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------------------------------------------------ Q6. If a = -35, b = 10 cm and c = -5, verify that: a * (b + c) = a * b + a * c.

Q7. In a class test containing 20 questions, 5 marks are awarded for each correct answer and 2 marks is deducted for each wrong answer. If Riya gets 15 correct answers out of all questions attempted, what is her total score?

Q8. The marks in a subject for 12 students are as follows: 31, 37, 35, 38, 42, 23, 17, 18, 35, 25, 35, 29. For the given data, find the mean, mode, and median.

We need to verify the distributive property, which says that for any real numbers [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex], the equation [tex]a \times (b + c) = a \times b + a \times c[/tex] should hold true.

Given: [tex]a = -35[/tex], [tex]b = 10[/tex], and [tex]c = -5[/tex].

Calculate [tex]b + c[/tex]:
[tex]b + c = 10 + (-5) = 5[/tex]
Calculate [tex]a \times (b + c)[/tex]:
[tex]a \times (b + c) = -35 \times 5 = -175[/tex]
Calculate [tex]a \times b + a \times c[/tex]:
[tex]a \times b = -35 \times 10 = -350[/tex]
[tex]a \times c = -35 \times (-5) = 175[/tex]
[tex]a \times b + a \times c = -350 + 175 = -175[/tex]

Both [tex]a \times (b + c)[/tex] and [tex]a \times b + a \times c[/tex] are equal to [tex]-175[/tex], verifying the distributive property.

Q7. Riya's Test Score

In the test with 20 questions:

5 marks for each correct answer.
2 marks are deducted for each wrong answer.

Riya answered 15 questions correctly.

Calculate the score from the correct answers:
[tex]15 \times 5 = 75[/tex]
Calculate the number of wrong answers:
[tex]20 - 15 = 5[/tex]
Calculate the penalty for wrong answers:
[tex]5 \times 2 = 10[/tex]
Calculate Riya's total score:
[tex]75 - 10 = 65[/tex]

Thus, Riya's total score is 65.

Q8. Mean, Mode, and Median of Given Marks

Given marks are: 31, 37, 35, 38, 42, 23, 17, 18, 35, 25, 35, 29.

Mean:
Calculate the mean by finding the sum of all marks and dividing by the total number of students.
[tex]\text{Sum} = 31 + 37 + 35 + 38 + 42 + 23 + 17 + 18 + 35 + 25 + 35 + 29 = 365[/tex]
[tex]\text{Mean} = \frac{365}{12} = 30.42 \text{ (approx)}[/tex]
Mode:
The mode is the number that appears most frequently.
- Here, 35 appears three times. Therefore, the mode is 35.
Median:
To find the median, first arrange the marks in ascending order:
17, 18, 23, 25, 29, 31, 35, 35, 35, 37, 38, 42.
Since there are 12 students (an even number), the median will be the average of the 6th and 7th numbers:
[tex]\text{Median} = \frac{31 + 35}{2} = 33[/tex]

The mean is approximately 30.42, the mode is 35, and the median is 33.