Algebra MCQ Questions and Solutions with Explanations | Math Aptitude

a + a² + a³ - 1 = 0
a³ = 1 - a² - a . . . . . . . . (i)
a + a² + a³ = 1
on dividing by a
a² + a + 1 = $$\frac{1}{a}$$ . . . . . . . . (ii)
Add equation (i) & (ii),
$${a^3} + \frac{1}{a}$$
= 1 - a² - a + a² + a + 1
= 2

$$\eqalign{ & x\left( {3 - \frac{2}{x}} \right) = \frac{3}{x} \cr & 3x - 2 = \frac{3}{x} \cr & 3\left( {x - \frac{1}{x}} \right) = 2 \cr & x - \frac{1}{x} = \frac{2}{3} \cr & {x^3} - \frac{1}{{{x^3}}} = {\left( {\frac{2}{3}} \right)^3} + 3 \times \frac{2}{3} \cr & {x^3} - \frac{1}{{{x^3}}} = \frac{8}{{27}} + 2 \cr & {x^3} - \frac{1}{{{x^3}}} = \frac{{62}}{{27}} \cr} $$

$$\eqalign{ & \frac{{{x^8} + 1}}{{{x^4}}} = 14 \cr & {x^4} + \frac{1}{{{x^4}}} = 14 \cr & {x^2} + \frac{1}{{{x^2}}} = 4 \cr & {x^6} + \frac{1}{{{x^6}}} = {4^3} - 3 \times 4 \cr & {x^6} + \frac{1}{{{x^6}}} = 64 - 12 \cr & \frac{{{x^{12}} + 1}}{{{x^6}}} = 52 \cr} $$

Take x = 3, y = 1, z = 2, satisfy the equation
$$\therefore \,\frac{y}{{y - z}} + \frac{x}{{x - z}} \Rightarrow \frac{1}{{ - 1}} + \frac{3}{1} = 2$$

$$\eqalign{ & x + \frac{{16}}{x} = 8 \cr & {\text{at }}x = 4{\text{ it satisfy}} \cr & {x^2} + \frac{{32}}{{{x^2}}} = {4^2} + \frac{{32}}{{{4^2}}} \cr & = 16 + 2 \cr & = 18 \cr} $$

$$\eqalign{ & x + 3y - \frac{{2z}}{4} = 6 \cr & 2x + 6y - z = 12\,........\,\left( {\text{i}} \right) \cr & x + \frac{2}{3}\left( {2y + 3z} \right) = 33 \cr & 3x + 4y + 6z = 99\,........\,\left( {{\text{ii}}} \right) \cr & \frac{1}{7}\left( {x + y + z} \right) + 2z = 9 \cr & x + y + z + 14z = 63 \cr & x + y + 15z = 63\,........\,\left( {{\text{iii}}} \right) \cr & {\text{Add}}\,\left[ {21 \times \left( {\text{i}} \right)} \right] + \left( {{\text{ii}}} \right) + \left( {{\text{iii}}} \right){\text{ }}\left( {{\text{remove }}z} \right) \cr & 21\left( {2x + 6y + z} \right) = 12 \times 21 \cr & 3x + 4y + 6z = 99 \cr & x + y + 15z = 63 \cr & 42x + 126y - 21z = 252 \cr & 3x + 4y + 6z = 99 \cr & \underline {x + y + 15z = 63\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \cr & 46x + 131y = 414 \cr} $$

$$\eqalign{ & \sqrt x + \frac{1}{{\sqrt x }} = 3 \cr & x + \frac{1}{x} = 7 \cr & {x^3} + \frac{1}{{{x^3}}} = {7^3} - 3 \times 7 \cr & = 343 - 21 \cr & = 322 \cr} $$

a + b = 5, ab = 3
(a³ + b³) = (a + b)[(a + b)² - 3ab]
= 5[5² - 3 × 3]
= 5(25 - 9)
= 80

(4a - 3b) = 1, ab = $$\frac{1}{2}$$
Squaring both side
(4a - 3b)² = (1)²
16a² + 9b² - 2(4a)(3b) = 1
16a² + 9b² - 24ab = 1
16a² + 9b² - 24$$\left( {\frac{1}{2}} \right)$$ = 1
16a² + 9b² - 12 = 1
16a² + 9b² = 13
Adding 12 both side
16a² + 9b² + 12 = 13 + 12
(4a)² +(3b)² + 2(4a)(3b) = 25
(4a + 3b)² = 25
4a + 3b = 5
Cubing both side
64a³ + 27b³ = 125 - 3(4a)(3b)(4a + 3b)
64a³ + 27b³ = 125 - 90
64a³ + 27b³ = 35

$$\eqalign{ & 5\sqrt 5 {x^3} - 81\sqrt 3 {y^3} \div \sqrt 5 x - 3\sqrt 3 y = \left( {A{x^2} + B{y^2} + Cxy} \right) \cr & \Rightarrow \frac{{{{\left( {\sqrt 5 x} \right)}^3} - {{\left( {3\sqrt 3 y} \right)}^3}}}{{\sqrt 5 x - 3\sqrt 3 y}} = A{x^2} + B{y^2} + Cxy \cr & \Rightarrow \frac{{\left( {\sqrt 5 x - 3\sqrt 3 y} \right)\left( {5{x^2} + 27{y^2} + 3\sqrt {15} xy} \right)}}{{\sqrt 5 x - 3\sqrt 3 y}} = A{x^2} + B{y^2} + Cxy \cr & \Rightarrow 5{x^2} + 27{y^2} + 3\sqrt {15} xy = A{x^2} + B{y^2} + Cxy \cr & \Rightarrow {\text{ Comprison Coefficient}} \cr & A = 5,\,B = 27,\,C = 3\sqrt {15} \cr & \left( {6A + B - \sqrt {15} C} \right) = 6 \times 5 + 27 - 45 \cr & = 30 + 27 - 45 \cr & = 12 \cr} $$