Assertion (A): The roots of the quadratic equation 8x² + 2x - 3 = 0 are -$\dfrac{1}{2}$ and $\dfrac{3}{4}$.

Reason (R): The roots of the quadratic equation ax² + bx + c = 0 are given by $x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$.

1. Both A and R are true, and R is the correct explanation of A.

2. Both A and R are true, but R is not the correct explanation of A.

3. A is true, but R is false.

4. A is false, but R is true.

Solve for x, if $\dfrac{5}{x} + 4\sqrt{3} = \dfrac{2\sqrt{3}}{x^2}$, x ≠ 0

Determine whether the following quadratic equation has real roots.

5𝑥² − 9𝑥 + 4 = 0

(a) Give reasons for your answer.

(b) If the equation has real roots, identify them.

Case Study II

Raman Lal runs a stationery shop in Pune. The analysis of his sales, expenditures and profits showed that for x number of notebooks sold, the weekly profit (in ₹) was P(x) = - 2x² + 88x - 680. Raman Lal found that:

• He has a loss if he does not sell any notebook in a week.
• There is no profit no loss for a certain value x₀ of x.
• The profit goes on increasing with an increase in x i.e. the number of notebooks sold. But he gets a maximum profit at a sale of 22 notebooks in a week.

Now answer the following questions :

1. What will be Raman Lal’s profit if he sold 20 notebooks in a week?

1. ₹ 144
2. ₹ 280
3. ₹ 340
4. ₹ 560

2. What is the maximum profit that Raman Lal can earn in a week?

1. ₹ 144
2. ₹ 288
3. ₹ 340
4. ₹ 680

3. What is Raman Lal’s loss if he does not sell any notebooks in a particular week?

1. ₹ 0
2. ₹ 340
3. ₹ 680
4. ₹ 960

4. Write a quadratic equation for the condition when Raman Lal does not have any profit or loss during a week.

1. 2x² - 44x + 340 = 0
2. x² + 44x - 340 = 0
3. x² - 88x + 340 = 0
4. x² - 44x + 340 = 0

5. What is the minimum number of notebooks x₀ that Raman Lal should sell in a week so that he does not incur any loss?

1. 0
2. 10
3. 11
4. 12