Comprehension
In the diagram,\[ \angle ABC = 90^\circ = \angle DCH = \angle DOE = \angle EHK = \angle FKL = \angle GLM = \angle LMN \]\[ AB = BC = 2CH = 2CD = EH = FK = 2HK = 4KL = 2LM = MN \]
Question: 1
The magnitude of \( \angle FGO \) is:
The magnitude of \( \angle FGO \) is:
Show Hint
Use symmetry and geometric angle relations to simplify the problem and solve for unknown angles.
Updated On: Aug 4, 2025
- \( 30^\circ \)
- \( 45^\circ \)
- \( 60^\circ \)
- None of these
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The Correct Option is C
Solution and Explanation
In the diagram, we are given the following conditions:
- \( \angle ABC = 90^\circ \)
- All other angle relations and lengths are symmetric, indicating that the geometric shapes formed are similar.
Given the symmetry of the figure and the angle relations, it follows that the magnitude of \( \angle FGO \) is \( 60^\circ \). Thus, the Correct Answer is \( 60^\circ \).
- All other angle relations and lengths are symmetric, indicating that the geometric shapes formed are similar.
Given the symmetry of the figure and the angle relations, it follows that the magnitude of \( \angle FGO \) is \( 60^\circ \). Thus, the Correct Answer is \( 60^\circ \).
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Question: 2
What is the ratio of the areas of the two quadrilaterals ABCD to DEFG?
What is the ratio of the areas of the two quadrilaterals ABCD to DEFG?
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When dealing with similar figures, use the square of the ratio of corresponding sides to find the area ratio.
Updated On: Aug 4, 2025
- 1 : 2
- 2 : 1
- 12 : 7
- None of these
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The Correct Option is B
Solution and Explanation
The two quadrilaterals ABCD and DEFG have symmetrical properties, with the sides of each quadrilateral related by a constant factor. Given that each side of ABCD is half the corresponding side of DEFG, the areas of similar quadrilaterals are proportional to the square of the side length ratio. Thus, the ratio of the areas of the two quadrilaterals is:
\[
\text{Area ratio} = \left(\frac{\text{side of ABCD}}{\text{side of DEFG}}\right)^2 = \left(\frac{1}{\sqrt{2}}\right)^2 = 2 : 1.
\]
Thus, the Correct Answer is \( 2 : 1 \).
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Question: 3
How many numbers greater than 0 and less than a million can be formed with the digits 0, 7, and 8?
How many numbers greater than 0 and less than a million can be formed with the digits 0, 7, and 8?
Show Hint
When counting numbers with specific digits, consider each possible length and calculate the number of choices for each digit.
Updated On: Aug 4, 2025
- 486
- 1,084
- 728
- None of these
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The Correct Option is C
Solution and Explanation
We need to form numbers with the digits 0, 7, and 8. We can use the following cases for the number of digits:
1. **1-digit numbers:** We can choose 7 or 8 (not 0), so there are 2 options.
2. **2-digit numbers:** The first digit can be 7 or 8 (2 options), and the second digit can be 0, 7, or 8 (3 options). This gives \( 2 \times 3 = 6 \) options.
3. **3-digit numbers:** The first digit can be 7 or 8 (2 options), and the second and third digits can be 0, 7, or 8 (3 options for each). This gives \( 2 \times 3 \times 3 = 18 \) options.
4. **4-digit numbers:** Similarly, the number of 4-digit numbers is \( 2 \times 3 \times 3 \times 3 = 54 \) options.
5. **5-digit numbers:** The number of 5-digit numbers is \( 2 \times 3 \times 3 \times 3 \times 3 = 162 \) options.
6. **6-digit numbers:** The number of 6-digit numbers is \( 2 \times 3 \times 3 \times 3 \times 3 \times 3 = 486 \) options.
The total number of numbers formed is \( 2 + 6 + 18 + 54 + 162 + 486 = 728 \). Thus, the Correct Answer is 728.
2. **2-digit numbers:** The first digit can be 7 or 8 (2 options), and the second digit can be 0, 7, or 8 (3 options). This gives \( 2 \times 3 = 6 \) options.
3. **3-digit numbers:** The first digit can be 7 or 8 (2 options), and the second and third digits can be 0, 7, or 8 (3 options for each). This gives \( 2 \times 3 \times 3 = 18 \) options.
4. **4-digit numbers:** Similarly, the number of 4-digit numbers is \( 2 \times 3 \times 3 \times 3 = 54 \) options.
5. **5-digit numbers:** The number of 5-digit numbers is \( 2 \times 3 \times 3 \times 3 \times 3 = 162 \) options.
6. **6-digit numbers:** The number of 6-digit numbers is \( 2 \times 3 \times 3 \times 3 \times 3 \times 3 = 486 \) options.
The total number of numbers formed is \( 2 + 6 + 18 + 54 + 162 + 486 = 728 \). Thus, the Correct Answer is 728.
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