Algebra Practice Questions Hard Level

Algebra questions basically involve modeling word problems into equations and then solving them.

Question 1: If a = 1 – 1/b and b = 1 – 1/c, then the value of c – 1/a is 

a = 1 – 1/b
=>ab = b - 1
=>1/a = b/(b - 1) ——–(1)

And
b = 1-1/c
b + 1/c = 1
bc + 1 = c
bc – c  =  -1
c(b – 1)  = -1
c  = 1/(1 – b) ———–(2)

putting the values of 1/a and c from above 1 and 2 in c – 1/a,

1/(1 – b)- b/(b-1) =    (b + 1)/(1 - b)    

Question 2: If a + b + c = 3, then the value of 1/(1 – a)(1 – b) + 1/(1 – b)(1 – c) + 1/(1 – c)(1 – a) 

1/(1 – a)(1 – b) + 1/(1 – b)(1 – c) + 1/(1 – c)(1 – a) 
[(1 – c) + (1 – a) + (1 – b)]/(1 – a)(1 – b)(1 – c) 
[3 – (a + b + c)]/(1 – a)(1 – b)(1 – c) 
3 – 3 /(1 – a)(1 – b)(1 – c) = 0

Question 3: If a + 1/a = √3, then the value of a¹⁸ + a¹² + a⁶ + 1 is 

 a³ + 1/a³ = (a + 1/a)³ – 3(a + 1/a) 
=> 3 √3 – 3 √3 
=> 0 
a³ + 1/a³ = 0 
a⁶ + 1 = 0 

Then, 
a¹⁸ + a¹² + a⁶ + 1 
a¹²(a⁶ + 1) + (a⁶ + 1) 
a¹² x 0 + 0 = 0 

Question 4: If a = √3 + 1 / √3 -1 and b = √3 -1 / √3 + 1, then find the value of (a² + ab + b²)/(a² – ab + b²) is 

 a = 1/b 
therefore ab = 1 
a + b = (√3 + 1) / ( √3 -1) + (√3 -1) / (√3 + 1) 
(3 + 1 + 2√3 + 3 + 1 – 2√3)/ (3 – 1) 
8/2  = 4 

a + b = 4 
a² + b² = 4² – 2 *(ab) 
a² + b² = 14 

Now, (a² + ab + b²)/(a² – ab + b²) 
(14 + 1)/(14 -1) 
15/13 

Question 5: If x = 8, then find value of x⁵ – 9x⁴ + 9x³ – 9x² + 9x¹ – 1  

We can write it as 
8⁵ – 8*x⁴ – 1*x⁴ + 8*x³ + 1*x³ – 8*x² – 1*x² +8*x¹+ 1*x¹ – 1 
Now put x = 8 
8⁵ – 8*8⁴ – 1*8⁴ + 8*8³ + 1*8³ – 8*8² – 1*8² +8*8¹+ 1*8¹ – 1 
8 – 1 = 7 

Question 6: If m=√7 + √7 + √7….. and n=√7 - √7 - √7……., then among the following relation between m and n holds is 

 m = √(7 + m) 
m² = 7 + m 
m² – m = 7…….(1) 
and n = √(7 – n) 
n² + n = 7…….(2) 

from (1) and (2) 
m² – m = n² + n 
m² – n² – (m + n) = 0 
(m + n)(m – n) – (m + n)= 0 
m – n – 1 = 0 

Question 7: If x² + y² + z² = 2(x + y -1), then the value of x³ + y³ + z³? 

 x² + y² + z² = 2x + 2y -2 
(x² + 1 -2x) +(y² + 1 -2y) + (z²) = 0 
(x – 1)² + (y – 1)² + (z)² = 0 
(x – 1)² = 0 
x = 1 
(y – 1)² = 0 
y=1 
(z)² = 0 
z = 0 

Put value in eq 
x³ + y³ + z³ 
1³ + 1³ + 0³  = 2 

Question 8: If (x¹² + 1 )/x⁶ = 6, then the value of (x³⁶ + 1 )/x¹⁸ ? 

 Given 
(x¹² + 1 )/x⁶ = 6 
x⁶ + 1 /x⁶ = 6 

Cubing both sides 
(x⁶ + 1 /x⁶)³ = 6³ 
x¹⁸ + 1/x¹⁸ + 3 (x⁶ + 1 /x⁶) = 216 
x¹⁸ + 1/x¹⁸ + 3 * 6 = 216 
x¹⁸ + 1/x¹⁸ = 198 
(x³⁶ + 1)/x¹⁸ = 198 

Practice Problems (Hard)

Question 1: If a = 1 − 1/b and b = 1 − 1/c​, find the value of c − 1/a.

Question 2: If a + b + c = 3, find the value of: 1/(1 − a)(1 − b)​ + 1/(1 − b) (1 − c) ​+ 1/(1 − c) (1 − a).

Question 3: If a + 1/a = √3, calculate a¹⁸ + a¹² + a⁶ + 1.

Question 4: If x = 8, calculate x⁵ − 9x⁴ + 9x³ − 9x² + 9x − 1.

Question 5: If x² + y² + z² = 2(x + y − 1), calculate x³ + y³ + z³.

Question 6: If x¹² + 1 /x⁶ = 6, find the value of x³⁶ + 1/x¹⁸.

Question 7: If x + y + z = 6, x² + y² + z² = 26, and xy + yz + zx = 14, find the value of x³ + y³ + z³ − 3xyz.

Question 8: if a = 2 + √3 and b = 2 − √3, find the value of: a⁵ - b⁵/a - b.

Answer Key:

1. b + 1​
2. 0
3. 0
4. 7
5. 2
6. 198
7. 36
8. 82

Practice More

• Algebra Practice Questions Easy Level
• Algebra Practice Questions Medium Level
• Quiz – Algebra Quiz