ncert cbse chapter 9 sequences and series miscellaneous exercise
32.
150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on the second day. 4 more workers dropped out on the third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was finished
Let k = original number of days to finish the work with 150 workers
With workers dropping out , the number of days changed to (k+8) days.
Assuming total work is 1 unit,
150 workers one day's work = [ 1/k ]
1 workers one day's work = [1 / (150k)]
on the first day, 150* [1 / (150k)] = [150 / (150k)] was finished
on the second day only 150-4 = 146 workers remained
so on the second day, 146 [1 / (150k)] = [146 / (150k)] was finished
on the second day only 150-8 = 142 workers remained
so on the second day, 142 [1 / (150k)] = [142 / (150k)] was finished
This went on for (k+8) days by which the entire 1unit of work was finished
therefore
[150 / (150k)] + [146 / (150k)] + [142 / (150k)] +...(k+8)terms = 1 [full work]
[1/(150k)]{150 + 146 +142 + ... (k+8)terms} = 1
[150 + 146 +142 + ... (k+8)terms ] = 150k
LHS is sum of (k+8)terms of an AP with
a =150 ,
d = t2 -t1 = 146-150 = (-4)
n = (k+8)
using formula for sum of n terms of an AP,
Sn = [n/2][ 2a +(n-1)d ] on the LHS
[(k+8)/2]*[ 2(150) +(k+8-1)(-4) ] = 150k
[(k+8)/2]* [300 + (k+7)(-4) ] = 150k
[(k+8)/2]* [272-4k] = 150k
[(k+8)/2] *[4(68-k)]=150k
cancelling 2
[(k+8)] *[2(68-k)]=150k
dividing 2
(k+8) (68-k) = 75k
68k- (k^2) +544 -8k =75k
(k^2) + 15k -544 = 0
(k+32)(k-17)=0
k =(-32) reject
or k = 17
Required number of days = (k+8) = (17+8) = 25
ncert cbse chapter 9 sequences and series miscellaneous exercise
32.
150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on the second day. 4 more workers dropped out on the third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was finished
chapter 8 binomial theorem miscellaneous exercise
1.Find a , b and n in the expansion of (a+b)^n if the first three terms in the expansion are 729, 7290, 30375
2. Find a if the coefficients of (x^2) & (x^3) in the expansion of {(3+ax)^9} are equal
3.find the coefficient of {x^5} in the expansion of{(1+2x)^6}{(1-x)^7}
5.evaluate { (sqrt(3) + sqrt(2))^6 } - { (sqrt(3) - sqrt(2))^6 }
6.find the value of [(a^2)+sqrt{(a^2)-1}]^4 + [(a^2)-sqrt{(a^2)-1}]^4
7.find an approximate value of (0.99^5) using the first three terms of its expansion
8.find n if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of [(fourth root of 2) + {1/(fourth root of 3)}]^n is (sqrt6):1
solution
exercise 8.2
5. find the 4th term in the expansion of (x-2y)^12
7. Find the middle terms in the expansion of [3 - ((x^3) / 6)]^7
Q8) Find the middle terms in the expansion of [(x/3)+9y)]^10
solution
10.The coefficients of the (r-1)th, rth, (r+1)th terms in the expansion of [(x+1)^n] is in the ratio 1:3:5. Find n and r.
exercise 8.1
8. evaluate (101)^4
13.show that 9^(n+1) - 8n -9 is divisible by 64 whenever n is a positive integer
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