Simplification - Study Mode

[#236] If the sum of the squares three consecutive natural numbers is 110, then the smallest of these natural numbers is = ?
Correct Answer

(D) 5

Explanation

Solution: Let the three consecutive numbers are x, x + 1, x + 2 According to question x 2 + (x + 1) 2 + (x + 2) 2 = 110 ⇒ x 2 + x 2 + 1 + 2x + x 2 + 4 + 4x = 110 ⇒ 3x 2 + 6x + 5 = 110 ⇒ 3x 2 + 6x - 105 = 0 ⇒ x 2 + 2x - 35 = 0 ⇒ x 2 + 7x - 5x - 35 = 0 ⇒ x(x + 7) -5(x + 7) = 0 ⇒ (x + 7)(x - 5) = 0 ⇒ x = 5 & -7 [∴ (-) value can not considered] ∴ Smallest number is = 5

[#237] $$
oot 3 of {{{left( {333}
ight)}^3} + {{left( {333}
ight)}^3} + {{left( {334}
ight)}^3} - 3 imes 333 imes 333 imes 334} $$ xa0 xa0 xa0 xa0 xa0 is equal to = ?
Correct Answer

(C) 10

Explanation

Solution: According to question $$
oot 3 of {{{left( {333}
ight)}^3} + {{left( {333}
ight)}^3} + {{left( {334}
ight)}^3} - 3 imes 333 imes 333 imes 334} $$ As we know that a 3 + b 3 + c 3 - 3abc = $$frac{1}{2}$$ (a + b + c)[(a - b) 2 + (b - c) 2 + (c - a) 2 ] = $$frac{1}{2}$$ (333 + 333 + 334)[(333 - 333) 2 + (333 - 334) 2 + (334 - 333) 2 ] = $$frac{1}{2}$$ × 1000[0 + 1 + 1] = $$frac{1}{2}$$ × 1000 × 2 = $$
oot 3 of {1000} $$ = 10

[#238] If 4x = p(x + 3) + q(x - 1) is an identity, then the values of p and q are?
Correct Answer

(B) 1, 3

Explanation

Solution: $$eqalign{
& { ext{4}}x = pleft( {x + 3}
ight) + qleft( {x - 1}
ight) cr
& Rightarrow 4x = px + 3p + qx - q cr
& Rightarrow 4x = left( {p + q}
ight)x + left( {3p - q}
ight) cr
& Rightarrow p + q = 4,and,3p - q = 0 cr
& Rightarrow p + q = 4,and,p = frac{q}{3} cr
& Rightarrow frac{q}{3} + q = 4 cr
& Rightarrow frac{{4q}}{3} = 4 cr
& Rightarrow q = 3 cr
& p + q = 4,,,and cr
& Rightarrow p = 1 cr
& Rightarrow q = 3 cr} $$

[#239] Simplify the following $${81^{frac{3}{4}}} + left[ {left( {20 div 5,{ ext{of }}3 imes 6}
ight) + left{ {left( {8 div 24,{ ext{of }}3}
ight) imes 4}
ight} - 10 div 5}
ight] - {left( {frac{1}{{32}}}
ight)^{ - frac{2}{5}}}$$
Correct Answer

(D) $$29frac{4}{9}$$

Explanation

Solution: $$eqalign{
& {81^{frac{3}{4}}} + left[ {left( {20 div 5,{ ext{of }}3 imes 6}
ight) + left{ {left( {8 div 24,{ ext{of }}3}
ight) imes 4}
ight} - 10 div 5}
ight] - {left( {frac{1}{{32}}}
ight)^{ - frac{2}{5}}} cr
& = 27 + left[ {left( {20 div 15 imes 6}
ight) + left{ {left( {8 div 72}
ight) imes 4}
ight} - 2}
ight] - 4 cr
& = 27 + left[ {8 + frac{4}{9} - 2}
ight] - 4 cr
& = 23 + frac{{58}}{9} cr
& = frac{{265}}{9} cr
& = 29frac{4}{9} cr} $$

[#240] If $$frac{1}{{4.263}} = 0.2346,$$ xa0 find the value of $$frac{1}{{0.0004263}}.$$
Correct Answer

(A) 2346

Explanation

Solution: $$eqalign{
& frac{1}{{4.263}} = 0.2346 cr
& frac{{1000}}{{4263}} = 0.2346 cr
& frac{1}{{0.0004263}} cr
& = frac{{10000000}}{{4263}} cr
& = 10000 imes 0.2346 cr
& = 2346 cr} $$