Find (i) the last digit, (ii) the last two digits, and (iii) the last three digits of `17^(256)dot`

Correct option is: C. 18

Number of three digits numbers = 900 If a number has exactly three factors, then it must be square of a prime number. Squares of 11, 13, 17, 19, 23,...

Correct Answer - B The prime digits are 2, 3, 5, 7. If we fix 2 at first place, then other 2n - 1 places are filled by all four digits....

Correct Answer - D Three-digit numbers are 100, 101, …, 999. Total number of such numbers is 900. The three-digit numbers (which have all same digits) are 111, 222, 333, …,...

Correct Answer - `(3-sqrt(5))/2` `int_(-1)^(1)[x^(2)+{x}]dx=int_(-1)^(0)[x^(2)+x+1]dx+int_(0)^(1)[x^(2)+x]dx` `=0+int_(0)^((sqrt(5)-1)/2)0dx+int_((sqrt(5)-1)/2)^(1)1dx` `=(3-sqrt(5))/2`

Correct Answer - `1-(1-p)^(8)` Probability for an incorrec digit is p. hence, probability for 8 correct digit is `1-p)^(8).` Hence, required probability is `1-(1-p)^(8).`

`(d^(n))/(dx^(n))[f(x)]=|{:((d^(n))/(dx^(n))(x^(n)),n!,2),((d^(n))/(dx^(n))(cos x),cos (npi)/(2),4),((d^(n))/(dx^(n))(sin x), sin (npi)/(2),8):}|` `=|{:(n!,n!,2),(cos(x+(npi)/(2)),cos""(npi)/(2),4),(sin(x+(npi)/(2)),sin""(npi)/(2),8):}|` `"or "(d^(n))/(dx^(n))[f(x)]_(x=0)=0`

Correct Answer - B `(b)` First two digits are identical. `x x__to" Number of ways"=9xx9=81` `__x xto"Number of ways"=9xx8=72` (when last two digits are non-zero) `__0 0 to "Number of ways"=9`...

Correct Answer - B `(b)` Least digit used `=4` `:.` We can use `4,5,6,7,8,9`. So each of the four places can be filled in `6` ways So number of numbers is...

Correct Answer - C `(c )` Case I : All six digits alike i.e. `111111`, `222222`……..etc. `=5` ways Case II : `2` alike `+2` other alike. Select any three in `"^(5)C_(3)`...