NCERT Solutions class-11 Maths Miscellaneous Part-2

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17. If  are in G.P., prove that  are in G.P.

Ans. Given:  are in G.P.

To prove:  are in G.P.

  

Let

  

  

And

  

Also

  

Now,

 

 

 

 

Therefore,  are in G.P.


18. If  and  are the roots  and  are roots of  where  form a G.P. Prove that  

Ans. Let

  

  

And

  

Also

  

   and  are the roots

  

 

   ……….(i)

And

 

   ……….(ii)

Also  are roots of

  

 

  ……….(iii)

And

 

   ……….(iv)

Dividing eq. (iii) by eq. (i), 

 

 

Now

=

Therefore,


19. The ratio of the A.M. and G.M. of two positive numbers  and  is  Show that  =  

Ans. Given:

  

By component and dividend,

 

 

Again by component and dividend,

 

 

 

 

 

Therefore,  =


20. If  are in A.P.;  are in G.P. and  are in A.P., prove that  are in G.P.

Ans. Since,  are in A.P.

  

  

 

Since,  are in G.P.   

  ……….(i)

 Also  are in A.P.

  

 

 

 

Putting values of  and  in eq. (i),

 

 

 

  which shows that  are in G.P.


21. Find the sum of the following series up to  terms:

(i) 5 + 55 + 555 + ……….

(ii) .6 + .66 + .666 + ………….

Ans. (i)  = 5 + 55 + 555 + ………. up to  terms

= 5 [1 + 11 + 111 + ………. up to  terms]

=

=

=

=

=

(ii)  = .6 + .66 + .666 + …………. up to  terms

= 6 [.1 + .11 + .111 + ………. up to  terms]

=

=

=

=

=

=

=


22. Find the 20th term of the series  terms.

Ans. Given:  terms

  

   =

  


23. Find the sum of the first  terms of the series: 3 + 7 + 13 + 21 + 21 + ……….

Ans. Given: ……….(i)

Also   ……….(ii)

Subtracting eq. (ii) from eq. (i),

  

  

  

  

  

   =

=

=

=

=

=

=


24. If are the sum of first  natural numbers, their squares and their cubes respectively, show that  

Ans. Here ,  and

Now,

  

  

    Proved.


25. Find the sum of the following series up to  terms:   

Ans. Given:  up to  terms

  

=

=

   =

=

=

=

=


26. Show that  

Ans. Given:

=  =

=

=

=

=  =  =


27. A farmer buys a used tractor for Rs. 12000. He pays Rs. 6000 cash and agrees to pay the balance in annual installments of Rs. 500 plus 12% interest on the unpaid amount. How much will the tractor cost him?

Ans. Total cost of the tractor = Rs. 12000, Cash amount paid = Rs. 6000

Balance amount = 12000 – 6000 = Rs. 6000

Interest of 1st installment =  = Rs. 720

Amount of 1st installment = 500 + 720 = Rs. 1220

Interest of 2nd installment =  = Rs. 660

Amount of 2nd installment = 500 + 660 = Rs. 1160

Interest of 3rd installment =  = Rs. 600

Amount of 3rd installment = 500 + 600 = Rs. 1100

  Sequence of installments is 1220, 1160, 1100, ……….

Here,  and

  

=

 = 6 [2440 – 660] = Rs. 10680

Therefore, the total cost of tractor is (10680 + 6000) = Rs. 16680.


28. Shams had Ali buys a scooter for Rs. 22000. He pays Rs. 4000 cash and agrees to pay the balance in annual installment of Rs. 1000 plus 10% interest on the unpaid amount. How much will the scooter cost him?

Ans. Total cost of the scooter = Rs. 22000, Cash amount paid = Rs. 4000

Balance amount = 22000 – 4000 = Rs. 18000

Interest of 1st installment =  = Rs. 1800

Amount of 1st installment = 1000 + 1800 = Rs. 2800

Interest of 2nd installment =  = Rs. 1700

Amount of 2nd installment = 1000 + 1700 = Rs. 2700

Interest of 3rd installment =  = Rs. 1600

Amount of 3rd installment = 1000 + 1600 = Rs. 2600

  Sequence of installments is 2800, 2700, 2600, ……….

Here,  and

   =

 = 9 [5600 – 1700] = Rs. 35100

Therefore, the total cost of tractor is (35100 + 4000) = Rs. 39100.


29. A person writes a letter to four of his friends. He asks each one of them to copy the letter and mail to four different persons with instruction that they move the chain similarly. Assuming that the chain is not broken and that it costs 50 paise to mail one letter. Find the amount spent on the postage when 8th set of letter is mailed.

Ans. Total letters in the first set = 4, Total letters in the second set =  = 16

Total letters in the third set =  = 64

  Sequence of letters is 4, 16, 64, ………. In G.P.

Here  and

  

=

=

=  = 87380

Since, the amount of postage on each letter is 50 paise.

Therefore, amount spent on postage = = Rs. 43690.


30. A man deposited Rs. 10000 in a bank at the rate of 5% simple interest annually. Find the amount in 15th year since he deposited the amount and also calculate the total amount after 20 years.

Ans. Total amount deposited = Rs. 10000, Rate of interest = 5% per annum

Interest of first year =  = Rs. 500

Here

  Amount in 15th year =  = 10000 + 7000 = Rs. 17000

Total amount after 20 years =

  = 10 (20000 + 9500) = Rs. 295000


31. A manufacturer reckons that the value of a machine, which cost him Rs. 15625 will depreciate each year by 20%. Find the estimated value at the end of 5 years.

Ans. After 1 year value of machine =  = 15625 – 3125 = Rs. 12500

After 2 year value of machine =  = 12500 – 2500 = Rs. 10000

After 3 year value of machine =  = 10000 – 2000 = Rs. 8000

  Sequence of values of machine is 12500, 10000, 8000, ……. in G.P.

Here   and

   =  = Rs. 5120

Therefore, the value of machine at the end of 5 years is Rs. 5120.


32. 150 workers were engaged to finish a job in a certain number of boys. 4 workers dropped out on second day, 4 more workers dropped out on third day and so on. It took 8 more days to finish the work find the number of days in which the work was completed.

Ans. Number of workers on the first day = 150

Number of workers on the second day = 150 – 4 = 146

Number of workers on the third day = 146 – 4 = 142

  Sequence of values of workers is 150, 146, 142, ……… in A.P.

Here

 Total number of workers required to finish the work in  days

=

=   ……….(i)

If no worker had dropped out, then

Total number of workers required to finish the work  days =   ……….(ii)

From eq. (i) and (ii),

  

  

  

 

  and  which is not possible

Therefore, the work is completed in 25 days.


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