Number System - Study Mode

[#301] $$sqrt 2 $$ xa0 is a/an -
Correct Answer

(C) irrational number

Explanation

Solution: Since, $$sqrt 2 $$ xa0is a non-terminating and non-repeating decimal, so it is an irrational number.

[#302] If a number is divisible by both 11 and 13, then it must be necessarily :
Correct Answer

(B) Divisible by (11 × 13)

Explanation

Solution: We know that 11 and 13 are co-prime So, a number divisible by both 11 and 13 will be divisible by (11 × 13)

[#303] What is sum of all natural numbers from 1 to 100 ?
Correct Answer

(A) 5050

Explanation

Solution: Required sum $$eqalign{
& = frac{{100}}{2}left[ {1 + 100}
ight] cr
& = 101 imes 50 cr
& = 5050 cr} $$

[#304] The digit in unit's place of the product - $${left( {2464}
ight)^{1793}}$$ xa0 $$ imes {left( {615}
ight)^{317}}$$ xa0 $$ imes {left( {131}
ight)^{491}}$$
Correct Answer

(A) 0

Explanation

Solution: $$eqalign{
& {left( {2464}
ight)^{1793}} imes {left( {615}
ight)^{317}} imes {left( {131}
ight)^{491}} cr
& {4^1} o 4 o 4 cr
& {4^2} o 16 o 6 cr
& {4^3} o 64 o 4 cr} $$ So, odd power of 4 will have 4 as unit digit and even power of 4 will have 6 as unit digit 5 and 1 have same unit digits respectively. (2464) 1793 have the odd power therefore unit digit is 4 (615) 317 have unit digit 5 and (131) 491 have unit digit 1. i.e Unit digit of the product will be 4 × 5 × 1 = 20. ∴ 0 is the unit digit

[#305] Thrice the square of a natural number decreased by four times the number is equal to 50 more than the number, the number is -
Correct Answer

(B) 5

Explanation

Solution: Let the number be x According to question $$eqalign{
& left( {3 imes {x^2}}
ight){ ext{ - }}left( {4 imes x}
ight) = 50 + x.....(i) cr
& Rightarrow 3{x^2} - 4x = 50 + x cr
& Rightarrow 3{x^2} - 5x - 50 = 0 cr
& Rightarrow 3{x^2} - 15x + 10x - 50 = 0 cr
& Rightarrow 3{x^{}}(x - 5) + 10(x - 5) = 0 cr
& Rightarrow (x - 5)(3x + 10) = 0 cr
& x = 5,,or,, - frac{{10}}{3} cr} $$ Since the natural number is x = 5 Shortcut method : $$ Rightarrow 3{x^2} - 4x = 50 + x.....(i)$$ Now put the value of x from option (b) $$eqalign{
& x = 5 cr
& 3 imes {(5)^2} - 4 imes 5 = 50 + 5 cr
& 75 - 20 = 55 cr
& 55 = 55 cr} $$ LHS = RHS (it satisfies the conditions) $${ ext{so, x = 5}}$$