Quadratic Equations Polynomials

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

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.

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

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

$$\frac{b}{m}$$ + $$\frac{c-2b}{n}$$ is

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.

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

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

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

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

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

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?

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

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

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

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.

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

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?

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

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.

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?

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 $$(x^{2}-7x+11)^{(x2-13x+42)}$$ = 1?

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?

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

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

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

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

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

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 number of integral solutions of the equation $$(x^{2}-10)^{x2-3x-10}$$ = 1 is

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

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

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.

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

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 :

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?

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

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