Logical Reasoning and Constraint Deduction Practice Questions for GATE DA: 35+ Solved Questions with Step-by-Step Solutions

Solve 35+ Logical Reasoning and Constraint Deduction practice questions for GATE DA with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Logical Reasoning and Constraint Deduction

Chapter Journey: Logical Reasoning and Constraint Deduction

Step 1: Monotonicity-Based Logical Deduction

Understanding Strict vs. Non-Strict Monotonicity
Reversing Inequalities (Output trends to Input bounds)
The Single Peak and Single Valley Patterns

Step 2: Constraint-Based Sequencing and Elimination

Building Ordering Chains (A is before B, B is after C)
The Elimination Grid Method
Handling "Not" and "Either-Or" Constraints

The Core of Monotonicity

What is Monotonicity?
A function

f (x)

is monotonic if it preserves or reverses the order of its inputs. It never changes direction.

1. Monotonically Increasing

x

gets larger,

f (x)

gets larger (or stays same).

2. Monotonically Decreasing

x

gets larger,

f (x)

gets smaller (or stays same).

The Logical Superpower:
If a function is strictly increasing, it creates a perfect two-way mirror between inputs and outputs.

x_{1} < x_{2} ⟺ f (x_{1}) < f (x_{2})

This means if you are given an inequality about the outputs, you can instantly translate it into an inequality about the inputs.

Logical Reasoning and Constraint Deduction: Solved Questions with Step-by-Step Explanations (5 Problems)

Question 1 · Analytical Aptitude MCQ

A function $h (x)$ monotonically increases up to $x = 10$ and then monotonically decreases. If $h (4) = h (y)$ for some $y \neq = 4$ , which of the following is a possible value for $y$ ?

A.
2
B.
8
C.
12
D.
10

Correct Answer:

Step-by-Step Solution

Key idea: This is a single peak pattern question, recognizable by the function increasing then decreasing, and an equality between outputs at different inputs.

Step 1: Understand the shape of the function. It increases up to $x = 10$ (the peak) and then decreases.

Step 2: Analyze the given condition. We have $h (4) = h (y)$ with $y \neq = 4$ . The input $x = 4$ is on the increasing side of the peak (since $4 < 10$ ).

Step 3: Apply the single peak rule. For a function with a single peak, if two different inputs yield the same output, one input must be on the increasing side (left of the peak) and the other must be on the decreasing side (right of the peak).

Step 4: Since $x = 4$ is on the left, $y$ must be on the right side of the peak. Therefore, $y$ must be strictly greater than 10.

Step 5: Evaluate the options. 2, 8, and 10 are all $\leq 10$ . Only 12 is $> 10$ .

Answer: C

Question 2 · Analytical Aptitude MCQ

Consider a function $W (a)$ representing weight vs age. It strictly increases until age 50 and strictly decreases thereafter. If two brothers have the same weight but different ages, which statement MUST be true about their ages $a_{1}$ and $a_{2}$ (with $a_{1} < a_{2}$ )?

A.
Both are less than 50.
B.
Both are greater than 50.
C.
One is less than 50 and the other is greater than 50.
D.
Both are exactly 50.

Correct Answer:

Step-by-Step Solution

Key idea: Single Peak Pattern (Card c004/c006).

Step 1: The function has a single peak at $a = 50$ .

Step 2: On the left side ( $a < 50$ ), the function is strictly increasing. Thus, every weight value is unique.

Step 3: On the right side ( $a > 50$ ), the function is strictly decreasing. Thus, every weight value is unique.

Step 4: For two different ages to have the same weight, one must be on the increasing slope and the other on the decreasing slope.

Step 5: Therefore, one age is $< 50$ and the other is $> 50$ .

Answer: C

Question 3 · Analytical Aptitude MCQ

Temperature $T (h)$ as a function of altitude $h$ strictly increases up to $h = 10$ km and strictly decreases thereafter. If two different altitudes $h_{1}$ and $h_{2}$ ( $h_{1} < h_{2}$ ) have the same temperature, what must be true?

A.
$h_{1} < 10 < h_{2}$
B.
$h_{1} < h_{2} < 10$
C.
$10 < h_{1} < h_{2}$
D.
$h_{1} = 10$ or $h_{2} = 10$

Correct Answer:

Step-by-Step Solution

Key idea: This is a Single Peak Pattern question (Card c006), recognizable by a function that increases then decreases.

Step 1: The function $T (h)$ has a single peak at $h = 10$ .

Step 2: For $h < 10$ , the function is strictly increasing. Thus, every temperature value is unique on this side.

Step 3: For $h > 10$ , the function is strictly decreasing. Thus, every temperature value is unique on this side.

Step 4: For two different altitudes to have the same temperature, one must be on the increasing slope and the other on the decreasing slope.

Step 5: Since $h_{1} < h_{2}$ , it must be that $h_{1} < 10$ and $h_{2} > 10$ .

Step 6: This matches the condition $h_{1} < 10 < h_{2}$ .

Answer: A

Question 4 · Analytical Aptitude MCQ

Let $f (x)$ be a strictly increasing function. If $f (a) < f (b)$ , which of the following must be true?

A.
$a > b$
B.
$a < b$
C.
$a = b$
D.
Cannot be determined

Correct Answer:

Step-by-Step Solution

Key idea: This is a direct definition question on strictly increasing functions, recognizable by the phrase "strictly increasing" and an inequality between outputs.

Step 1: Recall the definition of a strictly increasing function. As the input $x$ increases, the output $f (x)$ strictly increases.

Step 2: This means the order of the inputs is perfectly preserved in the outputs. If $x_{1} < x_{2}$ , then $f (x_{1}) < f (x_{2})$ .

Step 3: We are given $f (a) < f (b)$ . Because the function is strictly increasing, the inequality between the outputs directly reflects the inequality between the inputs.

Step 4: Therefore, it must be true that $a < b$ .

Answer: B

Question 5 · Analytical Aptitude MCQ

The function $f (x)$ is strictly decreasing for all real $x$ . If $f (2 x - 1) > f (5)$ , which of the following must be true?

A.
$2 x - 1 < 5$
B.
$2 x - 1 > 5$
C.
$2 x - 1 = 5$
D.
Cannot be determined

Correct Answer:

Step-by-Step Solution

Key idea: This is an inequality reversal question, recognizable by the phrase "strictly decreasing" and an inequality between outputs involving algebraic expressions.

Step 1: Recall the property of a strictly decreasing function. As the input increases, the output strictly decreases.

Step 2: This means the order of the inputs is reversed in the outputs. If $A > B$ , then $f (A) < f (B)$ .

Step 3: We are given $f (2 x - 1) > f (5)$ . Because the function is strictly decreasing, the inequality between the outputs is the reverse of the inequality between the inputs.

Step 4: Therefore, the input corresponding to the larger output must be the smaller input. This gives $2 x - 1 < 5$ .

Answer: A

A question from this chapter

Question 1

A function h (x) monotonically increases up to x = 10 and then monotonically decreases. If h (4) = h (y) for some y \neq = 4, which of the following is a possible value for y ?

Question 2

Consider a function W (a) representing weight vs age. It strictly increases until age 50 and strictly decreases thereafter. If two brothers have the same weight but different ages, which statement MUST be true about their ages a_{1} and a_{2} (with a_{1} < a_{2})?

Question 3

Temperature T (h) as a function of altitude h strictly increases up to h = 10 km and strictly decreases thereafter. If two different altitudes h_{1} and h_{2} (h_{1} < h_{2}) have the same temperature, what must be true?

Question 4

Let f (x) be a strictly increasing function. If f (a) < f (b), which of the following must be true?

Question 5

The function f (x) is strictly decreasing for all real x . If f (2 x - 1) > f (5), which of the following must be true?

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Logical Reasoning and Constraint Deduction Practice Questions for GATE DA: 35+ Solved Questions with Step-by-Step Solutions

Chapter Roadmap: Logical Reasoning and Constraint Deduction

Chapter Journey: Logical Reasoning and Constraint Deduction

The Core of Monotonicity

Logical Reasoning and Constraint Deduction: Solved Questions with Step-by-Step Explanations (5 Problems)

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

Step-by-Step Solution

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