Simplification - Study Mode

[#151] Number of digits in the square root of 62478078 is = ?
Correct Answer

(A) 4

Explanation

Solution: Number of digits in 62478078 = 8 ∴ Number of digits in its square root = 4

[#152] $${ ext{If }}left( {{n^r} - tn + frac{1}{4}}
ight)$$ xa0 xa0 be a perfect square, then the values of t are = ?
Correct Answer

(D) ± 1

Explanation

Solution: $$eqalign{
& { ext{If }}left( {{n^r} - tn + frac{1}{4}}
ight){ ext{be a perfect square}} cr
& r = 2t = pm 1 cr
& left( {{ ext{If }}t = 1}
ight),,{n^2} - n + frac{1}{4} cr
& = {n^2} - 2 imes n imes frac{1}{2} + frac{1}{4} cr
& = {left( {n - frac{1}{2}}
ight)^2} cr
& left( {{ ext{If }}t = - 1}
ight),,{n^2} + n + frac{1}{4} cr
& = {n^2} + 2 imes n imes frac{1}{2} + frac{1}{4} cr
& = {left( {n + frac{1}{2}}
ight)^2} cr} $$

[#153] If x = a + m, y = b + m, z = c + m, then the value of $$frac{{{x^2} + {y^2} + {z^2} - yz - zx - xy}}{{{a^2} + {b^2} + {c^2} - ab - bc - ca}}$$ xa0 xa0 xa0 is = ?
Correct Answer

(A) 1

Explanation

Solution: $$eqalign{
& frac{{{x^2} + {y^2} + {z^2} - yz - zx - xy}}{{{a^2} + {b^2} + {c^2} - ab - bc - ca}}{ ext{ }} cr
& = frac{{{{left( {a + m}
ight)}^2} + {{left( {b + m}
ight)}^2} + {{left( {c + m}
ight)}^2} - left( {b + m}
ight)left( {c + m}
ight) - left( {c + m}
ight)left( {a + m}
ight) - left( {a + m}
ight)left( {b + m}
ight)}}{{{a^2} + {b^2} + {c^2} - ab - bc - ca}} cr
& = frac{{{a^2} + {m^2} + 2am + {b^2} + {m^2} + 2bm + {c^2} + {m^2} + 2cm - bc - bm - cm - {m^2} - ca - cm - am - {m^2} - ab - am - bm - {m^2}}}{{{a^2} + {b^2} + {c^2} - ab - bc - ca}} cr
& = frac{{{a^2} + {b^2} + {c^2} - ab - bc - ca}}{{{a^2} + {b^2} + {c^2} - ab - bc - ca}} cr
& = 1 cr} $$

[#154] $$left( {frac{{785 imes 785 imes 785 + 435 imes 435 imes 435}}{{785 imes 785 + 435 imes 435 - 785 imes 435}}}
ight)$$ xa0 xa0 xa0 simplifies to = ?
Correct Answer

(C) 1220

Explanation

Solution: $$eqalign{
& { ext{Given expression,}} cr
& left( {frac{{{a^3} + {b^3}}}{{{a^2} + {b^2} - ab}}}
ight){ ext{ = }}left( {a + b}
ight) cr
& left[ {{ ext{where}},a = 785,b = 345}
ight] cr
& = 785 + 345 = 1220 cr} $$

[#155] $$frac{{38 imes 38 imes 38 + 34 imes 34 imes 34 + 28 imes 28 imes 28 - 38 imes 34 imes 84}}{{38 imes 38 + 34 imes 34 + 28 imes 28 - 38 imes 34 - 34 imes 28 - 38 imes 28}}$$ xa0 xa0 xa0 xa0 xa0xa0 is equal to = ?
Correct Answer

(D) 100

Explanation

Solution: $$eqalign{
& { ext{Given experssion ,}} cr
& frac{{{a^3} + {b^3} + {c^3} - 3abc}}{{{a^2} + {b^2} + {c^2} - ab - bc - ca}} cr
& left( {Let,,38 = a,,34 = b,,28 = c}
ight) cr
& = a + b + c cr
& = 38 + 34 + 28 cr
& = 100 cr} $$