Linear and Quadratic Equations Questions for CAT

Which of the following is a positive-integer solution of $2x + 7y = 20$?

Does the equation $4x + 6y = 25$ have any integer solutions?

Let x and y be integers such that x > y. If x^2 – xy + x = 24 and y^2 – xy – y = 32, then the value of (3x – 2y) is

Let the roots of the equation (x^{2} + kx + 1008 = 0) be positive even integers. Then, the number of possible values of (k) is:

If $\alpha$ and $\beta$ are the roots of the Quadratic equation $5 y^{2}-7 y+1=0$ then find the value of $\frac{1}{\alpha}+\frac{1}{\beta}$.

Find the maximum value of the expression $(x^{2}+8 x+20)$.

If $p,q$ are roots of $ax^2+bx+c=0$, which quadratic equation has roots $p^2, q^2$?

Let the roots of $x^2+7x+l=0$ be p and q, and the roots of $x^2+13x+m=0$ be q and r. If q is the arithmetic mean of p and r, find the value of $l*m$.

For how many integral values of $a$ does $(a+2)x^2+3(a−3)x+(a−1)=0$ not intersect the x-axis?

If $a, b \in \{1,2,3,4,5\}$, then the number of equations of the form $a x^2 + b x + 1 = 0$ having real roots is

A farm has hens and cows. There are 50 heads and 140 legs in total. How many cows are there?

The equation $x^2+6x+|r|=0$ has integer roots. How many integer values can $r$ take?

The number of integer pairs $(x,y)$ satisfying $xy−x−y=11$ is

If the equations $5 x+9 y+17 z=a, 4 x+8 y+12 z=b$ and $2 x+3 y+8 z=c$ have at least one solution for $x, y$ and $z$ and $a, b$, and $c \neq 0$, then which of the following is true?

How many solutions in positive integers $(x, y)$ does $3x + 4y = 60$ have?

For the cubic $x^3 - ax^2 + bx - a = 0$, suppose all roots are real. Which statement is correct?

If $p,q$ are positive integers such that both $3x^2+2px+12q=0$ and $x^2+8qx+4p=0$ have real roots, find minimum $p+q$.

Let $p,q$ be roots of $x^2-(\alpha-2)x-(\alpha+1)=0$. What is the minimum value of $p^2+q^2$ attained over real $\alpha$?

If $(1+\sqrt{2})$ is a root of $ax^2+bx+c=0$ and $(3+2\sqrt{5})$ is a root of $ay^2+my+n=0$ with integer coefficients, compute $\left(\tfrac{b}{m}+\tfrac{c-2b}{n}\right)$.

Solve for $x$ ($x \neq -1, -\frac{1}{2}$): $\frac{2x+3}{x+1} = \frac{4x+5}{2x+1}$.