Exercise: 9-C
Assertion- Reason Questions
Q1: Assertion (A): The fourth proportional to 7, 12, 28 is 48.
Reason (R): If a, b, c, d are in proportion, then, ac = bd.
Step 1: Recall definition of fourth proportional:
If \(a : b = c : d\), then \(d\) is the fourth proportional to \(a, b, c\), which means
\[
\frac{a}{b} = \frac{c}{d} \\
ad = bc
\]Step 2: Find the fourth proportional \(d\) to 7, 12, 28:
\[
\frac{7}{12} = \frac{28}{d} \\
\Rightarrow 7d = 12 \times 28 = 336 \\
\Rightarrow d = \frac{336}{7} = 48
\]Step 3: Check the Reason (R) statement:
Reason states \(ac = bd\), but from proportion, the correct relation is \(ad = bc\).
Since \(ad = bc\), the Reason (R) statement is false.
Answer: c. Assertion (A) is true but Reason (R) is false.
Q2: Assertion (A): Mean proportion between 8 and 98 is 28.
Reason (R): Three numbers a, b, c are said to be in continued proportion, if a : b :: b : c.
Step 1: Calculate mean proportion between 8 and 98:
\[
\text{Mean proportion} = \sqrt{8 \times 98} = \sqrt{784} = 28
\]
So, Assertion (A) is true.
Step 2: Check Reason (R) statement:
Three numbers \(a, b, c\) are in continued proportion if:
\[
\frac{a}{b} = \frac{b}{c}
\]
This is exactly the definition given.
So, Reason (R) is true.
Answer: b. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
Q3: Assertion (A): If we add 3 to each of the numbers 3, 7, 18, 32, then the resulting numbers will be in proportion.
Reason (R): The numbers a, b, c, d are in proportion, if ab : cd.
Step 1: Add 3 to each number:
\[
3 + 3 = 6, \quad 7 + 3 = 10, \quad 18 + 3 = 21, \quad 32 + 3 = 35
\]Check if these four numbers are in proportion:
\[
\frac{6}{10} = 0.6, \quad \frac{21}{35} = 0.6
\]
Since \(\frac{6}{10} = \frac{21}{35}\), the numbers 6, 10, 21, 35 are in proportion.
Step 2: Check Reason (R):
The correct condition for four numbers \(a, b, c, d\) to be in proportion is:
\[
\frac{a}{b} = \frac{c}{d} \quad \text{or equivalently} \quad ad = bc
\]
But Reason (R) states \(ab = cd\), which is incorrect.
Answer: c. Assertion (A) is true but Reason (R) is false.
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