CAT Quadratic Equations Questions with Answers

Quadratic Equations

Quadratic equations covers roots, discriminant, nature of roots, cubic equations and roots. Most of the times, one encounters a quadratic equation while solving questions of other type and having a good grasp on this topic is crucial to performing well in the exam. Solve the sample questions given below and go through the detailed solutions to improve your understanding of the topic.

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Instructions

For the following questions answer them individually

Question 1

How many integer values can ‘k’ take if the equation $$x^2+9x+|k|=0$$ has integer roots?

Question 2

Two quadratic equations are there such that the roots of the first equation are in the ratio 1:3 and the roots of the second equation are in the ratio 3:5. The sum of roots of both the equations is the same. What is the minimum possible difference of the product of the roots of both the equations, if it is given that the roots of both the equations are integers?

Question 3

If 'a' and 'b' are roots of the quadriatic equation $$x^2 - (t-3)x -2t+1=0$$, what is the minimum possible value of $$a^2+b^2$$?

Question 4

Given that a quadratic equation f(x) attains its minimum of -29 at x = -2 and f(1) = 25, find the sum of roots of the equation.

Question 5

Consider the equation $$x^2 + (k-4)x + (k+4) = 0$$ . What is the least value of the sum of the squares of the roots of the equation, if the roots are real?

Question 6

Given that 'a' and 'b' are integers and that the quadratic equation $$x^2 + ax + b$$ is positive for all values of x except when x=3. For how many integral values of x is the equation $$x^2 + 2ax + 4b$$ negative?

Question 7

If p and q are the roots of the equation $$ax^2+bx+c=0$$, find the equation whose roots are p/q and q/p.

Question 8

$$N^2$$ = 1 + 2014*2015*2016*2017. What is the value of $$N$$?

Question 9

Given the expression (x-5)(x-10)(x-15)..... and so on till (x-60). Find the coeffiecient of $$x^{11}$$.

Question 10

A polynomial "$$ax^3+ bx^2+ cx + d$$" intersects x-axis at 1 and -1, and y-axis at 2. The value of b is:

Question 11

What will be the minimum value of the sum of the squares of the roots of the equation $$x^2-(m-3)x+(m-8)=0$$, where m is a positive integer.

Question 12

The quadratic equation $$2x^2-13x+|a|=0$$ has real roots. How many integral values can ‘a’ take?

Question 13

When the constant term of a quadratic equation is taken wrongly, the roots obtained are 17 and -2. When the coefficient of x is taken wrongly, the roots obtained are 25 and 2. What actual roots of the quadratic equation?

Question 14

If x and y are the roots of the equation $$a^{2} - 5a -1 =0$$, what is the value of (4+x)*(4+y)?

Question 15

The sum of the reciporocals of the roots of the equation $$px^2+qx+r=0$$ is 10 and the sum of the roots of the equation $$qx^2+px+r=0$$ is 20. What is the product of the roots of the equation $$rx^2+qx+p=0$$?

Question 16

If p and q are the roots of the equation $$ax^2+bx+c=0$$, find the equation whose roots are $$p^2$$ and $$-q^2$$ given that p-q=1

Question 17

Two quadratic equations f(x) = 0 and g(x) = 0 have the same roots. If the maximum value that g(x) can take is 5, what is the minimum value that f(x) can take?

Question 18

Find the roots of the equation $$15x^3-4x^2-53x+30=0$$ if two of the roots of the equation are reciprocals

Question 19

If the roots of the equation $$x^2-x-P=0$$ are real and the sum of the fourth powers of the roots is 337, find P.

Question 20

If the sum of the reciprocals of the roots of the quadratic equation $$x^2-ax+b=0$$ is 2, what is the sum of the reciprocals of the roots of the quadratic equation $$x^2-bx+a=0$$?

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