email: [email protected]ÎCî¦ô¢Ù 17 ì÷Ùñô¢ª 2019 Long Answer Type Questions (4M) 1. Show that the square of any positive integer is of the form 3q or 3q + 1 using Euclid’s division lemma? A: Let a be any positive integer and b = 3 Then by Euclid’s algorithm a = 3q + r for q 0 and r = 0, 1, 2 a = 3q or 3q + 1 or 3q + 2 Case 1: a = 3q a 2 = (3q) 2 = 9q 2 = 3(3q 2 ) = 3k 1 Case 2: a = 3q + 1 a 2 = (3q + 1) 2 = 9q 2 + 6q + 1 = 3 (3q 2 + 2q) + 1 = 3k 2 + 1 Case 3: a = 3q + 2 a 2 = (3q + 2) 2 = 9q 2 + 12q + 4 = 3(3q 2 + 4q + 1) +1 = 3k + 1 Hence square of any positive integer is of the form 3q or 3q + 1 2. Prove that 3 + 5 is irrational A: Assume that 3 + 5 is rational Let 3 + 5 = p q (p, q Z, p, q co-primes, q 0) p 3 = 5 q Squaring both sides p 2 p 3 = + 5 2 5 q 2 q 2p p 2 p 2 5 = + 5 3 = + 2 q q 2 q 2 p 2 + 2q 2 = q 2 p 2 + 2q 2 q p 2 q + 2q 3 5 = ( )( ) = q 2 2p 2pq 2 p 2 q + 2q 3 p, q Z, is rational 2pq 2 5 is rational But this contradicts that 5 is irrational. Our assumption that 3 + 5 is rational is wrong 3 + 5 + is irrational. 3. Prove that 5 is irrational. A: Let us assume 5 that is rational. Then there exist a, b Z, b 0 such that a 5 = b b 5 = a Squaring on both sides 5b 2 = a 2 5 divides a 2 5 divides a (1) So, we can take, a = 5c b5 = 5c Squaring both sides 5b 2 = 5 2 c 2 b 2 = 5c 2 5 divides b 2 5 divides b (2) From (1) and (2) We can say that 5 is a common factor to a and b. This contradicts the fact that a and b are co primes. So, our assumption that 5 is rational is false. So, 5 is irrational. x + y 1 4. If log = (log x + log y), then 3 2 x y find the value of + y x x + y Sol: 2 log = (log x + log y) 3 x + y 2 log ( ) = log xy 3 x + y 2 ( ) = xy 3 x 2 + 2xy + y 2 = xy 9 x 2 + 2xy + y 2 = 9xy x 2 + y 2 = 9xy 2xy x 2 + y 2 = 7xy x 2 y 2 7xy + = xy xy xy x y + = 7 y x TSVS Suryanarayana Murthy Writer Additional Problems 1. Prove that 11 is an irrational number by the method of contradiction. 2. If (2.3) = (0.23) y = 1000 then find the 1 1 value of + x y 3. Prove that 3 + 2 is irrational Target-2020 Tenth Maths - Paper I 100 100 Real Numbers 7 1. Write the decimal form of 8 A: 0.875 2. Statement A: HCF of 17 and 23 is 1 Statement B: HCF of relatively prime numbers is always 1 (i) Only statement A is true (ii) Only statement B is true (iii) Statement A and B are true but state- ment B does not support statement A (iv) Statement A and B are true and state- ment B supports statement A. A: (iv) 3. What is the last digit of the expansion 6 2019 A: 6 4. What are the quotient and remainder when we find the HCF of 300 and 550. A: quotient = 5 and remainder = 0. 5. Which of the following are terminating decimals? 33 7 97 , , 7 80 17 7 A: 80 6. Which of the following cannot be p expressed in the form of ? q 43.123456789, 0.142857142857…., 0.12012001200012000….. A: 0.12012001200012000….. 7. Statement A: If 5 divides 625, then 5 divides 25. Statement B: Let ‘p’ be a prime. If p divides a 2 then p divides a (i) Only statement A is true (ii) Only statement B is true (iii) Both the statements A and B are true and statement B is correct explanation of statement A (iv) Both the statements A and B are true but statement B is not correct explanation of statement A. A: (iii) 8. Match the following: Group A Group B 1. Log xy = a. y Log x 2. Log x y = b. Log x Log y x 3. Log = c. Log x + Log y y A) 1-a, 2-b, 3-c B) 1-c, 2-a, 3-b C) 1-c, 2-b, 3-a D) 1-b, 2-a, 3-c A: B 128 1. Expand log ? 343 x A: We know that = x y y 128 log = log 128 log 343 343 = log 2 7 log 7 3 = 7log 2 3log7 [ . . . log a m = m log a] 2. Is 7 11 13 + 7 a composite number? Why? A: 7 11 13 + 7 = 7 (11 13 + 1) = 7 144 = 7 2 2 2 2 3 3 = 7 1 2 4 3 2 (7 11 13 + 7) can be written as the product of primes. It is a composite number. (Or) 7 11 13 + 7 = 7 (11 13 + 1) = 7 144 The number has factors 7 and 144 other than 1 and itself. So it is a composite number. 15 3. Find whether is terminating or 1600 non-terminating decimal without actual division. 15 3 A: = 1600 320 3 3 = = 2 2 2 2 2 2 5 2 6 5 1 . . . Denominator is of the form 2 m 5 n . 15 is a terminating decimal. 1600 1 4. Find ‘x’ if 2log5 + log9 log3 = logx 2 1 A: logx = 2log5 + log9 log3 2 = log5 2 + log9 1 2 log3 = log 25 + log9 log3 = log25 + log3 log3 = log25 x = 25 5. Determine the values of log 81 3 and log 32 8. 1 1 1 A: log 81 3 = log 3 4 3 = log 3 3 = 1 = 4 4 4 3 3 3 log 32 8 = log 2 5 2 3 = log 2 2 = 1 = 5 5 5 1 2 Mark Questions One Mark Questions Write the decimal form of... Two Marks Questions 1. Find LCM and HCF of 72 and 108 by prime factorization method. A: 72 = 2 36 = 2 2 18 = 2 2 2 9 = 2 2 2 3 3 108 = 2 54 = 2 2 27 = 2 2 3 9 = 2 2 3 3 3 LCM = 2 2 2 3 3 3 = 216 HCF = 2 2 3 3 = 36 2. Solve 2 x+1 = 3 1x A: Given 2 x+1 = 3 1x Taking logs on both sides log 2 x+1 = log 3 1x [ . . . log x m = m log x] (x + 1) log2 = (1 x) log3 xlog2 + log2 = log3 xlog3 xlog2 + xlog3 = log3 log2 x(log2 + log3) = log3 log2 log3 log2 x = log2 + log3 3. If x 2 + y 2 = 29xy Prove that 2log(x y) = 3log3 + logx + logy A: Given, x 2 + y 2 = 29xy x 2 + y 2 2xy = 29xy 2xy = 27xy (x y) 2 = 3 3 xy Taking logs on both sides Log (x y) 2 = log (3 3 xy) 2 log (x y) = log3 3 + logx + logy = 3log 3 + log x + log y 4. Write log10 + 3log5 2log5 as log N and find N A: log10 + 3log5 2log5 = log10 + log5 3 log 5 2 [ . . . log a m = m log a] = log (10 125) log25 [ . . . log x + log y = log xy] 1250 x = log ( ) [ . . . log x log y = log ] 25 y = log 50 = log N N = 50 Additional Questions 1. Find H.C.F of 50 and 70 by using Euclid division lemma? 2. Write 2log3 + 3log5 5log2 as a single logarithm? 3. Solve 3 x = 5 x2 4. Find the HCF of 220 and 284. 5. How will you show that (19 13 2) + (19 13 5) is a composite number? Explain. 6. Use Euclid’s division algorithm to find the HCF of 900 and 270? 7. Use Euclid’s division algorithm to find HCF of 870 and 225? 8. Use Euclid’s division algorithm to find the HCF of 96 and 72?
the method of contradiction.2. If (2.3) = (0.23)y = 1000 then find the
1 1value of + x y
3. Prove that 3 + 2 is irrational
Target-2020
TenthMaths - Paper I
100100
Real Numbers
71. Write the decimal form of 8
A: 0.8752. Statement A: HCF of 17 and 23 is 1
Statement B: HCF of relatively primenumbers is always 1(i) Only statement A is true(ii) Only statement B is true(iii) Statement A and B are true but state-
ment B does not support statement A(iv) Statement A and B are true and state-
ment B supports statement A.A: (iv)3. What is the last digit of the expansion 62019
A: 64. What are the quotient and remainder when
we find the HCF of 300 and 550.A: quotient = 5 and remainder = 0.5. Which of the following are terminating
decimals?33 7 97 , , 7 80 177A: 80
6. Which of the following cannot be p
expressed in the form of ?q43.123456789, 0.142857142857….,0.12012001200012000…..
A: 0.12012001200012000…..7. Statement A: If 5 divides 625, then 5
divides 25.Statement B: Let ‘p’ be a prime. If pdivides a2 then p divides a (i) Only statement A is true(ii) Only statement B is true(iii) Both the statements A and B are true
and statement B is correct explanationof statement A
(iv) Both the statements A and B aretrue but statement B is not correctexplanation of statement A.
A: (iii)8. Match the following:
Group A Group B1. Log xy = a. y Log x2. Log xy = b. Log x Log y
x3. Log = c. Log x + Log yyA) 1-a, 2-b, 3-c B) 1-c, 2-a, 3-bC) 1-c, 2-b, 3-a D) 1-b, 2-a, 3-c A: B
1281. Expand log ?
343x
A: We know that = x yy
128 log = log 128 log 343343
= log 27 log 73
= 7log 2 3log7 [... log am = m log a]2. Is 7 11 13 + 7 a composite number?
Why?A: 7 11 13 + 7 = 7 (11 13 + 1)
= 7 144 = 7 2 2 2 2 3 3 = 71 24 32
(7 11 13 + 7) can be written as theproduct of primes.
It is a composite number.
(Or)
7 11 13 + 7 = 7 (11 13 + 1)= 7 144
The number has factors 7 and 144 otherthan 1 and itself. So it is a composite number.
153. Find whether is terminating or
1600non-terminating decimal without actualdivision.
15 3A: =
1600 3203 3
= = 2 2 2 2 2 2 5 26 51
... Denominator is of the form 2m5n.
15 is a terminating decimal.1600
14. Find ‘x’ if 2log5 + log9 log3 = logx2
1A: logx = 2log5 + log9 log3 2
= log52 + log912 log3
= log 25 + log9 log3
= log25 + log3 log3 = log25
x = 255. Determine the values of log813 and log328.
1 1 1 A: log813 = log34 3 = log33 = 1 =
4 4 4
3 3 3log328 = log25 23 = log22 = 1 =
5 5 5
12 Mark Questions One Mark Questions
Write the decimal form of...
Two Marks Questions
1. Find LCM and HCF of 72 and 108 by primefactorization method.
3. If x2 + y2 = 29xy Prove that 2log(x y)= 3log3 + logx + logy
A: Given, x2 + y2 = 29xy
x2 + y2 2xy = 29xy 2xy = 27xy
(x y)2 = 33xy
Taking logs on both sidesLog (x y)2 = log (33xy)2 log (x y) = log33 + logx + logy
= 3log 3 + log x + log y4. Write log10 + 3log5 2log5 as log N and
find NA: log10 + 3log5 2log5
= log10 + log53 log 52
[ ... log am = m log a]= log (10 125) log25
[... log x + log y = log xy]
1250 x= log ( ) [ ... log x log y = log ]25 y
= log 50 = log N N = 50
Additional Questions
1. Find H.C.F of 50 and 70 by using Eucliddivision lemma?
2. Write 2log3 + 3log5 5log2 as a singlelogarithm?
3. Solve 3x = 5 x2
4. Find the HCF of 220 and 284.5. How will you show that (19 13 2) + (1913 5) is a composite number? Explain.
6. Use Euclid’s division algorithm to find theHCF of 900 and 270?
7. Use Euclid’s division algorithm to find HCFof 870 and 225?
8. Use Euclid’s division algorithm to find theHCF of 96 and 72?
Attitude is AltitudeReading ComprehensionI. Read the following passage.
'When I was 13, I read anewspaper article about adisabled man who hadmanaged to achieve greatthings and helped others',said Nick.
'I realised why God hadmade us like this - to givehope to others. It was soinspirational to me that I decided to use mylife to encourage other people and give themthe courage that the article had given me'.
'I decided to be thankful for what I dohave, not get angry about what I don't'.
'I looked at myself in the mirror and said:'You know what the world is right that I haveno arms or legs, but they'll never take awaythe beauty of my eyes'. I wanted to concen-trate on something good that I had'.
'The challenges in our lives are there tostrengthen our convictions. They are notthere to run us over', said Nick. In 1990 Nick
won the Australian young citizen of the yearaward for his bravery and perseverance.
'And once I was in a car and a girl at traf-fic lights was looking at me interestingly. Shecould only see my head so I decided to do a360 degree spin in the car seat to freak herout. Her face was like woooooooah what isgoing on? She sped off really quickly'.
Nick began travelling the world and in2008 he went to Hawaii and met surfingmaster Bethany Hamilton, who had her armbitten off by a shark when she was 12.
Choose the correct answer from theoptions given and write in your answerscript. 4 1 = 41. The inspiration that the article had given
Nick is .... ( )A) to achieve great thingsB) to feel sorry for his conditionC) to hate God for creating him without
limbs
D) to use his life to encourage other peo-ple
2. I realized why god had made us like thiswhat does the word 'us' in the sentencemean?
A) The disabled B) Nick's parents C) Nick's football fans D) All the people
3. Nick decided to do a 360 in car seatbecause
A) to frighten her B) to make her very excitedC) to overtake her D) to take revenge on her
4. What type of text is this?A) NarrativeB) DescriptiveC) Biographical sketch D) Autobiography
Answers: 1) D 2) A 3) B 4) C.
Answer the following questions in two orthree sentences. 3 2 = 6
5. "The challenges in our lives are there tostrengthen our convictions". How canyou justify this statement to Nick's life?
A: Nick Vujicic over come his disability bytaking every thing as challenge.
6. "I wanted to concentrate on somethinggood that I had". What does Nick men bysaying this?
A: Even though I have no legs and armssomething good that I had like beautifuleyes. On which I would concentrate andmake my life encourage other.
7. Who was Bethany Hamilton? Where didNick meet her?
A: Bethany Hamilton was a surfing master.Nick met her at Hawaii.
Read the passage below. Five sentencesin the passage are numbered (1 - 5) at thebeginning. Each of these sentences hasan error correct and rewrite them in theanswer booklet. 5 1 = 5
(1) Nick, who was teased and bullied, hadan electric wheel chair for mobility, and ateam of career for help him. (2) 'I was deepdepressed when I was eight years old' hesaid. (3) I went to my mom cried and toldher I wanted to kill myself. (4) I feel coldand bitter. (5) I hated God for doing this tome and was terrified of what would happens when my parents weren't there tolook for me.
Answers1) Nick who was teased and bullied had an
electric wheel chair for mobility, and a
team of career to help him.2) I was deeply depressed when I was
'eight years old' he said.3) I went to my mum crying and told her I
wanted to tell my self.4) I felt cold and bitter.5) I hated God for doing this to me and was
terrified of what would happen when myparents weren't there to look after me.
Complete the following passage choosing the right words from thosegiven in the box. Write the answers in
1 1your booklet. 5 = 2
2 2torso, stretch loved, around, born, imagine.
Imagine having no arms to …..(1) in themorning, to help you scratch that itch, toallow you to wrap your arms ….. (2) yourloved ones. Imagine having no legs to kickpebbles down the street, to walk or run tobicycle or skate board, or get you from pointA to point B. Then …... (3) both at once,which is what Nick Vujcic has faced hiswhole life have you heard something likethis before? Nick Vujicic was ….. (4) with noarms and legs - but he does not let thedetails stop him. The brave 26 year old whois mainly a ….. (5) plays football and golf,swims and surfs, despite having no limbs.
Answers
1) stretch 2) around 3) imagine 4) born 5) torso
Major discourses
1. In the lesson "Attitude is Altitude" youhave read Nick became great motiva-tional speaker. He travels all over theworld and gives motivational speechesand gives hope to millions of people. Hewon the Australian young citizen of theyear award to his bravery and persever-ance and many more awards.Now imagine that you are news reporterand write an imaginary interviewbetween you and Nick. Include followingpersonal details of Nick.l Hardships he facedl Achievements and awards
2. Describe the feelings of Nick Vujicicwhen he drew the inspiration from thenews paper article. Your descriptionshould have following ideas.l What was the news article?
l What hope did the article give him?l What was the commitment?l How his inspiration changed his life?
3. You have read the lesson "Attitude isAltitude". Describe the disadvantagesfaced by disabled man without arms andlegs taking references from the lesson"Attitude is Altitude''.
Creative Writing
The challenges in our lives
Vocabulary and Grammar
ZPHS KANKAL
NoticeNICK VUJICIC MOTIVATIONAL
CLASSAll the students are here by by informedthat there will be a "Nick Vujicic" motiva-tional class on 21.11.2019 at 10 am inour school auditorium. All must attendwithout fail. School leader
l In the lesson Attitude is Altitude Nickbecame a famous motivational speaker.Now imagine you are student leader ofyour school and arranged his motivationalclass in your school. Write a notice inform-ing date, time and venue to your students.
Answer :
Notice
Minor discourses1. In the lesson Attitude is Altitude Nick
sent to main stream school, where hewas teased and bullied. Now imagine asNick's close friend and make a diaryentry of your feelings towards him.
2. You have read the story Nick Vujicic.Who was born with no arms and legs.
l You have read the lesson "Attitude isAltitude". In the lesson when Nick waseight years old he went to his mother cry-ing and told her that he wanted to kill him-self. Now write the possible conversationbetween Nick and his mother.
Answer :
Conversation 10 Marks
Nick: Something is worrying me. I wanted tokill myself, mom.
Mom: Oh! my God! It is not good what madeyou thinks so.
Nick: I am disabled. How can I lead my life?Mom: You need not worry. We are here to
help you.Nick: What can I do without arms and legs.Mom: I am sure, you can do many things.
Many have over come their disability.We will take care of you.
Nick: Is it possible to lead a normal life with-out arms and legs.
Mom: Yes, it is possible. Don't worry. Manyways are there to over come disad-vantages of disability.
Nick: Is it Mom.
Imagine yourself as Nick's mother andmake a diary entry of your feelings onthe day gave both to Nick.
