Simplification - Study Mode

[#156] The greatest 4 digit number which is a perfect square, is = ?
Correct Answer

(C) 9801

Explanation

Solution: $$eqalign{
& { ext{Greatest 4 digit number}} cr
& { ext{ = 9999 }} cr
& ,,,,,,,,{ ext{9| }},{ ext{9999 | 99}} cr
& ,,,,,,,,,,,|,,,81 cr
& overline {189,,|,,,1899} cr
& ,,,,,,,,,,,|,,,1701 cr
& ,,,,,,,,,,,overline {|,,,,,198} cr} $$ Greatest four digit number is a perfect square = 999 - 198 = 9801

[#157] What number must be added to the expression 16a 2 - 12a to make a perfect square ?
Correct Answer

(A) $$frac{9}{4}$$

Explanation

Solution: $$eqalign{
& {left( {a - b}
ight)^2} = {a^2} + {b^2} + 2ab cr
& 16{a^2} - { ext{12}}a cr
& = {(4a)^2} - 2 imes 4a imes frac{3}{2} + {left( {frac{3}{2}}
ight)^2} cr
& { ext{Number be added}} cr
& { ext{ = }}{left( {frac{3}{2}}
ight)^2} cr
& = frac{9}{4} cr} $$

[#158] A teacher wants to arrange his students in an equal number of rows and columns. If there are 1369 students, the number of students in the last row are ?
Correct Answer

(A) 37

Explanation

Solution: Let the number of students in a row be 'x' According to question ⇒ x × x = 1369 ⇒ x 2 = 1369 ⇒ x = 37

[#159] The value of $$frac{{{{left( {x - y}
ight)}^3} + {{left( {y - z}
ight)}^3} + {{left( {z - x}
ight)}^3}}}{{{{left( {{x^2} - {y^2}}
ight)}^3} + {{left( {{y^2} - {z^2}}
ight)}^3} + {{left( {{z^2} - {x^2}}
ight)}^3}}}$$ xa0 xa0 xa0 is = ?
Correct Answer

(D) $${left[ {left( {x + y} ight)left( {y + z} ight)left( {z + x} ight)} ight]^{ - 1}}$$

Explanation

Solution: $$eqalign{
& { ext{Since }}left( {x - y}
ight) + left( {y - z}
ight) + left( {z - x}
ight) = 0 cr
& { ext{So,}}{left( {x - y}
ight)^3} + {left( {y - z}
ight)^3} + {left( {z - x}
ight)^3} cr
& = 3left( {x - y}
ight)left( {y - z}
ight)left( {z - x}
ight) cr
& { ext{Since}}left( {{x^2} - {y^2}}
ight) + left( {{y^2} - {z^2}}
ight) + left( {{z^2} - {x^2}}
ight) = 0 cr
& { ext{So,}}left( {{x^2} - {y^2}}
ight) + left( {{y^2} - {z^2}}
ight) + left( {{z^2} - {x^2}}
ight) cr
& = 3left( {{x^2} - {y^2}}
ight)left( {{y^2} - {z^2}}
ight)left( {{z^2} - {x^2}}
ight) cr
& herefore { ext{Given expression }} cr
& { ext{ = }}frac{{3left( {x - y}
ight)left( {y - z}
ight)left( {z - x}
ight)}}{{3left( {{x^2} - {y^2}}
ight)left( {{y^2} - {z^2}}
ight)left( {{z^2} - {x^2}}
ight)}} cr
& = frac{1}{{left( {x + y}
ight)left( {y + z}
ight)left( {z + x}
ight)}} cr
& = {left[ {left( {x + y}
ight)left( {y + z}
ight)left( {z + x}
ight)}
ight]^{ - 1}} cr} $$

[#160] What is the value of $$0.51overline {345} $$ xa0in vulgar fraction?
Correct Answer

(D) $$frac{{51294}}{{99900}}$$

Explanation

Solution: $$eqalign{
& 0.51overline {345} cr
& = frac{{51345 - 51}}{{99900}} cr
& = frac{{51294}}{{99900}} cr} $$