Mastering Algebraic Identities for CAT and Other Exams

The Significance of Algebraic Identities

Tackling the Common Admission Test (CAT) can be a daunting task. One essential concept that frequently appears in the exam is algebraic identities. Understanding how to apply these identities is crucial for achieving a high score in this section.

In this blog, we will explore various algebraic identities. These identities will challenge your problem-solving skills. They will also enhance your comprehension of this vital topic. Whether you are preparing for the test for the first time or looking to improve your score, these practice questions will provide you with the necessary tools for success.

Practice with Algebraic Identities

Let's dive into the world of algebraic identities and elevate your CAT exam preparation!

What are Algebraic Identities?

Algebraic identities are equations that hold true for all values of the variables involved. They are fundamental in simplifying expressions and solving equations. Familiarity with these identities can significantly enhance your problem-solving speed and accuracy.

Common Algebraic Identities

Here are some common algebraic identities you should know:

1. Square of a Binomial:
   \[(a + b)^2 = a^2 + 2ab + b^2\]
   \[(a - b)^2 = a^2 - 2ab + b^2\]
   
   
2. Product of a Sum and a Difference:
   \[(a + b)(a - b) = a^2 - b^2\]
   
   
3. Cube of a Binomial:
   \[(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\]
   \[(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\]
   
   
4. Sum of Cubes:
   \[a^3 + b^3 = (a + b)(a^2 - ab + b^2)\]
   
   
5. Difference of Cubes:
   \[a^3 - b^3 = (a - b)(a^2 + ab + b^2)\]
   
   

Understanding these identities will help you tackle complex problems more efficiently.

Algebraic Identities Questions for CAT

Question Set

Let's put your knowledge to the test with some practice questions.

Q1. A cubic equation \( ax^3 + bx^2 + cx = 0 \) has three real roots \( \alpha, \beta, \) and \( \gamma \), where \( a, b, c \) are rational. What is the value of \( \left(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}\right) \)?

• a. 1
• b. 0
• c. 3
• d. Undefined
  
  

Q2. The number of rational roots of \( x^3 + 5x^2 + 6x + 2 = 0 \) is:

• a. 0
• b. 1
• c. 2
• d. 3
  
  

Q3. What is the remainder when \( x^4 + 5x^3 - 3x^2 + 4x + 3 \) is divided by \( x + 2 \)?

• a. -18
• b. 31
• c. -31
• d. -41
  
  

Q4. If two roots of the equation \( x^3 - 2x^2 - 15x + 36 = 0 \) are equal, then the sum of one of the equal roots and the third root can be:

• a. 1
• b. -1
• c. 2
• d. 4
  
  

Q5. If \( 3x^2 + ax + 4 \) is perfectly divisible by \( x - 8 \), then the value of \( a \) is:

• a. 24.5
• b. 25.5
• c. -24.5
• d. -25.5
  
  

Q6. Let \( a \) and \( b \) be two real numbers. If \( 2 \) and \( -2 \) are roots of \( ax^3 + bx^2 - 7x + 8 = 0 \), then \( b \) equals:

• a. 2
• b. -9
• c. -7/4
• d. -2
  
  

Q7. If the roots of the equation \( x^5 + ax^4 + bx^3 + cx^2 + dx + 2023 = 0 \) are all distinct integers, then find \( a \).

• a. 7
• b. 1
• c. 5
• d. 11
  
  

Q8. Let ‘a’ be a non-zero real number. If \( \frac{5}{a} \) is a root of \( ax^3 - 5x^2 - 6x + 15 = 0 \), then the value of ‘a’ is ___________.

• a. 2
• b. \( \sqrt{2} \)
• c. \( \sqrt{3} \)
• d. 3
  
  

Q9. In a cubic expression with leading coefficient one, if we substitute the unknown variable with \( 2, 3, \) or \( 5 \), we get \( 1 \) in each case. Find the value that will be obtained if the unknown variable is substituted with \( 6 \).

• a. 13
• b. 11
• c. 15
• d. 18
  
  

Q10. If the equation \( x^3 - ax^2 + bx - a = 0 \) has three real roots, then it must be the case that:

• a. \( a = 1 \)
• b. \( a \neq 1 \)
• c. \( b = 1 \)
• d. \( b \neq 1 \)
  
  

Q11. The polynomial \( f(x) = x^4 + 2x^3 + 3x^2 - ax - b \) when divided by \( (x - 1) \) and \( (x + 1) \) leaves the remainder \( 5 \) and \( 19 \) respectively. The value of \( b \) is:

• a. -3
• b. -8
• c. -6
• d. 6
  
  

Q12. If the equation \( ax^3 + bx^2 + cx + d = 0 \) has two positive real roots and one negative real root, then which of the following is always true?

• a. \( a \) and \( d \) have opposite signs
• b. \( a \) and \( d \) have the same sign
• c. \( b \) and \( c \) have the same sign
• d. \( a \) and \( b \) have opposite signs
  
  

Q13. If \( x^4 - 8x^3 + ax^2 - bx + 16 = 0 \) has positive real roots, find \( a - b \).

• a. -6
• b. -8
• c. -12
• d. -14
  
  

Q14. \( x^3 - 18x^2 + bx - c = 0 \) has positive real roots \( p, q, \) and \( z \). If the geometric mean of the roots is \( 6 \), find \( b \).

• a. 36
• b. 72
• c. 108
• d. 216
  
  

Q15. Find the minimum/maximum value of a quadratic expression \( (ax)^2 + bx + c = 0 \) with roots as \( 3 \) and \( 7 \).

• a. -4
• b. +4
• c. 3
• d. -3
  
  

Q16. If \( x = 2 - 2^{1/3} + 2^{2/3} \), then the value of \( x^3 - 6x^2 + 18x + 18 \) is:

• a. 22
• b. 33
• c. 40
• d. 45
  
  

Q17. Find the sum of the reciprocals of the roots of the cubic equation \( x^3 + 5x^2 - 7x + 1 = 0 \).

• a. 4
• b. 8
• c. 6
• d. 7
  
  

Q18. If \( f(x) = ax^2 + bx + 2 \). \( f(1) = 3 \) and \( f(4) = 42 \), then what is the value of ‘b’?

• a. 3
• b. -2
• c. 1
• d. 7
  
  

Q19. The number of roots common between the two equations \( x^3 + 3x^2 + 4x + 5 = 0 \) and \( x^3 + 2x^2 + 7x + 3 = 0 \) is:

• a. 0
• b. 1
• c. 2
• d. 3
  
  

Q20. What is the minimum value of \( x^2 - 2 \)?

• a. 2
• b. 3
• c. -2
• d. -3
  
  

Q21. What is the maximum/minimum value of \( 6x - x^2 + 7 \)?

• a. 16
• b. 15
• c. 14
• d. 18
  
  

Answers

1	2	3	4	5	6	7	8	9	10
D	B	D	B	C	D	7	A	13	D
11	12	13	14	15	16	17	18	19	20
B	B	B	C	A	C	D	B	A	C
21