# Competency Based Questions Chapter 4 Quadratic Equations

Hint: Write Quadratic Equation in order to represent the given situation mathematically.

Question.1. A sum of ₹4000 was divided among x persons. Had there been 10 more persons, each would have got ₹80 less. Which of the following represents the above situation?

(a) x^{2}+10x-500=0
(b) 8x^{2}+10x-400=0
(c) x^{2}+10x+500=0
(d) 8x^{2}+10x+400=0

Question.2. The product of three consecutive integers is equal to 6 times the sum of the three integers. If the smallest integer is x , which of the following equations represent the above situation?

(a) 2x^{2}+x-9=0
(b) x^{2}+2x+18=0
(c) x^{2}+2x-18=0
(d) 2x^{2}-x+9=0

Ans.1. (a) x^{2}+10x-500=0
Ans.2. (c) x^{2}+2x-18=0

Hint: Rewrite the given equations in the standard form in order to check whether they are quadratic or not.

Question.3. Which of these is a quadratic equation?

(a) x^{2}-x=x^{2}+2
(b) x^{2}+\frac{1}{x^{2}}+1=0
(c) (x+1)(x+3)=(x-1)(x-4)
(d) x(x+1)=2x-3

Question.4. Consider the equation kx^{2}+2x=c(2x^{2}+b)
For the equation to be quadratic, which of these cannot be the value of k ?

(a) c
(b) 2c
(c) 3c
(d) 2c+2b

Ans.3. (d) x(x+1)=2x-3
Ans.4. (b) 2c

Hint: Solve quadratic equations through factorization in order to find its roots.

Question.5. What are the roots of the equation 4x^{2}-2x-20=x^{2}+9x ?

(a) \frac{-4}{3} and -5
(b) \frac{4}{3} and -5
(c) \frac{-4}{3} and 5
(d) \frac{4}{3} and 5

Question.6. What is the smallest positive integer value of k such that the roots of the equation x^{2}-9x+18+k=0 can be calculated by factoring the equation?

(a) 1
(b) 2
(c) 3
(d) 4

Ans.5. (c) \frac{-4}{3} and 5
Ans.6. (b) 2

Hint: Solve quadratic equations through middle term splitting in order to find its roots.

Question.7. A student is trying to find the roots of 3x^{2}-10x-8=0 by splitting the middle term as follows:
Step 1: 3x^{2}-10x-8=0
Step 2: 3x^{2}-mx+nx-8=0
What could be the values of m and n ?

(a) m=12 and n=2
(b) m=-12 and n=-2
(c) m=8 and n=2
(d) m=-8 and n=-2

Question.8. Rahul follows the below steps to find the roots of the equation 3x^{2}-11x-20=0 , by splitting the middle term.
Step 1: 3x^{2}-11x-20=0
Step 2: 3x^{2}-15x+4x-20=0
Step 3: 3x(x-5)+4(x-5)=0
Step 4: (3x-4)(x-5)=0
Step 5: x= \frac{4}{3} and 5
In which step did Rahul make the first error?

(a) Step 2
(b) Step 3
(c) Step 4
(d) Step 5

Ans.7. (a) m=12 and n=2
Ans.8. (c) Step 4

Hint: Solve quadratic equations by completing the square in order to find its roots.

Question.9. A student simplified the equation 5x^{2}+3x-1=0 as x^{2}+\frac{3}{5}x-\frac{1}{5}=0
Which of these could be the next step to solve the equation using the method of completing the square?

(a) x^{2}+5(2)x+\frac{3}{5}x-\frac{1}{5}=0
(b) x^{2}+5(2)x+\frac{3}{5}x-\frac{3}{5}x+\frac{1}{5}=0
(c) x^{2}+2(\frac{3}{5}x)+(\frac{2}{3})^{2}-\frac{1}{3}=0
(d) x^{2}+(\frac{3}{5})x+(\frac{3}{10})^{2}-\frac{9}{100}-\frac{1}{5}=0

Question.10. The roots of the equation a^{2}x^{2}-3abx-4b^{2}=0 by completing the square method is

(a) \frac{4b}{a} and \frac{-b}{a}
(b) \frac{b}{2a} and \frac{b}{a}
(c) \frac{2a}{b} and \frac{a}{b}
(d) \frac{2a}{b} and \frac{2b}{a}

Ans.9. (d) x^{2}+(\frac{3}{5})x+(\frac{3}{10})^{2}-\frac{9}{100}-\frac{1}{5}=0
Ans.10. (a) \frac{4b}{a} and \frac{-b}{a}

Hint: Use the most appropriate method in order to find the roots of quadratic equation.

Question.11. A teacher asked students to find the roots of the equation \frac{x}{x+1}+\frac{x+1}{x}-\frac{34}{15}=0. Two students, Ravi and Ankit gave following answers. Ravi said one of the roots is \frac{3}{2}. Ankit said one of the roots is \frac{-5}{2}.
Who is correct?

(a) Ravi
(b) Ankit
(c) Both Ravi and Ankit
(d) Neither Ravi nor Ankit

Question.12. Which is the best method to find the roots of the equation 2x^{2}+7x+5=0?

(a) splitting the middle term 7x to 5x+2x and then factorizing the equation
(b) splitting the middle term 7x to 10x-3x and then factorizing the equation
(c) using the method of completing the square to get x^{2}+\frac{7}{2}x+5=0
(d) using the method of completing the square to get x^{2}+7x+5=0

Ans.11. (c) Both Ravi and Ankit
Ans.12. (a) splitting the middle term 7x to 5x+2x and then factorizing the equation

Hint: Substitute the value of the roots of a given equation in order to verify the roots of that quadratic equation.

Question.13. Which is the correct way to verify that 2 and 3 are the roots of the equation x^{2}-5x+6=0?

(a) On substituting x=2 and x=3 on the left-hand side of the equation, the result should be 0.
(b) On substituting x^{2} with 2 and x with 3 on the left-hand side of the equation, the result should be 0.
(c) On substituting x=2 on the left-hand side of the equation, the result should be 3.
(d) On substituting x=3 on the left-hand side of the equation, the result should be 2.

Question.14. A student solved a quadratic equation and obtains the roots as -4 and 3. Part of the student’s work to verify the root is shown: (-4)^{2}+2(-4)-9=0. Based on the student’s work, which of these is correct?

(a) The student calculated the roots correctly.
(b) The student calculated the roots correctly but should replace 2(-4) in his work with 2(3).
(c) The student calculated the roots of the equation that can be obtained by adding 1 to the equation that the student solved.
(d) The student calculated the roots of the equation that can be obtained by adding -1 to the equation that the student solved.

Ans.13. (a) On substituting x=2 and x=3 on the left-hand side of the equation, the result should be 0.
Ans.14. (c) The student calculated the roots of the equation that can be obtained by adding 1 to the equation that the student solved.

Hint: Examine the discriminant of quadratic equation in order to find out the nature of its roots.

Question.15. The roots of ax^{2}+bx+c=0, a \neq 0 are real and unequal. Which of these is true about the value of discriminant, D?

(a) D<0
(b) D>0
(c) D=0
(d) D \leq 0

Question.16. Consider the equation px^{2}+qx+r=0. Which conditions are sufficient to conclude that the equation have real roots?

(a) p>0, r<0
(b) p>0, r>0
(c) p>0, q>0
(d) p>0, q<0

Ans.15. (b) D>0
Ans.16. (a) p>0, r<0

Hint: Describe the nature of the roots of a quadratic equation in order to determine that whether a given situation is possible or not.

Question.17. For what value of k, are the roots of the quadratic equation 3x^{2}+2kx+27=0 real and equal?

(a) k=\pm 3
(b) k=\pm 9
(c) k=\pm 6
(d) k=\pm 4

Question.18. If the equation x^{2}-mx+1=0 does not possess real roots, then

(a) -3<m<3
(b) -2<m<2
(c) m>2
(d) m<-2

Ans.17. (b) k=\pm 9
Ans.18. (b) -2<m<2