CAT Algebra Questions

Algebra is one of the most critical sections in the CAT quantitative aptitude exam. Mastering algebra can give you a significant edge in the exam, as it forms the basis of many problems that you will encounter. In this blog post, we will discuss essential algebra formulas, review some of the previous years’ CAT algebra questions, and walk through ten of the most important questions and their solutions. Additionally, we will explore how you can efficiently prepare for the algebra section using Entri’s CAT coaching course.

Importance of Algebra in CAT Exam

Algebra forms the backbone of the quantitative section in the CAT exam. A strong foundation in algebra not only helps in solving direct algebra questions but also assists in tackling other quantitative problems. The ability to manipulate equations, understand functions, and solve inequalities is critical for success in the CAT exam. Moreover, algebraic concepts are often integrated with other topics like arithmetic, geometry, and number systems, making it indispensable for a high score.

Important Algebra Formulas for CAT

Here are some essential algebra formulas that every CAT aspirant should be familiar with:

• Linear Equations:
  • Slope of a line: m = (y2 - y1) / (x2 - x1)
  • Equation of a line: y = mx + c (where m is slope and c is y-intercept)
• Quadratic Equations:
  • Standard form: ax^2 + bx + c = 0
  • Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a
  • Discriminant (determines nature of roots): b^2 - 4ac
• Inequalities:
  • Basic properties: a > b => a + c > b + c , a > b and b > c => a > c
  • Working with absolute values: |x| >= 0
• Logarithms:
  • Logarithm laws: log(a * b) = log(a) + log(b), log(a^n) = n * log(a)
  • Change of base formula: log_a(b) = (log_c(b)) / (log_c(a)) (where c is any base)
• Permutations and Combinations:
  • Permutation (order matters): nPr = n! / (n - r)!
  • Combination (order doesn’t matter): nCr = n! / (r! * (n - r)!)

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CAT Algebra Previous Year Questions

Previous years’ questions provide valuable insights into the types of algebra problems that appear in the CAT exam. Let’s review some of these questions.

Question 1:

If the sum of two numbers is 30 and their difference is 6, find the product of the numbers.

Solution: Let the numbers be x and y.

 Given, x + y = 30 and x – y = 6.

Adding both equations, we get 2x = 36, so x = 18.

Substituting x in the first equation, we get y = 12.

Therefore, the product of the numbers is 18 * 12 = 216.

Question 2:

If the roots of the equation x^2 – 5x + k = 0 are in the ratio 2:3, find the value of k.

Solution: Let the roots be 2a and 3a.

Sum of roots = 5a = 5, so a = 1.

Product of roots = 6a^2 = k.

Therefore, k = 6.

Question 3:

Find the value of x for which the expression x^2 – 6x + 11 is minimum.

Solution: The given expression is a quadratic equation.

The minimum value occurs at the vertex of the parabola represented by the equation.

The x-coordinate of the vertex is given by -b/2a, where a and b are the coefficients of x^2 and x, respectively.

In this case, a = 1 and b = -6. So, x = -(-6) / 2*1 = 3.

Therefore, the value of x for which the expression is minimum is 3.   

Question 4:

If log2(x) + log2(y) = 4 and log2(x/y) = 2, find the value of xy.

Solution: Using the properties of logarithms, we can write:

                log2(xy) = log2(x) + log2(y) = 4.

Therefore, xy = 2^4 = 16.

Question 5:

x is a positive real number such that x^8 + (1/x)^8 = 47. What is the value of x^9 + (1/x)^9?

Solution: We can’t solve directly for x.

Notice the symmetrical nature of the equation.

Multiplying both sides by x gives: x^9 + 1 + (1/x) = 47x.

This simplifies to x^9 + (1/x)^9 = 46. (The answer is 46)

Question 6:

A train travels x km at y km/hr. It then travels the same distance at (y + 2) km/hr. The total time taken is 12 hours. What is the value of xy?

Solution: We can set up two equations for time:

                 x/y + x/(y + 2) = 12.

Solve for x and substitute back to find xy.

(The answer can be any value satisfying the equation)

Question 7:

Simplify the expression x² – 1 / x – 1 for x ≠ 1.

Solution:

The expression can be simplified by factoring the numerator:

x² – 1 / x – 1 = (x – 1(x + 1)) / x – 1

For x ≠ 1, the x – 1 terms cancel out, leaving:

x + 1

So, the simplified expression is x + 1.

Question 8:

If x + y = 10 and xy = 21, find the value of x² + y².

Solution:

Using the identity:

x² + y² = (x + y)² – 2xy

Substituting the given values:

x² + y² = 10² – 2 × 21 = 100 – 42 = 58

So, the answer is 58.

Question 9:

The sum of the roots of the quadratic equation ax² + bx + c = 0 is 7, and the product of the roots is 12. Find the equation.

Solution:

The sum of the roots α + β = -b/a = 7, and the product αβ = c/a = 12. Assuming a = 1:

x² – 7x + 12 = 0

This is the required equation.

Question 10:

Solve x² – 4x + 4 = 0.

Solution:

This is a perfect square trinomial:

(x – 2)² = 0

So, the solution is x = 2.

Question 11:

If log₁₀x = 2, find the value of x.

Solution:

From the definition of logarithms:

log₁₀x = 2 implies x = 10² = 100

So, the value of x is 100.

Entri’s CAT Coaching Course

Preparing for the CAT exam can be a daunting task, especially when it comes to mastering algebra. Entri’s CAT coaching course is designed to help you overcome these challenges with ease. Our course offers:

• Expert-Led Classes: Learn from experienced educators who break down complex algebra concepts into easy-to-understand lessons.
• Comprehensive Study Materials: Access a wealth of resources, including video lectures, practice questions, and detailed solutions.
• Personalized Learning Path: Our course adapts to your learning pace, ensuring you strengthen your weak areas and build on your strengths.
• Mock Tests and Analytics: Regular mock tests with in-depth performance analytics help you track your progress and fine-tune your preparation.

Join Entri today and take the first step towards securing your spot in one of India’s top B-schools. Entri is your trusted partner in achieving CAT success!

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Conclusion

Mastering algebra is critical for cracking the CAT exam’s quantitative section. With a strong grasp of algebraic concepts and regular practice, you can significantly improve your CAT score. By focusing on key formulas, solving previous years’ questions, and understanding how to approach different types of algebraic problems, you’ll be well-prepared for the exam. Additionally, Entri’s CAT coaching course offers comprehensive resources and expert guidance to help you excel in the algebra section and overall CAT preparation.