Linear and Quadratic Equations CAT PYQs , 41 Questions with Solutions

The number of non-negative integer values of $k$ for which the quadratic equation $x^2 - 5x + k = 0$ has only integer roots is

If $a-6 b+6 c=4$ and $6 a+3 b-3 c=50$, where $a, b$ and $c$ are real numbers, the value of $2 a+3 b-3 c$ is

A value of $c$ for which the minimum value of $f(x) = x^2 - 4cx + 8c$ is greater than the maximum value of $g(x) = -x^2 + 3cx - 2c$, is

Stocks A, B and C are priced at rupees 120, 90 and 150 per share, respectively. A trader holds a portfolio consisting of 10 shares of stock A, and 20 shares of stocks B and C put together. If the total value of her portfolio is rupees 3300, then the number of shares of stock B that she holds, is

The equations $3x^2 - 5x + p = 0$ and $2x^2 - 2x + q = 0$ have one common root. The sum of the other roots of these two equations is

If $f(x) = \left(x^2 + 3x\right)\left(x^2 + 3x + 2\right)$, then the sum of all real roots of the equation $\sqrt{f(x) + 1} = 9701$, is

In a school with 1500 students, each student chooses any one of the streams out of science, arts, and commerce, by paying a fee of Rs 1100, Rs 1000, and Rs 800, respectively. The total fee paid by all the students is Rs 15,50,000. If the number of science students is not more than the number of arts students, then the maximum possible number of science students in the school is

If the equations $x^2 + m x + 9 = 0$, $x^2 + n x + 17 = 0$ and $x^2 + (m+n) x + 35 = 0$ have a common negative root, then the value of $(2 m + 3 n)$ is

If $x$ and $y$ are real numbers such that $4x^2 + 4y^2 - 4xy - 6y + 3 = 0$, then the value of $(4x + 5y)$ is

The roots $\alpha$, $\beta$ of the equation $3x^2 + \lambda x - 1 = 0$, satisfy $\dfrac{1}{\alpha^2} + \dfrac{1}{\beta^2} = 15$. The value of $(\alpha^3 + \beta^3)^2$, is

The equation $x^3 + (2r+1)x^2 + (4r−1)x+2=0$ has $-2$ as one of its roots. If the other two roots are real, what is the minimum possible non-negative integer value of $r$?

The number of integer solutions of the equation

is:

Let $\alpha$ and $\beta$ be the two distinct roots of the equation $2x^2 - 6x + k = 0$, such that $(\alpha + \beta)$ and $\alpha \beta$ are the distinct roots of the equation $x^2 + px + p = 0$. Then, the value of $8(k - p)$ is

If $x$ and $y$ are real numbers such that $x^2 + (x - 2y - 1)^2 = -4y(x + y)$, then value of $x - 2y$?

Let $k$ be the largest integer such that the equation $(x-1)^2+2 k x+11=0$ has no real roots. If $y$ is a positive real number, then the least possible value of $\frac{k}{4 y}+9 y$ is

If $p^2 + q^2 - 29 = 2pq - 20 = 52 - 2pq,$ then the difference between the maximum and minimum possible values of $p^3 - q^3$ is:

A quadratic equation $x^2 + bx + c = 0$ has two real roots. If the difference between the reciprocals of the roots is $13\frac{1}{3}$, and the sum of the reciprocals of the squares of the roots is $59\frac{5}{9}$, then the largest possible value of $(b + c)$ is:

Let $a, b, c$ be non-zero real numbers such that $b^2 \lt 4 a c$ , and $f(x)=a x^2+b x+c$ . If the set $S$ consists of al integers $m$ such that $f(m)\lt0$ , then the set $S$ must necessarily be

Let $a$ and $b$ be natural numbers. If $a^2+a b+a=14$ and $b^2+a b+b=28$ , then $(2 a+b)$ equals

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

Let $r$ and $C$ be real numbers. If $r$ and $-r$ are roots of $5x^3 + cx^2 - 10x + 9 = 0$, then $c$ equals

Let $f(x)$ be a quadratic polynomial in $x$ such that $f(x) \geq 0$ for all real numbers $x$. If $f(2)=0$ and $f(4)=6$, then $f(-2)$ is equal to

If $(3+2 \sqrt{2})$ is a root of the equation $a x^2+b x+c=0$, and $(4+2 \sqrt{3})$ is a root of the equation $a y^2+m y+n=0$, where $a, b, c, m$ and $n$ are integers, then the value of $\left(\frac{b}{m}+\frac{c-2 b}{n}\right)$ is

Suppose $k$ is any integer such that the equation $2x^2 + kx + 5 = 0$ has no real roots and the equation $x^2 + (k-5)x + 1 = 0$ has two distinct real roots for $x$. Then, the number of possible values of $k$ is

If $r$ is a constant such that $|x^2 - 4x - 13| = r$ has exactly three distinct real roots, then the value of $r$ is

Consider the pair of equations: $x^2 - xy - x = 22$ and $y^2 - xy + y = 34$. If $x > y$, then $x - y$ equals

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

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

Let f(x) = x² + ax + b and g(x) = f(x + 1) - f(x - 1). If f(x) ≥ 0 for all real x, and g(20) = 72, then the smallest possible value of b is

In how many ways can a pair of integers $(x, a)$ be chosen such that

Let $m$ and $n$ be positive integers. If $x^2 + mx + 2n = 0$ and $x^2 + 2nx + m = 0$ have real roots, then the smallest possible value of $m + n$ is

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

The product of the distinct roots of $|x^2 - x - 6| = x + 2.$ is

What is the largest positive integer $n$ such that

is also a positive integer?

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 $b^2 + c$?

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

If $u^2 + (u - 2v - 1)^2 = -4v(u + v)$, then what is the value of $u + 3v$?

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

If $f_1(x) = x^2 + 11x + n$ and $f_2(x) = x$, then the largest positive integer $n$ for which the equation $f_1(x) = f_2(x)$ has two distinct real roots, is:

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

The minimum possible value of the sum of the squares of the roots of the equation $x^2 + (a + 3)x - (a + 5) = 0$ is