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Question 1 / 20
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1
Determine the total count of triangles visible within the provided diagram.
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Solution: Step 1: Identify and count the simplest triangles (composed of one component each). There are 8 such triangles: GLK, DLJ, DJM, HMN, QRE, IRA, IPA, and FPO. Step 2: Identify and count triangles composed of two components each. There are 10 such triangles: BDO, CDQ, DLM, PRA, KFI, NEI, HJI, GJI, DKI, and DNI. Step 3: Identify and count triangles composed of four components each. There are 5 such triangles: DIE, DFI, DOA, DQA, and GHI. Step 4: Identify and count triangles composed of six components each. There are 2 such triangles: DCA and DBA. Step 5: Identify and count triangles composed of eight components. There is 1 such triangle: DEF. Step 6: Identify and count triangles composed of twelve components. There is 1 such triangle: ABC. Step 7: Sum all identified triangles: 8 + 10 + 5 + 2 + 1 + 1 = 27.
2
From the presented figures, select the one that is different from the others.
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Solution: Step 1: Examine the shaded portion in each figure. Step 2: Observe figures (2), (3), (4), and (5): In these figures, the lower-right quarter portion of the main figure is shaded. Step 3: Observe figure (1): In this figure, the shaded portion is not the lower-right quarter. Step 4: Conclude that figure (1) is the odd one out because its shaded quadrant differs from the pattern.
3
Choose a suitable figure from the Answer Figures to replace the question mark in the sequence of Problem Figures.
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Solution: Step 1: Observe the overall rotation of the figure. The figure rotates through 90 degrees anti-clockwise. Step 2: Observe the change in the arrowhead: it shifts closer to the black circle. Step 3: Apply these two combined transformations (overall rotation and arrowhead shift) to the figure preceding the question mark to find the correct answer figure.
4
From the set of figures (1), (2), (3), (4), and (5), identify the one that is dissimilar.
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Solution: Step 1: Observe the central pentagon and the 'cup' like shapes surrounding it in each figure. Step 2: For each figure, count how many of these 'cup' shapes are oriented to open towards the central pentagon, and how many open outwards. Step 3: In figures (1), (3), (4), and (5), a consistent pattern is observed: three cups open towards the pentagon, and two cups open outwards. Step 4: In figure (2), this pattern is broken; the number of inward and outward-opening cups is different from the consistent pattern observed in the other figures. Step 5: Therefore, figure (2) is the one that is different from the rest.
5
Following the established arithmetic pattern, determine the number that replaces the question mark.
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Solution: Step 1: Examine the first relationship: Sum the first pair of numbers (6 + 5) = 11. Sum the second pair of numbers (7 + 4) = 11. Subtract the second sum from the first: 11 - 11 = 0. Step 2: Examine the second relationship: Sum the first pair of numbers (7 + 6) = 13. Sum the second pair of numbers (8 + 4) = 12. Subtract the second sum from the first: 13 - 12 = 1. Step 3: The pattern is `(Sum of first two numbers) - (Sum of next two numbers) = result`. Step 4: Apply this rule to the final set: Sum the first pair of numbers (11 + 2) = 13. Sum the second pair of numbers (2 + 0) = 2. Step 5: Subtract the second sum from the first: 13 - 2 = 11. Step 6: Therefore, the missing number is 11.
6
Identify the figure that completes the given sequence.
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Solution: Step 1: Analyze the given sequence of figures. Step 2: Identify the pattern or rule governing the sequence. Step 3: Apply the pattern to select the next figure in the sequence. Step 4: Choose the correct option from the provided choices.
7
Determine the total number of parallelograms present in the given figure.
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Solution: Step 1: Count the simplest parallelograms (composed of one component each). There are 6: ABFE, BCGF, CDHG, EFJI, FGKJ, and GHLK. Step 2: Count parallelograms composed of two components each. There are 7: ACGE, BDHF, EGKI, FHLJ, ABJI, BCKJ, and CDLK. Step 3: Count parallelograms composed of three components each. There are 2: ADHE and EHLI. Step 4: Count parallelograms composed of four components each. There are 2: ACKI and BDLJ. Step 5: Count parallelograms composed of six components. There is 1: ADLI. Step 6: Sum all identified parallelograms: 6 + 7 + 2 + 2 + 1 = 18.
8
Identify the missing figure in the sequence, which follows the same pattern as the given figures.
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Solution: Step 1: Analyze the pattern in the given figures. Step 2: Observe that each figure is transformed by rotating and removing parts. Step 3: Apply the same transformation rules to find the missing figure. Step 4: Based on the pattern, the correct answer is option D.
9
What completes the pattern? 22 * 22 = 16, 33 * 33 = 36, 44 * 44 = ?
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Solution: Step 1: Identify the pattern in the given equations. Step 2: For 22 * 22, sum the digits of 22: 2 + 2 = 4, then square the sum: 4^2 = 16. Step 3: For 33 * 33, sum the digits of 33: 3 + 3 = 6, then square the sum: 6^2 = 36. Step 4: Apply the pattern to 44 * 44: sum the digits of 44: 4 + 4 = 8, then square the sum: 8^2 = 64. Step 5: Therefore, 44 * 44 according to this pattern equals 64.
10
Select the most suitable figure from the Answer Figures that completes the missing pattern indicated by the question mark.
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Solution: Step 1: Observe the transformation from the first problem figure to the second. Step 2: Identify that the initial figure is divided into eight equal parts. Step 3: Apply this division transformation to the figure preceding the question mark to find the correct answer.
11
From the provided figures (1), (2), (3), (4), and (5), select the one that is dissimilar to the others.
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Solution: Step 1: Observe the primary geometric element, which appears to be a triangle, within each figure. Step 2: Pay attention to the orientation or inclination of this triangle in relation to the overall figure or any implied reference axis. Step 3: Notice that in figures (1), (2), (3), and (5), the triangle's inclination or position follows a consistent pattern or can be rotated to match each other. Step 4: Identify that figure (4) has the triangle oriented in a different way compared to the others, making its inclination unique. Step 5: Therefore, figure (4) is the one that is different from the rest.
12
Identify the option figure that fully embeds the given figure (X) as one of its components.
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Solution: Step 1: Analyze Figure (X) to discern its unique shape and components. Step 2: Examine each of the alternative figures (1), (2), (3), (4) one by one. Step 3: Mentally superimpose or trace Figure (X) onto the alternative figures to check for its presence. Step 4: Confirm that one of the alternative figures contains Figure (X) exactly as a part of its structure. Step 5: The alternative figure 3 is found to contain Figure (X) as its part.
13
Two friends, Person A and Person B, start walking from the same point. Person A walks 6 km towards the East, while Person B walks 8 km towards the South. What is the distance between the two friends now?
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Solution: Step 1: Person A walks 6 km East and Person B walks 8 km South. Step 2: Drawing a diagram, we have a right-angled triangle with sides 6 km and 8 km. Step 3: Using Pythagoras' theorem: Distance^2 = 6^2 + 8^2 Step 4: Distance^2 = 36 + 64 Step 5: Distance^2 = 100 Step 6: Distance = 10 km
14
From the Answer Figures, choose the appropriate figure to substitute for the question mark in the Problem Figures.
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Solution: Step 1: Analyze the relationship between the upper element(s) and the lower element(s) in the given problem figures. Step 2: Identify the transformation rule: the upper element is converted into a form similar to the lower elements. Step 3: Simultaneously, observe that each of the lower elements is converted into a form similar to the upper element. Step 4: Apply this reciprocal conversion rule to the figure preceding the question mark to determine the correct answer figure.
15
Determine the missing value at the top of a pyramid given its base and pattern rules.
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Solution: Step 1: Analyze the given pyramid structure and identify the pattern. Step 2: The bottom row of the pyramid is: 3, 1, 4, 2, 5 Step 3: Derive the second row by adding adjacent numbers: 4, 5, 6, 7 Step 4: Derive the third row: 9, 11, 13 Step 5: Derive the fourth row: 20, 24 Step 6: The top number is the sum of the two numbers directly below it: 20 + 28 is not derived from previous steps, re-evaluate the third row to get 10, 12, 14 and then fourth row 24, 28. Step 7: The top number is 24 + 28 = 52.
16
How many rectangles are there in the given diagram?
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Solution: Step 1: Label the figure for systematic counting. Step 2: Count simplest rectangles (1 component): ABJI, BCKJ, IJFG, JKEF (Total: 4). Step 3: Count rectangles composed of two components: ACKI, BCEF, IKEG, ABFG (Total: 4). Step 4: Count rectangle composed of four components: ACEG (Total: 1). Step 5: Sum all counted rectangles: 4 + 4 + 1 = 9 rectangles.
17
An accurate clock displays 8 o'clock in the morning. By how many degrees will the hour hand have rotated when the clock indicates 2 o'clock in the afternoon?
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Solution: Step 1: Calculate the total time duration from 8 a.m. to 2 p.m. Step 2: The duration is 2 p.m. - 8 a.m. = 6 hours. Step 3: Recall the rotation rate of the hour hand: The hour hand completes a full circle (360 degrees) in 12 hours. Step 4: Therefore, the hour hand rotates at a rate of 360 degrees / 12 hours = 30 degrees per hour. Step 5: Calculate the total angle rotated by the hour hand during the 6-hour interval. Step 6: Total angle = 6 hours * 30 degrees/hour = 180 degrees. Step 7: The hour hand will rotate through 180 degrees.
18
Find the missing number in the given sequence: (8+12)-(4+7) = 9, (15+26)-(12+19) = 10, (32+18)-(26+6) = ?
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Solution: Step 1: Analyze the given equations to identify the pattern. Step 2: First equation: (8+12) - (4+7) = 20 - 11 = 9 Step 3: Second equation: (15+26) - (12+19) = 41 - 31 = 10 Step 4: Apply the pattern to the third equation: (32+18) - (26+6) = 50 - 32 = 18 Step 5: Therefore, the missing number in the sequence is 18.
19
Determine the numerical value that replaces the question mark, adhering to the established mathematical pattern.
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Solution: Step 1: Analyze the first pattern: Multiply the first two numbers (18 x 12) = 216. Divide the product by the third number (3): 216 / 3 = 72. Step 2: Analyze the second pattern: Multiply the first two numbers (32 x 16) = 512. Divide the product by the third number (4): 512 / 4 = 128. Step 3: The established pattern is: (first number * second number) / third number = result. Step 4: Apply this rule to the final set. Let the missing number be '?'. So, (24 x 14) / ? = 112. Step 5: Calculate the product: 24 x 14 = 336. Step 6: Set up the equation: 336 / ? = 112. Step 7: Solve for ?: ? = 336 / 112 = 3. Step 8: Thus, the missing number is 3.
20
Determine the missing number that should replace the question mark based on the given pattern.
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Solution: Step 1: Analyze the given examples to identify the underlying pattern. Step 2: From the first example, calculate the sum of the first pair: (16 + 13) = 29. Step 3: Calculate the sum of the second pair: (14 + 15) = 29. Step 4: Observe that the sums are equal, indicating the pattern: (First Number + Second Number) = (Third Number + Fourth Number). Step 5: Apply this pattern to the final set: (29 + 16) = (15 + ?). Step 6: Calculate the sum of the left side: 29 + 16 = 45. Step 7: Set up the equation: 45 = 15 + ?. Step 8: Solve for ?: ? = 45 - 15 = 30. Step 9: The missing number is 30.
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