Mathematics

Assertion (A): The quadratic equation 3kx² - 4kx + 4 = 0 has equal roots, if k = 3.
Reason (R): For equal roots of a quadratic equation, we must have D = 0.
A is true, R is false
A is false, R is true
Both A and R are true
Both A and R are false

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Given,

⇒ 3kx² - 4kx + 4 = 0

when k = 3

⇒ 3 × (3) × x² - 4 × (3) × x + 4 = 0

⇒ 9x² - 12x + 4 = 0

Comparing 9x² - 12x + 4 = 0 with ax² + bx + c = 0 we get,

a = 9, b = -12 and c = 4.

We know that,

Discriminant (D) = b² - 4ac

= (-12)² - 4 × (9) × (4)

= 144 - 144 = 0.

Therefore, the equation has rational and equal roots.

So, Assertion (A) is true.

The Discriminant is given by b² - 4ac, if the discriminant of any quadratic equation is zero. Then it is said have equal and real roots.

So, Reason (R) is true.

Thus, both A and R are true.

Hence, option 3 is the correct option.

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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?
₹ 144
₹ 280
₹ 340
₹ 560
2. What is the maximum profit that Raman Lal can earn in a week?
₹ 144
₹ 288
₹ 340
₹ 680
3. What is Raman Lal’s loss if he does not sell any notebooks in a particular week?
₹ 0
₹ 340
₹ 680
₹ 960
4. Write a quadratic equation for the condition when Raman Lal does not have any profit or loss during a week.
2x² - 44x + 340 = 0
x² + 44x - 340 = 0
x² - 88x + 340 = 0
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?
0
10
11
12
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