XAT Quadratic Equations Practice Questions With Solutions

Let x and y be two positive integers and p be a prime number. If x (x - p) - y (y + p) = 7p, what will be the minimum value of x - y?

The sum of the cubes of two numbers is 128, while the sum of the reciprocals of their cubes is 2.

What is the product of the squares of the numbers?

Wilma, Xavier, Yaska and Zakir are four young friends, who have a passion for integers. One day, each of them selects one integer and writes it on a wall. The writing on the wall shows that Xavier and Zakir picked positive integers, Yaska picked a negative one, while Wilma’s integer is either negative, zero or positive. If their integers are denoted by the first letters of their respective names, the following is true:

$W^{4}+X^{3}+Y^{2}+Z\leq4$
$X^{3}+Z\geq2$
$W^{4}+Y^{2}\leq2$
$Y^{2}+Z\geq3$
Given the above, which of these can $W^{2}+X^{2}+Y^{2}+Z^{2}$ possibly evaluate to?

Consider the real-valued function $f(x)=\frac{\log{(3x-7)}}{\sqrt{2x^{2}-7x+6}}$ Find the domain of f(x).

Let $f(x) = \frac{x^2 + 1}{x^2 - 1}$ if $x \neq 1, -1,$ and 1 if x = 1, -1. Let $g(x) = \frac{x + 1}{x - 1}$ if $x \neq 1,$ and 3 if x = 1.
What is the minimum possible values of $\frac{f(x)}{g(x)}$ ?

Find z, if it is known that:
a: $-y^2 + x^2 = 20$
b: $y^3 - 2x^2 - 4z \geq -12$ and
c: x, y and z are all positive integers

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?

Let C be a circle of radius $\sqrt{20}$ cm. Let L1, L2 be the lines given by 2x − y −1 = 0 and x + 2y−18 = 0, respectively. Suppose that L1 passes through the center of C and that L2 is tangent to C at the point of intersection of L1 and L2. If (a,b) is the center of C, which of the following is a possible value of a + b?

Consider the function f(x) = (x + 4)(x + 6)(x + 8) ⋯ (x + 98). The number of integers x for which f(x) < 0 is:

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