41.
If x * y = x2 + y2 - xy, then the value of 9 * 11 is?
(A) 93
(B) 103
(C) 113
(D) 121
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Solution:
$$\eqalign{ & x*y = {x^2} + {y^2} - xy \cr & {\text{ }}9*11 = {\left( 9 \right)^2} + {\left( {11} \right)^2} - 11 \times 9 \cr & \Rightarrow 9*11 = 81 + 121 - 99 \cr & \Rightarrow 9*11 = 103 \cr} $$
42.
If x2 - 3x + 1 = 0, then the value of ÃÂÃÂ ÃÂÃÂ is?
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Solution:
$$\eqalign{ & {x^2} - 3x + 1 = 0 \cr & \Rightarrow {x^2} + 1 = 3x \cr & \Rightarrow x + \frac{1}{x} = 3 \cr & {\text{Squaring both sides}} \cr & \Rightarrow {x^2} + \frac{1}{{{x^2}}} + 2 = 9 \cr & \Rightarrow {x^2} + \frac{1}{{{x^2}}} = 7 \cr & \therefore {x^2} + x + \frac{1}{x} + \frac{1}{{{x^2}}} \cr & \Rightarrow {x^2} + \frac{1}{{{x^2}}} + x + \frac{1}{x} \cr & \Rightarrow 7 + 3 \cr & \Rightarrow 10 \cr} $$
43.
If m - 5n = 2, then the value of (m3 - 125n3 - 30mn) is?
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Solution:
$$\eqalign{ & {\text{Given, }}m - 5n = 2 \cr & {\text{Find, }}{m^3} - 125{n^3} - 30mn \cr & \Rightarrow m - 5n = 2 \cr & \left( {{\text{Cubing both sides}}} \right) \cr & \Rightarrow {\left( {m - 5n} \right)^3} = {2^3} \cr & \Rightarrow {m^3} - 125{n^3} - 3m \times 5n\left( {m - 5n} \right) = 8 \cr & \Rightarrow {m^3} - 125{n^3} - 15mn \times 2 = 8 \cr & \Rightarrow {m^3} - 125{n^3} - 30mn = 8 \cr} $$
44.
If x : y = 2 : 1, then (5x2 - 13xy + 6y2 ) is equal to ?
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Solution:
$$\eqalign{ & x:y = 2:1 \cr & {\text{then, }}5{x^2} - 13xy + 6{y^2} \cr & \Rightarrow 5 \times 4 - 13 \times 2 \times 1 + 6 \times {1^2} \cr & \Rightarrow 20 - 26 + 6 \cr & \Rightarrow 0 \cr} $$
45.
If ÃÂÃÂ ÃÂÃÂ then the value of ÃÂÃÂ ÃÂÃÂ is?
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Solution:
$$\eqalign{ & \frac{{5x}}{{4{x^2} + 10x + 1}} \cr & = \frac{5x}{{x\left( {4x + 10 + \frac{1}{x}} \right)}} \cr & = \frac{5}{{4x + \frac{1}{x} + 10}} \cr & = \frac{5}{{5 + 10}} \cr & = \frac{5}{{15}} \cr & = \frac{1}{3} \cr} $$
46.
If . . . . . ÃÂÃÂ then the value of x is?
(A) 31
(B) 32
(C) 36
(D) 37
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Solution:
$$\frac{1}{4} \times $$ $$\frac{2}{6} \times $$ $$\frac{3}{8} \times $$ $$\frac{4}{{10}} \times $$ $$\frac{5}{{12}} \times $$ . . . . . $$ \times \frac{{31}}{{64}}$$ $$ = \frac{1}{{{2^x}}}$$ Denominator of first term is cut by the numerator of next terms, second term to the next one and so on. $$\eqalign{ & \Rightarrow {\left( {\frac{1}{2}} \right)^{30}} \times {\left( {\frac{1}{2}} \right)^6} = \frac{1}{{{2^x}}} \cr & \Rightarrow {\left( {\frac{1}{2}} \right)^{30 + 6}} = \frac{1}{{{2^x}}} \cr & \Rightarrow \frac{1}{{{2^{36}}}} = \frac{1}{{{2^x}}} \cr & \Rightarrow x = 36 \cr} $$
47.
If a = 1 + ÃÂÃÂÃÂâÃÂÃÂÃÂÃÂ3, b = 1 - ÃÂÃÂÃÂâÃÂÃÂÃÂÃÂ3, then what is the value of a + b?22
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Solution:
a = 1 + √3 ∴ a2 = 1 + 3 + 2√3 b = 1 - √3 ∴ b2 = 1 + 3 - 2√3 ∴ a2 + b2 = 8
48.
The value of
(A) -1
(B) 1
(C) 3
(D) -3
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Solution:
$$\eqalign{ & {a^3} + {b^3} + {c^3} = 3abc \cr & {\text{If }}a + b + c = 0 \cr & a = 0.324 \cr & b = 0.221 \cr & c = - 0.545 \cr & \frac{{\left( {0.545} \right)\left( {0.081} \right)\left( {0.51} \right)\left( {5.2} \right)}}{{3abc}} \cr & = - \frac{{0.545 \times 0.081 \times 0.51 \times 5.2}}{{3 \times 0.324 \times 0.221 \times 0.545}} \cr & = - \frac{{81 \times 510 \times 5.2}}{{3 \times 18 \times 18 \times 13 \times 17}} \cr & = - 1 \cr} $$
49.
If x = 0.5 and y = 0.2, then the value of ÃÂÃÂ is equal to?
(A) 1.0
(B) 0.5
(C) 0.6
(D) 1.1
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Solution:
$$\eqalign{ & x = 0.5 \cr & y = 0.2 \cr & \sqrt {0.6} \times {\left( {3y} \right)^x}{\text{ }} \cr & = \sqrt {0.6} \times {\left( {3 \times 0.2} \right)^{0.5}}{\text{ }} \cr & = \sqrt {0.6} \times \sqrt {0.6} \cr & = 0.6 \cr} $$
50.
If 3x2 - 9x + 3 = 0, then what is the value of
(A) 9
(B) 729
(C) 81
(D) 27
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Solution:
$$\eqalign{ & 3{x^2} - 9x + 3 = 0 \cr & 3x\left( {x - 3 + \frac{1}{x}} \right) = 0,\,x + \frac{1}{x} = 3 \cr & \therefore \,{\left( {x + \frac{1}{x}} \right)^3} = {\left( 3 \right)^3} = 27 \cr} $$