CAT Algebra Short Notes: Chapter-wise Weightage and Solved Questions

CAT Algebra: 6 chapters, 0 practice questions and one solved question from each chapter.

Algebra Chapter Matrix

Chapter	PYQs	Share of unit PYQs	Practice questions
Functions, Domains, Ranges and Optimization	0	0%	0
Inequalities, Modulus and Absolute Value	0	0%	0
Linear Systems, Integer Solutions and Algebraic Expressions	0	0%	0
Logarithms, Exponents and Surds	0	0%	0
Polynomial, Quadratic and Root Relations	0	0%	0
Sequences, Series and Progressions	0	0%	0

One Solved Question from Each Algebra Chapter

Question 1 · Quantitative Ability NAT

Let $f (x) = \frac{x ^{2} - x + 1}{x ^{2} + x + 1}$ for all real $x$ . If $M$ and $m$ are the maximum and minimum values of $f (x)$ respectively, what is the value of $3 (M + m)$ ?

Correct Answer:

Step-by-Step Solution

Key idea: This is an optimization via discriminant question. When finding the range (max/min values) of a rational function where both numerator and denominator are quadratics, the most efficient method is to use the discriminant.

Step 1: Set the function equal to $y$ .

$y = \frac{x ^{2} - x + 1}{x ^{2} + x + 1}$ .

Step 2: Cross-multiply and rearrange into a standard quadratic equation in terms of $x$ .

$y (x^{2} + x + 1) = x^{2} - x + 1$

$y x^{2} + y x + y = x^{2} - x + 1$

$(y - 1) x^{2} + (y + 1) x + (y - 1) = 0$ .

Step 3: Apply the condition for real roots.

Since the domain is all real $x$ , for any valid output $y$ , there must exist at least one real $x$ that satisfies this equation. Therefore, the discriminant $Δ$ must be greater than or equal to zero.

$Δ = b^{2} - 4 a c \geq 0$

$(y + 1)^{2} - 4 (y - 1) (y - 1) \geq 0$ .

Step 4: Solve the inequality for $y$ .

Expand the terms:

$(y^{2} + 2 y + 1) - 4 (y^{2} - 2 y + 1) \geq 0$

$y^{2} + 2 y + 1 - 4 y^{2} + 8 y - 4 \geq 0$

$- 3 y^{2} + 10 y - 3 \geq 0$ .

Multiply by $- 1$ and flip the inequality sign:

$3 y^{2} - 10 y + 3 \leq 0$ .

Factor the quadratic:

$(3 y - 1) (y - 3) \leq 0$ .

The roots are $y = 1/3$ and $y = 3$ . Since the parabola opens upwards and is $\leq 0$ , the solution is the interval between the roots:

$1/3 \leq y \leq 3$ .

Step 5: Identify $M$ and $m$ and calculate the final answer.

The maximum value $M = 3$ .

The minimum value $m = 1/3$ .

The question asks for $3 (M + m)$ :

$3 (3 + 1/3) = 3 (10/3) = 10$ .

Answer: 10

chapter

CAT Algebra Short Notes: Chapter-wise Weightage and Solved Questions

CAT Algebra: 6 chapters, 0 practice questions and one solved question from each chapter.

A question from this chapter

Question 1

Let f (x) = \frac{x ^{2} - x + 1}{x ^{2} + x + 1} for all real x . If M and m are the maximum and minimum values of f (x) respectively, what is the value of 3 (M + m) ?

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