model 2 based on simplification Practice Questions Answers Test with Solutions & More Shortcuts

Answer: (a)

Expression=$√{8 - 2 √{15}}$

= $√{5+3-2√{5}×√{3}}$

= $√{(√5)^2+(√3)^2-2√5×√3}$

= $√{(√5-√3)^2} = √5-√3$

Answer: (b)

$√5/{√3 + √2} - {3√3}/{√5 + √2} + {2√2}/{√5 + √3}$

$√5/{√3 + √2} = {√5(√3 - √2)}/{(√3 + √2)(√3 - √2)}$

= ${√{15} - √{10}}/{3 - 2} = √{15} - √{10}$

${3√3}/{√5 + √2} = {3√3}/{√5 + √2} × {√5 - √2}/{√5 - √2}$

= ${3√3(√5 - √2)}/{5-2} = √{15} - √6$

${2√2}/{√5 + √3} = {2√2(√5 - √3)}/{(√5 + √3)(√5 - √3)}$

= ${2√2(√5 -√3)}/{5 - 3} = √{10} - √6$

Expression

= $(√{15} - √{10}) - (√{15} - √6) + (√{10} - √6)$

= $√{15} - √{10} - √{15} + √6 + √{10} - √6 = 0$

Answer: (b)

$1/{√{3.25} + √{2.25}} + 1/{√{4.25} + √{3.25}} + 1/{√{5.25} + √{4.25}} + 1/{√{6.25} + √{5.25}}$

$1/{√{3.25} + √{2.25}}$

= $1/(√{3.25} + √{2.25})×{{√{3.25} - √{2.25}}/{√{3.25} - √{2.25}}}$

= ${√{3.25} - √{2.25}}/{3.25 - 2.25} = √{3.25} - √{2.25}$

Similarly

$1/{√{4.25} + √{3.25}}= √{4.25} - √{3.25}$

$1/{√{5.25} + √{4.25}}= √{5.25} - √{4.25}$

$1/{√{6.25} + √{5.25}}= √{6.25} - √{5.25}$

Expression

=$√{3.25} - √{2.25} +√{4.25} - √{3.25} +√{5.25} - √{4.25} +√{6.25} - √{5.25}$

= $√{6.25} - √{2.25}$

= 2.5 - 1.5 =1

Answer: (d)

Expression

= $12/{3 + √5 + 2√2}$

= ${12(3+√5-2√2)}/{[(3+√5)+2√2][(3+√5)-2√2]}$

[Rationalising the denominator]

= ${12(3+√5-2√2)}/{(3+√5)^2- (2√2)^2}$

= ${12(3+√5-2√2)}/{9+5+6√5-8}$

= ${12(3√5-2√2)}/{6√5+6}$

= ${2(3+√5-2√2)}/{√5+1}$

= ${2(3+√5-2√2)(√5-1)}/{(√5+1)(√5-1)}$

= ${2(3√5+5-2√{10}-3-√5+2√2)}/{5-1}$

= ${2(2√5+2√2-2√{10}+2)}/4$

= ${2×2(√5+√2-√{10}+1)}/4$

= $1+√5+√2-√{10}$

Answer: (d)

$(16)^{0.16} × (16)^{0.04} × (2)^{0.2}$

= $(2^4)^{0.16} × (2^4)^{0.04} × (2)^{0.2}$

= $2^{0.64} × 2^{0.16} × 2^{0.2}$

= $(2)^{0.64+0.16+0.2}$ = 2