CAT Questions | CAT Algebra Questions

CAT Quantitative Aptitude | CAT Algebra: Linear Equations; Quadratic Equations

CAT Algebra questions from Linear equations and Quadratic equations that appear in the Quantitative Aptitude section of the CAT Exam consists of concepts from Equations and Algebra. Get as much practice as you can in these two topics because the benefits of being good at framing equations can be enormous and useful in other CAT topics as well. In CAT Exam, one can generally expect to get 1~2 questions from Linear Equations and Quadratic Equations. Make use of 2IIMs Free CAT Questions, provided with detailed solutions and Video explanations to obtain a wonderful CAT score. If you would like to take these questions as a Quiz, head on here to take these questions in a test format, absolutely free.

CAT Linear Equations - Counting

3x + 4|y| = 33. How many integer values of (x, y) are possible?
1. 6
2. 3
3. 4
4. More than 6
Choice D
More than 6
CAT Linear Equations - Counting

(|x| - 3) (|y| + 4) = 12. How many pairs of integers (x, y) satisfy this equation?
1. 4
2. 6
3. 10
4. 8
Choice C
10
CAT Linear Equations - Integer Values; Modulus

x + |y| = 8, |x| + y = 6. How many pairs of x, y satisfy these two equations?
1. 2
2. 4
3. 0
4. 1
Choice D
1
CAT Quadratic Equations - Counting.

What is the number of real solutions of the equation x² - 7|x| - 18 = 0?
1. 2
2. 4
3. 3
4. 1
Choice A
2
CAT Quadratic Equations - Counting

x² - 9x + |k| = 0 has real roots. How many integer values can 'k' take?
1. 40
2. 21
3. 20
4. 41
Choice D
41
CAT Quadratic Equations - Integer Roots

x² - 11x + |p| = 0 has integer roots. How many integer values can 'p' take?
1. 6
2. 4
3. 8
4. More than 8
Choice D
More than 8.
CAT Linear Equations - Integer Solutions

2x + 5y = 103. Find the number of pairs of positive integers x and y that satisfy this equation.
1. 9
2. 10
3. 12
4. 20
Choice B
10
CAT Linear Equations - Maximum, Minimum

Consider three numbers a, b and c. Max (a,b,c) + Min (a,b,c) = 13. Median (a,b,c) - Mean (a,b,c) = 2. Find the median of a, b, and c.
1. 11.5
2. 9
3. 9.5
4. 12
Choice C
9.5
CAT Linear Equations - Unique Solutions

a₁x + b₁y + c₁z = d₁, a₂x + b₂y + c₂z = d₂, a₃x + b₃y + c₃z = d₃.

Which of the following statements if true would imply that the above system of equations does not have a unique solution?
i. \\frac{a_{1}}{a_{2}}\\) = \\frac{b_{1}}{b_{2}}\\) = \\frac{c_{1}}{c_{2}}\\) ≠ \\frac{d_{1}}{d_{2}}\\)
ii. \\frac{a_{1}}{a_{2}}\\) = \\frac{ a_{2} }{ a_{3} }\\) ; \\frac{ b_{1} }{ b_{2} }\\) = \\frac{ b_{2} }{ b_{3} }\\)
iii. a₁, a₂, a₃ are integers; b₁, b₂, b₃ are rational numbers, c₁, c₂, c₃ are irrational numbers
1. Statement i
2. Statement ii
3. Statement iii
4. None
Statement i
CAT Quadratic Equations - Roots of Equation

Equation x² + 5x – 7 = 0 has roots a and b. Equation 2x² + px + q = 0 has roots a + 1 and b + 1. Find p + q.
1. 6
2. 0
3. -16
4. 2
Choice C
-16
CAT Quadratic Equations - Sum and product of the roots

Sum of the roots of a quadratic equation is 5 less than the product of the roots. If one root is 1 more than the other root, find the product of the roots?
1. 6 or 3
2. 12 or 2
3. 8 or 4
4. 12 or 4
Choice B
12 or 2
CAT Quadratic Equations - Real and imaginary Roots

How many real solutions are there for the equation x² – 7|x| - 30 = 0?
1. 3
2. 1
3. 2
4. none
Choice C
2
CAT Quadratic Equations - Sum of the variables

If (3x+2y-22)² + (4x-5y+9)² = 0 and 5x-4y = 0. Find the value of x+y.
1. 7
2. 9
3. 11
4. 13
Choice B
9
CAT Cubic Equations - AP and roots of an equation

Let x³- x² + bx + c = 0 has 3 real roots which are in A.P. which of the following could be true
1. b=2,c=2
2. b=1,c=1
3. b= -1,c = 1
4. b= -1,c= -1
Choice B
b=1,c=1
CAT Linear Quadratic Equations - Roots of an Equation

(3 + 2√2)^{(x² - 3)} + (3 - 2√2)^{(x² - 3)} = b which of the following can be the value of b?
1. 2
2. √2
3. -√2
4. All of the above
Choice A
2
CAT Quadratic Equations - Real roots

If f(y) = x² + (2p + 1)x + p² - 1 and x is a real number, for what values of ‘p' the function becomes 0?
1. p > 0
2. p > -1
3. p ≥ \\frac{-5}{4}\\)
4. p ≤ \\frac{3}{4}\\)
Choice C
p ≥ \\frac{-5}{4}\\)
CAT Linear Quadratic Equations - Medium

A merchant decides to sell off 100 articles a week at a selling price of Rs. 150 each. For each 4% rise in the selling price he sells 3 less articles a week. If the selling price of each article is Rs x, then which of the below expression represents the number of articles sold by the merchant in that week?
1. 175 - \\frac{x}{2}\\)
2. \\frac{x}{3}\\) + 156
3. 350 - \\frac{x^2}{2}\\)
4. x² - 2x + 75
Choice A
175 - \\frac{x}{2}\\)

The Questions that follow, are from actual CAT papers. If you wish to take them separately or plan to solve actual CAT papers at a later point in time, It would be a good idea to stop here.

CAT 2024 Slot 3 - QA

If \((a+b \sqrt{3})^2=52+30 \sqrt{3}\), where \(a\) and \(b\) are natural numbers, then \(a+b\) equals
1. 7
2. 10
3. 8
4. 9
Choice C
8
CAT 2024 Slot 3 - QA

For some constant real numbers \(p, k\) and \(a\), consider the following system of linear equations in \(x\) and \(y\) : \[ \begin{aligned} & p x-4 y=2 \\ & 3 x+k y=a \end{aligned} \] A necessary condition for the system to have no solution for \((x, y)\), is
1. 2a+k≠0
2. ap−6=0
3. ap+6=0
4. kp+12≠0
Choice A
2a+k≠0
CAT 2024 Slot 2 - QA

When Rajesh's age was same as the present age of Garima, the ratio of their ages was 3 : 2. When Garima's age becomes the same as the present age of Rajesh, the ratio of the ages of Rajesh and Garima will become
1. 4 : 3
2. 2 : 1
3. 3 : 2
4. 5 : 4
Choice D
5 : 4
CAT 2024 Slot 2 - QA

If \(x\) and \(y\) are real numbers such that \(4 x^2+4 y^2-4 x y-6 y+3=0\), then the value of \((4 x+5 y)\) is
7
CAT 2024 Slot 2 - QA

The roots \(\alpha, \beta\) of the equation \(3 x^2+\lambda x-1=0\), satisfy \(\frac{1}{\alpha^2}+\frac{1}{\beta^2}=15\). The value of \(\left(\alpha^3+\beta^3\right)^2\), is
1. 9
2. 16
3. 4
4. 1
Choice C
4
CAT 2024 Slot 2 - QA

If \((x+6 \sqrt{2})^{\frac{1}{2}}-(x-6 \sqrt{2})^{\frac{1}{2}}=2 \sqrt{2}\), then \(x\) equals
11
CAT 2024 Slot 1 - QA

If the equations \(x^2+m x+9=0, x^2+n x+17=0\) and \(x^2+(m+n) x+35=0\) have a common negative root, then the value of \((2 m+3 n)\) is
38
CAT 2024 Slot 1 - QA

A shop wants to sell a certain quantity (in kg) of grains. It sells half the quantity and an additional 3 kg of these grains to the first customer. Then, it sells half of the remaining quantity and an additional 3 kg of these grains to the second customer. Finally, when the shop sells half of the remaining quantity and an additional 3 kg of these grains to the third customer, there are no grains left. The initial quantity, in kg, of grains is
1. 18
2. 36
3. 50
4. 42
Choice D
42
CAT 2024 Slot 1 - QA

Let \(x, y\), and \(z\) be real numbers satisfying \[ \begin{aligned} & 4\left(x^2+y^2+z^2\right)=a \\ & 4(x-y-z)=3+a \end{aligned} \] Then \(a\) equals
1. 3
2. \(1 \frac{1}{3}\)
3. 1
4. 4
Choice A
3
CAT 2024 Slot 1 - QA

If \((a+b \sqrt{n})\) is the positive square root of \((29-12 \sqrt{5})\), where \(a\) and \(b\) are integers, and \(n\) is a natural number, then the maximum possible value of \((a+b+n)\) is
1. 22
2. 6
3. 18
4. 4
Choice C
18
CAT 2023 Slot 3 - QA

For some real numbers \(a\) and \(b\), the system of equations \(x+y=4\) and \((a+5) x+\left(b^2-15\right) y=8 b\) has infinitely many solutions for \(x\) and \(y\). Then, the maximum possible value of \(a b\) is
1. 15
2. 33
3. 55
4. 25
Choice B
33
CAT 2023 Slot 3 - QA

A quadratic equation \(x^2+b x+c=0\) has two real roots. If the difference between the reciprocals of the roots is \(\frac{1}{3}\), and the sum of the reciprocals of the squares of the roots is \(\frac{5}{9}\), then the largest possible value of \((b+c)\) is
9
CAT 2023 Slot 2 - QA

The sum of all possible values of \(x\) satisfying the equation \(2^{4 x^2}-2^{2 x^2+x+16}+2^{2 x+30}=0\), is
1. \(\frac{3}{2}\)
2. \(\frac{5}{2}\)
3. \(\frac{1}{2}\)
4. 3
Choice C
\(\frac{1}{2}\)
CAT 2023 Slot 2 - QA

Let \(k\) be the largest integer such that the equation \((x-1)^2+2 k x+11=0\) has no real roots. If \(y\) is a positive real number, then the least possible value of \(\frac{k}{4 y}+9 y\) is
6
CAT 2023 Slot 1 - QA

If \(\sqrt{5 x+9}+\sqrt{5 x-9}=3(2+\sqrt{2})\), then \(\sqrt{10 x+9}\) is equal to
1. \(4 \sqrt{5}\)
2. \(2 \sqrt{7}\)
3. \(3 \sqrt{31}\)
4. \(3 \sqrt{7}\)
Choice D
\(3 \sqrt{7}\)
CAT 2023 Slot 1 - QA

Let \(\alpha\) and \(\beta\) be the two distinct roots of the equation \(2 x^2-6 x+k=0\), such that \((\alpha+\beta)\) and \(\alpha \beta\) are the distinct roots of the equation \(x^2+p x+p=0\). Then, the value of \(8(k-p)\) is
6
CAT 2023 Slot 1 - QA

The equation \(x^3+(2 r+1) x^2+(4 r-1) x+2=0\) has -2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of \(r\) is
2
CAT 2022 Slot 3 - QA

Suppose \(k\) is any integer such that the equation \(2 x^2+k x+5=0\) has no real roots and the equation \(x^2+(k-5) x+1=0\) has two distinct real roots for \(x\). Then, the number of possible values of \(k\) is
1. 7
2. 8
3. 9
4. 13
Choice C
9
CAT 2022 Slot 3 - QA

If \((3+2 \sqrt{2})\) is a root of the equation \(a x^2+b x+c=0\), and \((4+2 \sqrt{3})\) is a root of the equation \(a y^2+m y+n=0\), where \(a, b, c, m\) and \(n\) are integers, then the value of \(\left(\frac{b}{m}+\frac{c-2 b}{n}\right)\) is
1. 3
2. 1
3. 4
4. 0
Choice C
4
CAT 2022 Slot 3 - QA

A donation box can receive only cheques of ₹100, ₹250, and ₹500. On one good day, the donation box was found to contain exactly 100 cheques amounting to a total sum of ₹15250. Then, the maximum possible number of cheques of ₹500 that the donation box may have contained, is
12
CAT 2022 Slot 2 - QA

Let \(r\) and \(c\) be real numbers. If \(r\) and \(-r\) are roots of \(5 x^3+c x^2-10 x+9=0\), then \(c\) equals
1. \(-\frac{9}{2}\)
2. \(\frac{9}{2}\)
3. \(-4\)
4. \(4\)
Choice A
\(-\frac{9}{2}\)
CAT 2022 Slot 2 - QA

The number of integer solutions of the equation \(\left(x^2-10\right)^{\left(x^2-3 x-10\right)}=1\) is
4
CAT 2022 Slot 1 - QA

Let \(a, b, c\) be non-zero real numbers such that \(b^2 \lt 4 a c\), and \(f(x)=a x^2+b x+c\). If the set \(S\) consists of al integers \(m\) such that \(f(m)\lt0\), then the set \(S\) must necessarily be
1. the set of all integers
2. either the empty set or the set of all integers
3. the empty set
4. the set of all positive integers
Choice B
either the empty set or the set of all integers
CAT 2021 Slot 3 - QA

If \(3 x+2|y|+y=7\) and \(x+|x|+3 y=1\), then \(x+2 y\) is
1. 0
2. 1
3. \(-\frac{4}{3}\)
4. \(\frac{8}{3}\)
Choice A
0

Correct: 11.05%
Incorrect: 22.99%
Unattempted: 65.96%
CAT 2021 Slot 2 - QA

Suppose one of the roots of the equation a x² - b x + c = 0 is 2 + √3, where a, b and c are rational numbers and a ≠ 0. If b = c³ then |a| equals
1. 2
2. 3
3. 4
4. 1
Choice A
2

Correct: 25.75%
Incorrect: 9.3%
Unattempted: 64.95%
CAT 2020 Question Paper Slot 3 - Linear & quadratic equations

Let k be a constant. The equations kx + y = 3 and 4x + ky = 4 have a unique solution if and only if
1. |k| = 2
2. k ≠ 2
3. |k| ≠ 2
4. k = 2
CAT 2020 Question Paper Slot 3 - Linear & quadratic equations

Let m and n be positive integers, If x²+ mx + 2n = 0 and x²+ 2nx + m = 0 have real roots, then the smallest possible value of m + n is
1. 8
2. 6
3. 5
4. 7
CAT 2020 Question Paper Slot 2 - Linear & quadratic equations

The number of pairs of integers(x,y) satisfying x ≥ y ≥ -20 and 2x + 5y = 99 is

Correct Answer: 17
CAT 2020 Question Paper Slot 2 - Linear & quadratic equations

The number of integers that satisfy the equality (x² - 5x + 7)^{x + 1} = 1 is
1. 5
2. 4
3. 3
4. 2
CAT 2020 Question Paper Slot 2 - Linear & quadratic equations

In how many ways can a pair of integers (x , a) be chosen such that x² − 2 | x | + | a - 2 | = 0 ?
1. 7
2. 6
3. 4
4. 5
CAT 2020 Question Paper Slot 2 - Linear & quadratic equations

Aron bought some pencils and sharpeners. Spending the same amount of money as Aron, Aditya bought twice as many pencils and 10 less sharpeners. If the cost of one sharpener is 2 more than the cost of a pencil, then the minimum possible number of pencils bought by Aron and Aditya together is
1. 33
2. 27
3. 30
4. 36
CAT 2020 Question Paper Slot 1 - Linear & quadratic equations

The number of distinct real roots of the equation
(x + \\frac{1}{x}))² - 3(x + \\frac{1}{x})) + 2 = 0 equals

Correct Answer: 1
CAT 2020 Question Paper Slot 1 - Linear & quadratic equations

How many distinct positive integer-valued solutions exist to the equation (x² - 7x + 11)^{(x² - 13x + 42)} = 1?
1. 6
2. 2
3. 4
4. 8
CAT 2019 Question Paper Slot 2 - Number theory

Let a, b, x, y be real numbers such that a² + b² = 25 , x² + y² = 169 and ax + by = 65. If k = ay - bx, then
1. k = 0
2. k > \\frac{5}{13})
3. k = \\frac{5}{13})
4. 0 < k ≤ \\frac{5}{13})
Choice A
k = 0
CAT 2019 Question Paper Slot 2 - Linear & quadratic equations

What is the largest positive integer such that \\frac{n^2+7n+12}{n^2-n-12}) is also positive integer?
1. 6
2. 8
3. 16
4. 12
Choice D
12
CAT 2019 Question Paper Slot 2 - Linear & quadratic equations

Let A be a real number. Then the roots of the equation x² - 4x - log₂A = 0 are real and distinct if and only if
1. A < \\frac{1}{16})
2. A > \\frac{1}{8})
3. A > \\frac{1}{16})
4. A < \\frac{1}{8})
Choice C
\\frac{1}{16})
CAT 2019 Question Paper Slot 2 - Linear & quadratic equations

The quadratic equation x² + bx + c = 0 has two roots 4a and 3a, where a is an integer. Which of the following is a possible value of b² + c?
1. 3721
2. 549
3. 361
4. 427
Choice B
549
CAT 2019 Question Paper Slot 1 - Number theory

The product of two positive numbers is 616. If the ratio of the difference of their cubes to the cube of their difference is 157 : 3, then the sum of the two numbers is
1. 50
2. 85
3. 95
4. 58
Choice A
50
CAT 2019 Question Paper Slot 1 - Quadratic equations

The number of solutions of the equation |x|(6x² + 1) = 5x² is [TITA]
5
CAT 2019 Question Paper Slot 1 - Quadratic equations

The product of the distinct roots of |x² - x - 6| = x + 2 is
1. -4
2. -16
3. -8
4. -24
Choice B
-16
CAT 2018 Question Paper Slot 1 - Linear and Quadratic Equations

If u² + (u−2v−1)² = −4v(u + v), then what is the value of u + 3v?
1. \\frac{1}{4})
2. \\frac{1}{2})
3. 0
4. -\\frac{1}{4})
Choice D
-\\frac{1}{4})
CAT 2017 Question Paper Slot 2 - Quadratic Equations

The minimum possible value of the sum of the squares of the roots of the equation x² + (a + 3)x - (a + 5) = 0 is
1. 1
2. 2
3. 3
4. 4
Choice C
3
CAT 2017 Question Paper Slot 1 - Algebra

If x + 1 = x² and x > 0, then 2x⁴ is:
1. 6 + 4√5
2. 3 + 5√5
3. 5 + 3√5
4. 7 + 3√5
Choice D
7 + 3√5

The Questions that follow, are from actual XAT papers. If you wish to take them separately or plan to solve actual XAT papers at a later point in time, It would be a good idea to stop here.

XAT 2018 Question Paper - QADI
Two different quadratic equations have a common root. Let the three unique roots of the two equations be A, B and C - all of them are positive integers. If (A + B + C) = 41 and the product of the roots of one of the equations is 35, which of the following options is definitely correct?
1. The common root is 29.
2. The smallest among the roots is 1.
3. One of the roots is 5.
4. Product of the roots of the other equation is 5.
5. All of the above are possible, but none are definitely correct.
Choice C
One of the roots is 5.

The Questions that follow, are from actual IPMAT papers. If you wish to take them separately or plan to solve actual IPMAT papers at a later point in time, It would be a good idea to stop here.

IPMAT 2019 Question Paper - IPM Indore Quants
If \|x| < 100\\) and \|y| < 100\\), then the number of integer solutions of (x, y) satisfying the equation 4x + 7y = 3 is
29
IPMAT 2019 Question Paper - IPM Indore Quants
If a, b, c are real numbers a² + b² + c² = 1, then the set of values ab + bc + ca can take is:
1. [-1,2]
2. [-\\frac{1}{2}\\), 2]
3. [-1,1]
4. [-\\frac{1}{2}\\), 1]
Choice D
[-\\frac{1}{2}\\), 1]
IPMAT 2019 Question Paper - IPM Indore Quants
Let \\alpha, \beta\\) be the roots of x² - x + p = 0 and \\gamma, \delta\\) be the roots of x² - 4x + q = 0 where p and q are integers. If \\alpha, \beta, \gamma, \delta\\) are in geometric progression then p + q is
1. -34
2. 30
3. 26
4. -38
Choice A
-34