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Algebra • Algebraic Integral Solutions Practice Drills (53 Qs)• Official Answer Keys

CAT Integral Solutions (Algebra) Practice Questions (50+ Questions)

Integer Roots, Factor Splitting for 1/x + 1/y = 1/N, and Diophantine Equations in Real Algebra

53 Total Questions
MCQ: 41 (+3 / -1)
TITA: 12 (0 Negative Penalty)
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Core Formulas & Shortcut Matrix: Algebraic Integral Solutions Practice Drills

Reciprocal Pair IdentityFormula #1
$1x\frac{1}{x} + 1y\frac{1}{y} = 1N\frac{1}{N} \iff (x - N)(y - N) = N^2

Number of positive integral pairs = number of factors of N^2.

Exam Hall Traps & Speedbreakers to Avoid
  • •Forgetting to distinguish between positive integral solutions vs total (including negative) integral solutions.

Bite-Sized Practice Sets (5 Modules Available)

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Set 01: Integral Solutions

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Set 02: Integral Solutions

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Set 03: Integral Solutions

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Set 04: Integral Solutions

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Set 05: Integral Solutions

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Official Exam Questions & Explanations (25 of 53)

Sorted in official convenor sequence
Question 1 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsEasy
Find the number of integral solutions for the equation x+y=10x + y = 10 where xx and yy are both positive integers.
TITA Answer:
Practice More Questions
Official Correct Answer: 45. For each positive integer value of $x$ from 1 to 9, there exists a corresponding positive integer value of $y$ from 9 to 1. Thus, there are 9 solutions in total, which can be calculated as the sum of the first 9 positive integers: $1 + 2 + 3 + ... + 9 = \frac{9 \cdot (9 + 1)}{2} = 45$.
Question 2 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsModerate
How many integral solutions does the equation 2x+3y=122x + 3y = 12 have where xx and yy are both non-negative integers?
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Official Correct Answer: B. We need to find pairs $(x, y)$ such that $2x + 3y = 12$. Testing non-negative integer values, we get: $x = 0 \implies y = 4 , x = 3 \implies y = 2 , x = 6 \implies y = 0$. These are the only solutions, giving us 3 pairs.
Question 3 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsHard
Find the number of integral solutions for the equation x2−5x+6=0x^2 - 5x + 6 = 0 where xx is a non-negative integer.
TITA Answer:
Practice More Questions
Official Correct Answer: 2. Solving the quadratic equation $x^2 - 5x + 6 = 0 , we factorize it as $(x-2)(x-3) = 0$. Thus, the solutions are $x = 2$ and $x = 3$. Both are non-negative integers, giving us 2 solutions.
Question 4 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsEasy
Find the number of integral solutions for the equation x−y=5x - y = 5 where xx and yy are both positive integers.
TITA Answer:
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Official Correct Answer: 4. For each positive integer value of $y$ from 1 to 4, there exists a corresponding positive integer value of $x$ from 6 to 9. Thus, there are 4 solutions in total.
Question 5 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsModerate
How many integral solutions does the equation x2−4x+3=0x^2 - 4x + 3 = 0 have where xx is a non-negative integer?
Practice More Questions
Official Correct Answer: C. Solving the quadratic equation $x^2 - 4x + 3 = 0 , we factorize it as $(x-1)(x-3) = 0$. Thus, the solutions are $x = 1$ and $x = 3$. Both are non-negative integers, giving us 2 solutions.
Question 6 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsHard
How many integral solutions does the equation 2x+5y=202x + 5y = 20 have where xx and yy are both non-negative integers?
Practice More Questions
Official Correct Answer: A. We need to find pairs $(x, y)$ such that $2x + 5y = 20$. Testing non-negative integer values, we get: $x = 0 \implies y = 4 , x = 5 \implies y = 2 , x = 10 \implies y = 0$. These are the only solutions, giving us 4 pairs.
Question 7 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsEasy
How many integral solutions does the equation x+2y=12x + 2y = 12 have?
TITA Answer:
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Official Correct Answer: 6. Step 1: Express y in terms of x, $y = \frac{12 - x}{2}$. Step 2: Determine the range of x for which y is an integer. Step 3: The values of x can be 0, 2, 4, 6, 8, 10. Step 4: Therefore, there are 6 integral solutions.
Question 8 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsModerate
Find the number of integral solutions for 3x+5y=303x + 5y = 30.
TITA Answer:
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Official Correct Answer: 6. Step 1: Express y in terms of x, $y = \frac{30 - 3x}{5}$. Step 2: Determine the range of x for which y is an integer. Step 3: The values of x can be 0, 5, 10, 15, 20, 25. Step 4: Therefore, there are 6 integral solutions.
Question 9 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsHard
How many integral solutions does the equation 4x−3y=124x - 3y = 12 have?
TITA Answer:
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Official Correct Answer: 4. Step 1: Express y in terms of x, $y = \frac{4x - 12}{3}$. Step 2: Determine the range of x for which y is an integer. Step 3: The values of x can be 3, 6, 9, 12. Step 4: Therefore, there are 4 integral solutions.
Question 10 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsEasy
How many integral solutions does the equation x+y=10x + y = 10 have?
TITA Answer:
Practice More Questions
Official Correct Answer: 11. Step 1: Determine the range of x for which y is an integer. Step 2: The values of x can be 0, 1, 2, 3, ..., 10. Step 3: Therefore, there are 11 integral solutions.
Question 11 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsModerate
Find the number of integral solutions for 2x−5y=152x - 5y = 15.
TITA Answer:
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Official Correct Answer: 7. Step 1: Express y in terms of x, $y = \frac{2x - 15}{5}$. Step 2: Determine the range of x for which y is an integer. Step 3: The values of x can be 5, 10, 15, 20, 25, 30, 35. Step 4: Therefore, there are 7 integral solutions.
Question 12 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsHard
How many integral solutions does the equation 5x−2y=105x - 2y = 10 have?
TITA Answer:
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Official Correct Answer: 5. Step 1: Express y in terms of x, $y = \frac{5x - 10}{2}$. Step 2: Determine the range of x for which y is an integer. Step 3: The values of x can be 2, 4, 6, 8, 10. Step 4: Therefore, there are 5 integral solutions.
Question 13 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsEasy
The equation x2−5x+6=0x^2 - 5x + 6 = 0 has integral solutions. Find the sum of these solutions.
Practice More Questions
Official Correct Answer: C. Factorize the quadratic equation as $(x - 2)(x - 3) = 0$. The integral solutions are $x = 2$ and $x = 3$. Their sum is $2 + 3 = 5$.
Question 14 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsModerate
The equation 2x2−5x−3=02x^2 - 5x - 3 = 0 has integral solutions. Find the product of these solutions.
Practice More Questions
Official Correct Answer: D. Factorize the quadratic equation as $(2x + 1)(x - 3) = 0$. The integral solutions are $x = -0.5$ and $x = 3$. Their product is - 0.5 \times 3 = -1.5$.
Question 15 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsHard
The equation x2−4x−21=0x^2 - 4x - 21 = 0 has integral solutions. Find the sum of these solutions.
Practice More Questions
Official Correct Answer: D. Factorize the quadratic equation as $(x - 7)(x + 3) = 0$. The integral solutions are $x = 7$ and $x = -3$. Their sum is $7 + (-3) = 4$.
Question 16 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsEasy
The equation x2−3x+2=0x^2 - 3x + 2 = 0 has integral solutions. Find the product of these solutions.
Practice More Questions
Official Correct Answer: B. Factorize the quadratic equation as $(x - 1)(x - 2) = 0$. The integral solutions are $x = 1$ and $x = 2$. Their product is $1 \times 2 = 2$.
Question 17 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsModerate
The equation 2x2−7x+3=02x^2 - 7x + 3 = 0 has integral solutions. Find the sum of these solutions.
Practice More Questions
Official Correct Answer: C. Factorize the quadratic equation as $(2x - 1)(x - 3) = 0$. The integral solutions are $x = 0.5$ and $x = 3$. Their sum is $0.5 + 3 = 3.5$.
Question 18 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsHard
The equation x2−6x+8=0x^2 - 6x + 8 = 0 has integral solutions. Find the sum of these solutions.
Practice More Questions
Official Correct Answer: C. Factorize the quadratic equation as $(x - 2)(x - 4) = 0$. The integral solutions are $x = 2$ and $x = 4$. Their sum is $2 + 4 = 6$.
Question 19 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsEasy
How many integral solutions does the equation x2−4x+3=0x^2 - 4x + 3 = 0 have?
Practice More Questions
Official Correct Answer: C. Solve the quadratic equation: $(x - 1)(x - 3) = 0$. Thus, $x = 1$ or $x = 3$. There are 2 integral solutions.
Question 20 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsModerate
How many integral solutions does the equation 2x2+3x−2=02x^2 + 3x - 2 = 0 have?
Practice More Questions
Official Correct Answer: C. Factorize the quadratic equation: $(2x - 1)(x + 2) = 0$. Thus, $x = \frac{1}{2}$ or $x = -2$. Only $x = -2$ is an integral solution.
Question 21 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsHard
How many integral solutions does the equation x2+5x+6=0x^2 + 5x + 6 = 0 have?
Practice More Questions
Official Correct Answer: C. Factorize the quadratic equation: $(x + 2)(x + 3) = 0$. Thus, $x = -2$ or $x = -3$. There are 2 integral solutions.
Question 22 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsEasy
How many integral solutions does the equation x2−3x+2=0x^2 - 3x + 2 = 0 have?
TITA Answer:
Practice More Questions
Official Correct Answer: 2. Factorize the quadratic equation: $(x - 1)(x - 2) = 0$. Thus, $x = 1$ or $x = 2$. There are 2 integral solutions.
Question 23 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsModerate
How many integral solutions does the equation x2−5x+6=0x^2 - 5x + 6 = 0 have?
Practice More Questions
Official Correct Answer: C. Factorize the quadratic equation: $(x - 2)(x - 3) = 0$. Thus, $x = 2$ or $x = 3$. There are 2 integral solutions.
Question 24 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsHard
How many integral solutions does the equation 2x2+x−6=02x^2 + x - 6 = 0 have?
Practice More Questions
Official Correct Answer: C. Factorize the quadratic equation: $(2x - 3)(x + 2) = 0$. Thus, $x = \frac{3}{2}$ or $x = -2$. Only $x = -2$ is an integral solution.
Question 25 of 53
ThinkCAT Practice SetQAAlgebra • Integral SolutionsEasy
For which of the following values of xx is 2x+3=92x + 3 = 9 true?
Practice More Questions
Official Correct Answer: B. Step 1: Subtract 3 from both sides to get $2x = 6$. Step 2: Divide both sides by 2 to get $x = 3$. Step 3: Verify by substituting $x = 3$ into the original equation.

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Frequently Asked Questions about CAT Integral Solutions (Algebra) Practice Questions (50+ Questions)

What concepts are covered in the Algebraic Integral Solutions Practice Drills practice module?

This module covers Integral Solutions with 53 verified problems ranging from core foundation to high-difficulty CAT exam hall level.

What is the recommended solving time for Algebraic Integral Solutions Practice Drills questions?

The ideal target pace is 2.0 to 2.5 mins / question. Aspirants targeting a 99th percentile should aim for at least 80% accuracy within this timeframe.

Where can I find official past year CAT papers for Algebraic Integral Solutions Practice Drills?

Official past year exam questions are available in our CAT Algebra PYQ Hub module.

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