UPSC CAPF Archives

21. In the above diagram square represents boys, circle represents the tal

In the above diagram square represents boys, circle represents the tall persons, triangle represents tennis players, and rectangle represents the swimmers. Which one of the following numbers represents tall boys who are swimmers, but don’t play tennis ?

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This question was previously asked in

UPSC CAPF – 2009

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The correct option is B) 3.

The diagram uses shapes to represent different groups of people:
Square represents boys.
Circle represents tall persons.
Triangle represents tennis players.
Rectangle represents swimmers.
We are looking for the number representing “tall boys who are swimmers, but don’t play tennis”. This translates to the region that is the intersection of the Circle (tall persons), the Square (boys), and the Rectangle (swimmers), while being outside the Triangle (tennis players).
In set notation: (Circle ∩ Square ∩ Rectangle) \ Triangle.
Looking at the standard Venn diagram labelling for 4 sets using these shapes (as found in the source CSAT 2015 paper), the region representing (Circle ∩ Square ∩ Rectangle) \ Triangle is labelled with the number ‘2’.
However, the options provided are A) 4, B) 3, C) 6, D) 5. The number 2 is not among the options.
Based on the official answer key for CSAT 2015 Set B, the answer to this question (Question 18) is B, which corresponds to the number 3.
The region labelled ‘3’ in the diagram represents the intersection of the Square, Circle, and Triangle, but outside the Rectangle. This corresponds to “Boys, Tall, and Tennis players, but NOT Swimmers”. This contradicts the criteria given in the question (“swimmers, but don’t play tennis”).
There appears to be an inconsistency between the question criteria, the diagram labeling, and the official answer key. However, following the provided correct option, we state that 3 is the answer, acknowledging that this does not logically follow from the diagram based on the stated criteria. Assuming the official key is correct despite the apparent conflict, the answer is 3. *Note: Based on the standard interpretation of the diagram and the question text, region 2 (not option 2) represents the desired group.* Given the discrepancy, providing a step-by-step logical derivation to arrive at option B (3) from the problem statement is not possible without assuming an error in the question or diagram.

This is a standard type of Venn diagram problem involving multiple overlapping categories. The key is to identify the region that satisfies all the given conditions (inclusion in certain sets, exclusion from others). The conflict between the problem statement and the provided answer highlights a potential issue with the question itself in the source material.

22. From the above graph, who out of the four persons A, B, C and D, saves

From the above graph, who out of the four persons A, B, C and D, saves the least percentage of his monthly income ?

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UPSC CAPF – 2009

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The correct option is C) C.

To find who saves the least percentage of his monthly income, we need to calculate the saving percentage for each person (A, B, C, and D). The percentage of saving is calculated as (Savings / Income) * 100%.
From the bar graph:
Person A: Income = 1000, Savings = 400. Saving Percentage = (400 / 1000) * 100% = 0.4 * 100% = 40%.
Person B: Income = 1500, Savings = 500. Saving Percentage = (500 / 1500) * 100% = (1/3) * 100% ≈ 33.33%.
Person C: Income = 2000, Savings = 600. Saving Percentage = (600 / 2000) * 100% = (6/20) * 100% = (3/10) * 100% = 30%.
Person D: Income = 2500, Savings = 800. Saving Percentage = (800 / 2500) * 100% = (8/25) * 100% = 0.32 * 100% = 32%.
Comparing the percentages: A=40%, B≈33.33%, C=30%, D=32%.
The least saving percentage is 30%, which corresponds to Person C.

This is a data interpretation question requiring calculation of percentages from a bar graph. The bar graph visually represents two data series (Income and Savings) for different categories (Persons A, B, C, D). Calculating the percentage of one value relative to another is a common type of quantitative reasoning task.

23. Distribution of work hours in a factory is shown in the below given ta

Distribution of work hours in a factory is shown in the below given table :
Number of workers | Number of hours worked
—|—
20 | 45—50
15 | 40—44
25 | 35—39
16 | 30—34
04 | 00—29
What is the percentage of workers worked for 40 or more hours ?

33.33

43.75

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The correct option is D) 43.75.

To find the percentage of workers who worked for 40 or more hours, first, we need to find the total number of workers.
Total number of workers = Sum of workers in all categories = 20 + 15 + 25 + 16 + 04 = 80 workers.
Next, identify the categories of workers who worked for 40 or more hours. These are the categories “40—44” and “45—50”.
Number of workers in the “40—44” category = 15.
Number of workers in the “45—50” category = 20.
Total number of workers who worked for 40 or more hours = 15 + 20 = 35 workers.
Finally, calculate the percentage:
Percentage = (Number of workers who worked 40+ hours / Total number of workers) * 100
Percentage = (35 / 80) * 100
Percentage = (35/80) * 100 = (7/16) * 100
Percentage = 700 / 16 = 350 / 8 = 175 / 4 = 43.75.

This is a simple data interpretation problem from a frequency table. It requires calculating a part-to-whole percentage. The categories in the table are class intervals for the number of hours worked.

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24. Consider the following statements: Every square is a rectangle. Eve

Consider the following statements:

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Every square is a rectangle.
Every rectangle is a parallelogram.
Every parallelogram is not necessarily a square.

Which one of the following conclusions can be drawn on the basis of the above statements ?

All parallelograms are either squares or rectangles.

A non-parallelogram figures cannot be either a square or a rectangle.

All rectangles are either squares or parallelograms.

Squares and rectangles are the only parallelograms.

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The correct option is B) A non-parallelogram figures cannot be either a square or a rectangle.

Let the statements be:
1. Every square is a rectangle. (Square -> Rectangle)
2. Every rectangle is a parallelogram. (Rectangle -> Parallelogram)
3. Every parallelogram is not necessarily a square. (Parallelogram -/> Square)

From statement 1 and 2, we can form a chain: Square -> Rectangle -> Parallelogram.
This implies that every square is a parallelogram.

Now let’s analyze the conclusions:
A) All parallelograms are either squares or rectangles. This is false. A rhombus is a parallelogram but is not necessarily a square or a rectangle.
B) A non-parallelogram figures cannot be either a square or a rectangle. This is true.
From Rectangle -> Parallelogram, the contrapositive is (Not Parallelogram) -> (Not Rectangle).
From Square -> Rectangle, the contrapositive is (Not Rectangle) -> (Not Square).
Combining these, if a figure is not a parallelogram, then it is not a rectangle. If it is not a rectangle, then it is not a square. Therefore, if a figure is a non-parallelogram, it cannot be a rectangle, and thus cannot be a square. This conclusion is logically derived from the first two statements.
C) All rectangles are either squares or parallelograms. This statement is technically true because all rectangles *are* parallelograms (Statement 2), and the set of squares is a subset of rectangles. So, a rectangle is always in the set of parallelograms, and sometimes in the set of squares. However, the wording “either…or” often implies mutual exclusion or at least that being a parallelogram is not always the case for a rectangle, which contradicts statement 2. Conclusion B is a more direct and stronger inference from the combined statements.
D) Squares and rectangles are the only parallelograms. This is false. Rhombuses and other non-rectangular parallelograms exist.

Conclusion B is the only valid and direct conclusion that can be drawn based on the transitive property implied by the statements.

This question tests understanding of the hierarchy of quadrilaterals. The relationships are: Square ⊂ Rhombus, Square ⊂ Rectangle, Rhombus ⊂ Parallelogram, Rectangle ⊂ Parallelogram. Square is the most specific type, being a rectangle with equal sides, and a rhombus with right angles. Parallelogram is a broader category.

25. Which is the next figure in the sequence given above ?

Which is the next figure in the sequence given above ?

(figure represented by (a))

(figure represented by (b))

(figure represented by (c))

(figure represented by (d))

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The correct option is A) (figure represented by (a)).

Observe the sequence of figures:
Figure 1: Square, axis-aligned, filled with horizontal lines.
Figure 2: Square, axis-aligned, filled with vertical lines.
Figure 3: Square, rotated 45 degrees (diamond shape), filled with lines that were horizontal in the original orientation.
Figure 4: Square, rotated 45 degrees (diamond shape), filled with lines that were vertical in the original orientation.
The pattern alternates between axis-aligned and 45-degree rotated squares, and within each pair, it alternates between horizontal and vertical lines (relative to the axis-aligned orientation).
Sequence: (Axis-aligned, Horizontal) -> (Axis-aligned, Vertical) -> (Rotated, Horizontal) -> (Rotated, Vertical).
Following this pattern, the next figure should be (Axis-aligned, Horizontal).
Looking at the options (a), (b), (c), (d) which represent figures:
Figure (a) is an axis-aligned square filled with horizontal lines. This matches the expected next figure in the sequence.
Figure (b) is an axis-aligned square filled with vertical lines.
Figure (c) is a rotated square filled with lines parallel to one of its sides.
Figure (d) is a rotated square filled with diagonal lines.
Therefore, option A represents the figure that follows the pattern.

Figure sequence problems test pattern recognition skills. The patterns can involve rotation, reflection, scaling, changes in filling, changes in number of elements, or alternation of attributes. In this case, the pattern combines rotation and internal filling style.

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26. The below Venn diagram shows a city population on which read three pop

The below Venn diagram shows a city population on which read three popular daily newspapers Hindustan Times (HT), The Times of India (TI) and Navbharat Times (NT) :
If a person is randomly selected from the city population and it is found that he reads at least one of the three newspapers then the person belongs to which part of the population ?

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a + b + c

P-h

P-g

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The correct option is C) P-h.

The Venn diagram represents the population (P) of a city and three sets: Hindustan Times (HT), The Times of India (TI), and Navbharat Times (NT). Each region within the diagram represents a subset of the population based on which newspapers they read. The regions are typically labelled:
a: Reads HT only
b: Reads HT and TI only
c: Reads TI only
d: Reads HT and NT only
e: Reads HT, TI, and NT
f: Reads TI and NT only
g: Reads NT only
h: Reads none of the three newspapers (outside all circles)
The question asks to identify the part of the population that reads “at least one of the three newspapers”. This corresponds to the union of the three sets (HT ∪ TI ∪ NT). In terms of the labelled regions, this union includes all regions within the three circles: a + b + c + d + e + f + g. The total population is represented by P, which includes all regions inside and outside the circles (a + b + c + d + e + f + g + h). Therefore, the population that reads at least one newspaper is the total population (P) minus the population that reads none (h). This is represented as P – h.

In set theory terms, if A, B, and C are sets representing readers of HT, TI, and NT respectively, the set of people who read at least one newspaper is the union A ∪ B ∪ C. If U is the universal set representing the total population, and h represents the set of people who read none (U \ (A ∪ B ∪ C)), then A ∪ B ∪ C = U \ h.

27. The following figure shows the displacement time (x-t) graph of a body

The following figure shows the displacement time (x-t) graph of a body in motion. The ratio of the speed in first second and that in next two seconds is :

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1 : 2

1 : 3

3 : 1

2 : 1

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The correct option is D) 2 : 1.

The speed of the body is given by the magnitude of the slope of the displacement-time (x-t) graph.
In the first second (from t=0 to t=1), the displacement changes from 0 to 3. The velocity is `(3 – 0) / (1 – 0) = 3/1 = 3` units/second. The speed in the first second (v1) is `|3| = 3`.
In the next two seconds (from t=1 to t=3), the displacement changes from 3 to 0. The velocity is `(0 – 3) / (3 – 1) = -3/2` units/second. The speed in the next two seconds (v2) is `|-3/2| = 3/2`.
The ratio of the speed in the first second and that in the next two seconds is `v1 : v2 = 3 : (3/2)`. To express this ratio in integers, we can multiply both parts by 2: `(3 * 2) : (3/2 * 2) = 6 : 3`. Dividing by 3, we get the simplified ratio `2 : 1`.

The displacement-time graph provides information about the position of the body over time. Velocity is the rate of change of displacement, and speed is the magnitude of velocity. A positive slope indicates movement in one direction, while a negative slope indicates movement in the opposite direction. In this graph, the body moves from x=0 to x=3 in the first second and then back from x=3 to x=0 in the subsequent two seconds.

28. A square is drawn inside the circle as shown in the figure above. If t

A square is drawn inside the circle as shown in the figure above. If the area of the shaded portion is 32/7 units then the radius of the circle is :

√2 units

2 units

3 units

4 units

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The correct option is B) 2 units.

The problem involves a square inscribed in a circle. Let the radius of the circle be `r`. The diameter of the circle is `2r`. When a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. Let the side of the square be `s`. The diagonal of the square is `s√2`. Thus, `s√2 = 2r`, which means `s = 2r/√2 = r√2`. The area of the square is `s² = (r√2)² = 2r²`. The area of the circle is `πr²`. The shaded portion is the area of the circle minus the area of the square. Given the area of the shaded portion is 32/7 units, we have `πr² – 2r² = 32/7`. Factoring out `r²`, we get `r²(π – 2) = 32/7`. Using the approximation `π ≈ 22/7`, we have `r²(22/7 – 2) = 32/7`. This simplifies to `r²((22 – 14)/7) = 32/7`, which is `r²(8/7) = 32/7`. Multiplying both sides by 7/8, we get `r² = (32/7) * (7/8) = 32/8 = 4`. Taking the square root, `r = √4 = 2` (since radius must be positive). The radius of the circle is 2 units.

For a square inscribed in a circle of radius `r`, the side length is `r√2` and the area is `2r²`. The ratio of the area of the inscribed square to the area of the circle is `2r² / (πr²) = 2/π`. The shaded area calculation relies on the difference between the circle’s area and the square’s area.

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29. A person moves along a circular path by a distance equal to half the c

A person moves along a circular path by a distance equal to half the circumference in a given time. The ratio of his average speed to his average velocity is :

0.5

0.5π

0.75π

1.0

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The correct option is B.

Let the circular path have radius R. The circumference is $C = 2\pi R$.
The distance covered by the person is half the circumference, $d = \frac{1}{2} C = \pi R$.
Let the time taken be $t$.
Average speed is defined as the total distance traveled divided by the total time taken.
Average speed = $\frac{d}{t} = \frac{\pi R}{t}$.

The person moves along a circular path by a distance equal to half the circumference. This means the person starts at one point on the circle and ends at the diametrically opposite point.
Let the starting point be A and the ending point be B, where AB is a diameter of the circle.
The displacement is the shortest straight-line distance from the initial position to the final position. In this case, the displacement is the length of the diameter.
Displacement = $2R$.

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Average velocity is defined as the total displacement divided by the total time taken.
Average velocity = $\frac{\text{Displacement}}{t} = \frac{2R}{t}$.

The ratio of average speed to average velocity is:
Ratio = $\frac{\text{Average speed}}{\text{Average velocity}} = \frac{\pi R / t}{2R / t} = \frac{\pi R}{t} \times \frac{t}{2R} = \frac{\pi}{2}$.
The value $\frac{\pi}{2}$ is equivalent to $0.5\pi$.

This question highlights the difference between speed (scalar, based on distance) and velocity (vector, based on displacement). Distance is the path length, while displacement is the change in position vector. For motion along a curved path, the distance is generally greater than the magnitude of the displacement. For a half circle, the distance is $\pi R$ and the displacement magnitude is $2R$.

30. From the following Venn diagram identify the number of persons who are

From the following Venn diagram identify the number of persons who are either good speakers or post graduates or doctors.

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The correct option is D.

The Venn diagram shows the number of persons in different categories and their overlaps.
Good Speakers (S) circle includes regions with numbers 3, 2, 4, 1.
Post Graduates (P) circle includes regions with numbers 5, 2, 4, 3.
Doctors (D) circle includes regions with numbers 4, 1, 3, 4.

The question asks for the number of persons who are either good speakers or post graduates or doctors. This corresponds to the union of the three sets (S $\cup$ P $\cup$ D).
The number of elements in the union of sets is the sum of the numbers in all the distinct regions covered by the circles.
These regions are:
– Only Good Speakers (S only): 3
– Only Post Graduates (P only): 5
– Only Doctors (D only): 4
– Good Speakers and Post Graduates only (S $\cap$ P only): 2
– Good Speakers and Doctors only (S $\cap$ D only): 1
– Post Graduates and Doctors only (P $\cap$ D only): 3
– Good Speakers, Post Graduates, and Doctors (S $\cap$ P $\cap$ D): 4

Total number of persons in the union = Sum of numbers in all these regions
Total = 3 + 5 + 4 + 2 + 1 + 3 + 4 = 22.

The question asks for the size of the union of the three sets. The Venn diagram provides the size of each disjoint region formed by the intersections of the sets. Summing the numbers in all regions within the circles gives the total number of elements in the union.

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