XAT Algebra: Equations, Functions, Progressions Short Notes: Chapter-wise Weightage and Solved Questions

XAT Algebra: Equations, Functions, Progressions: 3 chapters, 0 practice questions and one solved question from each chapter.

Algebra: Equations, Functions, Progressions Chapter Matrix

Chapter	PYQs	Share of unit PYQs	Practice questions
Algebraic Word Problems and Number Properties	0	0%	0
Equations, Functions and Logs	0	0%	0
Sequences, Progressions and Patterns	0	0%	0

One Solved Question from Each Algebra: Equations, Functions, Progressions Chapter

Question 1 · Quantitative Aptitude and Data Interpretation (QA & DI) NAT

Let $n$ be a positive integer such that $n^{3} + 2 n^{2} + 3 n + 4$ is a perfect cube. What is the sum of all possible values of $n$ ?

Correct Answer:

Step-by-Step Solution

Key idea: This is a perfect power bounding problem. For large $n$ , the expression lies strictly between consecutive cubes, so only small $n$ need checking. Modular arithmetic eliminates remaining candidates.

Step 1: Compare with nearby cubes. Note $(n + 1)^{3} = n^{3} + 3 n^{2} + 3 n + 1$ .

Our expression: $E (n) = n^{3} + 2 n^{2} + 3 n + 4$ .

Difference: $(n + 1)^{3} - E (n) = n^{2} - 3$ .

For $n \geq 2$ , $n^{2} - 3 > 0 \Rightarrow E (n) < (n + 1)^{3}$ .

Step 2: Compare with $n^{3}$ . Clearly $E (n) > n^{3}$ for $n > 0$ .

So for $n \geq 2$ , $n^{3} < E (n) < (n + 1)^{3}$ . Thus $E (n)$ cannot be a perfect cube for $n \geq 2$ .

Step 3: Check $n = 1$ : $E (1) = 1 + 2 + 3 + 4 = 10$ , not a cube.

Step 4: No positive integer $n$ satisfies the condition. Sum of all possible values is 0.

Step 5: Verify no edge cases missed. For $n = 0$ (not positive), $E (0) = 4$ , not cube. Negative $n$ excluded by problem.

Answer: 0

chapter

XAT Algebra: Equations, Functions, Progressions Short Notes: Chapter-wise Weightage and Solved Questions

XAT Algebra: Equations, Functions, Progressions: 3 chapters, 0 practice questions and one solved question from each chapter.

A question from this chapter

Question 1

Let n be a positive integer such that n^{3} + 2 n^{2} + 3 n + 4 is a perfect cube. What is the sum of all possible values of n ?

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