**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