Number System - Study Mode
[#281] If a + b + c = 6 and ab + bc + ca = 10, then value of a 3 + b 3 + c 3 - 3abc is :
Correct Answer
(A) 36
Explanation
Solution: Given , a + b + c = 6 ab + bc + ca = 10 ∴ (a + b + c) 2 = 36 ⇒ a 2 + b 2 + c 2 + 2ab + 2bc + 2ca = 36 ⇒ a 2 + b 2 + c 2 + 2(ab + bc + ca) = 36 ⇒ a 2 + b 2 + c 2 + 2 × 10 = 36 ⇒ a 2 + b 2 + c 2 = 16 As we know : $$ Rightarrow frac{{{a^3} + {b^3} + {c^3} - 3abc}}{{{a^2} + {b^2} + {c^2} - ab - bc - ca}}$$ xa0 xa0 xa0 $$ = left( {a + b + c}
ight)$$ $$eqalign{
& Rightarrow frac{{{a^3} + {b^3} + {c^3} - 3abc}}{{16 - left( {ab + bc + ca}
ight)}} = 6 cr
& Rightarrow frac{{{a^3} + {b^3} + {c^3} - 3abc}}{{16 - 10}} = 6 cr
& Rightarrow {a^3} + {b^3} + {c^3} - 3abc = 6 imes 6 cr
& Rightarrow {a^3} + {b^3} + {c^3} - 3abc = 36 cr} $$
[#282] The smallest number of 5 digits beginning with 3 and ending with 5 will be :
Correct Answer
(C) 30005
Explanation
Solution: Required number = 30005
[#283] On multiplying a number by 7, all the digits in the product appear as 3’s. The smallest such number is :
Correct Answer
(A) 47619
Explanation
Solution: We keep on dividing 33333... by 7 till we get 0 as remainder. ∴ Required number = 47619
[#284] Between two distinct rational numbers a and b, there exists another rational number which is :
Correct Answer
(D) $$frac{a + b}{2}$$
Explanation
Solution: If a and b are two rational numbers, then $$frac{a + b}{2}$$ xa0 is a rational number lying between a and b.
[#285] Find the remainder when 65 203 is divided by 7.
Correct Answer
(C) 1
Explanation
Solution: $$eqalign{
& frac{{{{65}^{203}}}}{7} cr
& Or,,frac{{{{left( {63 + 2}
ight)}^{203}}}}{7} cr
& 63,{ ext{is}},{ ext{divisible}},{ ext{by}},7, cr
& { ext{so}},{ ext{remainder}},{ ext{will}},{ ext{depend}},{ ext{on}},{ ext{the}},{ ext{powers}},{ ext{of}},2 cr
& Or,,frac{{{2^{203}}}}{7} cr
& { ext{Its}},{ ext{remainder}},{ ext{would}},{ ext{be}},{ ext{same}},{ ext{as}} cr
& frac{{{2^3}}}{7} cr
& { ext{Now,}},{ ext{Required}},{ ext{Remainder}},{ ext{wold}},{ ext{be}}, cr
& frac{8}{7} = 1cr
& { ext{Required}},{ ext{remainder}} = 1 cr} $$ Note: We have manipulated the powers in the form of (4x + n). It means 203 is taken as, 203 = 4x + n = 4 × 50 + 3. We neglect power which is in the multiple of 4.