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  • Which of the following statements are true?Option: 1 The digit at the unit place in the number 171995 + 111995 - 71995</su

Which of the following statements are true?

Option: 1

The digit at the unit place in the number 171995 + 111995 - 71995- 1 is 1.


Option: 2

The digit at the unit place in the number 171995 + 111995 - 71995- 1 is 2.


Option: 3

If \left(1+2 x-3 x^{2}\right)^{2020}=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{4040} x^{4040}

then, a_{0}+a_{2}+a_{4}+a_{6}+\dots+a_{4020} is a negative Integer.


Option: 4

None of these is true.


Answers (1)

best_answer

(A)     (17)1995 + 111995 - 71995 - 1

    = (10 + 7)1995 + (10 + 1)1995 - 71995 - 1

 = (10^{1995} + ^{1995} C_1 . 10 + ^{1995} C_2 . 10^2 + .... + 7^{1995}) + (10^{1995} + ^{1995} C_1 . 10 + ^{1995} C_2 . 10^2 + .... + 1^{1995}) - 7^{1995} - 1\\\\ = (10 k_1 + 7^{1995}) + ( 10 k_2 + 1) - 7^{1995} - 1\\\\ = 10 (k_1 + k_2)\\
    So, it is a multiple of 10, and hence the digit at the unit place is 0.

(B)     It is also wrong as unit digit is 0.

(C)     Put x = 1 and –1 and add.
        0 + ( - 4 )^{2020} = a_0 + a_ 2 + a_ 4 + a_ 6 + + a_{4040} which is a +ve Integer.

Posted by

sudhir.kumar

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