Algebra MCQ Questions and Solutions with Explanations | Math Aptitude

a² + b² + c² + $$\frac{1}{{{a^2}}} + \frac{1}{{{b^2}}} + \frac{1}{{{c^2}}} = 6,$$
Put a = b = c = 1
∴ a² + b² + c² = 1 + 1 + 1 = 3

x² + 4y² - 4xy + 4y - 2x - 8
= (x - 2y)² - 2(x - 2y) - 8
put y = (x - 2y)
= y² - 2y - 8
= y² - 4y + 2y - 8
= y(y - 4) + 2(y - 4)
= (y - 4)(y + 2)
= (x - 2y - 4)(x - 2y + 2)

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

27x³ - 64y³
= (3x)³ - (4y)³
= (3x - 4y)[(3x)² + (4y)² + 3x × 4y]
= (3x - 4y)[9x² + 16y² + 12xy]
= (Ax + By) (Cx² - Dy² + 12xy)
Compare value A, B, C, D
A = 3; B = -4; C = 9; D = -16
4A + B + 3C + 2D
= 4 × 3 + (-4) + 3 × 9 + 2 × (-16)
= 12 - 4 + 27 - 32
= 3

If a + 3b = 12
ab = 9
(a - 3b)² = a² + 9b² - 6ab
(a - 3b)² = (a + 3b)² - 12ab
(a - 3b)² = 144 - 108
(a - 3b)² = 36
(a - 3b) = 6

$$\eqalign{ & a = 943\,\,\therefore 2a = 1886 \cr & b = 864\,\,\therefore 2b = 1728 \cr & \therefore {\text{Expression}} \cr & = \frac{{{{\left( {a + b} \right)}^2} - {{\left( {a - b} \right)}^2}}}{{2a \times 2b}} \cr & = \frac{{4ab}}{{4ab}} \cr & = 1 \cr} $$

$$\eqalign{ & \frac{{48.3 \times \left[ {{{\left( {4.95} \right)}^2} + 4.95 \times 13.25} \right]}}{{\left[ {{{\left( {12.55} \right)}^2} - {{\left( {5.65} \right)}^2}} \right] \times 19.8}} \cr & = \frac{{48.3 \times 4.95\left[ {4.95 + 13.25} \right]}}{{\left( {12.55 + 5.65} \right)\left( {12.55 - 5.65} \right) \times 19.8}} \cr & = \frac{{48.3 \times 4.95 \times 18.2}}{{18.2 \times 6.9 \times 19.8}} \cr & = \frac{7}{4} \cr & = 1.75 \cr} $$

$$\eqalign{ & {a^2} + \frac{2}{{{a^2}}} = 16 \cr & \frac{{72{a^2}}}{{{a^4} + 2 + 8{a^2}}} \cr & = \frac{{72}}{{{a^2} + \frac{2}{{{a^2}}} + 8}} \cr & = \frac{{72}}{{16 + 8}} \cr & = 3 \cr} $$

a² + b² = 4b + 6a - 13
a² + 9 - 3a + b² + 4 - 4b = 0
(a - 3)² + (b - 2)² = 0
a = 3, b = 2
a + b = 5

$$\eqalign{ & {\text{If }}a + b + c = 0, \cr & {\text{then }} \Rightarrow {a^3} + {b^3} + {c^3} = 3abc \cr & = \frac{{{a^2}}}{{bc}} + \frac{{{b^2}}}{{ca}} + \frac{{{c^2}}}{{ab}} \cr & = \frac{{{a^3} + {b^3} + {c^3}}}{{abc}} \cr & = \frac{{3abc}}{{abc}} \cr & = 3 \cr} $$