CAT Algebra Previous Year Questions (PYQs): 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

Functions, Domains, Ranges and Optimization Previous Year Questions (PYQs) (11 questions)
Inequalities, Modulus and Absolute Value Previous Year Questions (PYQs) (14 questions)
Linear Systems, Integer Solutions and Algebraic Expressions Previous Year Questions (PYQs) (13 questions)
Logarithms, Exponents and Surds Previous Year Questions (PYQs) (20 questions)
Polynomial, Quadratic and Root Relations Previous Year Questions (PYQs) (11 questions)
Sequences, Series and Progressions Previous Year Questions (PYQs) (14 questions)

One Solved Question from Each Algebra Chapter

Question 1 · Quantitative Ability MCQ

For all real values of x, the range of the function $f (x) = \frac{x ^{2} + 2 x + 4}{2 x ^{2} + 4 x + 9}$ is:

A.
$[\frac{4}{9}, \frac{8}{9}]$
B.
$[\frac{3}{7}, \frac{8}{9})$
C.
$(\frac{3}{7}, \frac{1}{2})$
D.
$[\frac{3}{7}, \frac{1}{2})$

Correct Answer:

Step-by-Step Solution

Key idea: This is a rational function range question where the numerator and denominator quadratics are proportional. This signals that algebraic substitution is much faster than the discriminant method.

Step 1: Spot the proportional terms and substitute.

Let $u = x^{2} + 2 x + 4$ .

Notice that the denominator is $2 x^{2} + 4 x + 9 = 2 (x^{2} + 2 x + 4) + 1 = 2 u + 1$ .

The function simplifies to $y = \frac{u}{2 u + 1}$ .

Step 2: Find the valid range of $u$ .

Complete the square for $u$ : $u = (x^{2} + 2 x + 1) + 3 = (x + 1)^{2} + 3$ .

Since $(x + 1)^{2} \geq 0$ for all real $x$ , the minimum value of $u$ is $3$ . Thus, $u \geq 3$ .

Step 3: Analyze the behavior of $y$ with respect to $u$ .

Rewrite $y$ to make the limits obvious:

$y = \frac{1}{2} (\frac{2 u}{2 u + 1}) = \frac{1}{2} (\frac{2 u + 1 - 1}{2 u + 1}) = \frac{1}{2} (1 - \frac{1}{2 u + 1})$ .

Step 4: Map the domain of $u$ to the range of $y$ .

Since $u \geq 3$ , we have $2 u + 1 \geq 7$ .

Therefore, $0 < \frac{1}{2 u + 1} \leq \frac{1}{7}$ .

Multiplying by $- 1$ reverses the inequalities: $- \frac{1}{7} \leq - \frac{1}{2 u + 1} < 0$ .

Adding $1$ : $1 - \frac{1}{7} \leq 1 - \frac{1}{2 u + 1} < 1 ⟹ \frac{6}{7} \leq 1 - \frac{1}{2 u + 1} < 1$ .

Finally, multiply by $\frac{1}{2}$ : $\frac{3}{7} \leq \frac{1}{2} (1 - \frac{1}{2 u + 1}) < \frac{1}{2}$ .

Step 5: Write the final interval.

The range is $[\frac{3}{7}, \frac{1}{2})$ .

Answer: [\frac{3}{7}, \frac{1}{2})

chapter

CAT Algebra Previous Year Questions (PYQs): 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

For all real values of x, the range of the function f (x) = \frac{x ^{2} + 2 x + 4}{2 x ^{2} + 4 x + 9} is:

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