Algebra MCQ Questions and Solutions with Explanations

Section 1 Section 2 Section 3 Section 4 Section 5 Section 6

11.
If t² - 4t + 1 = 0, then the value of $${t^3} + \frac{1}{{{t^3}}}$$ is?

A. 44

B. 48

C. 52

D. 64

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12.
If $$\root 3 \of a + \root 3 \of b = \root 3 \of c {\text{,}}$$ then the simplest value of $${\left( {a + b - c} \right)^3}$$ + $$27abc$$ is?

A. -1

B. 3

C. -3

D. 0

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13.
If $$x = \root 3 \of {a + \sqrt {{a^2} + {b^3}} } $$ + $$\root 3 \of {a - \sqrt {{a^2} + {b^3}} } $$ then $${x^3} + 3bx$$ is equal to?

A. 0

B. a

C. 2a

D. 1

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14.
If $$\frac{{{x^{24}} + 1}}{{{x^{12}}}} = 7,$$ then the value of $$\frac{{{x^{72}} + 1}}{{{x^{36}}}} = \,?$$

A. 343

B. 433

C. 432

D. 322

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15.
If x = 2 then the value of x³ + 27x² + 243x + 631 = ?

A. 1321

B. 1233

C. 1231

D. 1211

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16.
If 5x + 9y = 5 and 125x³ + 729y³ = 120, then the value of the product of x and y is?

A. 45

B. $$\frac{1}{9}$$

C. $$\frac{1}{{135}}$$

D. 135

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17.
x² + y² + z² = 2(x + z - 1), then the value of x³ + y³ + z³ = ?

A. -1

B. 2

C. 0

D. 1

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18.
If $$x + \frac{1}{x} = 1,$$ then the value of $$\frac{2}{{{x^2} - x + 2}} = \,?$$

A. $$\frac{2}{3}$$

B. 2

C. 1

D. 4

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19.
If $$x = \frac{{\sqrt 5 - \sqrt 3 }}{{\sqrt 5 + \sqrt 3 }}$$ and $$y = \frac{{\sqrt 5 + \sqrt 3 }}{{\sqrt 5 - \sqrt 3 }}$$ then the value of $$\frac{{{x^2} + xy + {y^2}}}{{{x^2} - xy + {y^2}}} = ?$$

A. $$\frac{{65}}{{63}}$$

B. $$\frac{{67}}{{65}}$$

C. $$\frac{{69}}{{67}}$$

D. $$\frac{{63}}{{61}}$$

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Answer & Solution

Answer: Option D

Solution:

$$\eqalign{ & {\text{Given,}} \cr & x = \frac{{\sqrt 5 - \sqrt 3 }}{{\sqrt 5 + \sqrt 3 }}{\text{ , }}y = \frac{{\sqrt 5 + \sqrt 3 }}{{\sqrt 5 - \sqrt 3 }} \cr & {\text{Find, }}\frac{{{x^2} + xy + {y^2}}}{{{x^2} - xy + {y^2}}} = ? \cr & \Rightarrow {\text{ }}\frac{{{x^2} + {y^2} + 2xy - xy}}{{{x^2} + {y^2} - 2xy + xy}} \cr & \Rightarrow {\text{ }}\frac{{{{\left( {x + y} \right)}^2} - xy}}{{{{\left( {x - y} \right)}^2} + xy}} = ? \cr & {\text{Now,}} \cr & {\text{ }}x + y = \frac{{\sqrt 5 - \sqrt 3 }}{{\sqrt 5 + \sqrt 3 }} + \frac{{\sqrt 5 + \sqrt 3 }}{{\sqrt 5 - \sqrt 3 }} \cr & \Rightarrow x + y = \frac{{{{\left( {\sqrt 5 - \sqrt 3 } \right)}^2} + {{\left( {\sqrt 5 + \sqrt 3 } \right)}^2}}}{{{{\sqrt 5 }^2} - {{\sqrt 3 }^2}}} \cr & \Rightarrow {\text{ }}x + y = \frac{{2\left( {{{\sqrt 5 }^2} + {{\sqrt 3 }^2}} \right)}}{{5 - 3}} \cr & \Rightarrow x + y = 8\,.......(i) \cr & Again, \cr & x - y = \frac{{\sqrt 5 - \sqrt 3 }}{{\sqrt 5 + \sqrt 3 }} - \frac{{\sqrt 5 + \sqrt 3 }}{{\sqrt 5 - \sqrt 3 }} \cr & \Rightarrow {\text{ }}x - y = \frac{{4 \times \sqrt 5 \times \sqrt 3 }}{2} \cr & \Rightarrow x - y = 2\sqrt {15} ..............(ii) \cr & {\text{And, }}xy = \frac{{\sqrt 5 - \sqrt 3 }}{{\sqrt 5 + \sqrt 3 }} \times \frac{{\sqrt 5 + \sqrt 3 }}{{\sqrt 5 - \sqrt 3 }} \cr & \Rightarrow {\text{ }}xy = 1 \cr & {\text{Substitutes values in the question}}{\text{.}} \cr & \Rightarrow \frac{{{{\left( {x + y} \right)}^2} - xy}}{{{{\left( {x - y} \right)}^2} + xy}} \cr & \Rightarrow \frac{{{8^2} - 1}}{{{{\left( {2\sqrt {15} } \right)}^2} + 1}} \cr & \Rightarrow \frac{{63}}{{61}} \cr} $$

20.
Simplified value of $$\left[ {\,\left( {1 + \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\left( {1 + \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\, - \,\,\left( {1 - \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\left( {1 - \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,} \right]$$ $$ ÷ $$ $$\left[ {\,\left( {1 + \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,\, + \,\,\left( {1 - \frac{1}{{10 + \frac{1}{{10}}}}} \right)\,} \right]$$ = ?

A. $$\frac{{20}}{{101}}$$

B. $$\frac{{100}}{{101}}$$

C. 2

D. $$\frac{{90}}{{101}}$$

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Algebra MCQ Questions and Solutions with Explanations | Math Aptitude

11. If t2 - 4t + 1 = 0, then the value of $${t^3} + \frac{1}{{{t^3}}}$$ is?

Answer & Solution

12. If $$\root 3 \of a + \root 3 \of b = \root 3 \of c {\text{,}}$$ then the simplest value of $${\left( {a + b - c} \right)^3}$$ + $$27abc$$ is?

Answer & Solution

13. If $$x = \root 3 \of {a + \sqrt {{a^2} + {b^3}} } $$ + $$\root 3 \of {a - \sqrt {{a^2} + {b^3}} } $$ then $${x^3} + 3bx$$ is equal to?

Answer & Solution

14. If $$\frac{{{x^{24}} + 1}}{{{x^{12}}}} = 7,$$ then the value of $$\frac{{{x^{72}} + 1}}{{{x^{36}}}} = \,?$$

Answer & Solution

15. If x = 2 then the value of x3 + 27x2 + 243x + 631 = ?

Answer & Solution

16. If 5x + 9y = 5 and 125x3 + 729y3 = 120, then the value of the product of x and y is?

Answer & Solution

17. x2 + y2 + z2 = 2(x + z - 1), then the value of x3 + y3 + z3 = ?

Answer & Solution

18. If $$x + \frac{1}{x} = 1,$$ then the value of $$\frac{2}{{{x^2} - x + 2}} = \,?$$

Answer & Solution

19. If $$x = \frac{{\sqrt 5 - \sqrt 3 }}{{\sqrt 5 + \sqrt 3 }}$$ and $$y = \frac{{\sqrt 5 + \sqrt 3 }}{{\sqrt 5 - \sqrt 3 }}$$ then the value of $$\frac{{{x^2} + xy + {y^2}}}{{{x^2} - xy + {y^2}}} = ?$$

Answer & Solution

Answer & Solution

Read More Section(Algebra)

11.
If t² - 4t + 1 = 0, then the value of $${t^3} + \frac{1}{{{t^3}}}$$ is?

12.
If $$\root 3 \of a + \root 3 \of b = \root 3 \of c {\text{,}}$$ then the simplest value of $${\left( {a + b - c} \right)^3}$$ + $$27abc$$ is?

13.
If $$x = \root 3 \of {a + \sqrt {{a^2} + {b^3}} } $$ + $$\root 3 \of {a - \sqrt {{a^2} + {b^3}} } $$ then $${x^3} + 3bx$$ is equal to?

14.
If $$\frac{{{x^{24}} + 1}}{{{x^{12}}}} = 7,$$ then the value of $$\frac{{{x^{72}} + 1}}{{{x^{36}}}} = \,?$$

15.
If x = 2 then the value of x³ + 27x² + 243x + 631 = ?

16.
If 5x + 9y = 5 and 125x³ + 729y³ = 120, then the value of the product of x and y is?

17.
x² + y² + z² = 2(x + z - 1), then the value of x³ + y³ + z³ = ?

18.
If $$x + \frac{1}{x} = 1,$$ then the value of $$\frac{2}{{{x^2} - x + 2}} = \,?$$

19.
If $$x = \frac{{\sqrt 5 - \sqrt 3 }}{{\sqrt 5 + \sqrt 3 }}$$ and $$y = \frac{{\sqrt 5 + \sqrt 3 }}{{\sqrt 5 - \sqrt 3 }}$$ then the value of $$\frac{{{x^2} + xy + {y^2}}}{{{x^2} - xy + {y^2}}} = ?$$