Quadratic Equations — CAT Previous-Year Questions with Solutions

CAT 2024 Slot 1 · QA

Q1.

If the equations x² + mx + 9 = 0, x² + nx + 17 = 0 and x² + (m + n)x + 35 = 0 have a common negative root, then the value of (2m + 3n) is

CAT 2024 Slot 2 · QA

Q2.

The roots of α, β of the equation 3x² + λx - 1 = 0, satisfy $\frac{1}{α^{2}} + \frac{1}{β^{2}}$ = 15.

The value of (α³ + β³)², is

CAT 2024 Slot 3 · QA

Q3.

The sum of all distinct real values of x that satisfy the equation 10^x + $\frac{4}{10^{x}}$ = $\frac{81}{2}$ , is

CAT 2023 Slot 1 · QA

Q4.

The number of integral solutions of equation 2|x|(x² + 1) = 5x² is?

CAT 2023 Slot 1 · QA

Q5.

Let α and β be two distinct root of the equation 2x² – 6x + k = 0, such that (α + β) and αβ are the two roots of the equation x² + px + p = 0. Then the value of 8(k - p)?

CAT 2023 Slot 1 · QA

Q6.

The equation x³ + (2r + 1)x² + (4r - 1)x + 2 = 0 has -2 as one of the roots. If the other roots are real, then the minimum possible non-negative integer value of r is?

CAT 2023 Slot 2 · QA

Q7.

Let k be the largest integer such that the equation (x - 1)² + 2kx + 11 = 0 has no real roots. If y is a positive real number, then the least possible value of k/4y + 9y is?

CAT 2023 Slot 3 · QA

Q8.

If x is a positive real number such that x⁸ + ${(\frac{1}{x})}^{8}$ = 47, then the value of x⁹ + ${(\frac{1}{x})}^{9}$ is

CAT 2023 Slot 3 · QA

Q9.

A quadratic equation x² + bx + c = 0 has two real roots. If the difference between the reciprocals of the roots is 1/3, and the sum of the reciprocals of the squares of the roots is 5/9, then the largest possible value of (b + c) is

CAT 2022 Slot 1 · QA

Q10.

Let a, b and c be non-zero real numbers such that b² < 4ac, and f(x) = ax² + bx + c. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be

CAT 2022 Slot 2 · QA

Q11.

Let f(x) be a quadratic ploynomial in x such that f(x) ≥ 0 for all real numbers x. If f(2) = 0 and f(4) = 6, then f(-2) is equal to

CAT 2022 Slot 2 · QA

Q12.

Let r and c be real numbers. If r and -r are roots of 5x³ + cx² - 10x + 9 = 0, then c equals

CAT 2022 Slot 2 · QA

Q13.

The number of integral solutions of the equation ${(x^{2} - 10)}^{(x^{2} - 3 x - 10)}$ = 1 is

CAT 2022 Slot 3 · QA

Q14.

Suppose k is any integer such that the equation 2x² + kx + 5 = 0 has no real roots and the equation x² + (k - 5)x + 1 = 0 has two distinct real roots for x. Then, the number of possible values of k is

CAT 2022 Slot 3 · QA

Q15.

If (3 + 2√2) is a root of the equation ax² + bx + c = 0, and (4 + 2√3) is a root of the equation ay² + my + n = 0, where a, b, c, m and n are integers, then the value of $(\frac{b}{m} + \frac{c - 2 b}{n})$ is

CAT 2021 Slot 1 · QA

Q16.

If r is a constant such that |x² – 4x - 13| = r has exactly three distinct real roots, then the value of r is?

CAT 2021 Slot 2 · QA

Q17.

Suppose one of the roots of the equation ax² – bx + c = 0 is 2 + √3, where a, b and c are rational numbers and a ≠ 0. If b = c³ then |a| equals

CAT 2021 Slot 2 · QA

Q18.

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

CAT 2021 Slot 3 · QA

Q19.

A tea shop offers tea in cups of three different sizes. The product of the prices, in INR, of three different sizes is equal to 800. The prices of the smallest size and the medium size are in the ratio 2 : 5. If the shop owner decides to increase the prices of the smallest and the medium ones by INR 6 keeping the price of the largest size unchanged, the product then changes to 3200. The sum of the original prices of three different sizes, in INR, is

CAT 2021 Slot 3 · QA

Q20.

A shop owner bought a total of 64 shirts from a wholesale market that came in two sizes, small and large. The price of a small shirt was INR 50 less than that of a large shirt. She paid a total of INR 5000 for the large shirts, and a total of INR 1800 for the small shirts. Then, the price of a large shirt and a small shirt together, in INR, is

CAT 2020 Slot 1 · QA

Q21.

How many distinct positive integer-valued solutions exist to the equation ${(x^{2} - 7 x + 11)}^{(x^{2} - 13 x + 42)}$ = 1?

CAT 2020 Slot 1 · QA

Q22.

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

CAT 2020 Slot 2 · QA

Q23.

In how many ways can a pair of integers (x , a) be chosen such that x² − 2|x| + |a - 2| = 0?

CAT 2020 Slot 2 · QA

Q24.

The number of integers that satisfy the equality (x² - 5x + 7)^x+1 = 1 is

CAT 2020 Slot 3 · QA

Q25.

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

CAT 2019 Slot 1 · QA

Q26.

The number of solutions to the equation |x|(6 $x^{2}$ + 1) = 5 $x^{2}$ is

CAT 2019 Slot 1 · QA

Q27.

The product of the distinct roots of ∣x² − x − 6∣ = x + 2 is

CAT 2019 Slot 2 · QA

Q28.

The real root of the equation 2^6x + 2^3x+2 - 21 = 0 is

CAT 2019 Slot 2 · QA

Q29.

Let A be a real number. Then the roots of the equation x² - 4x - log₂A = 0 are real and distinct if and only if

CAT 2019 Slot 2 · QA

Q30.

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?

CAT 2018 Slot 1 · QA

Q31.

If u² + (u−2v−1)² = −4v(u + v), then what is the value of u + 3v?

CAT 2018 Slot 2 · QA

Q32.

If a and b are integers such that 2x² − ax + 2 > 0 and x² − bx + 8 ≥ 0 for all real numbers x, then the largest possible value of 2a − 6b is

CAT 2018 Slot 2 · QA

Q33.

The smallest integer n such that n³ – 11n²+ 32n – 28 > 0 is

CAT 2017 Slot 1 · QA

Q34.

If f₁(x) = x² + 11x + n and f₂(x) = x, then the largest positive integer n for which the equation f₁(x) = f₂(x) has two distinct real roots, is :

CAT 2017 Slot 2 · QA

Q35.

The minimum possible value of the squares of the roots of the equation: x² + (a + 3)x – (a + 5) = 0 is

CAT 2008 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Let f(x) = ax² + bx + c, where, a, b and c are certain constants and a ≠ 0. It is known that f(5) = −3f(2) and that 3 is a root of f(x) = 0.

Q36.

What is the other root of f(x) = 0?

CAT 2008 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Let f(x) = ax² + bx + c, where, a, b and c are certain constants and a ≠ 0. It is known that f(5) = −3f(2) and that 3 is a root of f(x) = 0.

Q37.

What is the value of a + b + c?

CAT 2008 · QA

Passage / Data

Directions for next 2 questions:

The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.

Q38.

If the roots of the equation x³− ax² + bx – c = 0 are three consecutive integers, then what is the smallest possible value of b?

CAT 2007 · QA

Q39.

A quadratic function f(x) attains a maximum of 3 at x = 1. The value of the function at x = 0 is 1. What is the value of f(x) at x = 10?

CAT 2007 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Mr. David manufactures and sells a single product at a fixed price in a niche market. The selling price of each unit is Rs. 30. On the other hand, the cost, in rupees, of producing x units is 240 + bx + cx², where b and c are some constants. Mr. David noticed that doubling the daily production from 20 to 40 units increases the daily production cost by 66.66%. However, an increase in daily production from 40 to 60 units results in an increase of only 50% in the daily production cost. Assume that demand is unlimited and that Mr. David can sell as much as he can produce. His objective is to maximize the profit.

Q40.

How many units should Mr. David produce daily?

CAT 2007 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Mr. David manufactures and sells a single product at a fixed price in a niche market. The selling price of each unit is Rs. 30. On the other hand, the cost, in rupees, of producing x units is 240 + bx + cx², where b and c are some constants. Mr. David noticed that doubling the daily production from 20 to 40 units increases the daily production cost by 66.66%. However, an increase in daily production from 40 to 60 units results in an increase of only 50% in the daily production cost. Assume that demand is unlimited and that Mr. David can sell as much as he can produce. His objective is to maximize the profit.

Q41.

What is the maximum daily profit, in rupees, that Mr. David can realize from his business?

CAT 2006 · QA

Passage / Data

Answer the next 2 questions based on the information given below:

A punching machine is used to punch a circular hole of diameter two units from a square sheet of aluminium of width 2 units, as shown below. The hole is punched such that the circular hole touches one corner P of the square sheet and the diameter of the hole originating at P is in line with a diagonal of the square.

Q42.

What values of x satisfy $x^{\frac{2}{3}} + x^{\frac{1}{3}} - 2 \leq 0$ ?

CAT 2005 · QA

Passage / Data

Answer the next 2 questions based on the information given below.

Ram and Shyam run a race between points A and B, 5 km apart. Ram starts at 9 a.m. from A at a speed of 5 km/hr, reaches B, and returns to A at the same speed. Shyam starts at 9:45 a.m. from A at a speed of 10 km/hr, reaches B and comes back to A at the same speed.

Q43.

For which value of k does the following pair of equations yield a unique solution for x such that the solution is positive?

x² – y² = 0

(x – k)² + y² = 1

CAT 2003 Slot 1 · QA

Passage / Data

Each question is followed by two statements, A and B. Answer each question using the following instructions

Choose 1 if the question can be answered by using one of the statements alone but not by using the other statement alone.
Choose 2 if the question can be answered by using either of the statements alone.
Choose 3 if the question can be answered by using both statements together but not by either statement alone.
Choose 4 if the question cannot be answered on the basis of the two statements.

Q44.

What are the unique values of b and c in the equation 4x² + bx + c = 0 if one of the roots of the equation is (−1/2)?

A. The second root is 1/2
B. The ratio of c and b is 1

CAT 2003 Slot 1 · QA

Passage / Data

Each question is followed by two statements, A and B. Answer each question using the following instructions

Choose 1 if the question can be answered by using one of the statements alone but not by using the other statement alone.
Choose 2 if the question can be answered by using either of the statements alone.
Choose 3 if the question can be answered by using both statements together but not by either statement alone.
Choose 4 if the question cannot be answered on the basis of the two statements.

Q45.

Let p and q be the roots of the quadratic equation x²− (α − 2)x − α − 1 = 0. What is the minimum possible value of p² + q²?

CAT 2003 Slot 2 · QA

Passage / Data

Answer the following question based on the information given below.

Consider three circular parks of equal size with centres at A₁, A₂ and A₃ respectively. The parks touch each other at the edge as shown in the figure (not drawn to scale). There are three paths formed by the triangles A₁A₂A₃, B₁B₂B₃ and C₁C₂C₃, as shown. Three sprinters A, B, and C begin running from points A₁, B₁ and C₁ respectively. Each sprinter traverses her respective triangular path clockwise and returns to her starting point.

Q46.

The number of roots common between the two equations x³+ 3x²+ 4x + 5 = 0 and x³+ 2x²+ 7x + 3 = 0 is:

CAT 2003 Slot 2 · QA

Passage / Data

Answer the following question based on the information given below.

A string of three English letters is formed as per the following rules:

The first letter is any vowel.
The second letter is m, n or p.
If the second letter is m, then the third letter is any vowel which is different from the first letter.
If the second letter is n, then the third letter is e or u.
1. If the second letter is p, then the third letter is the same as the first letter.

Q47.

If both a and b belong to the set {1, 2, 3, 4}, then the number of equations of the form ax² + bx + 1 = 0 having real roots is:

CAT 2002 · QA

Q48.

The number of roots $\frac{A^{2}}{x} + \frac{B^{2}}{x - 1} = 1$ is

CAT 2002 · QA

Passage / Data

Answer the following question based on the information given below.

There are 11 alphabets A, H, I, M, O, T, U, V, W, X, Y. They are called symmetrical alphabets. The remaining alphabets are known as asymmetrical alphabets.

Q49.

Davji shop sells samosas in boxes of different sizes. The samosas are priced at Rs. 2 per samosa upto 200 samosas. For every additional 20 samosas, the price of the whole lot goes down by 10 paisa per samosa. What should be the maximum size of the box that would maximize the revenue?

CAT 2001 · QA

Q50.

m is the smallest positive integer such that for any integer n > m, the quantity n³ – 7n² + 11n – 5 is positive. What is the value of m?

CAT 2001 · QA

Passage / Data

Answer the following question based on the information given below.

The batting average (BA) of a test batsman is computed from runs scored and innings played-completed innings and incomplete innings (not out) in the following manner:

r₁ = number of runs scored in completed innings; n₁ = number of completed innings

r₂ = number of runs scored in incomplete innings; n₂ = number of incomplete innings

BA = $\frac{r_{1} + r_{2}}{n_{1}}$

To better assess batsman's accomplishments, the ICC is considering two other measures MBA₁ and MBA₂ defined as follows:

MBA₁ = $\frac{r_{1}}{n_{1}} + \frac{n_{2}}{n_{1}} \times m a x [0, (\frac{r_{2}}{n_{2}} - \frac{r_{1}}{n_{1}}])$

MBA₂ = $\frac{r_{1} + r_{2}}{n_{1} + n_{2}}$

Q51.

Ujakar and Keshab attempted to solve a quadratic equation. Ujakar made a mistake in writing down the constant term. He ended up with the roots (4, 3). Keshab made a mistake in writing down the coefficient of x. He got the root as (3, 2). What will be the exact roots of the original quadratic equation?

CAT 2000 · QA

Passage / Data

Answer the following question based on the information given below.

Sixteen teams have been invited to participate in the ABC Gold Cup cricket tournament. The tournament is conducted in two stages. In the first stage, the teams are divided into two groups. Each group consists of eight teams, with each team playing every other team in its group exactly once. At the end of the first stage, the top four teams from each group advance to the second stage while the rest are eliminated. The second stage comprises of several rounds. A round involves one match for each team. The winner of a match in a round advances to the next round, while the loser is eliminated. The team that remains undefeated in the second stage is declared the winner and claims the Gold Cup.

The tournament rules are such that each match results in a winner and a loser with no possibility of a tie. In the first stage, a team earns one point for each win and no points for a loss. At the end of the first stage teams in each group are ranked on the basis of total points to determine the qualifiers advancing to the next stage. Ties are resolved by a series of complex tie-breaking rules so that exactly four teams from each group advance to the next stage.

Q52.

If the equation x³ – ax² + bx – a = 0 has three real roots, then it must be the case that,

CAT 1997 · QA

Passage / Data

Answer the next 3 questions based on the following information.

There are 60 students in a class. These students are divided into three groups A, B and C of 15, 20 and 25 students each. The groups A and C are combined to form group D.

Q53.

If the roots x₁ and x₂ of the quadratic equation x² − 2x + c = 0 also satisfy the equation 7x₂ – 4x₁ = 47, then which of the following is true?

CAT 1996 · QA

Passage / Data

Direction: Answer the questions based on the following information.

A salesman enters the quantity sold and the price into the computer. Both the numbers are two-digit numbers. But, by mistake, both the numbers were entered with their digits interchanged. The total sales value remained the same, i.e. Rs. 1,148, but the inventory reduced by 54.

Q54.

Given the quadratic equation x² – (A – 3)x – (A – 2), for what value of A will the sum of the squares of the roots be zero?

CAT 1995 · QA

Passage / Data

Direction: Answer the questions based on the following information.
Four sisters — Suvarna, Tara, Uma and Vibha are playing a game such that the loser doubles the money of each of the other players from her share. They played four games and each sister lost one game in alphabetical order. At the end of fourth game, each sister had Rs.32.

Q55.

One root of x² + kx – 8 = 0 is square of the other. Then the value of k is

CAT 1995 · QA

Passage / Data

Direction: Answer the questions based on the following information.
Four sisters — Suvarna, Tara, Uma and Vibha are playing a game such that the loser doubles the money of each of the other players from her share. They played four games and each sister lost one game in alphabetical order. At the end of fourth game, each sister had Rs.32.

Q56.

Two positive integers differ by 4 and sum of their reciprocals is $\frac{10}{21}$ . Then one of the numbers is

CAT 1994 · QA

Passage / Data

Answer the next 3 questions based on the information given below:

Ghoshbabu is staying at Ghosh Housing Society, Aghosh Colony, Dighospur, Calcutta. In Ghosh Housing Society 6 persons read daily Ganashakti and 4 read Anand Bazar Patrika; in his colony there is no person who reads both. Total number of persons who read these two newspapers in Aghosh Colony and Dighospur is 52 and 200 respectively. Number of persons who read Ganashakti in Aghosh Colony and Dighospur is 33 and 121 respectively; while the persons who read Anand Bazar Patrika in Aghosh Colony and Dighospur are 32 and 117 respectively.

Q57.

If one root of x² − px + 12 = 0 is 4, while the equation x² − px + q = 0 has equal roots, then the value of q is

Quadratic Equations — CAT Previous-Year Questions

Quadratic Equations · CAT PYQs