The question asks to identify the correct statements regarding the reflection of light by spherical mirrors. We need to analyze each statement based on the properties of convex and concave mirrors, specifically their ability to converge or diverge light rays and form real or virtual images.
D) Only (b) and (d)
Simplify the following.
0.\(\overline{06}\) × 8.\(\overline{3}\)
0.\(\overline{050}\)
0.\(\overline{05}\)
0.\(\overline{50}\)
0.1\(\overline{50}\)
To simplify the given expression, we need to convert the recurring decimals into fractions. Then, we can perform the multiplication of these fractions and convert the result back into a recurring decimal if necessary.
Convert the first recurring decimal to a fraction:
Let \(x = 0.\overline{06}\)
This means \(x = 0.060606...\)
Since two digits are repeating, multiply by \(100\):
\[100x = 6.060606...\]Subtract the first equation from the second:
\[100x - x = 6.060606... - 0.060606...\] \[99x = 6\] \[x = \frac{6}{99} = \frac{2}{33}\]Convert the second recurring decimal to a fraction:
Let \(y = 8.\overline{3}\)
This means \(y = 8.3333...\)
We can write this as \(y = 8 + 0.\overline{3}\)
Let \(z = 0.\overline{3}\)
Multiply by \(10\):
\[10z = 3.3333...\]Subtract:
\[10z - z = 3.3333... - 0.3333...\] \[9z = 3\] \[z = \frac{3}{9} = \frac{1}{3}\]So, \[y = 8 + \frac{1}{3} = \frac{24+1}{3} = \frac{25}{3}\]
Perform the multiplication:
Now, multiply the two fractions:
\[0.\overline{06} \times 8.\overline{3} = \frac{2}{33} \times \frac{25}{3}\] \[= \frac{2 \times 25}{33 \times 3} = \frac{50}{99}\]Convert the result back to a recurring decimal:
To convert \(\frac{50}{99}\) to a decimal, divide \(50\) by \(99\).
When the denominator is \(99\), a two-digit numerator repeats. So, \(\frac{50}{99} = 0.\overline{50}\).
Alternatively, perform long division:
0.5050...
99|50.0000
-0
---
50 0
-49 5
-----
5 00
-4 95
------
5
Thus, \(0.\overline{06} \times 8.\overline{3} = 0.\overline{50}\).
C) 0.\overline{50}
The question asks to identify the forest officer responsible for initiating a new strategy in the Arabari forest range of Midnapore district, which marked a significant change in forest management by involving local communities. This is a factual question related to the history of Joint Forest Management (JFM) in India.
Correct Option: C) AK Banerjee was the forest officer who initiated the change in strategy in the Arabari forest range, leading to the successful implementation of community-based forest management, which later became a model for Joint Forest Management (JFM) in India.
This problem involves arranging people in a linear row based on given positional relationships. We will use a step-by-step approach to place each person relative to others and then count the total number of positions occupied.
P _ QP _ Q _ _ _ SP _ Q _ _ _ S _ _ _ TC) 11 is the total number of people seated in the row, as derived from the step-by-step placement.
The question asks to identify a language that is a descendant of Sanskrit and has been significantly influenced by Dravidian, Arabic, Portuguese, English, Persian, and Turkish languages. We need to evaluate each option based on its linguistic origin and historical influences.
Correct Option: B) Hindi — Hindi is an Indo-Aryan language, a direct descendant of Sanskrit. Over centuries, it has been significantly influenced by Persian and Arabic due to various invasions and cultural exchanges. Turkish words entered through Persian. Portuguese and English influences came through trade and colonial rule. Due to geographical and cultural interactions, Hindi has also borrowed words from Dravidian languages, making it a language that fits all the described influences.
Which of the following terms will replace the question mark (?) in the given series to make it logically complete?
YSL 22, WQJ 25, UOH 28, SMF 31,?
The problem presents a series of terms, each consisting of three letters and a number. To find the missing term, we need to identify the pattern in the letters and the numbers separately.
Y (25th letter), W (23rd letter), U (21st letter), S (19th letter)
The pattern for the first letter is a decrease of 2 positions in the alphabetical order (Y \(-2\) = W, W \(-2\) = U, U \(-2\) = S). Following this pattern, the next letter will be S \(-2\) = Q (17th letter).
S (19th letter), Q (17th letter), O (15th letter), M (13th letter)
The pattern for the second letter is also a decrease of 2 positions (S \(-2\) = Q, Q \(-2\) = O, O \(-2\) = M). Following this pattern, the next letter will be M \(-2\) = K (11th letter).
L (12th letter), J (10th letter), H (8th letter), F (6th letter)
The pattern for the third letter is a decrease of 2 positions (L \(-2\) = J, J \(-2\) = H, H \(-2\) = F). Following this pattern, the next letter will be F \(-2\) = D (4th letter).
22, 25, 28, 31
The pattern for the number is an increase of 3 (22 \(+3\) = 25, 25 \(+3\) = 28, 28 \(+3\) = 31). Following this pattern, the next number will be 31 \(+3\) = 34.
The next term in the series will have the letters Q, K, D and the number 34. So, the term is QKD 34.
D) QKD 34 — This option matches the derived pattern for both the letters and the number.
The nominal marks obtained by students who appeared in a test are given below. Find the median marks of the students.
| Marks | 11 | 20 | 8 | 16 | 29 | 22 |
| No. of students | 14 | 5 | 11 | 9 | 3 | 8 |
To find the median marks, we first need to arrange the data in ascending order of marks and then calculate the cumulative frequency. The median is the value that divides the data into two equal halves. For grouped data, the median position is found using the total number of observations, and then the median value is determined from the cumulative frequency distribution.
Organize the data: Create a frequency distribution table with marks in ascending order and their corresponding number of students (frequency).
| Marks (x) | No. of students (f) |
|---|---|
| 8 | 11 |
| 11 | 14 |
| 16 | 9 |
| 20 | 5 |
| 22 | 8 |
| 29 | 3 |
Calculate the cumulative frequency (cf): Add a column for cumulative frequency.
| Marks (x) | No. of students (f) | Cumulative Frequency (cf) |
|---|---|---|
| 8 | 11 | 11 |
| 11 | 14 | \(11 + 14 = 25\) |
| 16 | 9 | \(25 + 9 = 34\) |
| 20 | 5 | \(34 + 5 = 39\) |
| 22 | 8 | \(39 + 8 = 47\) |
| 29 | 3 | \(47 + 3 = 50\) |
Find the total number of observations (N): The total number of students is the last value in the cumulative frequency column.
\[ N = \sum f = 50 \]Determine the median position: Since N is an even number, the median is the average of the \((N/2)\)-th and \((N/2 + 1)\)-th observations.
\[ \text{Median position} = \frac{N}{2} \text{ and } \left(\frac{N}{2} + 1\right) \] \[ \frac{50}{2} = 25 \text{ and } \left(\frac{50}{2} + 1\right) = 26 \] So, we need to find the 25th and 26th observations.Identify the values at the median positions:
Calculate the median: The median is the average of these two values.
\[ \text{Median} = \frac{\text{25th observation} + \text{26th observation}}{2} \] \[ \text{Median} = \frac{11 + 16}{2} \] \[ \text{Median} = \frac{27}{2} \] \[ \text{Median} = 13.5 \]Correct Option: C) 13.5
Select the option that is related to the fifth number in the same way as the second number is related to the first number and the fourth number is related to the third number.
15 : 230 :: 13 : 174 :: 11 :?
The problem asks us to find the missing number in a given analogy. We need to identify the relationship between the first and second numbers, and the third and fourth numbers, and then apply the same relationship to the fifth number to find the sixth number.
Let's try to find a pattern between 15 and 230.
Let's check if the same pattern applies to the second pair.
Now, we apply the established pattern to the third number, 11, to find the missing number.
Therefore, the missing number is 126.
D) 126
The question asks for the Indian state where the Cardamom Hills are located. This requires knowledge of the geographical distribution of major hill ranges in India.
A) Kerala is the correct answer. The Cardamom Hills are a mountain range forming the southern part of the Western Ghats in southeastern Kerala and southwestern Tamil Nadu. They are known for cardamom, coffee, and tea plantations.
To find the missing number in the series, we need to identify the pattern or rule governing the sequence of numbers. We will examine the differences between consecutive terms to uncover this pattern.
Thus, the missing number is 123.
Correct Option: B) 123