Quadratic Equations — CAT Previous-Year Questions

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

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

The value of (α³ + β³)², is

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

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

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)?

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?

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?

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

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

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

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

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

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

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

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 $\left(\frac{b}{m}+\frac{c-2b}{n}\right)$ is

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

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

For all real values of x, the range of the function f(x) = $\frac{x^2+2x+4}{2x^2+4x+9}$ is

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

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

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

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

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

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

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

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

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

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

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

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?

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

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

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

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 :

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

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

What is the value of a + b + c?

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

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?

How many units should Mr. David produce daily?

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

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

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

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

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?

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

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:

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

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?

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?

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?

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

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?

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?

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

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

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