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 \( \angle FGO \) =
The magnitude of \( \angle FGO \) =
Show Hint
Place the points on a coordinate grid using the given lengths and right angles, then find the angle at G between GF and GO.
Updated On: Nov 4, 2026
- \( 30^\circ \)
- \( 45^\circ \)
- None of these
- \( 60^\circ \)
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The Correct Option is C
Solution and Explanation
Step 1: Choose a scale:
Let \(AB = BC = 4\). Then \(CH = CD = 2\), \(EH = FK = 4\), \(HK = 2\), \(KL = 1\), \(LM = 2\), \(MN = 4\).
Step 2: Coordinates:
Put B = (0, 0), C = (4, 0), H = (6, 0), K = (8, 0), L = (9, 0), M = (11, 0). The right angles give D = (4, 2), E = (6, 4), F = (8, 4). O lies on EH with DO perpendicular to EH, so O = (6, 2). From the figure, D, O, I and G lie on one horizontal line (the quadrilateral DEFG has DG horizontal), so G = (9, 2), directly above L.
Step 3: Angle at G:
Vectors from G: \(\vec{GF} = (-1, 2)\) and \(\vec{GO} = (-3, 0)\). The angle between GF and the horizontal line GO has \(\tan\theta = \dfrac{2}{1} = 2\), so \(\theta = \tan^{-1} 2 \approx 63.4^\circ\).
Step 4: Compare:
63.4 degrees is not 30, 45 or 60 degrees, so the answer is "None of these".
Final Answer:
The correct option is (3): None of these
Let \(AB = BC = 4\). Then \(CH = CD = 2\), \(EH = FK = 4\), \(HK = 2\), \(KL = 1\), \(LM = 2\), \(MN = 4\).
Step 2: Coordinates:
Put B = (0, 0), C = (4, 0), H = (6, 0), K = (8, 0), L = (9, 0), M = (11, 0). The right angles give D = (4, 2), E = (6, 4), F = (8, 4). O lies on EH with DO perpendicular to EH, so O = (6, 2). From the figure, D, O, I and G lie on one horizontal line (the quadrilateral DEFG has DG horizontal), so G = (9, 2), directly above L.
Step 3: Angle at G:
Vectors from G: \(\vec{GF} = (-1, 2)\) and \(\vec{GO} = (-3, 0)\). The angle between GF and the horizontal line GO has \(\tan\theta = \dfrac{2}{1} = 2\), so \(\theta = \tan^{-1} 2 \approx 63.4^\circ\).
Step 4: Compare:
63.4 degrees is not 30, 45 or 60 degrees, so the answer is "None of these".
Final Answer:
The correct option is (3): None of these
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Question: 2
In the following diagram, \(\angle ABC = 90^\circ = \angle DCH = \angle DOE = \angle EHK = \angle FKL = \angle GLM = \angle LMN\) and \(AB = BC = 2CH = 2CD = EH = FK = 2HK = 4KL = 2LM = MN\).
What is the ratio of the areas of the two quadrilaterals ABCD to DEFG?
In the following diagram, \(\angle ABC = 90^\circ = \angle DCH = \angle DOE = \angle EHK = \angle FKL = \angle GLM = \angle LMN\) and \(AB = BC = 2CH = 2CD = EH = FK = 2HK = 4KL = 2LM = MN\).
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?
Show Hint
Take CH as one unit and write every other length in that unit. Then find the area of the trapezium ABCD, and split DEFG into two triangles and a rectangle.
Updated On: Nov 4, 2026
- 1 : 2
- 12 : 7
- 2 : 1
- None of these
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The Correct Option is B
Solution and Explanation
Step 1: Put every length in terms of CH = u:
AB = BC = 2u, CD = u, EH = 2u, FK = 2u, HK = u, KL = u/2, LM = u, MN = 2u. All the marked angles are right angles, so AB, DC, EH, FK, GL and MN are vertical, and D, O, I, G lie on one horizontal line at height u above BM.
Step 2: Area of ABCD:
AB and DC are parallel (both vertical) with AB = 2u, DC = u, and the distance between them is BC = 2u. Area = \(\tfrac12(2u+u)\times 2u = 3u^2\).
Step 3: Area of DEFG in three pieces:
Triangle DOE: DO = CH = u and OE = EH - OH = 2u - u = u, so its area is \(\tfrac12 u^2\).
Rectangle OEFI: OI = HK = u and EO = FI = u, so its area is \(u^2\).
Triangle FIG: FI = FK - IK = u and IG = KL = u/2, so its area is \(\tfrac12\times u\times\tfrac u2 = \tfrac14 u^2\).
Area of DEFG = \(\tfrac12 u^2 + u^2 + \tfrac14 u^2 = \tfrac74 u^2\).
Step 4: Take the ratio:
\(\dfrac{ABCD}{DEFG} = \dfrac{3u^2}{\tfrac74 u^2} = \dfrac{12}{7}\), that is 12 : 7.
Final Answer:
The correct option is (2): 12 : 7
AB = BC = 2u, CD = u, EH = 2u, FK = 2u, HK = u, KL = u/2, LM = u, MN = 2u. All the marked angles are right angles, so AB, DC, EH, FK, GL and MN are vertical, and D, O, I, G lie on one horizontal line at height u above BM.
Step 2: Area of ABCD:
AB and DC are parallel (both vertical) with AB = 2u, DC = u, and the distance between them is BC = 2u. Area = \(\tfrac12(2u+u)\times 2u = 3u^2\).
Step 3: Area of DEFG in three pieces:
Triangle DOE: DO = CH = u and OE = EH - OH = 2u - u = u, so its area is \(\tfrac12 u^2\).
Rectangle OEFI: OI = HK = u and EO = FI = u, so its area is \(u^2\).
Triangle FIG: FI = FK - IK = u and IG = KL = u/2, so its area is \(\tfrac12\times u\times\tfrac u2 = \tfrac14 u^2\).
Area of DEFG = \(\tfrac12 u^2 + u^2 + \tfrac14 u^2 = \tfrac74 u^2\).
Step 4: Take the ratio:
\(\dfrac{ABCD}{DEFG} = \dfrac{3u^2}{\tfrac74 u^2} = \dfrac{12}{7}\), that is 12 : 7.
Final Answer:
The correct option is (2): 12 : 7
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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: Jul 4, 2026
- 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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