Advanced Algebra Quiz - Quantitative Aptitude for SSC & Police | Concept of Maths

Question 1

1. If $x + \frac{1}{x} = 5$, what is the value of $x^5 + \frac{1}{x^5}$?

A) (A) 2515

B) (B) 2520

C) (C) 2500

D) (D) 2525

Answer

Correct Answer:

Solution: Formula: $x^5 + \frac{1}{x^5} = (x^2 + \frac{1}{x^2})(x^3 + \frac{1}{x^3}) - (x + \frac{1}{x})$ $x^2 + \frac{1}{x^2} = 5^2 - 2 = 23$ $x^3 + \frac{1}{x^3} = 5^3 - 3(5) = 110$ $x^5 + \frac{1}{x^5} = (23 \times 110) - 5 = 2530 - 5 = 2525$.

Question 2

2. If $x^4 + x^2y^2 + y^4 = 133$ and $x^2 - xy + y^2 = 7$, find the value of $xy$.

A) (A) 3

B) (B) 6

C) (C) 12

D) (D) 19

Answer

Correct Answer:

Solution: Identity: $x^4 + x^2y^2 + y^4 = (x^2 - xy + y^2)(x^2 + xy + y^2)$ $133 = 7 \times (x^2 + xy + y^2) \implies x^2 + xy + y^2 = 19$ Subtracting the two equations: $(x^2 + xy + y^2) - (x^2 - xy + y^2) = 19 - 7$ $2xy = 12 \implies xy = 6$.

Question 3

3. If $a^3 + b^3 + c^3 - 3abc = 432$ and $a + b + c = 12$, find the value of $(a-b)^2 + (b-c)^2 + (c-a)^2$.

A) (A) 12

B) (B) 36

C) (C) 72

D) (D) 144

Answer

Correct Answer:

Solution: Formula: $a^3 + b^3 + c^3 - 3abc = \frac{1}{2}(a+b+c)[(a-b)^2 + (b-c)^2 + (c-a)^2]$ $432 = \frac{1}{2}(12) \times [(a-b)^2 + (b-c)^2 + (c-a)^2]$ $432 = 6 \times [(a-b)^2 + (b-c)^2 + (c-a)^2]$ $432 / 6 = 72$.

Question 4

4. If $x = \sqrt{3} + \sqrt{2}$, find the value of $x^3 - \frac{1}{x^3}$.

A) (A) $22\sqrt{2}$

B) (B) $22\sqrt{3}$

C) (C) $18\sqrt{2}$

D) (D) $18\sqrt{3}$

Answer

Correct Answer:

Solution: If $x = \sqrt{3} + \sqrt{2}$, then $\frac{1}{x} = \sqrt{3} - \sqrt{2}$ (by rationalizing). $x - \frac{1}{x} = (\sqrt{3} + \sqrt{2}) - (\sqrt{3} - \sqrt{2}) = 2\sqrt{2}$ $x^3 - \frac{1}{x^3} = (x - \frac{1}{x})^3 + 3(x - \frac{1}{x})$ $= (2\sqrt{2})^3 + 3(2\sqrt{2}) = 16\sqrt{2} + 6\sqrt{2} = 22\sqrt{2}$.

Question 5

5. If $\sqrt{x} + \frac{1}{\sqrt{x}} = \sqrt{7}$, find the value of $x^3 + \frac{1}{x^3}$.

A) (A) 140

B) (B) 120

C) (C) 110

D) (D) 90

Answer

Correct Answer:

Solution: Square both sides: $(\sqrt{x} + \frac{1}{\sqrt{x}})^2 = 7 \implies x + \frac{1}{x} + 2 = 7 \implies x + \frac{1}{x} = 5$. $x^3 + \frac{1}{x^3} = (x + \frac{1}{x})^3 - 3(x + \frac{1}{x})$ $= 5^3 - 3(5) = 125 - 15 = 110$.

Question 6

6. If $a + b + c = 0$, find the value of $\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}$.

A) (A) 3

B) (B) 1

C) (C) 0

D) (D) -1

Answer

Correct Answer:

Solution: Take the LCM of the expression: $\frac{a^3 + b^3 + c^3}{abc}$. According to the algebraic identity, if $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$. Substituting this back: $\frac{3abc}{abc} = 3$.

Question 7

7. If $8x^3 - 27y^3 = (Ax + By)(Cx^2 - Dxy + Ey^2)$, then what is the value of $A + B + C - D + E$?

A) (A) 12

B) (B) 18

C) (C) 24

D) (D) 30

Answer

Correct Answer:

Solution: Use $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$. $8x^3 - 27y^3 = (2x)^3 - (3y)^3 = (2x - 3y)(4x^2 + 6xy + 9y^2)$. Compare coefficients: $A=2, B=-3, C=4, -D=6 \implies D=-6, E=9$. $A + B + C - D + E = 2 - 3 + 4 - (-6) + 9 = 18$.

Question 8

8. If $x = 11$, find the value of $x^5 - 12x^4 + 12x^3 - 12x^2 + 12x - 1$.

A) (A) 10

B) (B) 11

C) (C) 12

D) (D) 1

Answer

Correct Answer:

Solution: Break $12$ into $(11 + 1)$ and substitute $11$ with $x$: $x^5 - (x+1)x^4 + (x+1)x^3 - (x+1)x^2 + (x+1)x - 1$ $= x^5 - x^5 - x^4 + x^4 + x^3 - x^3 - x^2 + x^2 + x - 1$ $= x - 1$. Since $x = 11$, the answer is $11 - 1 = 10$.

Question 9

9. Let $a, b, c$ be real numbers such that $a^2 + b^2 + c^2 = 2(a - b - c) - 3$. Find the value of $a + b + c$.

A) (A) 1

B) (B) 0

C) (C) -1

D) (D) 3

Answer

Correct Answer:

Solution: Rearrange: $a^2 - 2a + b^2 + 2b + c^2 + 2c + 3 = 0$. Split the $3$ into $1+1+1$ to form perfect squares: $(a^2 - 2a + 1) + (b^2 + 2b + 1) + (c^2 + 2c + 1) = 0$ $(a - 1)^2 + (b + 1)^2 + (c + 1)^2 = 0$. Squares sum to $0$ only if each is $0$. So, $a=1, b=-1, c=-1$. $a + b + c = 1 - 1 - 1 = -1$.

Question 10

10. If $x(x-3) = -1$, find the value of $x^3(x^3 - 18)$.

A) (A) 1

B) (B) -1

C) (C) 18

D) (D) 0

Answer

Correct Answer:

Solution: $x(x-3) = -1 \implies x^2 - 3x + 1 = 0 \implies x + \frac{1}{x} = 3$. We need $x^6 - 18x^3$. $x^3 + \frac{1}{x^3} = 3^3 - 3(3) = 18$. Multiply by $x^3$: $x^6 + 1 = 18x^3 \implies x^6 - 18x^3 = -1$.

Question 11

11. What is the minimum value of $(x-2)(x-9)$?

A) (A) -12

B) (B) -12.25

C) (C) -12.5

D) (D) -11.75

Answer

Correct Answer:

Solution: Expand: $x^2 - 11x + 18$. This is a parabola opening upwards. Minimum value occurs at $x = -\frac{b}{2a} = \frac{11}{2} = 5.5$. Substitute $x = 5.5$: $(5.5 - 2)(5.5 - 9) = (3.5)(-3.5) = -12.25$.

Question 12

12. If $a^4 + \frac{1}{a^4} = 119$ and $a > 1$, find the value of $a^3 - \frac{1}{a^3}$.

A) (A) 32

B) (B) 34

C) (C) 36

D) (D) 38

Answer

Correct Answer:

Solution: Add 2: $a^4 + \frac{1}{a^4} + 2 = 121 \implies (a^2 + \frac{1}{a^2})^2 = 121 \implies a^2 + \frac{1}{a^2} = 11$. Subtract 2: $a^2 + \frac{1}{a^2} - 2 = 9 \implies (a - \frac{1}{a})^2 = 9 \implies a - \frac{1}{a} = 3$ (since $a > 1$). $a^3 - \frac{1}{a^3} = 3^3 + 3(3) = 27 + 9 = 36$.

Question 13

13. If $x + y + z = 19$ and $xy + yz + zx = 114$, then find the value of $\sqrt{x^3 + y^3 + z^3 - 3xyz}$.

A) (A) 114

B) (B) 133

C) (C) 361

D) (D) 19

Answer

Correct Answer:

Solution: First, find $x^2 + y^2 + z^2 = (x+y+z)^2 - 2(xy+yz+zx) = 19^2 - 2(114) = 361 - 228 = 133$. Identity: $x^3 + y^3 + z^3 - 3xyz = (x+y+z)(x^2 + y^2 + z^2 - (xy+yz+zx))$. $= 19 \times (133 - 114) = 19 \times 19 = 361$. Therefore, $\sqrt{361} = 19$.

Question 14

14. If $\frac{x^2}{y^2} + \frac{y^2}{x^2} = 7$, find the value of $\frac{x^3}{y^3} + \frac{y^3}{x^3}$ (assuming $x, y > 0$).

A) (A) 18

B) (B) 21

C) (C) 24

D) (D) 27

Answer

Correct Answer:

Solution: Let $t = \frac{x}{y}$. Then $t^2 + \frac{1}{t^2} = 7$. Add 2: $(t + \frac{1}{t})^2 = 9 \implies t + \frac{1}{t} = 3$ (since $x, y > 0$). $t^3 + \frac{1}{t^3} = 3^3 - 3(3) = 27 - 9 = 18$.

Question 15

15. If $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0$ and $x+y+z = 9$, what is the value of $x^2 + y^2 + z^2$?

A) (A) 0

B) (B) 81

C) (C) 27

D) (D) 9

Answer

Correct Answer:

Solution: $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{xy+yz+zx}{xyz} = 0 \implies xy+yz+zx = 0$. $(x+y+z)^2 = x^2 + y^2 + z^2 + 2(xy+yz+zx)$. $9^2 = x^2 + y^2 + z^2 + 2(0) \implies x^2 + y^2 + z^2 = 81$.

Question 16

16. If $2x + \frac{1}{3x} = 4$, find the value of $9x^2 + \frac{1}{4x^2}$.

A) (A) 31

B) (B) 32

C) (C) 33

D) (D) 36

Answer

Correct Answer:

Solution: To get $3x$ and $\frac{1}{2x}$, multiply the entire equation by $\frac{3}{2}$. $\frac{3}{2}(2x + \frac{1}{3x}) = 4 \times \frac{3}{2} \implies 3x + \frac{1}{2x} = 6$. Square both sides: $(3x)^2 + (\frac{1}{2x})^2 + 2(3x)(\frac{1}{2x}) = 36$ $9x^2 + \frac{1}{4x^2} + 3 = 36 \implies 9x^2 + \frac{1}{4x^2} = 33$.

Question 17

17. If $x = 2 - \sqrt{3}$, find the value of $x^3 - 2x^2 - 7x + 5$.

A) (A) 3

B) (B) 0

C) (C) 5

D) (D) 7

Answer

Correct Answer:

Solution: $x - 2 = -\sqrt{3}$. Square both sides: $(x-2)^2 = 3 \implies x^2 - 4x + 1 = 0$. Divide the given expression $x^3 - 2x^2 - 7x + 5$ by $(x^2 - 4x + 1)$. $x^3 - 2x^2 - 7x + 5 = x(x^2 - 4x + 1) + 2x^2 - 8x + 5$ $= x(0) + 2(x^2 - 4x + 1) + 3 = 0 + 2(0) + 3 = 3$.

Question 18

18. If $P = 99$, find the value of $P(P^2 + 3P + 3)$.

A) (A) 99999

B) (B) 999999

C) (C) 1000000

D) (D) 999998

Answer

Correct Answer:

Solution: Multiply $P$ inside: $P^3 + 3P^2 + 3P$. Add and subtract 1: $(P^3 + 3P^2 + 3P + 1) - 1$ $= (P + 1)^3 - 1$ $= (99 + 1)^3 - 1 = 100^3 - 1 = 1,000,000 - 1 = 999,999$.

Question 19

19. If $a + b + c = 6$ and $a^2 + b^2 + c^2 = 14$, find the value of $ab + bc + ca$.

A) (A) 10

B) (B) 14

C) (C) 11

D) (D) 22

Answer

Correct Answer:

Solution: $(a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab+bc+ca)$ $6^2 = 14 + 2(ab+bc+ca)$ $36 - 14 = 2(ab+bc+ca) \implies 22 = 2(ab+bc+ca)$ $ab+bc+ca = 11$.

Question 20

20. Solve for $x$: $\sqrt{x+5} + \sqrt{x-16} = 7$.

A) (A) 16

B) (B) 25

C) (C) 11

D) (D) 20

Answer

Correct Answer:

Solution: Let $\sqrt{x+5} = a$ and $\sqrt{x-16} = b$. Then $a + b = 7$. Also, $a^2 - b^2 = (x+5) - (x-16) = 21$. $(a-b)(a+b) = 21 \implies (a-b)(7) = 21 \implies a-b = 3$. We have $a+b=7$ and $a-b=3$. Adding them gives $2a=10 \implies a=5$. $a^2 = x+5 \implies 25 = x+5 \implies x = 20$.

Master Advanced Algebra for SSC & Police Exams | Concept of Maths

Master Advanced Algebra for SSC & Police Exams | Concept of Maths

Discussion 0

Enjoyed this content?

4. If $x = \sqrt{3} + \sqrt{2}$, find the value of $x^3 - \frac{1}{x^3}$.

5. If $\sqrt{x} + \frac{1}{\sqrt{x}} = \sqrt{7}$, find the value of $x^3 + \frac{1}{x^3}$.

6. If $a + b + c = 0$, find the value of $\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}$.

7. If $8x^3 - 27y^3 = (Ax + By)(Cx^2 - Dxy + Ey^2)$, then what is the value of $A + B + C - D + E$?

8. If $x = 11$, find the value of $x^5 - 12x^4 + 12x^3 - 12x^2 + 12x - 1$.

9. Let $a, b, c$ be real numbers such that $a^2 + b^2 + c^2 = 2(a - b - c) - 3$. Find the value of $a + b + c$.

10. If $x(x-3) = -1$, find the value of $x^3(x^3 - 18)$.

11. What is the minimum value of $(x-2)(x-9)$?

12. If $a^4 + \frac{1}{a^4} = 119$ and $a > 1$, find the value of $a^3 - \frac{1}{a^3}$.

13. If $x + y + z = 19$ and $xy + yz + zx = 114$, then find the value of $\sqrt{x^3 + y^3 + z^3 - 3xyz}$.

14. If $\frac{x^2}{y^2} + \frac{y^2}{x^2} = 7$, find the value of $\frac{x^3}{y^3} + \frac{y^3}{x^3}$ (assuming $x, y > 0$).

15. If $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0$ and $x+y+z = 9$, what is the value of $x^2 + y^2 + z^2$?

16. If $2x + \frac{1}{3x} = 4$, find the value of $9x^2 + \frac{1}{4x^2}$.

17. If $x = 2 - \sqrt{3}$, find the value of $x^3 - 2x^2 - 7x + 5$.

18. If $P = 99$, find the value of $P(P^2 + 3P + 3)$.

19. If $a + b + c = 6$ and $a^2 + b^2 + c^2 = 14$, find the value of $ab + bc + ca$.

20. Solve for $x$: $\sqrt{x+5} + \sqrt{x-16} = 7$.

Advanced Algebra Quiz - Quantitative Aptitude for SSC & Police | Concept of Maths

1. If $x + \frac{1}{x} = 5$, what is the value of $x^5 + \frac{1}{x^5}$?

2. If $x^4 + x^2y^2 + y^4 = 133$ and $x^2 - xy + y^2 = 7$, find the value of $xy$.

3. If $a^3 + b^3 + c^3 - 3abc = 432$ and $a + b + c = 12$, find the value of $(a-b)^2 + (b-c)^2 + (c-a)^2$.

Master Advanced Algebra for SSC & Police Exams | Concept of Maths

Master Advanced Algebra for SSC & Police Exams | Concept of Maths

Share This Quiz

Discussion 0

Enjoyed this content?

Related Content

4. If $x = \sqrt{3} + \sqrt{2}$, find the value of $x^3 - \frac{1}{x^3}$.

5. If $\sqrt{x} + \frac{1}{\sqrt{x}} = \sqrt{7}$, find the value of $x^3 + \frac{1}{x^3}$.

6. If $a + b + c = 0$, find the value of $\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}$.

7. If $8x^3 - 27y^3 = (Ax + By)(Cx^2 - Dxy + Ey^2)$, then what is the value of $A + B + C - D + E$?

8. If $x = 11$, find the value of $x^5 - 12x^4 + 12x^3 - 12x^2 + 12x - 1$.

9. Let $a, b, c$ be real numbers such that $a^2 + b^2 + c^2 = 2(a - b - c) - 3$. Find the value of $a + b + c$.

10. If $x(x-3) = -1$, find the value of $x^3(x^3 - 18)$.

11. What is the minimum value of $(x-2)(x-9)$?

12. If $a^4 + \frac{1}{a^4} = 119$ and $a > 1$, find the value of $a^3 - \frac{1}{a^3}$.

13. If $x + y + z = 19$ and $xy + yz + zx = 114$, then find the value of $\sqrt{x^3 + y^3 + z^3 - 3xyz}$.

14. If $\frac{x^2}{y^2} + \frac{y^2}{x^2} = 7$, find the value of $\frac{x^3}{y^3} + \frac{y^3}{x^3}$ (assuming $x, y > 0$).

15. If $\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = 0$ and $x+y+z = 9$, what is the value of $x^2 + y^2 + z^2$?

16. If $2x + \frac{1}{3x} = 4$, find the value of $9x^2 + \frac{1}{4x^2}$.

17. If $x = 2 - \sqrt{3}$, find the value of $x^3 - 2x^2 - 7x + 5$.

18. If $P = 99$, find the value of $P(P^2 + 3P + 3)$.

19. If $a + b + c = 6$ and $a^2 + b^2 + c^2 = 14$, find the value of $ab + bc + ca$.

20. Solve for $x$: $\sqrt{x+5} + \sqrt{x-16} = 7$.