17. Two cubes each of volume 27 cm3 are joined end to end to form a solid. Find the, surface area of the resulting cuboid., , 55

Page 56 : MATHEMATICS / X / 2021 – 22, , Q18. Find the number of plates 1.5 cm in diameter and 0.2 cm thick can be fitted, completely inside a right circular cylinder of height 10 cm and diameter 4.5 cm, Q19. A cylindrical glass tube with radius 10 cm has water up to a height of 9 cm. A metal, cube of 8 cm edge is immersed completely. By how much water level will rise, , in, , the glass tube?, Q20. A solid metallic object is shaped like a double cone as shown in figure. Radius of base, of both cones is same but their heights are different. If this cone is immersed in water,, find the quantity of water it will disperse, , Q21. If the areas of three adjacent faces of a cuboid are X, Y and Z respectively, then find, the volume of the cuboid., Q22. Find the volume (in cm3) of the largest right circular cone that can be cut off from a, cube of edge 4.2 cm., Q23. A wooden article was made by scooping out a hemisphere of radius 7 cm, from each, end of a solid cylinder of height 10 cm and diameter 14 cm. Find the total surface area, of the article (use  , , 22, ), 7, , Q24. A heap of rice is in the form of a cone of base diameter 24 m and and height 3.5 m., Find how much canvas cloth is required to just cover the heap?, Q25. The sum of the radius of base and height of a solid right circular cylinder is 37 cm. If, the total surface area of the solid cylinder is 1628 cm2, find the volume of the cylinder., (use  , , 22, ), 7, , Q26. A toy is in the shape of a solid cylinder surmounted by a conical top. If the height and, diameter of the cylindrical part are 21 cm and 40 cm respectively, and the height of, cone is 15 cm, then find the total surface area of the toy.(use   3.14 ), , 56

Page 57 : MATHEMATICS / X / 2021 – 22, , Q27. The dimensions of a metallic cuboid are 1m x 0.8m x 0.64m. It is melted and recast into, a cube. Find the surface area of the cube., Q28. Three cubes of iron whose edges are 3 cm, 4 cm and 5 cm respectively, are melted and, formed into a single cube, what will be the edge of the new cube so formed ?, Q29. A solid sphere of radius 10.5 cm is melted and recast into smaller solid cone, ach of, radius 3.5 cm and height 3 cm. Find the number of cones so formed., Q30. A cubical block of side 0.07 m is surmounted by a hemisphere. What is the greatest, diameter the hemisphere can have? Find the surface area of the solid, LONG ANSWER QUESTIONS, SECTION C ( 4 MARK QUESTIONS), Q31. A tent is in the shape of a right circular cylinder up to a height of 300 cm and conical, above it. The total height of the tent is 1350 cm and radius of its base is 1400cm. Find, the cost of cloth required to make the tent at the rate of `80 per square metre. (Take π=, 22, ), 7, , Q32. A hemispherical bowl of internal diameter 0.36 m contains liquid. This liquid is filled, into 72 cylindrical bottles of diameter 6cm.Find the height of each bottle, if 10% l, iquid is wasted in this transfer., Q33. From a cuboidal solid metallic block of dimensions 15cm X 10cm X 5cm a cylindrical, hole of diameter 0.07m is drilled out. Find the surface area of the remaining block. (π=, 22, ), 7, , Q34. A metallic cylinder has radius 0.03cm and height 0.05cm. To reduce its weight a, conical hole is drilled in the cylinder. The conical hole has a radius of 3/2cm and, its depth is 8/ 9cm. calculate the ratio of the volume of metal left in the cylinder to the, volume of metal taken out in conical shape., 57

Page 58 : MATHEMATICS / X / 2021 – 22, , Q35. A hollow cylindrical pipe is made up of copper. It is 21 dm long. The outer and inner, diameters of the pipe are 10cm and 6cm respectively. Find the volume of copper used, in making the pipe (π=, , 22, ), 7, , Q36. A farmer connects a pipe of internal diameter 20cm. from a canal into a cylindrical tank, which is 10m in diameter and 2cm deep. If the water flows through the pipe at the rate, of 4km per hour, in how much time will the Tank be filled completely?, Q37. A solid is in the shape of a cone mounted on a hemisphere of same base radius. If the, curved surface areas of the hemispherical part and the conical part are equal, then find, the ratio of the radius and the height of the conical part., Q38. A hollow sphere of internal and external diameter 4cm and 8cm respectively is melted, to form a cone of base diameter 8cm. find the height and the slant height of the cone., Q39. A hemispherical tank, full of water is emptied by a pipe at the rate of, , 25, liters/sec., 7, , How much time will it, take to empty half of the tank, if the diameter of the base of the, tank is 3m?, Q40. Water running in a cylindrical pipe of inner diameter 7cm, is collected in a container at, the rate of 192.5 liter per minute. Find the rate of flow of water in the pipe in km/h., Q41. A well of diameter 4cm is dug 14m deep. The earth taken out is spread evenly all around, the well to form a 40cm high embankment. Find the width of the embankment., Q42. A vessel full of water is in the form of an inverted cone of height 0.08m and the radius, of its top, which is open is 5cm. 100 spherical lead balls are dropped into the vessel., One-fourth of the water flows out of the vessel. Find the radius of the spherical ball., Q43. The radius of two right circular cylinders are in the ratio 2:3 and their heights are in the, ratio of 5:4. Calculate the ratio of their curved surface areas and ratio of their volumes., Q44. A container shaped right circular cylinder having base radius 6cm and heights 15cm. is, full of ice cream. The ice cream is to be filled into cones of height 12cm. and radius, 3cm, having, , a hemispherical shape on the top. Find the number of such cones which, , can be filled with ice cream., , 58

Page 59 : MATHEMATICS / X / 2021 – 22, , Q45. A solid copper sphere of surface area 1386 sq.cm. is melted and drawn into a wire of, uniform cross. Section if the length of the wire is 31.5m, find the diameter of the wire., (π=, , 22, ), 7, , CASE STUDY BASED QUESTIONS, Case study question 1, During Covid times people prefer using homogenized milk, UHT Processed and, aseptically packed in an exceptional six layer, tamper-proof Tetra Packaging with 0%, bacteria and 100% pure health. This new six layer interfere proof, prevents air and, freshness, light and bacteria from entering the pack. As an effect, the milk stays fresh and, pure for a minimum of 180 days until opened, even without refrigeration.The 500ml milk, is packed in cuboidal containers of dimensions 15 x 8 x 5 . These milk packets are then, packed in cuboidal cartons of dimension 30x 32 x 15 .(All dimensions are in cm), , Based on the above given information answer the following questions, 1., 2., , How many liters of milk will a carton contain?, How much cardboard is needed to make the carton if 10% of wastage is taken, into account?, , Case study question 2, An antique box and its dimensions excluding the stand is given below., dome, , 59

Page 60 : MATHEMATICS / X / 2021 – 22, , 1. Considering the thickness of the box to be negligible,, How much velvet cloth will be needed to cover the cuboidal inner area?, 2. How many gold coins of diameter 2cm and thickness 0.5cm will fill, , 𝟏𝒕𝒉, 𝟕, , of, , the volume of the dome of jewelry box., Case study question 3, Gulab jamun is a milk-solid-based sweet, originating in India and a type of mithai popular, in India, Nepal, Pakistan, the Maldives, and Bangladesh, as well as Myanmar. It is also, declared as the national dessert of Pakistan officially by Government of Pakistan. For, preparing gulab jamun the dough is divided into small balls, deep fried and then soaked in, sugar syrup., A dough is made in the shape of a sphere of radius 4.2cm. A gulab jamun contains sugar, syrup up to about 70% of its volume, , Based on the above given information answer the following questions, 1., , How much sugar syrup will be left out after soaking all the jamuns, if one makes, quarter liter syrup, , 2. How much silver foil will we need to coat one third of all the Gulab jamun surface?, , Answer Key, SHORT ANSWER (SECTION A), Question Answer, , Question Answer, , 1, , h=3cm, , 9, , 2, , 5, , 10, , 3, , 6cm, , 11, , 21 cm, , 4, , 2156/3= 718.666…, , 12, , 572 cm2, , 5, , 160 cm2, , 13, , 225.625cm2, , 126 cones, , 60

Page 61 : MATHEMATICS / X / 2021 – 22, , 6, , 572 cm2, , 14, , 2cm, , 7, , 163.33…, , 15, , 2m, , 8, , 5cm, , Short answer, , Long Answer, , 16. 1cm, , 31. Cost of cloth required = Rs 82,720, , 17. Surface area = 90 cm2, , 32. 5.4 cm, , 18. No. of plates = 450, , 33. Total surface area = 583 cm2, , 19. Water will rise = 1.629 cm, , 34. (Hint: Don’t put the value of ) 133: 2, , 20. Water displaced =, 21. Volume =, , 1 2, r h  H , 3, , XYZ, , 22. Volume = 19.4cm, , 35. Put 10 cm =1 dm, volume = 10560 cm3, 36. Time = 1 hour 15 minute, 37. Ratio = 1:, , 3, , 3, , 23. Surface area = 1056 cm2, 24. Curved surface area = 471.42sq.cm, 25. Volume = 4620 cm, , 2, , 26. Total Surface area = 5463.6 cm2, 27. Surface area = 38400 cm2, 28. Edge = 6 cm, 29. No. of cones = 126, 30. a) Greatest diameter = 7 cm, b) Surface area =332.5 cm2, , 38. Height = 14 cm, slant height =2 53 cm, 39. (Hint : 1m3 = 1000 l) Time = 16 min 30 sec, 40. (Hint : 1l = 1000 cm3 ), Rate of flow =, 3km/hr, 41. Width of embankment = 10 m, 42. Radius = 0.5 cm, 43. a) Ratio of curved surface areas = 5 : 6, b) Ratio of volumes = 5 : 9, 44. No. of cones = 10, 45. Diameter = 1.4 cm, , Case study questions, Case study -1, 1. (30×32×15) ∕ (15×8×5) = 2 x 4 x 3 = 24 boxes., So 24 x 500ml = 12 liters., 2. TSA + 10% of TSA, TSA = 2(30x15 + 32 x1 5 + 15x8), = 2(450+ 480 + 120)=2100., Cardboard needed = 2100+210=3310cm2, , 61

Page 62 : MATHEMATICS / X / 2021 – 22, , Case study -2, 1. CSA + BA = 2h(l+b) + lb, = 2 x 10 (14 + 30) + 14 x 30, = 880 + 420 = 1300cm2, 2. n x volume of one coin = ( volume of box), n= 210, , Case study -3, 1. 70% of total volume = 310.5×0.7, = 217.35, Syrup left = 250-217.35, =32.65ml, 2. One eighth of the surface area of 64 gulab jamuns= ⅛ x 13.86 x 64, = 110.88 cm2, , 62

Page 63 : MATHEMATICS / X / 2021 – 22, , STATISTICS, SYLLABUS, Mean , Median& Mode of grouped data, Mean by Direct Method & Assumed Mean Method, DELETED TOPICS, Step deviation method for finding the mean& Cumulative Frequency Graph, MIND MAP, , ARITHMETIC MEAN,  Direct Method, , 𝑥̅ =, , ∑, ,  Assumed Mean Method, , ∑, , 𝑥̅ = 𝑎 +, , ∑, ∑, , MODE, COMPUTATION OF MODE FOR A CONTINOUS FREQUENCY DISTRIBUTION, Algorithm, 1. Obtain the continuous frequency distribution, 63

Page 65 : MATHEMATICS / X / 2021– 22, , Short Answer Questions, SECTION A (2 mark questions), Q1. Find the mean of the following distribution using assumed mean method, Class, , 0-10, , Frequency, , 7, , 10-20, , 20-30, , 12, , 13, , 30-40, , 40-50, , 10, , 8, , Q2. The mean and median of 100 observations are 50 and 52 respectively. The value of the, largest observation is 100. It was later found that it is 110 not 100. Find the true mean, and median., Q3. From the following distribution, find the lower limit of the median class, Class interval 85-89, Frequency, 10, , 90-94, 12, , 95-99, 11, , 100-104, 5, , 105-109, 30, , Q4. Find the unknown values in the following table., Class Interval Frequency Cumulative, Frequency, 0-10, , 5, , 10-20, , 7, , 20-30, 30-40, 40-50, Q5., , 5, 𝑎, 𝑏, , 18, , 5, , 𝐶, 𝑑, , 30, , For the following distribution find the modal class, Marks, , Numberof Students, , Below 10 3, Below 20 12, Below 30 27, Below 40 57, Below 50 75, Below 60 80, , 65

Page 66 : MATHEMATICS / X / 2021– 22, , Q6. Find the value of 𝑝, if the arithmetic mean of the following distribution is 25, CI, F, , 0-10, , 10-20, , 20-30, , 5, , 8, , 15, , 30-40, , 40-50, 6, , 𝑃, , Q7. Find the mode of the following data, CI, F, , 1-3, 7, , 3-5, 8, , 5-7, 2, , 7-9, 2, , 9-11, 1, , Q8. Find 𝑥̅ , if 𝑑 = 𝑥 − 25 ; ∑ 𝑓 𝑑 = 20 ; ∑ 𝑓 = 100, Q9. Find mode, using an empirical relation, when it is given that mean and median are 10.5, and 9.6 respectively, Q10. Change the following distribution in to a ‘𝑚𝑜𝑟𝑒 𝑡ℎ𝑎𝑛 𝑡𝑦𝑝𝑒′ distribution table, Classes, , 0-10 10-20 20-30 30-40 40-50 50-60 60-70, , Frequency 5, , 15, , 20, , 23, , 17, , 11, , 9, , Q11. The frequency distribution table showing daily income of 100 workers of a factory is, given below. Convert this table to a frequency distribution table of ‘ 𝑙𝑒𝑠𝑠 𝑡ℎ𝑎𝑛 𝑡𝑦𝑝𝑒 ’., Daily income (₹) 200-300, No of workers, 12, , 300-400, 18, , 400-500, 35, , 500-600, 20, , 600-700, 15, , SHORT ANSWER QUESTIONS, SECTION B (3 MARK QUESTIONS), Q1. Daily wages of a factory workers are recorded as follows. Find the mode of the given, distribution, Daily wages, No of workers, , 131-136 137-142 143-148 149-154 155-160, 5, , 27, , 20, , 18, , 12, , Q2. Find the median of the following distribution, , 66

Page 67 : MATHEMATICS / X / 2021– 22, , Marks obtained, , 0-10 10-20 20-30 30-40 40-50 50-60, , Number of Students 8, , 10, , 12, , 22, , 30, , 18, , Q3. The median of the following data is 525. Find the missing frequency 𝑥, CLASS, , FREQUENCY, , 0-100, , 2, , 100-200, , 5, , 200-300, , x, , 300-400, , 12, , 400-500, , 17, , 500-600, , 20, , 600-700, , 15, , 700-800, , 9, , 800-900, , 7, , 900-1000 4, Q4. The following data gives the information on the observed life times (in hours), , of, , 150 electrical components. Find the mode of the distribution, Life time (in hours) 0-20 20-40 40-60 60-80 80-100, Frequency, , 15, , 10, , 35, , 50, , 40, , Q5. Determine the missing frequency 𝑥, from the following data, when mode is 67., Class, , 40-50 50-60 60-70 70-80 80-90, , Frequency 5, , 𝑥, , 15, , 12, , 7, , Q6. The lengths of 40 leaves of a plant are measured correct to the nearest millimetre and, the data obtained is represented in the following table. Find the median length of the, leaves, Length of leaf in (mm), , No of leaves, , 118-126, , 3, , 127-135, , 5, , 136-144, , 9, , 67

Page 68 : MATHEMATICS / X / 2021– 22, , 145-153, , 12, , 154-162, , 5, , 163-171, , 4, , 172-180, , 2, , Q7. The mean of the following distribution is 48 and the sum of all frequencies is 50. Find, the missing frequencies., Class, , 20-30 30-40 40-50 50-60 60-70, , Frequency, Q8., , 8, , 6, , 11, , 𝑥, , 𝑦, , Find the mean of the following distribution by appropriate method, Class, , 20-30 30-40 40-50 50-60 60-70, , Frequency, , 25, , 40, , 42, , 33, , 10, , LONG ANSWER QUESTIONS, SECTION C ( 4 MARK QUESTIONS, Q1. If the median of the data is 32.5, find the value of, Class, Frequency, , 𝑥 and 𝑦, , 0-10 10-20 20-30 30-40 40-50 50-60 60-70 Total, 5, , 𝑥, , 9, , 12, , 𝑦, , 3, , 2, , 40, , Q2. Find the median of the following data, Class, , Less, , Less, , Less, , Less, , Less, , Less, , Less, , Less, , than 10, , than 30, , than, , than, , than, , than, , than, , than, , 50, , 70, , 90, , 110, , 130, , 150, , 25, , 43, , 65, , 87, , 96, , 100, , Frequency, , 0, , 10, , Q3. The median of the following data is 50. Find the values of 𝑝 and 𝑞 if sum of all, frequencies is 90 ., Mark, , 20-30, , 30-40, , 40-50, , 50-60, , 60-70, , 70-80, , 80-90, , Frequency, , 𝑝, , 15, , 25, , 20, , 𝑞, , 8, , 10, , 68

Page 69 : MATHEMATICS / X / 2021– 22, , Q4. The table below shows the salaries of 280 persons., Calculate the median and mode of the given data, , SALARY, , Number of, , (In thousand Rupees ), , Persons, , 5-10, , 49, , 10 -15, , 133, , 15-20, , 63, , 20-25, , 15, , 25-30, , 6, , 30-35, , 7, , 35-40, , 4, , 40-45, , 2, , 45-50, , 1, , Q5. In the following frequency distribution. the frequency of a class interval is missing. It, is known that the mean of the distribution is 52. Find the missing frequency X., Wages (in, Rs), Frequency, , 10-20, 5, , 20-30, , 30-40, , 40-50, , 3, , 4, , X, , 50-60, 2, , 60-70, , 70-80, , 6, , 13, , Q6. The daily wages of 110 workers, obtained in a survey are tabulated below. Compute the, mean daily wages and modal daily wages of these workers., Daily, , 100-, , 120-, , 140-, , 160-, , 180-, , 200-, , 220-, , wages(₹), , 120, , 140, , 160, , 180, , 200, , 220, , 240, , No.of, , 10, , 15, , 20, , 22, , 18, , 12, , 13, , workers, , Q7. Find the median of the following data, if the total frequency is 400, Class, , 50-52 53-55 56-58 59-61 62-64, , Frequency 15, , 110, , 135, , 115, , 25, , 69

Page 70 : MATHEMATICS / X / 2021– 22, , CASE STUDY QUESTIONS, CASE STUDY I, Q8., A group of students decided to make a project on statistics. They are collecting, the heights (in cm) of 51 girls of class X A, B, C of their school. After collecting the, data, they arranged the data in the following less than cumulative frequency, distribution table form:, , Height (in cm), Less than 140, Less than 145, Less than 150, Less than 155, Less than 160, Less than 165, , No of girls, 4, 11, 29, 40, 46, 51, , ANSWER THE QUESTIONS BASED ON THE ABOVE INFORMATION, , Q9., , 1., , What is the mean of lower limits of median and modal class?, , 2., , Calculate Median of the above data, , CASE STUDY II, The following tables shows the age distribution of case admitted during a day in two, different hospitals, , 70

Page 71 : MATHEMATICS / X / 2021– 22, , Table 1, Age (in years), , 5-15, , 15-25, , 25-35, , 35-45, , 45-55, , 55-65, , No. of cases, , 6, , 11, , 21, , 23, , 14, , 5, , Age (in years), , 5-15, , 15-25, , 25-35, , 35-45, , 45-55, , 55-65, , No. of cases, , 8, , 16, , 10, , 42, , 24, , 12, , Table 2, , Based on the above data answer the following questions, 1. From table 1, find mean of the given data, 2. From table 2, find mode of the given data, , Q10., , CASE STUDY III, Stopwatch was used to find the time that it took a group of students to run 100m, Time (in sec), , 0-20 20-40 40-60 60-80 80-100, , No.of students 8, , 10, , 13, , 6, , 3, , Answer the following Questions, 1) Estimate the mean time taken by a student to finish the race, 2) Find the Sum of upper limits of median class and modal class., , 71

Page 72 : MATHEMATICS / X / 2021– 22, , ANSWER KEY, SECTION A, , SECTION B, , SECTION C, , 1, , 25, , 1 Mode=141.05, , 1, , 𝑥 = 3,𝑦 = 6, , 2, , Mean=50.10, , 2 Median = 39.09, , 2, , Median = 76.36, , 3 x=9, , 3, , 𝑝 = 5, 𝑞 = 7, , 4 Mode = 72, , 4, , Median=13421, , Median = 52, 3, , Lower limit of, median class = 99.5, , 4, , a=12 , b=6, c=23,d=7, , mode = 12727, 5, , Modal class = 30 -40, , 6, , 𝑝=6, , 5 x=3, , 5, , x=7, , 6 Median = 146.75, , 6, , Mean=170.18, mode= 166.67, , 7, 8, , Mode= 3.28, , 7, , 𝑥 = 12 , 𝑦 = 13, , 8 Mean = 42.5, , 𝑥̅ = 25.2, , 7, , median = 57.16, , 8, , a ) 145 ,, b) 149.03, , 9, , Mode = 7.8, , 9, , a ) 35.37 ,, b) 41.4, , 10, , 11, , more than 0, , 100, , more than10, , 95, , more than 20, , 80, , more than 30, , 60, , more than 40, , 37, , more than 50, , 20, , more than 60, , 9, , Less than 300, , 10 a) 43 seconds,, b) 120, , 12, , Less than 400, 30, Less than 500, 65, Less than 600, 85, Less than 700, 100, , 72

Page 73 : MATHEMATICS / X / 2021– 22, , CBSE Sample Question Paper, Mathematics- Basic (241), Class- X, Session: 2021-22, TERM II, Maximum Marks: 40, , Time Allowed: 2 hours, General Instructions:, 1., 2., , The question paper consists of 14 questions divided into 3 sections A, B, C., Section A comprises of 6 questions of 2 marks each. Internal choice has been provided in two, questions., Section B comprises of 4questions of 3 marks each. Internal choice has been provided in one, question., Section C comprises of 4 questions of 4 marks each. An internal choice has been provided in one, question. It contains two case study-based questions., , 3., 4., , SECTION A, Q.No, 1, , Marks, 2, , 2, , Find the value of 𝑘 for which quadratic equation 3𝑥 − 7𝑥 − 6 = 0., OR, Find the values of k for which the quadratic equation 3𝑥2 + 𝑘𝑥 + 3 = 0 has real and, equal roots., , 2, , Three cubes each of volume 64cm3 are joined end to end to form a cuboid. Find the, , 2, , total surface area of the cuboid so formed?, 3, , An inter house cricket match was organized by a school. Distribution of runs made, , 2, , by the students is given below. Find the median runs scored., , 4, , Runs Scored, , 0 -20, , 20 -40, , 40-60, , 60-80, , 80-100, , Number of students, , 4, , 6, , 5, , 3, , 4, , Find the common difference of the AP 4,9,14, … If the first term changes to 6 andthe, , 2, , common difference remains the same then write the new AP, 5, , The mode of the following frequency distribution is 38. Find the value of x., , 2, , Class Interval, , 0-10, , 10-20, , 20-30, , 30-40, , 40-50, , 50-60, , 60-70, , Frequency, , 7, , 9, , 12, , 16, , x, , 6, , 11, , 73

Page 74 : MATHEMATICS / X / 2021– 22, , 6, , XY and MN are the tangents drawn at the end points of the diameter DE of the circle, , 2, , with centre O. Prove that 𝑋𝑌 ∥ 𝑀𝑁., , X, , D, , Y, , O, M, , E, , N, OR, , In the given figure, a circle is inscribed in the quadrilateral ABCD. Given AB=6cm,, BC=7cm and CD=4cm. Find AD, , SECTION B, 7, , An AP 5, 8, 11…has 40 terms. Find the last term. Also find the sum of the last 10, , 3, , terms., 8, , A tree is broken due to the storm in such a way that the top of the tree touches the, , 3, , ground and makes an angle of 300 with the ground. Length of the broken upper part of, the tree is 8 meters. Find the height of the tree before it was broken., OR, Two poles of equal height are standing opposite each other on either side of the road, 80m wide. From a point between them on the road the angles of elevation of, the top of the two poles are respectively 60 0 and 300. Find the distance of the pointfrom, the two poles., 9, , PA and PB are the tangents drawn to a circle with centre O. If PA= 6 cm and , , 3, , APB=600, then find the length of the chord AB., , 74

Page 75 : MATHEMATICS / X / 2021– 22, , 10, , The sum of the squares of three positive numbers that are consecutive multiples of 5 is, , 3, , 725. Find the three numbers., SECTION C, 11, , Construct two concentric circles of radii 3cm and 7cm. Draw two tangents to the, , 4, , smaller circle from a point P which lies on the bigger circle., OR, Draw a pair of tangents to a circle of radius 6cm which are inclined to each other at an, angle of 600. Also find the length of the tangent., 12, , The following age wise chart of 300 passengers flying from Delhi to Pune is prepared, , 4, , by the airline staff, Age, No of, , Less, , Less, , than 10, , than 20 than 30, , 14, , Less, , 44, , 82, , Less, , Less, , than 40 than 50, , 134, , 184, , Less, , Less, , Less, , than 60, , than 70, , than 30, , 245, , 287, , 300, , passengers, Find the mean age of the passengers, 13, , A lighthouse is a tall tower with light near the top. These are often built on islands,, coasts or on cliffs. Lighthouses on water surface act as a navigational aid to the, mariners and send warning to boats and ships for dangers. Initially wood, coal would be, used as illuminators. Gradually it was replaced by candles, lanterns, electric lights., Nowadays they are run by machines and remote monitoring., Prongs Reef lighthouse of Mumbai was constructed in 1874 -75. It is approximately 40, meters high and its beam can be seen at a distance of 30 kilometres. A ship and a boat, are coming towards the lighthouse from opposite directions. Angles of depression of, flash light from the lighthouse to the boat and the ship are 30 0 and 600respectively, , 75

Page 76 : MATHEMATICS / X / 2021– 22, , 2, 2, I., , Which of the two, boat or the ship is nearer to the light house. Find its distance, from the lighthouse?, , II., , Find the time taken by the boat to reach the light house if it is moving atthe rate, of 20 km per hour., , 14, , Krishnanagar is a small town in Nadia District of West Bengal. Krishnanagar clay dolls, are unique in their realism and quality of their finish. They are created by modelling, coils of clay over a metal frame. The figures are painted in natural colours and their hair, is made either by sheep’s wool or jute. Artisans make models starting from fruits,, animals, God, goddess, farmer, fisherman, weavers to Donald Duck and present comic, characters. These creations are displayed in different national and international, museums., , The ratio of diameters of red spherical apples in Doll-1 to that of spherical oranges, in Doll-2 is 2:3. In Doll-3, male doll of blue colour has cylindrical body and a, spherical head. The spherical head touches the cylindrical body. The radius of both, the spherical head and the cylindrical body is 3cm and the height of the cylindrical, body is 8cm. Based on the above information answer the following questions:, 76

Page 77 : MATHEMATICS / X / 2021– 22, , 1) What is the ratio of the surface areas of red spherical apples in Doll -1 to, , 2, , that of spherical oranges in Doll-2.?, , 2) The blue doll of Doll-3 is melted and its clay is used to make the cylindrical, , 2, , drum of Doll - 4. If the radius of the drum is also 3cm, find the height of the, drum., , 𝑴𝒂𝒓𝒌𝒊𝒏𝒈 𝑺𝒄𝒉𝒆𝒎𝒆, Mathematics –, Basic(241)Class- X, Session- 2021-22, TERM II, Q.N., 1, , HINTS/SOLUTION, , 3𝑥2 − 7𝑥 − 6 = 0, ⇒ 3𝑥2 − 9𝑥 + 2𝑥 − 6 = 0, ⇒ 3𝑥 (𝑥 − 3) + 2(𝑥 − 3) = 0, ⇒ (𝑥 – 3)(3𝑥 + 2) = 0, 𝑥 = 3 ,−, , 1/2, 1/2, OR, , 2, , 3, , Since the roots are real and equal, ∴ 𝐷 = 𝑏2 − 4𝑎𝑐 = 0, ⇒k2 – 4×3×3 = 0 (∵ 𝑎 = 3, 𝑏 = 𝑘, 𝑐 = 3), ⇒k2 = 36, ⇒k = 6 𝑜𝑟 −6, Let 𝑙 be the side of the cube and L, B, H be the dimensions of the cuboid, Since 𝑙3 = 64 𝑐𝑚3 ∴ 𝑙 = 4 𝑐𝑚, Total surface area of cuboid is 2[𝐿𝐵 + 𝐵𝐻 + 𝐻𝐿], Where L=12, B=4 and H=4, =2(12 × 4 + 4 × 4 + 4 × 12) 𝑐𝑚2 = 224𝑐𝑚2, Runs scored, 0-20, 20-40, 40-60, 60-80, 80-100, , Frequency, 4, 6, 5, 3, 4, , Marks, , Cumulative Frequency, 4, 10, 15, 18, 22, , 1, 1, 1/2 +1/2, 1/2, 1/2, 1, , 1/2, , 77

Page 78 : MATHEMATICS / X / 2021– 22, , 𝑁, , 2, , Total frequency (N) = 22, = 11; So 40-60 is the median class., , 1/2, , 𝑁, , Median = 𝑙 +, , ( )−𝑐𝑓, , 2, , = 40 +, , ×ℎ, , 𝑓, 11−10, , 1/2, , x 20, , 5, , = 44 runs, 4, , 5, , 1/2, , The common difference is 9 - 4=5, If the first term is 6 and common difference is 5, then new AP is,6,, 6+5, 6+10…, =6,11,16…., ∵ Mode = 38., ∴ The modal class is 30-40., Mode = 𝑙 +, , 𝑓1− 𝑓0, , 1, 1, 1/2, , ×ℎ, , 1/2, , 2𝑓1−𝑓0−𝑓2, , 6, , ∵XY is the tangent to the circle at the point D, ∴ OD  XY   ODX = 900   EDX = 900, Also, MN is the tangent to the circle at E, ∴ OE  MN  OEN = 900   DEN = 900, ⇒  EDX =  DEN (𝑒𝑎𝑐ℎ 900). which are alternate interior angles., ∴ XY  MN, , 1/2, , 1/2, 1, , OR, , 78

Page 79 : MATHEMATICS / X / 2021– 22, , ∵Tangent segments drawn from, an external point to a circle are equal, ∴ BP=BQCR=CQDR=DSAP=AS, ⇒BP+CR+DR+AP = BQ+CQ+DS+AS, ⇒ AB+DC = BC+AD, ∴ AD= 10-7= 3 cm, 7, , Last term = 𝑎40 = a+(40-1) d, = 5 + 39 × 3 = 122, , 1, , Also 𝑎31 = 𝑎 + 30𝑑 = 5 + 30 × 3 = 95, , 1, , Sum of last 10 terms = 𝑛 (𝑎, 2, , =, , 8, , Section-B, , First Term of the AP(a) = 5, Common difference (d) = 8-5=3, , 31, , 10, 2, , +𝑎 ), 40, , (95 + 122), = 5 × 217 = 1085, , Let, AB be the tree broken at C,, Also let 𝐴𝐶 = 𝑥, In ∆ CAD, sin300 =, ⇒, , 1, 2, , =, , 𝑥, , 𝐴𝐶, , 1, , 1, 1/2, 1/2, 1(CORREC, , 𝐷𝐶, , 8, , ⇒𝑥 =4𝑚, , T FIG), , ⇒the length of the tree is = 8+4 =12, OR, Let AB and CD be two poles of height h meters also let P be a point between them on the road, which is x meters away from foot of first pole AB,, PD= (80-x) meters., In ∆ABP, 𝑡𝑎𝑛60° = ⇒ ℎ = 𝑥√3. … (1), In ∆CDP, 𝑡𝑎𝑛 30𝑜 =, , ⇒ ℎ =, , √, , 1, 1/2, , ….(2), 79

Page 81 : MATHEMATICS / X / 2021– 22, , OR, , Draw a circle of radius 6cm, Draw OA and Construct ∠ 𝐴𝑂𝐵 = 1200, Draw ∠ 𝑂𝐴𝑃 = ∠ 𝑂𝐵𝑃 = 900, PA and PB are required tangents, Join OP and apply tan∠𝐴𝑃𝑂 = tan 30° =, ⇒ Length of tangent = 6√3 cm, , 6, , 𝑃𝐴, , 12, , 2, , 1, , 1, , 81

Page 83 : MATHEMATICS / X / 2021– 22, , CBSE Sample Question Paper, Mathematics- Standard (041), Class- X, Session: 2021-22, TERM II, Time Allowed: 2 hours, , Maximum Marks: 40, , General Instructions:, 1., 2., 3., 4., 5., , The question paper consists of 14 questions divided into 3 sections A, B, C., All questions are compulsory., Section A comprises of 6 questions of 2 marks each. Internal choice has been provided in two, questions., Section B comprises of 4questions of 3 marks each. Internal choice has been provided in one, question., Section C comprises of 4 questions of 4 marks each. An internal choice has been provided in one, question. It contains two case study based questions., Section A, , Q No, 1, , Find the value of a25 - a15 for the AP: 6, 9, 12, 15, …………, , Marks, 2, , OR, If 7 times the seventh term of the AP is equal to 5 times the fifth term, thenfind, the value of its 12th term., , 2, , Find the value of 𝑚 so that the quadratic equation 𝑚𝑥 (5𝑥 − 6) = 0 has twoequal, , 2, , roots., , 3, , From a point P, two tangents PA and PB are drawn to a circle C (O, r). If OP = 2r, then, , 2, , find ∠APB. What type of triangle is APB?, , 4, , The curved surface area of a right circular cone is 12320 cm². If the radius of its, , 2, , base is 56cm, then find its height., , 83

Page 84 : MATHEMATICS / X / 2021– 22, , 5, , Mrs. Garg recorded the marks obtained by her students in the following table. She, , 2, , calculated the modal marks of the students of the class as 45. While printing the data, a, blank was left. Find the missing frequency in the table givenbelow, , 6, , Marks obtained, , 0 -20, , 20 -40, , 40-60, , 60-80, , 80-100, , Number of students, , 5, , 10, , ----, , 6, , 3, , If Ritu were younger by 5 years than what she really is, then the square of her age would, , 2, , have been 11 more than five times her present age. What is her present age?, OR, Solve for x: 9x² - 6px + (p² - q²) = 0, , Section-B, 7, , Following is the distribution of the long jump competition in which 250 students, , 3, , participated. Find the median distance jumped by the students. Interpret the median, , 8, , Distance in (m), , 0 -1, , 1-2, , 2-3, , 3-4, , 4-5, , Number of students, , 40, , 80, , 62, , 38, , 30, , 3, , Construct a pair of tangents to a circle of radius 4cm, which are inclined toeach, other at an angle of 60°., , 9, , The distribution given below shows the runs scored by batsmen in one-day cricket, , 3, , matches. Find the mean number of runs., , 10, , Runs Scored, , 0 - 40, , 40 - 80, , 80 - 120, , 120 - 160, , 160 - 200, , Number of Batsmen, , 12, , 20, , 35, , 30, , 23, , Two vertical poles of different heights are standing 20m away from each other on the, , 3, , level ground. The angle of elevation of the top of the first pole from the foot of the, second pole is 60° and angle of elevation of the top of the second pole from the foot of, the first pole is 30°. Find the difference between the heightsof two poles. (Take √3 = 1.73), OR, A boy 1.7 m tall is standing on a horizontal ground, 50 m away from a building. The, angle of elevation of the top of the building from his eye is 60°. Calculate, the height of the building. (Take √3 = 1.73), , Section-C, 84

Page 85 : MATHEMATICS / X / 2021– 22, , 11, , The internal and external radii of a spherical shell are 3cm and 5cm respectively. It is, , 4, , melted and recast into a solid cylinder of diameter 14cm, findthe height of the cylinder., Also find the total surface area of the cylinder., (𝑇𝑎𝑘𝑒 𝜋 =, , 12, , 22, 7, , Prove that the angle between the two tangents drawn from an external point toa circle is, , 4, , supplementary to the angle subtended by the line segment joining thepoints of contact to, the centre., OR, Two tangents TP and TQ are drawn to a circle with centre O from an external point T., Prove that ∠𝑃𝑇𝑄 = 2∠𝑂𝑃𝑄, , 13, , Case Study-1, Trigonometry in the form of triangulation forms the basis of navigation, whether it is by, land, sea or air. GPS a radio navigation system helps to locate our position on earth with, the help of satellites., A guard, stationed at the top of a 240m tower, observed an unidentified boat coming, towards it. A clinometer or inclinometer is an instrument used for measuring angles or, slopes(tilt). The guard used the clinometer to measure theangle of depression of the boat, coming towards the lighthouse and found it to be 30°., , 85

Page 86 : MATHEMATICS / X / 2021– 22, , (Lighthouse of Mumbai Harbour. Picture credits - Times of India Travel), I., , Make a labelled figure on the basis of the given information and calculate the, , 2, , distance of the boat from the foot of the observation tower., II., , After 10 minutes, the guard observed that the boat was approaching thetower and, its distance from tower is reduced by 240(√3 - 1) m. He immediately raised the, , 2, , alarm. What was the new angle of depression ofthe boat from the top of the, observation tower?, , 14, , Case Study-2, Push-ups are a fast and effective exercise for building strength. These are helpful in, almost all sports including athletics. While the push-up primarily targets the muscles of, the chest, arms, and shoulders, support required from other muscles helps in toning up, the whole body., , Nitesh wants to participate in the push-up challenge. He can currently make 3000 pushups in one hour. But he wants to achieve a target of 3900 push-ups in 1 hour for which, he practices regularly. With each day of practice, he is ableto make 5 more push-ups in, one hour as compared to the previous day. If on first day of practice he makes 3000 pushups and continues to practice regularly till his target is achieved. Keeping the above, situation in mind answer the following questions:, , i) Form an A.P representing the number of push-ups per day and hence find the, minimum number of days he needs to practice before the day his goal is, accomplished?, ii) Find the total number of push-ups performed by Nitesh up to the day his goal is, achieved., , 2, 2, , 86

Page 91 : MATHEMATICS / X / 2021– 22, , 14, , 1, , 3000, 3005, 3010, ...,3900., 𝑎𝑛 = 𝑎 + (𝑛 − 1)𝑑, 3900 = 3000 + (n - 1)5, ⇒ 900 = 5𝑛 − 5 ⇒ 5𝑛 = 905 ⇒ 𝑛 = 181, Minimum number of days of practice = 𝑛 − 1 = 180 𝑑𝑎𝑦𝑠, , 1, 1, , 2) 𝑆 = (𝑎 + 𝑙), =, , × (3000 + 3900) = 624450 pushups, , 1, , PRACTICE PAPERS, Sample Question Paper- 1, Mathematics- Basic (241), Class- X Session: 2021-22, TERM II, Time Allowed: 2 hours, Maximum Marks: 40, General Instructions:, 1. The question paper consists of 14 questions divided into 3 sections A, B, C., 2. Section A comprises of 6 questions of 2 marks each. Internal choice has been provided in, two questions., 3. Section B comprises of 4questions of 3 marks each. Internal choice has been provided in, one question., 4. Section C comprises of 4 questions of 4 marks each. An internal choice has been provided, in one question. It contains two case study based questions., SECTION A, Q:No, 1., , 2., 3., 4., , A ladder 15 m long just reaches the top of a vertical wall. If the ladder, makes an angle of 60° with the wall, then calculate the height of the wall, OR, A ladder, leaning against a wall, makes an angle of 60° with the horizontal., If the foot of the ladder is 2.5 m away from the wall, find the length of the, ladder, A solid metallic spherical ball of diameter 6 cm is melted and recast into a, cone with diameter of the base as 12 cm, Find height of the cone ?, If a tower 30 m high, casts a shadow 10 √3 m long on the ground, then what, is the angle of elevation of the sun?, The arithmetic mean of the following distribution is 50. Find the missing, frequency p., Class Interval, Frequency, 0 – 20, 7, 20 – 40, 6, 40 – 60, 9, 60 – 80, 13, 80 – 100, p, , MARKS, 2, , 2, 2, 2, , 91

Page 92 : MATHEMATICS / X / 2021– 22, , 5., , 6., , 7., 8., 9., , 10., 11., , 12., , 13., , AB is a chord of the circle and AOC is its diameter such that angle ACB =, 50°. If AT is the tangent to the circle at the point A, then find ∠ BAT, , OR, Prove that a parallelogram circumscribing a circle is a rhombus., Find the mode of the following data, Class, 20 – 25 25 – 30 30 – 35 35 – 40 40 – 45 45 – 50, Interval, Frequency 9, 13, 35, 20, 15, 8, SECTION B, (a)For what values of k , the quadratic equation kx (x – 2) + 6 = 0 have, two equal roots?, (b)Find the roots of the quadratic equation x2 + 6x +5 =0, Prove that the lengths of the tangents from an external point to a circle are, equal. Using this result Prove that a Parallelogram circumscribing a circle, is a Rhombus, The angle of elevation of the top of a building for the foot of the tower is, 30o and the angle of elevation of the top of a tower from the foot of the, building is 60o . If the tower is 50m high, find the height of the building., OR, A vertical tower stands on the ground is surmounted by a flag-staff of, height 5 m. From a point on the ground, the angle of elevation of the, bottom of the flag staff is 45o and that of the top of the flag-staff is 60o ., Find the height of the tower. (√3 = 1.732), The sum of the squares of two consecutive odd numbers is 394. Find the, numbers, SECTION C, Draw a circle of radius 4 cm. From a point P, 9 cm away from the centre of, the circle, draw two tangents to the circle. Also measure the length of the, tangents, OR, Draw a pair of tangents to a circle of radius 5 cm which are inclined to each, other at an angle of 60°., Find the missing frequencies in the following frequency distribution table if, n = 100 and median is 32, Class, 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50 50 – 60, Interval, Frequency 10, x, 25, 30, y, 10, The historical monument Gol Gumbaz is the tomb of king Muhammad Adil, Shah, Adil Shah Dynasty. It is located in Vijayapura, Bijapur, Karnataka., Construction of the tomb was started in 1626 and completed in 1656.In this, , 2, , 2, , 3, 3, 3, , 3, 4, , 4, , 92

Page 93 : MATHEMATICS / X / 2021– 22, , monument one can find combination of solid figures. There are cubical, bases & hemispherical domes at the top., , 2, Find the total surface area of the hemispherical dome having, radius 7 cm, (ii), A block of the Gol Gumbaz is in the shape of a cylinder of, diameter 0.5 cm with two hemispheres stuck to each of its ends., 2, The length of the shape is 2cm. Find the volume of the block, (Use 𝜋 = 3.14), Ashin is a plant lover. She has a wide range of plant collection. She decides, to open a nursery. The planted pots had to be arranged to make it, impressive for a buyer on seeing the beautiful flowering plants. She makes, an arrangement in such a way that the number of pots in the first row is 3,, second row is 5, third row is 7 and so on…, (i), , 14., , 2, (i), (ii), , If Ashin wants to place a total of 120 pots, how many rows, should be made in this arrangement?, Find the difference in number of pots placed in 8 th row and 3rd, row, , 2, , 93

Page 94 : MATHEMATICS / X / 2021– 22, , MARKING SCHEME, CLASS- X SESSION- 2021-22, TERM 2, SUBJECT- MATHEMATICS (BASIC), Qn no:, 1., , SECTION A, HINT/SOLUTION, , MARKS, , 1, ∠BAC = 180° – (90° + 60o ) = 30°, sin 30° =, ½ =BC/15, 2BC = 15, BC = 15/2 m or 7.5 m, OR, Let AC be the ladder, , 1, , 1, , ∴ Length of ladder, AC = 5 m 2.5 m, 2., , Volume of the spherical ball Vs=, , 3, , 3, , 𝜋r =36πcm, , 1, 1, , Volume of the cone made from sphere Vc= πr2h =12hπcm3, ⇒Vs= Vc⇒ 36π =12hπ ⇒h= 36/12 = 3cm, , 1, , 3., 1, In rt triangle ABC , tan 𝜃 =, , 4., , √, , = √3, , But tan 600 = √3 , Therefore 𝜃 = 600 , Hence sun’s elevation is, 600, Class, Frequency(fi) Class, fixi, Interval, mark, (xi), 0 - 20, 12, 10, 120, , 1, , 1, , 94

Page 95 : MATHEMATICS / X / 2021– 22, , 20 – 40, 40 – 60, 60 – 80, 80 –, 100, , 15, 32, P, 13, , 30, 50, 70, 90, , ∑xi, = 72 + p, 53 =, 5., , 6., , 7., , 450, 1600, 70p, 1170, . ∑𝑓𝑖𝑥𝑖 = 3340 + 70𝑝, , , p = 28, 0, , ∠𝐴𝐵𝐶 = 90 ( Angle in a semicircle), In ∆ ABC , 900 + 500+∠𝐵𝐴𝐶 = 1800(Angle sum), ∠𝐵𝐴𝐶 = 400, ∠𝐶𝐴𝑇 = 900 ( Tangent perpendicular to radius), But ∠𝐶𝐴𝑇 = ∠𝐶𝐴𝐵 + ∠𝐵𝐴𝑇, ∠𝐵𝐴𝑇 = 900 – 400 = 500, OR, Correct figure,given ,to prove ,correct proof, Modal class 30 = 35, l = 30,f1=35,f0=13,f2=20,h=5, Mode = l + (, )h, On substituting , mode = 32.97, (a) kx(x−2)+6=0, , , kx2−2kx+6=0, , Since the roots are equal, b2 - 4ac = 0, (−2k)2=4(k)(6) , 4k2=4k(6) ∴k=6, , 8., , 1, , ½, ½, 1, 1, ½, ½, 1, ½, ½, ½, , (b) Appling Quadratic formula, x = - 1 , - 5, , 1½, , Correct figure , given ,to prove , correct proof, , 1½x2, =3, ½, , 9., , Let the height of the tower be AB and the height of the building, , ½, , be CD, In ΔABC,, , ½, , tan 60° = AB/BC, , 1, , √3 = 50/BC , BC = 50/√3 ....(i), , ½, , In ΔBCD, tan 30° = CD / BC, , 95

Page 96 : MATHEMATICS / X / 2021– 22, , 1/√3 = CD / BC , 1/√3 = CD / 50/√3 [from (i)], CD = 1/√3 × 50/√3 , CD = 50/3 Height of the building CD =, 50/3 m., OR, , ½, , Given that the Height of Flagstaff =5 m, Now, Let the Height of the Tower be q and distance of a point from the, ½, , Tower be x, In △ABC, 𝟓, tan 600=, , 𝒒, , , 5 + q = x√𝟑 ........(1), , 𝒙, , ½, , In △DBC, tan300 = q/x ⇒ x= q. √𝟑 ...............(2), From (1) and (2), ⇒5+q=3q ⇒2q=5⇒q=2.5, Therefore, Height of Tower =2.5 m, , 10., , let the two numbers be 2x−1 and 2x+1., Given that the sum of their squares is 394, (2x−1)2+(2x+1)2=394, 4x2+1−4x+4x2+1+4x=394, Solving for x, x=7 ,The two odd numbers,2n−1 and 2n+1 are 13 and 15, , 11., 12., , Correct construction, , ½, ½, , ½, ½, 1, 1½, 3, , class, , f, , cf, , 0-10, , 10, , 10, , 10-20, , x, , 10+x, , 20-30, , 25, , 35+x, , 30-40, , 30, , 65+x, , ½, , 96

Page 97 : MATHEMATICS / X / 2021– 22, , 40-50, , y, , 65+x+y, , 50-60, , 10, , 75+x+y, , 75+x+y, , ½, , 75+x+y=100 , x+y=25, f=30,h=10,cf=35+x, =50, Median=l+(, , 1, , )×h , 32=(3050−35−x)×10, , 1, , 6=15−x , x=9 ∴y=16, 13., , (i) Radius of the hemisphere is 7 cm., Total surface area of the hemisphere =3πr2 =3× ×72 =462 cm2, (ii) Volume of block= Volume of both the hemispherical parts + Volume of, , 2, ½, , cylindrical part, The hemisphere and the cylinder will have the same radius r =0.25cm, , ½, , Since total length of the shape is 2 cm, the length of the cylindrical part will, be 2−0.25−0.25=1.5cm, Hence, Volume of the block = 2 x, 14., , 𝜋 r3 + πr2h, , On substituting the values , Volume = 0.36 cm3, (i) Sn = n/2 ( 2a + (n- 1 )d), 120 = n/2 ( 2x3 + ( n – 1 ) 2), Solving for n, n= 10, Hence there should be 10 rows of this, arrangement, (ii) a8 – a3 = a + 7d –( a + 2d), a = 3 , d = 2 , on substituting, Difference in number of pots placed in 8th row and 3rd row = 10, , 1, , ½, ½, 1, ½, 1½, , 97

Page 98 : MATHEMATICS / X / 2021– 22, , Sample Question Paper - 2, MATHEMATICS-BASIC (241), CLASS-X SESSIOIN-2021-22, TERM II, Time Allowed:- 2 hours, General Instructions:, , Maximum Marks: 40, , 1. The question paper consists of 14 questions divided into 3 sections A , B , C, 2. Section A comprises of 6 questions of 2 marks each. Internal choice has been provided in, two questions., 3. Section B comprises of 4 questions of 3 marks each. Internal choice has been provided in, one questions., 4. Section C comprises of 4 questions of 4 marks each. An internal choice has been provided, in one questions. It contains two case study based questions, SECTION A, Marks, , Q.No, 1, , Find the value of 𝑘 for which quadratic equation 9𝑥 +8kx+16=0, , 2, , has equal roots ?, OR, Find the roots of quadratic equation 9𝑥 − 15𝑥 + 6 = 0, 2, , A solid sphere of radius 3 cm is melted and then cast into smaller, , 2, , spherical balls each of diameter 0.6 cm . Find the number of balls, thus obtained, 3, , Find the mode of the following distribution of marks obtained by 80, , 2, , students, Marks obtained, , 0-10, , 10-20, , 20-30, , 30-40, , 40-50, , Number of students, , 6, , 10, , 12, , 32, , 20, , 4, , Which term of AP:21,42,63,…..is 420 ?, , 2, , 5, , Two concentric circles of centre O are of radii 5 cm and 3 cm. Find, , 2, , the length of the chord of the larger circle AB, , 98

Page 99 : MATHEMATICS / X / 2021– 22, , OR, The incircle of ΔABC touches the sides BC, CA and AB at D , E , and F, respectively. If AB = AC ,prove that BD = CD, , 6, , If the mean of the following frequency distribution is 27. Find the, , 2, , value of p, Class interval, Frequency, , 0-10, , 10-20, , 20-30, , 30-40, , 40-50, , 8, , P, , 12, , 13, , 10, , SECTION B, 7, , The first term of an AP is 5 , the last term is 45 and the sum is, , 3, , 400.Find the number of terms and the common difference, 8, , The height of a tower is 10 m. Calculate the length of shadow when, , 3, , the sun’s altitude is 450, OR, A ladder 15 meters long just reaches the top of the vertical wall. If, the ladder makes an angle of 600 with the wall , find the height of, the wall, 9, , In the given figure, find the perimeter of ΔABC. If AP = 10 cm, , 3, , 10, , The product of two successive multiples of 3 is 180. Determine the, , 3, , numbers, SECTION C, , 99

Page 100 : MATHEMATICS / X / 2021– 22, , 11, , Draw a pair of tangents to a circle of radius 6cm which are inclined, , 4, , to each other at an angle of 600. Also find the length of the tangent., OR, Draw a circle of radius 6 cm. From a point 10 cm away from its, centre, construct the pair of tangents to the circle and measure its, length, 12, , The median of the distribution given below is 35.Find the values of, , 4, , 𝑥 and 𝑦 , if the sum of all frequencies is 170, , 13, , Variable, , 0-10, , Frequency, , 10, , 10-20 20-30, 20, , x, , 30-40 40-50 50-60 60-70, 40, , y, , 25, , 15, , Tower cranes are a common fixture at any major construction site., They are pretty hard to miss-they often raise hundreds of feet into, the air , and can reach out just as far. The construction crew uses the, tower crane to lift steel , concrete , large tools like acetylene, torches and generators and a wide variety of other building, materials, , A crane stands on a level ground. It is represented by a tower AB of, height 24 m and a jib BR. The jib is of length 16 m and can rotate in, a vertical plane about B.A vertical cable, RS, carries a load S. The, diagram above shows current position of jib, cable and load, 1.Find the distance BS, , 2, , 100

Page 101 : MATHEMATICS / X / 2021– 22, , 2.Find the angle that the jib BR makes with the horizontal, 14, , 2, , Ramesh a juice seller has set three types of glasses with inner, diameter 5 cm to serve customers. The height of the glasses is 10 cm, Type A-A glass with plane bottom, Type B-A glass with hemispherical raised bottom, Type C-A glass with conical raised bottom of height 1.5 cm, , 1) Find which glass has maximum capacity and which has, , 2, , minimum capacity? (Use π=3.14), 2) If vessel type A is melted to form spheres of radius 0.5 cm, , 2, , .How many spheres can be obtained from it ?, , 101

Page 102 : MATHEMATICS / X / 2021– 22, , Q.No, 1, , 2, , 3, , 𝑴𝒂𝒓𝒌𝒊𝒏𝒈 𝑺𝒄𝒉𝒆𝒎𝒆, Mathematics –Basic(241), Class- X Session- 2021-22, TERM II, HINTS/SOLUTION, 𝑏 − 4𝑎𝑐 = 0, (8𝑘) − 4𝑥9𝑥16 = 0, 𝑘 =, =9, k=±3, OR, 9𝑥 − 15𝑥 + 6 = 0, 9𝑥 − 9𝑥 − 6𝑥 + 6 = 0, (9x-6)(x-1) = 0, 2, 𝑥 = ,𝑥 = 1, 3, Number of balls =, , Marks, 0-10, 10-20, 20-30, 30-40, 40-50, 𝑓 = 32 ,𝑓, , =, ., ., =125, No. of students, 6, 10, 12, 32, 20, = 12 , 𝑓 = 20, , Mode = 𝑙 +, , 𝑥ℎ, , =30 +, = 30 +, , ., , 1, 1, 1, 1, , 1, , 1, , 1/2, , 𝑥 10, x 10, , =, =36.25, 4, , MARKS, 1, , 𝑎 = 𝑎 + (𝑛 − 1)𝑑, 420 = 21 (n-1) 21, 420-21 = (n-1) 21, 399/29 = n-1, 19 = n-1, n = 20, , 1/2, 1, , 1, , 102

Page 104 : MATHEMATICS / X / 2021– 22, , 8, , 1, , Let AB be the 10 m tower and BC be the length of shadow, tan 450 =, , 1, , 1=, BC = 10 m, Length of shadow = 10 m, OR, , 1, , Let AB be the height of wall and AC be the length of ladder ,, , 9, , 10, , Perimeter of ΔABC =AB + BC + AC, = AB+BX+CX+AC, = (AB+B)P+(AC+CQ) (BX=BP , CX =CQ), =AP+AQ (AP=AQ), = 2 AP =2 X 10 = 20 cm, Let one of the multiple of 3 be 3x, Successive multiple of 3 = 3x + 3, 3x(3x+3) = 180, 9x2 + 9x -180 = 0, X2 + x – 20 = 0, (x-4)(x+5), X = 4 , x=-5, Multiples of 3 are 12,15 and -15 ,-12, , 1, 1, 1, 1, 1, , 1, , SECTION C, , 104

Page 105 : MATHEMATICS / X / 2021– 22, , 11, , Draw a circle of radius 6 cm, Draw OA and construct  AOB= 1200, Draw  OAP =  OBP = 900, PA and PB are required tangents, Length of tangents = 10.4 cm (approx.), OR, , 1, 1, 1, , Draw a circle of radius 6 cm with centre O, Draw OP 10 cm and bisect it at M, With M as the centre draw a circle intersecting previous circle at A and B, Join PA and PB, PA=PB= 8 cm, , 1, 1, 1, , 1, , 1, , 105

Page 106 : MATHEMATICS / X / 2021– 22, , 12, , Variable, 0-10, 10-20, 20-30, 30-40, 40-50, 50-60, 60-70, , Frequency, 10, 20, X, 40, y, 25, 15, , Cumm Frequency(cf), 10, 30, 30+x, 70+x, 70+x+y, 95+x+y, 110+x+y, , N=170, 110+x+y=170, X + y =60, = 85 , l=30 ,cf = 30+x , f=40 , l=30, Median = 𝑙, , +, , 35 = 30 +, , 1, , 1, 1, , 𝑥ℎ, (, , ), , 𝑥 10, 1, , X= 35 , y= 25, , 13, , 1), In right ΔRSB , by Pythagoras theorem, 162 = 82 + BS2, 162- 82 = BS2, BS = 8√3 cm, , 2, , 2), sin θ =, =, , 2, , sin θ =, θ = 300, 1), , r =2.5 cm, Height = 10 cm, Volume of glass type A = πr2h= 3.14 x 2.5 x 2.5 x 10, = 196.25 cm3, Volume of glass type B = Volume of cylinder - Volume of, hemispherical base, = 196.25 + 2/3 πr3, = 196.25 – 32.71 = 163.54 cm 3, Volume of glass type C = Volume of cylinder - Volume of conical, base, = 196.25 + 1/3 πr2, = 196.25 – 9.81 = 163.54 cm 3, Type A is maximum volume and Type B has minimum volume, , 1, , 1, , 106

Page 107 : MATHEMATICS / X / 2021– 22, , 2), Number of spheres =, , 1, = 375, , 14, , 1, , 1), , r =2.5 cm, Height = 10 cm, Volume of glass type A = πr2h= 3.14 x 2.5 x 2.5 x 10, = 196.25 cm3, Volume of glass type B = Volume of cylinder - Volume of, hemispherical base, = 196.25 + 2/3 πr3, = 196.25 – 32.71 = 163.54 cm 3, Volume of glass type C = Volume of cylinder - Volume of conical, base, = 196.25 + 1/3 πr2, = 196.25 – 9.81 = 163.54 cm 3, Type A is maximum volume and Type B has minimum volume, 2), Number of spheres =, = 375, , 2, , 2, , 107

Page 108 : MATHEMATICS / X / 2021– 22, , Sample Question Paper, Mathematics- Standard (041), Class- X, Session: 2021-22, TERM II, Time Allowed: 2 hours, , Maximum Marks: 40, BLUE PRINT, , S., , NAME OF CHAPTER, , NO, , SA I, , SA II, , LA, , 2Marks, , 3Marks, , 4Marks, , TOTAL, , 1, , QUADRATIC EQUATIONS, , 4(2), , -, , -, , 4(2), , 2, , ARITHMETIC PROGRESSION, , 2(1), , -, , 4(1)*, , 6(2), , 3, , CIRCLES, , 2(1), , -, , 4(1), , 6(2), , 4, , CONSTRUCTIONS, , -, , 3(1), , -, , 3(1), , 5, , SOME APPLICATIONS OF, , -, , 3(1), , 4(1) *, , 7(2), , TRIGONOMETRY, 6, , SURFACE AREA AND VOLUMES, , 2(1), , -, , 4(1), , 6(2), , 7, , STATISTICS, , 2(1), , 6(2), , -, , 8(3), , 12(6), , 12(4), , 16(4), , 40(14), , TOTAL, *CASE STUDY BASED QUESTIONS, , 108

Page 109 : MATHEMATICS / X / 2021– 22, , Sample Question Paper, Mathematics- Standard (041), Class- X, Session: 2021-22, TERM II, Time Allowed: 2 hours, Maximum Marks: 40, General Instructions:, 1. The question paper consists of 14 questions divided into 3 sections A, B, C., 2. All questions are compulsory., 3. Section A comprises of 6 questions of 2 marks each. Internal choice has been, provided in two questions., 4. Section B comprises of 4questions of 3 marks each. Internal choice has been provided, in one question., 5. Section C comprises of 4 questions of 4 marks each. An internal choice has been, provided in one question. It contains two case study based questions., SECTION A, Q NO., 1, , MARKS, Find the 4th term from the end of the AP -11, -8, -5, .........., 49., , 2, , OR, Find the value of the middle most term (s) of the AP :, –11, –7, –3,..., 49., 2, , Find the values of k for the following quadratic equation, so that it has, , 2, , two equal roots., kx(x - 2) + 6 = 0., 3, , In figure, PQ and PR are tangents to the circle with center O and S is a point, , 2, , on the circle such that ∠SQL =500 and ∠SRM =600. Find ∠QSR., , 109

Page 110 : MATHEMATICS / X / 2021– 22, , 4, , The sum of the radius of base and height of a solid right circular, , 2, , cylinder is 37 cm. If the total surface area of the solid cylinder is 1628, sq. cm, find the radius and height of the cylinder. ( 𝜋 =, 5, , ), , The mode of the following frequency distribution is 36. Find the, , 2, , missing frequency f ., Class, , Frequency, , 6, , 0-10, , 10-20, , 20-30, , 8, , 10, , f, , 30-40 40-50, , 16, , 12, , 50-60, , 60-70, , 6, , 7, , Had Salma scored 10 more marks in her mathematics test out of 30, , 2, , marks, 9 times these marks would have been the square of her actual, marks. How many marks did she get in the test?, OR, 2, , Solve the quadratic equation, 2x + ax - a2 = 0 for x ., SECTION B, 7, , Find the mean of the following distribution :, , 3, , Height (in cm), , No. of students, , Less than 75, , 5, , Less than 100, , 11, , Less than 125, , 14, , Less than 150, , 18, , Less than 175, , 21, , Less than 200, , 28, , Less than 225, , 33, , Less than 250, , 37, , Less than 275, , 45, , 110

Page 111 : MATHEMATICS / X / 2021– 22, , Less than 300, 8, , 50, , Construct a pair of tangents to a circle of radius 5 cm which are, , 3, , inclined to each other at an angle of 60°., 9, , The following table shows the weights (in gms) of a sample of 100, , 3, , apples, taken from a large consignment., Weight (in gms), , No. of Apples, , 50 – 60, , 8, , 60 – 70, , 10, , 70 – 80, , 12, , 80 – 90, , 16, , 90 – 100, , 18, , 100 – 110, , 14, , 110 – 120, , 12, , 120 – 130, , 10, , Find the median weight of apples., 10, , The angle of elevation of the top of a vertical tower from a point on, , 3, , the ground is 60°. From another point 10 m vertically above the first,, its angle of elevation is 45°. Find the height of the tower., OR, From a point on a bridge across a river, the angles of depression of the, banks on opposite sides of the river are 30° and 45°, respectively. If, the bridge is at a height of 4 m from the banks, find the width of the, river. ( Take √3 =1.732), SECTION C, 11, , 504 cones, each of diameter 3.5 cm and height 3 cm, are melted and, , 4, , recast into a metallic sphere. Find the diameter of the sphere and, hence find its surface area., Use 𝜋 =, , ., , 111

Page 112 : MATHEMATICS / X / 2021– 22, , 12, , In the given figure, O is the centre of the circle Determine ∠𝐴𝑃𝐶 , if, , 4, , DA and DC are tangents and ∠𝐴𝐷𝐶 = 50°., , OR, Prove that opposite sides of a quadrilateral circumscribing a circle, subtend supplementary angles at the centre of the circle., 13, , CASE STUDY 1, , 2+2=4, , A road roller (sometimes called a roller-compactor, or just roller) is a, compactor-type engineering vehicle used to compact soil, gravel,, concrete, or asphalt in the construction of roads and foundations., Similar rollers are used also at landfills or in agriculture. Road rollers, are frequently referred to as steamrollers, regardless of their method of, propulsion., , RCB Machine Pvt Ltd started making road roller 10 year ago., Company increased its production uniformly by fixed number every, year. The company produces 800 roller in the 6th year and 1130 roller, in the 9th year., , 112

Page 113 : MATHEMATICS / X / 2021– 22, , (i), , How many road rollers the company might have produced, in its first year ? What was the company’s production in, the 8th year ?, , (ii), , Find the total number of road rollers produced by the, company till now?, , 14, , CASE STUDY 2, , 2+2=4, , A hot air balloon is a type of aircraft. It is lifted by heating the air, inside the balloon, usually with fire. Hot air weighs less than the same, volume of cold air (it is less dense), which means that hot air will rise, up or float when there is cold air around it, just like a bubble of air in a, pot of water. The greater the difference between the hot and the cold,, the greater the difference in density, and the stronger the balloon will, pull up., , Lakshman is riding on a hot air balloon. After reaching at height x at, point P , he spots a lorry parked at B on the ground at an angle of, depression of 300. The balloon rises further by 50 metres at point Q, and now he spots the same lorry at an angle of depression of 45 0 and a, car parked at C at an angle of depression of 300., (i), , What is the relation between the height x of the balloon at, point P and distance d between point A and B? When, balloon rises further 50 metres, then what is the relation, between new height y and d ?, , 113

Page 114 : MATHEMATICS / X / 2021– 22, , (ii), , Find the distance between the lorry and the car., , ****************************, , Sample Question Paper- Marking Scheme, Mathematics- Standard (041), Class- X, Session: 2021-22, TERM II, Time Allowed: 2 hours, Maximum Marks: 40, MARKING SCHEME, SECTION A, Q NO, 1, , MARKS, a = −11, d = 3 , an = 49, an = a + (n−1)d, 49 = −11 + (n − 1) × 3, n −1 = 20, n = 21, Fourth term from the end =18th term, a18 = a+(18−1)d = − 11 + 17 × 3 = −11+51 = 40, , 1, 1, , [ Alternate Method can also be adopted ], OR, Here, a = –11, d = –7 – (–11) = 4, an = 49, We have an = a + (n – 1) d, So, 49 = –11 + (n – 1) × 4, i.e., 60 = (n – 1) × 4, i.e., n = 16, , 1, , As n is an even number, there will be two middle terms which are, ( 16/2)th and ( 16/2 )+1 )th term, 8th term and the 9th term., a8 = a + 7d = –11 + 7 × 4 = 17, , 114

Page 115 : MATHEMATICS / X / 2021– 22, , a9 = a + 8d = –11 + 8 × 4 = 21, So, the values of the two middle most terms are 17 and 21, respectively., 2, , 1, , Given quadratic equation., kx(x - 2) + 6 = 0., i.e:; kx2 – 2kx + 6 = 0, Since the equation has two equal roots , b2 – 4ac = 0, , ½, ½, , (-2k )2 – 4 x k x 6 = 0, 4 k ( k – 6) = 0, k=0 or k = 6, k cannot be 0 , so k = 6, 3, , 1, , Join OQ and OR, , ∠ORP = ∠OQP = 900 (The tangent to the circle is perpendicular to the radius, of the circle at the point of contact. ), , ½, , ∠OQS = 90∘ − 50∘ ⇒ ∠OQS = 40∘, ∠ORS = 90∘ − 60∘ ⇒ ∠ORS = 30∘, OS=OQ=OR= radius..............(1), , ½, , ∠OSQ=∠OQS........(2), ∠OSR=∠ORS.........(3) [angles opposite to equal sides of an isosceles triangle, , ½, , are equal.], , 115

Page 116 : MATHEMATICS / X / 2021– 22, , 4, , ∠QSR = ∠OSQ + ∠OSR ⇒ ∠QSR = 40∘ + 30∘ ⇒ ∠QSR = 70∘, , ½, , We have r + h = 37 …………(1), , ½, , and 2𝜋𝑟 (𝑟 + ℎ ) = 1628 ………………….(2), , ½, , Thus 2𝜋𝑟 × 37 = 1628, 2𝜋𝑟 =, , ½, , & r = 7 cm, , Substituting r = 7 in (1) we have, h = 30 cm., 5, , ½, , The mode = 36 ., Class, , 0-10, , 10-20, , 20-30, , 8, , 10, , f, , Frequency, , 30-40 40-50, , 16, , 50-60, , 60-70, , 6, , 7, , 12, , Modal class = 30-40, l=30, f0 = f , f1 = 16 , f2 =12 , h = 10, mode = 𝑙 +, , ×ℎ, , 36 = 30 +, 6=, , ×, , × 10, , 1, , x10, , f= 10, 6, , ½, , ½, , Let Salma’s actual marks be x, Therefore, 9 (x +10) = x2, i.e., x2 – 9x – 90 = 0, , 1, , i.e., x2 – 15x + 6x – 90 = 0, i.e., x(x – 15) + 6(x –15) = 0, i.e., (x + 6) (x –15) = 0, Therefore, x = – 6 o r x =15, Since x is the marks obtained, x ≠ – 6. Therefore, x = 15., , 1, , 116

Page 117 : MATHEMATICS / X / 2021– 22, , So, Ajita got 15 marks in her mathematics test., OR, 2x2+ ax - a2 = 0, By quadratic formula ,, 𝑥=, , =, , −𝑏 ± √𝑏 − 4𝑎𝑐, 2𝑎, , 1, , −𝑎 ± √𝑎 + 4 × 2 × 𝑎, 2×2, −𝑎 ± √9𝑎, 4, , =, , =, , −𝑎 ± 3𝑎, 4, , =, , ,, 1, , 𝑎, = −𝑎 ,, 2, SECTION B, 7, , We prepare following table to find mean., Class, , Frequency xi, , di=xi - a, , fidi, , Interval, , fi, , 50-75, , 5, , 62.5, , - 125, , - 625, , 75-100, , 6, , 87.5, , -100, , - 600, , 100- 125, , 3, , 112.5, , -75, , -225, , 125- 150, , 4, , 137.5, , -50, , -200, , 150 175, , 3, , 162.5, , -25, , -75, , 175 - 200, , 7, , a=187.5, , 0, , 0, , 117

Page 118 : MATHEMATICS / X / 2021– 22, , 200 – 225, , 5, , 212.5, , 25, , 125, , 225 – 250, , 4, , 237.5, , 50, , 200, , 250 – 275, , 8, , 262.5, , 75, , 600, , 275 - 300, , 5, , 287.5, , 100, , 500, , total, , 50, , Mean = a +, , ∑, ∑, , -300, , 2, , = 187.5 – (300/50) = 181.5, 1, , 8, , Correct construction, , 3, , 9, , Cumulative, Weight (in, , No. of Apples, , Frequency(cf), , gm ), 50 – 60, , 8, , 8, , 60 – 70, , 10, , 18, , 70 – 80, , 12, , 30, , 80 – 90, , 16, , 46, , 90 – 100, , 18, , 64, , 100 – 110, , 14, , 78, , 110 – 120, , 12, , 90, , 120 – 130, , 10, , 100, , 1, , We have n = 100 , n/2 =50, Median class = 90 – 100, l=90 , cf = 46 , f = 18 , h = 10, , ½, , 118

Page 119 : MATHEMATICS / X / 2021– 22, , Median = 𝑙 +, , (, , ), , = 90 +, , ×ℎ, × 10 = 90 + 40/18 = 92.2 g, , 1½, , Median weight of apples = 92.2 g, 10, , Let OT be the tower., , 1, (correct fig), 0, , In ∆POT , tan 60 =, In ∆ABT , tan 450 =, ⟹ x=, , √, , ⟹ √3 =, , ⟹ 𝐻 = 𝑥√3 ………………(1), , ½, ½, , ⟹ 𝑥 = 𝐻 − 10 ⟹ x = 𝑥 √3 − 10, , ⟹1 =, , ⟹ x = 5 (√3 + 1) m, , Height of tower = H = 5√3 (√3 + 1) m, , 1, , OR, Given : Height of bridge from river AO = 4 m, 1, , 119

Page 120 : MATHEMATICS / X / 2021– 22, , BO = x, and CO = y, , ½, , From right angled ∆AOB,, AO/BO = tan 45° ⇒ 4/x = 1 ⇒ x = 4 m ……(i), ½, , Again from right angled ∆AOC,, AO/CO = tan 30° ⇒ 4/y = 1/√3 ⇒ y = 4√3 m ……(ii), From equation (i) and (ii), , 1, , Width of river = x + y = 4√3 + 4, = 4(√3 + 1) = 4(1.732 + 1) = 4(2.732) = 10.92 m, Hence, width of river is 10.92 m, SECTION C, 11, , Volume of sphere is equal to the volume of 504 cones., , ½, , 4, 1, 𝜋𝑟𝑅 = 504 × 𝜋𝑟 ℎ, 3, 3, , 1, , ⟹ 4𝑅 = 504 × 𝑟 ℎ, ., , ⟹ 𝑅 = 504 ×, ⟹ 𝑅 = 504 x, , ., , ×, , ×3, ., , ×, , ., , x 3 = 9 x 3.5 x 3.5x 3.5 x 3, , 𝑅 = 3 𝑥3.5 = 10.5 𝑐𝑚, Diameter of sphere = 21 cm, Surface area = 4𝜋𝑟 = 4 ×, 12, , × 10.5 × 10.5 = 1386 𝑐𝑚 ., , 1, ½, 1, , Join OA and OC, , Figure with correct construction ………………, Since DA and DC are tangents from point D to the circle with centre O, and, radius is always perpendicular to tangent, thus, ∠DAO = ∠DCO = 900 and, , ½, , 120

Page 121 : MATHEMATICS / X / 2021– 22, , ∠ADC +∠DAO +∠DCO +∠AOC = 3600, 500 + 900 + 900 +∠AOC = 3600, , 1, , 230c +∠AOC = 3600, ∠AOC = 3600 - 2300 = 1300, Now, Reflex ∠AOC = 3600- 1300 = 2300, ∠APC= ½ reflex ∠AOC, , 1, , = 1150, OR, , ½, 1, , Correct figure …………………………………………, A circle centre O is inscribed in a quadrilateral ABCD as shown in figure.., Since OE and OF are radius of circle,, OE = OF, , 1, , Tangent drawn at any point of a circle is perpendicular to the radius through, the point contact., Thus ∠OEA = ∠OFA = 900, Now in ∆ AEO and ∆ AFO,, OE = OF, ∠OEA = ∠OFA = 900, , ½, , OA = OA (Common side), Thus ∆ AEO ≅ ∆AFO (SAS congruency), ∠7 =∠8 (CPCT), Similarly, ∠1 =∠2, ∠3 = ∠4, ∠5 = ∠6, Since angle around a point is 3600, ∠1 +∠2 +∠3 +∠4 +∠5 +∠6 +∠7 +∠8 = 3600, , 1½, , 121

Page 122 : MATHEMATICS / X / 2021– 22, , 2∠1 + 2∠8 + 2∠4 + 2∠5 = 3600, ∠1 +∠8 +∠4 +∠5 = 1800, ∠AOB +∠COD = 1800, 1, , Hence Proved., 13, , CASE STUDY 1, (i) Let a be the production in first year and d be the increase every year in, production., We have a6 = 800 and a9 = 1130, a + 5d = 800 ...(1), a + 8d = 1130 ...(2), Solving (1) and (2), we get, , 1, , d = 110, a = 800 – 5x110 = 250, The company produced 250 road rollers in its first year, Since, a = 250 and d = 110, a8 = a + (8 - 1)d, , 1, , = 250 + 7x110 = 1020, The company produced 250 road rollers in its 8th year., (ii), , The total number of road rollers produced by the company till now = S 10, =, , 14, , [2 x 250 + 9 x 110] = 7450, , 2, , CASE STUDY 2, , (i), , In ∆𝐴𝑃𝐵 , tan 300 = AP /AB, , 122

Page 124 : MATHEMATICS / X / 2021– 22, , SAMPLE QUESTION PAPER - 2, MATHEMATICS – STANDARD (041), CLASS X SESSION 2021-22, TERM – II, Time Allowed : 2 Hrs, Maximum Marks: 40, General Instructions:, , 1. The question paper consists of 14 questions divided into three sections A, B and C., 2. All questions are compulsory., 3. Section A comprises of 6 questions of 2 marks each. Internal choice has been, provided in two questions., 4. Section A comprises of 4 questions of 3 marks each. Internal choice has been, provided in one question., 5. Section A comprises of 4 questions of 4 marks each. An internal choice has been, provided in one question. It contains two case study based questions., SECTION – A, 1. Find the value of k if the discriminant of the equation kx2 – 3√2x + 4√2 = 0 is 10., 2. Find the 12th term from the end of the A.P – 2, – 4, – 6, ……., – 100., OR, Find the sum of first seven numbers which are multiples of 2 as well as 9., 3. In the given figure, AB and AC are tangents to the circle with centre o such that, ∠BAC = 40°. Then calculate ∠BOC and ∠OBC, , 4. If the volumes of two spheres are in the ratio 64 : 27, find the ratio of their surface, areas., 5. If the mean of the following data is 18.75. find the value of p., xi, , 10, , 15, , 20, , 25, , 30, , fi, , 5, , 10, , p, , 8, , 2, , 6. Solve the following quadratic equation for x:, , x2 – 2ax – (4b2 – a2) = 0, , OR, , 124

Page 125 : MATHEMATICS / X / 2021– 22, , Three consecutive natural numbers are such that the square of the middle number, exceeds the difference of the squares of the other two by 60. Find the numbers., SECTION – B, 7. Draw a circle of radius 4 cm. From a point 6cm from its centre, construct a pair of, tangents to the circle and measure their lengths., 8. Given below is the distribution of weekly pocket money received by students of a, class., Calculate the pocket money that is received by most of the students., Pocket, , 0 - 10, , 20 - 40, , Money (in, , 40 -, , 60 -, , 80 –, , 100 –, , 120 –, , 60, , 80, , 100, , 120, , 140, , 3, , 12, , 18, , 5, , 2, , ₹), No. of, , 2, , 2, , students, , 9. A survey regarding the heights ( in cm) of 51 boys of class X of a school was, conducted and the following data was obtained., Height (in cm), , Number of boys, , Less than 140, , 4, , Less than 145, , 11, , Less than 150, , 29, , Less than 155, , 40, , Less than 160, , 46, , Less than 165, , 51, , 10. The angle of elevation of an aeroplane from a point on the ground is 60°. After a, flight of 30 seconds the angle of elevation becomes 30°. If the aeroplane is flying, at a constant height of3000√3 m, find the speed of the aeroplane., OR, From the top of a hill, the angle of depression of two consecutive kilometer stones, due East are found to be 300 and 450. Find the height of the hill. ( Use √3 = 1.73), , 125

Page 126 : MATHEMATICS / X / 2021– 22, , SECTION – C, 11. In the figure, l and m are two parallel tangents to a circle with centre O, touching, the circle, at A and B respectively. Another tangent at C intersects the line I at D and m at E., Prove that ∠DOE = 90°., , OR, Prove that opposite sides of a quadrilateral circumscribing a circle subtend, supplementary angles at the centre of the circle., 12. A model of a rocket is in the form of a right circular cylinder closed at the lower, end and surmounted, by a cone with the same radius as that of the cylinder. The diameter and height of, the cylinder are, 6 cm and 12 cm respectively. If the slant height of the conical portion is 5 cm, then, find the total surface area of the rocket., 13. Case Study – 1, One fine evening, Surabhi was standing on the balcony of her house watching her, brother Sonu play ball. She observes the ball at an angle of depression 30 o. The, ball is now approaching the foot of the buiding in a straight line with a uniform, speed. Six second’s later, the angle of depression of the ball is found to be 60 o., Now the ball is at a point 25m away from the foot of the building.., , 126

Page 127 : MATHEMATICS / X / 2021– 22, , Based on the above information, answer the following questions:, (a) Find the distance between the two positions of the ball., (b) Find the speed of the ball and total time taken to reach the foot of the, building., 14. Case Study – 2, In a potato race, a bucket is placed at the starting point, which is 5 m from the first, potato and the other potatoes are placed 3 m apart in a straight line. There are ten, potatoes in lines (see below figure). A competitor starts from the bucket, picks up, the nearest potato, runs back with it, drops it in the bucket, runs back to pick up, the next potato, runs to the bucket to drop it in and she continues in the same way, until all the potatoes are in the bucket.., , Keeping the above situation in mind, answer the following questions:, (a) What is the distance run to pick up the 6th potato?, (b) What is the total distance run by the competitor?, ANSWERS:, , 1., , √, , 2. – 78, , OR 504., , 3. 140O and 20O, 4. 16 : 9, 5. p = 7, 6. x = a – 2b , a + 2b., , OR 9 ,10 ,11., , 7. 4.48 cm, 8. Rs. 86.32 ( approx.), , 127

Page 128 : MATHEMATICS / X / 2021– 22, , 9. 149.03 m, 10. 1.365 km OR 720 km., 11. ., 12. 301.44 cm2, 13. (a) 50m, , (b), , 𝑚/s , 7.8 seconds., , 14. 40m, 370m., , 128