Practice MCQ Questions and Answer on Algebra

51.

If $$x + \left( {\frac{1}{x}} \right) = 2{\text{,}}$$   then the value of $${x^7}{\text{ + }}\left( {\frac{1}{{{x^5}}}} \right)$$   is?

• (A) 25
• (B) 212
• (C) 2
• (D) 27

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 $$\eqalign{ & x + \frac{1}{x} = 2 \cr & {\text{Find , }}{x^7} + \frac{1}{{{x^5}}} \cr & \,\,\,\,\,\,\,x + \frac{1}{x} = 2 \cr & \Rightarrow \mathop \downarrow \limits_1 {\text{ }}\,\,\,\,{\text{ }}\mathop \downarrow \limits_1 \,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow {\text{Let }}x = 1 \cr & {\text{To put value in question,}} \cr & \Rightarrow {x^7} + \frac{1}{{{x^5}}} \cr & \Rightarrow {{\text{1}}^7} + \frac{1}{{{1^5}}} \cr & \Rightarrow 1 + 1 \cr & \Rightarrow 2 \cr} $$

52.

If 2, 0 is a solution of the linear equation 2x + 3y = k, then the value of k is?

• (A) 4
• (B) 6
• (C) 5
• (D) 2

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 $$\eqalign{ & 2x + 3y = k\left( {2 = x,{\text{ }}0 = y} \right) \cr & \therefore 2 \times 2 + 3 \times 0 = k \cr & \Leftrightarrow k = 4 \cr} $$

53.

If a + b = 12, ab = 22, then (a² + b²) is equal to?

• (A) 188
• (B) 144
• (C) 34
• (D) 100

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 $$\eqalign{ & a + b = 12\,.....{\text{(i)}} \cr & ab = 22\,.....{\text{(ii)}} \cr} $$ Squaring both sides of equation (i) $$\eqalign{ & \Rightarrow {a^2} + {b^2} + 2ab = 144 \cr & \Rightarrow {a^2} + {b^2} + 2 \times 22 = 144 \cr & \Rightarrow {a^2} + {b^2} = 144 - 44 \cr & \Rightarrow {a^2} + {b^2} = 100 \cr} $$

54.

If $$n = 7 + 4\sqrt 3 {\text{,}}$$   then the value of $$\left( {\sqrt n + \frac{1}{{\sqrt n }}} \right)$$   is:

• (A) $${\text{2}}\sqrt 3 $$
• (B) 4
• (C) -4
• (D) $$ - 2\sqrt 3 $$

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 $$\eqalign{ & n = 7 + 4\sqrt 3 \cr & \Rightarrow n = 4 + 3 + 4\sqrt 3 \cr & \Rightarrow n = {\left( 2 \right)^2} + {\left( {\sqrt 3 } \right)^2} + 2 \times 2 \times \sqrt 3 \cr & \Rightarrow n = {\left( {2 + \sqrt 3 } \right)^2} \cr & \Rightarrow \sqrt n = 2 + \sqrt 3 \cr & \Rightarrow \frac{1}{{\sqrt n }} = 2 - \sqrt 3 \cr & \therefore \sqrt n + \frac{1}{{\sqrt n }} \cr & = 2 + \sqrt 3 + 2 - \sqrt 3 \cr & = 4 \cr} $$

55.

If $$\frac{4}{3}\left( {{x^2} + \frac{1}{{{x^2}}}} \right) = 110\frac{2}{3},$$     find $$\frac{1}{9}\left( {{x^3} - \frac{1}{{{x^3}}}} \right),$$   where x > 0.

• (A) 84
• (B) 85
• (C) 74
• (D) 76

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 $$\eqalign{ & \frac{4}{3}\left( {{x^2} + \frac{1}{{{x^2}}}} \right) = 110\frac{2}{3} \cr & {x^2} + \frac{1}{{{x^2}}} = \frac{{332}}{3} \times \frac{3}{4} \cr & {x^2} + \frac{1}{{{x^2}}} = 83 \cr & {\left( {x + \frac{1}{x}} \right)^2} = {x^2} + \frac{1}{{{x^2}}} - 2 \times x \times \frac{1}{x} \cr & {\left( {x + \frac{1}{x}} \right)^2} = 83 - 2 \cr & {\left( {x + \frac{1}{x}} \right)^2} = 81 \cr & x + \frac{1}{x} = 9 \cr & {\text{Hence,}} \cr & \frac{1}{9}\left( {{x^3} - \frac{1}{{{x^3}}}} \right) \cr & = \frac{1}{9}\left[ {{{\left( {x - \frac{1}{x}} \right)}^3} + 3 \times \left( {x - \frac{1}{x}} \right)} \right] \cr & = \frac{1}{9}\left[ {729 + 3 \times 9} \right] \cr & = 84 \cr} $$

56.

If x + y = z, then the expression x³ + y³ - z³ + 3xyz will be equal to?

• (A) 0
• (B) 3xyz
• (C) -3xyz
• (D) 3z

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 $$\eqalign{ & x + y = z \cr & x + y - z = 0 \cr & {\text{If }}a + b + c = 0 \cr & {\text{then }}{a^3} + {b^3} + {c^3} - 3abc = 0 \cr & \Rightarrow {x^3} + {y^3} - {z^3} = - 3xyz \cr & \therefore {x^3} + {y^3} - {z^3} + 3xyz = 0 \cr & \Rightarrow 3xyz - 3xyz = 0 \cr} $$

57.

If $$\sqrt {0.04 \times 0.4 \times a} $$    = 0.004 × 0.4 × $$\sqrt b ,$$  then the value of $$\frac{a}{b}\,{\text{is?}}$$

• (A) $${\text{16}} \times {\text{1}}{{\text{0}}^{ - 3}}$$
• (B) $${\text{16}} \times {\text{1}}{{\text{0}}^{ - 4}}$$
• (C) $${\text{16}} \times {\text{1}}{{\text{0}}^{ - 5}}$$
• (D) $${\text{16}} \times {\text{1}}{{\text{0}}^{ - 6}}$$

Show Answer

 $$\eqalign{ & {\text{ }}\sqrt {0.04 \times 0.4 \times a} = 0.004 \times 0.4 \times \sqrt b \cr & \Rightarrow \sqrt {\frac{4}{{100}} \times \frac{4}{{10}} \times a} = \frac{4}{{1000}} \times \frac{4}{{10}} \times \sqrt b \cr & \Rightarrow \frac{4}{{10}}\sqrt {\frac{a}{{10}}} = \frac{4}{{10}} \times \frac{4}{{1000}} \times \sqrt b \cr & \Rightarrow \frac{a}{{10}} = \frac{{16}}{{1000000}} \times b \cr & \Rightarrow \frac{a}{b} = \frac{{16 \times 10}}{{1000000}} \cr & \Rightarrow \frac{a}{b} = \frac{{16}}{{100000}} \cr & \Rightarrow \frac{a}{b} = 16 \times {10^{ - 5}} \cr} $$

58.

If a² + b² = 5ab, then the value of $$\left( {\frac{{{a^2}}}{{{b^2}}}{\text{ + }}\frac{{{b^2}}}{{{a^2}}}} \right)$$   is?

• (A) 32
• (B) 16
• (C) 23
• (D) -23

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 $$\eqalign{ & {a^2} + {b^2} = 5ab \cr & \Rightarrow \frac{{{a^2}}}{{ab}} + \frac{{{b^2}}}{{ab}} = 5 \cr & \Rightarrow \frac{a}{b}{\text{ + }}\frac{b}{a}{\text{ = 5}} \cr & {\text{Squaring the both sides}} \cr & \Rightarrow {\left( {\frac{a}{b}} \right)^2}{\text{ + }}{\left( {\frac{b}{a}} \right)^2} + 2 \times \frac{a}{b} \times \frac{b}{a} = 25 \cr & \Rightarrow \frac{{{a^2}}}{{{b^2}}}{\text{ + }}\frac{{{b^2}}}{{{a^2}}} = 25 - 2 \cr & \Rightarrow \frac{{{a^2}}}{{{b^2}}}{\text{ + }}\frac{{{b^2}}}{{{a^2}}} = 23 \cr} $$

59.

If $$x = \sqrt 3 + \sqrt 2 {\text{,}}$$    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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 $$\eqalign{ & {\text{ }}x = \sqrt 3 + \sqrt 2 \cr & \frac{1}{x} = \frac{1}{{\sqrt 3 + \sqrt 2 }} \times \frac{{\sqrt 3 - \sqrt 2 }}{{\sqrt 3 - \sqrt 2 }} \cr & \frac{1}{x} = \sqrt 3 - \sqrt 2 \cr & \therefore x + \frac{1}{x} \cr & = \sqrt 3 + \sqrt 2 + \sqrt 3 - \sqrt 2 \cr & = 2\sqrt 3 \cr} $$

60.

If $$x + \frac{1}{x} = 99{\text{,}}$$   find the value of   $$\frac{{100x}}{{2{x^2} + 2 + 102x}}$$    is?

• (A) $$\frac{1}{6}$$
• (B) $$\frac{1}{2}$$
• (C) $$\frac{1}{3}$$
• (D) $$\frac{1}{4}$$

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 $$\eqalign{ & x + \frac{1}{x} = 99 \cr & \therefore {x^2} + 1 = 99x \cr & \Rightarrow 2\left( {{x^2} + 1} \right) = 2 \times 99x \cr & \Rightarrow 2{x^2} + 2 = 198x \cr & = \frac{{100x}}{{2{x^2} + 2 + 102x}} \cr & = {\text{ }}\frac{{100x}}{{198x + 102x}} \cr & = \frac{{100x}}{{300x}} \cr & = \frac{1}{3} \cr} $$