SSLC റിവിഷന്‍ - രേഖീയ സംഖ്യകള്‍
>> Monday, January 25, 2010
രേഖീയ സംഖ്യകളെ അപഗ്രഥനം ചെയ്യാനുള്ള ശേഷി നേടുന്നതിനുള്ള ആദ്യ ചുവടുവെപ്പാണ് പത്താം ക്ലാസിലെ രേഖീയ സംഖ്യകള്‍ എന്ന യൂണിറ്റ്. വളരെ ചെറുത് എന്ന് തോന്നിക്കുന്ന പാഠഭാഗത്ത് രണ്ട് അടിസ്ഥാന ആശയങ്ങളാണ് പ്രധാനമായും ഉള്ളത്. 'അകലം' എന്ന ആശയം. അത് സംഖ്യാ ഗണിതവും ബീജഗണിതവുമൊക്കെയായി ബന്ധിപ്പിച്ചു കൊണ്ട് കേവലവില എന്ന ആശയം അവതരിപ്പിക്കുന്നു. കേവലവിലയുടെ ബീജഗണിതം ഇവിടെ ചര്‍ച്ച ചെയ്യപ്പെടുന്നു. വിവിധ സന്ദര്‍ഭങ്ങളില്‍ കേവലവില ഉപയോഗിക്കുന്നതിന് അനേകം ഉദാഹരണങ്ങള്‍ കൂടി അഭ്യസിക്കേണ്ടതുണ്ട്. പരമാവധി സന്ദര്‍ഭങ്ങള്‍ ഉള്‍പ്പെടുത്തിക്കൊണ്ടാണ് ചോദ്യങ്ങള്‍ തയ്യാറാക്കിയിരിക്കുന്നത്. താഴെയുള്ള ലിങ്കില്‍ നിന്നും പി.ഡി.എഫ് ഡോക്യുമെന്റായി ഒരുക്കിയിട്ടുള്ള ചോദ്യങ്ങള്‍ ഡൗണ്‍ലോഡ് ചെയ്തെടുക്കാം.
ഈ പാഠത്തില്‍ ചര്‍ച്ച ചെയ്യപ്പെടുന്ന പഠനലക്ഷ്യങ്ങള്‍ എന്തെല്ലാം
Click here to download the Questions of Real Numbers
44 comments:
I think thr 9 th question is repeating.
Dear Chenthamarakshan sir
Second Part of 9 th question
What is the value of (mod x)/x when x is negative
Shall i remember O.K Sir,
O KUNHIKRISHNAN
An interesting problem
Let us discus this problem:
Almost all natural numbers can be expressed as the sum of two or more consecutive natural numbers.But there is only one number between 8000 and 16000 which cannot be expressed like this. Find that number.It is an easy problem though it seems to be difficult
To
Maths blog Team
Thank you very much for giving me the correct answer. Can i give that sir's e-mail id.I had some doubts in my Engineering Maths question papers.I need quick ways to simplify the problems.Once again thanks to all Blog team members.
Murali Kumar
Kottayam
Dear Vijayan Sir
Is the answer 8192?
@ Vijayan Sir
Hello sir sorry for the delay ……Project finalisation is going on …little bit busy
Sir
Let the terms be a-3d, a-d, a+d, a+3d
Here sum is 16
a-3d +a-d+ a+d+ a+3d=16
4a= 16
a= 4
Multiplying
(a-3d)(a-d)(a+d)(a+3d)=105
(a^2-d^2) (a^2-9d^2)=105
a^4-9a^2d^2-a^2d^2+9d^4=105
Substituting value of a
256-144d^2-16d^2+9d^4=105
9d^4-160d^2+151=0
I think for 10th std 4th power solutions is tedious
So we can take d^2=x
Take d^2=x
Then 9x^2-160x+151=0
On solving this quadratic eqn
We get x=16.77 and x=1
Here 16.77 is not possible
So Value of x=1
So d^2=1
From this d=+1, -1
So the A.P is either 1,3,5,7(when d=+1)
Or 7,5,3,1(when d=-1)
@kannan sir,though you are busy ,you are ready with correct solution.thanks from blogteam.
@azees. about O.K SIR's problem. I simply copied a qn from his orcut site ,and i am little bit doubtful about that qn. so i don't know it is correct or not.let others comment about qn AND ANSWER.
@ Azeez sir
Palindrom Numbers
1^8+ 2^6+ 3^9+ 4^1+ 5^4+ 6^7+7^5
+8^3+ 9^2 = 317713
1^7 + 2^8 + 3^2 + 4^9 + 5^6 + 6^1 + 7^3 + 8^5 + 9^4 =317713
1^5 + 2^6 + 3^2 + 4^1 + 5^8 + 6^4 + 7^3 + 8^7 + 9^9 = 389909983
i think every perfect square no.s can't express the sum of some consecutive +ve integers
sorry
every natural numbrs in the form of 2^n can't express as the sum of concecutive natural no.s
ie 2^13 =8192 can'be expressed as the sum of two or more consecutive natural numbers.
Which are the Smallest and Greatest perfect square numbers containing all nine digits(1 to 9)?
Azeez sir
smallest and largest square numbers containing the digits 1 to 9 are
11826^2 = 139 854 276
30384^2 = 923 187 456
smallest and largest square numbers containing the digits 0 to 9 are
32043^2 = 1026 753 849
99066^2 = 9814 072 356
This question appeared in Federal Bank Quiz competition last year
ഒന്നു മുതല്‍ ഒമ്പതു വരെയുള്ള സംഖ്യകള്‍ ഉള്ള ഏറ്റവും ചെറിയ പൂര്‍ണവര്‍ഗസംഖ്യ 11826 ന്റെ വര്‍ഗമായ 139854276 ഉം
വലിയ പൂര്‍ണവര്‍ഗസംഖ്യ 30384 ന്റെ വര്‍ഗമായ 923187456 ഉം ആണ്.
Dear Kannan Sir
You are so fast.
Dear Hari Sir
You are 1 minute late.
Now Find this one.
6205 & 3869 are friends as shown below
6205 = 38^2 + 69^2 and 3869 = 62^2 + 05^2
Find one more 4 digit friends.
Brother of AZEES
" 12^2+33^2=1233.
88^2+33^2=8833.
4 digit friends
5965 = 77^2 + 06^2
7706 = 59^2 + 65^2
2010:
0^7+1^9+2^8+3^6+4^5=2010.
12+34*56+7+89=2010.
9+8*7+6*548*3*2+1=2010.
36*57-42=2010.
2345*6/7=2010.
2/3*45*67=2010.
find the next letter of the following series
o,t,t,f,f,s,s,----
These are the first letters of One,Two,Three,Four,Five,Six,Seven.....
So next letter is e represents Eight
o,t,t,f,f,s,s,e,......
Isn't it Muraleedharan sir
These are the first letters of One,Two,Three,Four,Five,Six,Seven.....
So next letter is e represents Eight
o,t,t,f,f,s,s,e,......
Isn't it Muraleedharan sir
e,n,t,e,t,t,f,f,s,s,
I drew 4 triangles with integer sides .two were right and two were not.the right ones had area numerically equal to there perimeter,
and the others areas were two thirds of their perimeters . what were the sides ,perimeters and areas of 4 triangles?
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Memories in our Souls.
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@Vijayan Sir
Let the base of right angled triangle be ‘b’ , altitude ‘a ‘ and hypotenuse ‘c’
Then by Pythagorean theorem c = (a^2 + b^2)^1/2
The right triangles have perimeter P = a + b + c = a + b + (a^2 + b^2)^1/2
and area A = a b / 2.
ab - 2a - 2b = 2 (a^2 + b^2)^1/2 or (ab - 2a - 2b)^2 = 4(a^2 + b^2).
Simplifying the sides of the first right triangle are 6,8,10
AREA = PERIMETER = 24
In the Same way sides of second right angled triangle are5,12,13
AREA = PERIMETER = 30.
The others have perimeter P = a + b + c THEN, area A = [s(s-a)(s-b)(s-c)] ^1/2
S= P / 2.
On solving
a=b=c=100 reveals the sides of the first triangle are 5,5,8, Area = 12 , Perimeter = 18
The sides of the second triangle are 3,25,26 .practically it is a small angles triangle Area = 36 , Perimeter = 54
1)For right angled triangle
Area =Perimeter = 24
2)For Second right angled triangle
Area =Perimeter = 30
For other triangles
1)Area=12 and Perimeter = 18
2)Area=36,Perimeter = 54
The sides of the right triangles are (6,8,10) & (5,12,13)
ജോണ്‍ സര്‍ നാലു പട്ടണങ്ങളുടെ ഉത്തരം നോക്കിയോ .നല്ല രസമുള്ള ചോത്യം ആയിരുന്നു അത് .ഉത്തരം കിട്ടുന്ന വരെ നല്ല ത്രില്‍ ഉണ്ടാക്കിയ ച്യോത്യം
Is my answer correct or not
Actually bhama teacher's answer guided me to get my answer
If my answer is correct
50% Credit to Bhama teacher also
50% Credit to the clue from John sir (The question is based on Trignometry )
എങ്ങിനെ ഒരു ആവറേജ് കുട്ടിക്ക് അംശബന്ധവും അനുപാതവും പഠിപ്പിക്കാമെന്ന് ആരെങ്കിലും വിശദീകരിക്കാമോ?
നിസാര്‍ സാര്‍.
ഞാന്‍ ശ്രമിക്കാം.ഏതാനും T M മാത്യക തയ്യാറാക്കട്ടെ.
@kannan sir,bhama:
i wonder how did you get enough time to go through the qns.
thank you.
The area of the triangle with sides 205,188 and 85 is GREATER THAN the area of the triangle with sides 225,188 and 85, Even though here one side is larger. WHY?
@ Azeez sir
Most probably it is because the height will decrease if we increase a side that’s why the area decreases
Mostly this is because an obtuse angled triangle will be formed.
I AM NOT VERY MUCH SURE ABOUT THE ANSWER .
Is it correct or not ?
The area of triangles with sides 25,25,30 & 25,25,40 are same.perhaps you may wonder. many are in the list(perimeter different and area same). make a list of such triangles and find out any speciality
25, 25 30 triangle is the sum of 2 rht. triangle whose sides are 25, 15 2o ( if these triangles joins with the side 20 we get 25 25 30 triangle)
lllrly 25, 25 40 triangle is the sum of 2 rht. triangle whose sides are 25, 15 2o (if these triangles joins with the side 15 we get 25 25 40 triangle)
Also If we join 2 rht. triangles whose sides are 5 12 13 with the side 5 we get 13,13 24 triangle
If we join 2 rht. triangles whose sides are 5 12 13 with the side 12 we get 13 13 10 traingle
ie area same perimeter different
സമപാര്‍ശ്വത്രികോണങ്ങളെ തുല്യവിസ്തീണ്ണമുള്ള ജോടികളായി കാണാം.ഈ അഭിപ്രായത്തോട് പ്രതികരിക്കുക
DEAR MURALISIR,JOHN SIR,AZEES,
ABOUT TRIANGLE;
you are travelling in same boat and sitting in different places.your destination is same.(am i right?)
the area of 25,25,30&25,25,40 are same.so many examples.since 30,40,50 a rt triangle we get the result.
17,17,16&17,17,30 have the same area 120 .both are isoceles and part of same rt triangle 34,30,16.
take any rt triangle,divide it into two parts,the area is same and it will be isoceles( geometric proof: take a rt triangle.draw a semicircle with radius half of hyp.and the mid point of hyp. as centre. the area of the two triangles will be same.
so i am getting down from the boat.
തൃശ്ശൂരില്‍ ഇപ്പോള്‍ മാത്സ് ക്ലാസ്റ്റര്‍ ആണ്. എസ,എച് സി ജി എച് എസ് എസ്സില്‍ . ബ്ലോഗ്‌ പരിചയപ്പെടലാണ്.
പക്ഷെ അവിടത്തെ എച്. എം ലാബ് തരില്ലെന്ന്. !!!!!!!!ഇങ്ങനെയും എച്.എമോ
four terms in AP:
@kannan sir,perhaps you may went through the comment of jessy at 8.16/29.01.10
so shall i add one thing in your solution regardig four terms in AP.
X=16.77 IS ADMISSIBLE
then the terms are "-8.3,-.1,+8.1,+16.3" if we take sqroot (151/9) is 4.1.
here the correct value of d=sqroot(151/9).
then the product is 105 AND THE SUM IS 16.
@ Vijayan Sir
Really I never go through that value sir .It’s my mistake.
Thanks to point out my mistake
Really it is a big mistake.I know… I am sorry sir
Once again thanks to Vijayan sir & Jessy Sir to point out mistake .
Sir,Can u give the correctnees of
ഒോര െപാകമള രണ െകടിെങള ഒര ൈമതാനതിെെ രണറതമണ്. ൈമതാനതിെെ ഒര
സാനതനിന ോനാകോോാള അവയെെ മകളറം 60˚, 30˚എനീ ോമലോകാണകളില കാണന. ഓോരാ
െകടിെതിനം എത െപാകമണ്? ോനാകന സാനം നിരണയികക.
and please give some hints
SAJIKUMAR
Sir,
51) ഒര സംഖയയെെയം അതിെെ വയലകമതിെെയം തക 20 1/30 (20&1/30) ആയാല സംഖയ
കാണക
is above question has a perfect answer
Please give the answer also
SAJIKUMAR P
Sir
96.
ഒോര െപാകമള രണ െകടിെങള ഒര ൈമതാനതിെെ രണറതമണ്. ൈമതാനതിെെ ഒര
സാനതനിന ോനാകോോാള അവയെെ മകളറം 60˚, 30˚എനീ ോമലോകാണകളില കാണന. ഓോരാ
െകടിെതിനം എത െപാകമണ്? ോനാകന സാനം നിരണയികക.
please make sure that this question is correct?
Dear Saji
Please correct it as 2 and 1/30
Kindly go through the comments below that post. I think some other corrections are also made in the comments
Sir,My question is regarding no.96 in your 100 questions.is it necessary to give any additinal information in that question
96.
ഒോര െപാകമള രണ െകടിെങള ഒര ൈമതാനതിെെ രണറതമണ്. ൈമതാനതിെെ ഒര
സാനതനിന ോനാകോോാള അവയെെ മകളറം 60˚, 30˚എനീ ോമലോകാണകളില കാണന. ഓോരാ
െകടിെതിനം എത െപാകമണ്? ോനാകന സാനം നിരണയികക.
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