by Naveen Neredimalli Wed 05 Jun 2024 05:26 PM 166

#### Question 1

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?

#### Question 2

If the roots of the equation $$x^3 - ax^2 + bx - c = 0$$ are three consecutive integers, then what is the smallest possible value of b?

[CAT 2008]

#### Question 3

Let p and q be the roots of the quadratic equation $$x^2 - (\alpha - 2) x - \alpha -1= 0$$ . What is the minimum possible value of $$p^2 + q^2$$?

#### Question 4

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

$$x^2 - y^2 = 0$$

$$(x-k)^2 + y^2 = 1$$

#### Question 5

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 roots as (3, 2). What will be the exact roots of the original quadratic equation?

#### Question 6

If the equation $$x^3 - ax^2 + bx - a = 0$$ has three real roots, then it must be the case that,

#### Question 7

The number of real roots of the equation $$A^2/x + B^2/(x-1) = 1$$ , where A and B are real numbers not equal to zero simultaneously, is

#### Question 8

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

#### Question 9

The value of $$\frac{(1-d^3)}{(1-d)}$$ is

#### Question 10

The roots of the equation $$ax^{2} + 3x + 6 = 0$$ will be reciprocal to each other if the value of a is

#### Question 11

If $$xy + yz + zx = 0$$, then $$(x + y + z)^2$$ equals

#### Question 12

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

#### Question 13

If the roots $$x_1$$ and $$x_2$$ are the roots of the quadratic equation $$x^2 -2x+c=0$$ also satisfy the equation $$7x_2 - 4x_1 = 47$$, then which of the following is true?

#### Question 14

If $$x+1=x^{2}$$ and $$x>0$$, then $$2x^{4}$$  is

#### Question 15

Consider a function $$f(x) = x^4 + x^3 + x^2 + x + 1$$, where x is a positive integer greater than 1. What will be the remainder if $$f(x^5)$$ is divided by f(x)?

#### Question 16

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?

#### Question 17

If $$U^{2}+(U-2V-1)^{2}$$= −$$4V(U+V)$$ , then what is the value of $$U+3V$$ ?

#### Question 18

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

#### Question 19

Given that a and b are integers and that $$5x+2\sqrt{7}$$ is a root of the polynomial $$x^2 - ax + b + 2\sqrt{7}$$ in $$x$$, what is the value of b?

#### Question 20

If $$x^2 + x + 1 = 0$$, then $$x^{2018} + x^{2019}$$ equals which of the following:

#### Question 21

Let A be a real number. Then the roots of the equation $$x^2 - 4x - log_{2}{A} = 0$$ are real and distinct if and only if

#### Question 22

The quadratic equation $$x^2 + bx + c = 0$$ has two roots 4a and 3a, where a is an integer. Which of the following is a possible value of $$b^2 + c$$?

#### Question 23

The product of the distinct roots of $$\mid x^2 - x - 6 \mid = x + 2$$ is

#### Question 24

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

#### Question 25

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

#### Question 26

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

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