21.
If xy(x + y) = m, then the value of x3 + y3 + 3m is?
(A) $$\frac{{{m^3}}}{{xy}}$$
(B) $$\frac{{{m^3}}}{{{{\left( {x + y} \right)}^2}}}$$
(C) $$\frac{{{m^3}}}{{{x^3}{y^3}}}$$
(D) $$\frac{m}{{{x^3}{y^3}}}$$
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Solution:
$$\eqalign{ & xy\left( {x + y} \right) = m \cr & {\text{find }}{x^3} + {y^3} + 3m \cr & xy\left( {x + y} \right) = m\, . . . . . (i) \cr & \left( {x + y} \right) = \frac{m}{xy} \cr &{\text{Cubing both side}} \cr & \Rightarrow {x^3} + {y^3} + 3.xy\left( {x + y} \right) = \frac{{{m^3}}}{{{x^3}{y^3}}} \cr & {\text{From equation (i)}} \cr & \Rightarrow {x^3} + {y^3} + 3.m = \frac{{{m^3}}}{{{x^3}{y^3}}} \cr & \Rightarrow {x^3} + {y^3} + 3m = \frac{{{m^3}}}{{{x^3}{y^3}}} \cr} $$
22.
If $${a^{\frac{1}{3}}} = 11,$$ ÃÂÃÂ then the value of a2 - 331a is?
(A) 1331331
(B) 1331000
(C) 1334331
(D) 1330030
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Solution:
$$\eqalign{ & {\text{ }}{a^{\frac{1}{3}}} = 11 \cr & \Leftrightarrow a = {11^3} = 1331 \cr & {a^2} - 331a \cr & = a\left( {a - 331} \right) \cr & = 1331\left( {1331 - 331} \right) \cr & = 1331\left( {1000} \right) \cr & = 1331000 \cr} $$
23.
$${\text{If }}\frac{{4\sqrt 3 + 5\sqrt 2 }}{{\sqrt {48} + \sqrt {18} }} = a + b\sqrt 6 {\text{,}}$$ ÃÂÃÂ ÃÂÃÂ ÃÂÃÂ then the value of a and b are respectively?
(A) $$\frac{9}{{15}}, - \frac{4}{{15}}$$
(B) $$\frac{3}{{11}},\frac{4}{{33}}$$
(C) $$\frac{9}{{10}},\frac{2}{5}$$
(D) $$\frac{3}{5},\frac{4}{{15}}$$
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Solution:
$$\eqalign{ & \frac{{4\sqrt 3 + 5\sqrt 2 }}{{\sqrt {48} + \sqrt {18} }} = a + b\sqrt 6 \cr & \Rightarrow \frac{{4\sqrt 3 + 5\sqrt 2 }}{{\sqrt {16 \times 3} + \sqrt {9 \times 2} }} = a + b\sqrt 6 \cr & \Rightarrow \frac{{4\sqrt 3 + 5\sqrt 2 }}{{4\sqrt 3 + 3\sqrt 2 }} = a + b\sqrt 6 \cr & \Rightarrow \frac{{4\sqrt 3 + 5\sqrt 2 }}{{4\sqrt 3 + 3\sqrt 2 }} \times \frac{{4\sqrt 3 - 3\sqrt 2 }}{{4\sqrt 3 - 3\sqrt 2 }} = a + b\sqrt 6 \cr & \Rightarrow \frac{{\left( {4\sqrt 3 + 5\sqrt 2 } \right)\left( {4\sqrt 3 - 3\sqrt 2 } \right)}}{{48 - 18}} = a + b\sqrt 6 \cr & \Rightarrow \frac{{8\sqrt 6 + 18}}{{30}} = a + b\sqrt 6 \cr & \Rightarrow \frac{{8\sqrt 6 }}{{30}} + \frac{{18}}{{30}} = a + b\sqrt 6 \cr & \Rightarrow \frac{4}{{15}}\sqrt 6 + \frac{3}{5} = a + b\sqrt 6 \cr & \Rightarrow \frac{3}{5} + \frac{4}{{15}}\sqrt 6 = a + b\sqrt 6 \cr} $$ By comparing coefficients of rational and irrational parts. $$\eqalign{ & \Rightarrow a = \frac{3}{5}{\text{ , }}b = \frac{4}{{15}} \cr & \therefore \left( {\frac{3}{5},\frac{4}{{15}}} \right) \cr} $$
24.
If x : y = 7 : 3 then the value of $$\frac{{xy + {y^2}}}{{{x^2} - {y^2}}}{\text{ is?}}$$
(A) $$\frac{3}{4}$$
(B) $$\frac{4}{3}$$
(C) $$\frac{3}{7}$$
(D) $$\frac{7}{3}$$
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Solution:
$$\eqalign{ & x:y \cr & 7:3{\text{ }} \cr & \therefore {\text{ }}\frac{{xy + {y^2}}}{{{x^2} - {y^2}}} \cr & = \frac{{21 + 9}}{{49 - 9}} \cr & = \frac{{30}}{{40}} \cr & = \frac{3}{4} \cr} $$
25.
If $$\frac{{2a + b}}{{a + 4b}} = 3{\text{,}}$$ ÃÂÃÂ then find the value of $$\frac{{a + b}}{{a + 2b}} = ?$$
(A) $$\frac{5}{9}$$
(B) $$\frac{2}{7}$$
(C) $$\frac{{10}}{9}$$
(D) $$\frac{{10}}{7}$$
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Solution:
$$\eqalign{ & \frac{{2a + b}}{{a + 4b}} = 3{\text{ }}\left( {{\text{Given}}} \right) \cr & \Rightarrow 2a + b = 3\left( {a + 4b} \right) \cr & \Rightarrow 2a + b = 3a + 12b \cr & \Rightarrow - a = 11b \cr & \Rightarrow a = - 11b \cr & \therefore \frac{{a + b}}{{a + 2b}} \cr & \Rightarrow \frac{{ - 11b + b}}{{ - 11b + 2b}} \cr & \Rightarrow \frac{{ - 10b}}{{ - 9b}} \cr & \Rightarrow \frac{{10}}{9} \cr} $$
26.
If x : y = 4 : 15 then the value of $$\left( {\frac{{x - y}}{{x + y}}} \right)$$ ÃÂÃÂ is?
(A) $$\frac{{11}}{{19}}$$
(B) $$\frac{{19}}{{11}}$$
(C) $$\frac{4}{{11}}$$
(D) $$\frac{{15}}{{19}}$$
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Solution:
$$\eqalign{ & y:x = 4:15 \cr & \therefore \frac{y}{x} = \frac{4}{{15}} \cr & \therefore \frac{{x - y}}{{x + y}} \cr & \Rightarrow \frac{{x\left( {1 - \frac{y}{x}} \right)}}{{x\left( {1 + \frac{y}{x}} \right)}} \cr & {\text{Taking }}x{\text{ common}} \cr & \Rightarrow \frac{{1 - \frac{4}{{15}}}}{{1 + \frac{4}{{15}}}} \cr & \Rightarrow \frac{{11}}{{15}} \times \frac{{15}}{{19}} \cr & \Rightarrow \frac{{11}}{{19}} \cr} $$
27.
If $$x = \sqrt 3 + \sqrt 2 {\text{,}}$$ ÃÂÃÂ ÃÂÃÂ then the value of $$\left( {x + \frac{1}{x}} \right)\,{\text{is?}}$$
(A) $${\text{2}}\sqrt 2 $$
(B) $${\text{2}}\sqrt 3 $$
(C) 2
(D) 3
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Solution:
$$\eqalign{ & {\text{ }}x = \sqrt 3 + \sqrt 2 \cr & \frac{1}{x} = \frac{1}{{\sqrt 3 + \sqrt 2 }} \times \frac{{\sqrt 3 - \sqrt 2 }}{{\sqrt 3 - \sqrt 2 }} \cr & \frac{1}{x} = \sqrt 3 - \sqrt 2 \cr & \therefore x + \frac{1}{x} \cr & = \sqrt 3 + \sqrt 2 + \sqrt 3 - \sqrt 2 \cr & = 2\sqrt 3 \cr} $$
28.
If $$x = 5 - \sqrt {21} {\text{,}}$$ ÃÂÃÂ then the value of $$\frac{{\sqrt x }}{{\sqrt {32 - 2x} - \sqrt {21} }}$$ ÃÂÃÂ ÃÂÃÂ is?
(A) $$\frac{1}{{\sqrt 2 }}\left( {\sqrt 3 - \sqrt 7 } \right)$$
(B) $$\frac{1}{{\sqrt 2 }}\left( {\sqrt 7 - \sqrt 3 } \right)$$
(C) $$\frac{1}{{\sqrt 2 }}\left( {\sqrt 7 + \sqrt 3 } \right)$$
(D) $$\frac{1}{{\sqrt 2 }}\left( {7 + \sqrt 3 } \right)$$
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Solution:
$$\eqalign{ & x = 5 - \sqrt {21} \cr & 2x = 10 - 2\sqrt {21} \,......(i) \cr & \Rightarrow 2x = {\left( {\sqrt 7 } \right)^2} + {\left( {\sqrt 3 } \right)^2} - 2\left( {\sqrt 7 } \right)\left( {\sqrt 3 } \right) \cr & \Rightarrow 2x = {\left( {\sqrt 7 - \sqrt 3 } \right)^2} \cr & \Rightarrow x = \frac{1}{2}{\left( {\sqrt 7 - \sqrt 3 } \right)^2} \cr & \Rightarrow \sqrt x = \frac{1}{{\sqrt 2 }}\sqrt {{{\left( {\sqrt 7 - \sqrt 3 } \right)}^2}} \cr & \Rightarrow \sqrt x = \frac{1}{{\sqrt 2 }}\left( {\sqrt 7 - \sqrt 3 } \right) \cr & \therefore \frac{{\sqrt x }}{{\sqrt {32 - 2x} - \sqrt {21} }} \cr & = \frac{{\sqrt 7 - \sqrt 3 }}{{\sqrt 2 \left[ {\sqrt {32 - \left( {10 - 2\sqrt {21} } \right)} - \sqrt {21} } \right]}} \cr & = \frac{{\sqrt 7 - \sqrt 3 }}{{\sqrt 2 \left[ {\sqrt {22 + 2\sqrt {21} } - \sqrt {21} } \right]}} \cr & = \frac{{\sqrt 7 - \sqrt 3 }}{{\sqrt 2 \left[ {\sqrt {{{\left( {\sqrt {21} + 1} \right)}^2}} - \sqrt {21} } \right]}} \cr & = \frac{{\sqrt 7 - \sqrt 3 }}{{\sqrt 2 \left[ {\sqrt {21} + 1 - \sqrt {21} } \right]}} \cr & = \frac{{\sqrt 7 - \sqrt 3 }}{{\sqrt 2 }} \cr & = \frac{1}{{\sqrt 2 }}\left( {\sqrt 7 - \sqrt 3 } \right) \cr} $$
29.
The minimum value of (x - 2)(x - 9) is?
(A) $$ - \frac{{11}}{4}$$
(B) $$\frac{{49}}{4}$$
(C) 0
(D) $$ - \frac{{49}}{4}$$
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Solution:
$$\eqalign{ & \left( {x - 2} \right)\left( {x - 9} \right) \cr & = {x^2} - 9x - 2x + 18 \cr & = {x^2} - 11x + 18 \cr & = a{x^2} + bx + c = 0 \cr & {\text{For minimum value}} \cr & = \frac{{4ac - {b^2}}}{{4a}} \cr & = \frac{{4 \times 1 \times 18 - {{\left( { - 11} \right)}^2}}}{{4 \times 1}} \cr & = \frac{{72 - 121}}{4} \cr & = \frac{{ - 49}}{4} \cr & = - \frac{{49}}{4} \cr} $$
30.
If x + y + z = 22 and xy + yz + zx = 35, then what is the value of (x - y)2 + (y - z)2 + (z - x)2 ?
(A) 793
(B) 681
(C) 758
(D) 715
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Solution:
x + y + z = 22 xy + yz + zx = 35 (x + y + z)2 = x2 + y2 + z2 + 2(xy + yz + zx) (22)2 = x2 + y2 + z2 + 2 × 35 484 - 70 = x2 + y2 + z2 x2 + y2 + z2 = 414 (x - y)2 + (y - z)2 + (z - x)2 = 2(x2 + y2 + z2 - xy - yz - zx) = 2(414 - 35) = 2 × 379 = 758