UPSC CAPF Archives

21. Which one of the following statements about blood elements is correct?

Which one of the following statements about blood elements is correct?

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Red blood cells’ number is same as that of white blood cells and platelets.

The number of white blood cells is more than red blood cells.

The number of red blood cells is the highest.

The number of red blood cells is lower than platelets.

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UPSC CAPF – 2018

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Blood is composed of plasma and blood cells (also called formed elements). The main types of blood cells are Red Blood Cells (RBCs or erythrocytes), White Blood Cells (WBCs or leukocytes), and Platelets (thrombocytes). Their relative numbers per unit volume of blood are vastly different. RBCs are by far the most numerous, with typical counts in millions per microliter (around 4-6 million/µL). Platelets are the second most numerous, with counts in the hundreds of thousands per microliter (around 150,000-450,000/µL). WBCs are the least numerous, with counts in the thousands per microliter (around 4,000-11,000/µL). Therefore, the number of red blood cells is the highest among the blood elements.

The relative abundance of formed elements in blood follows the order: Red Blood Cells > Platelets > White Blood Cells.

Each type of blood cell has distinct functions: RBCs transport oxygen, WBCs are involved in the immune system, and Platelets are crucial for blood clotting. Abnormal counts of any of these can indicate various health conditions.

22. On simplification the product (x + y)(x² + y²)(x⁴ + y⁴)(x⁸ + y⁸) … h

On simplification the product (x + y)(x² + y²)(x⁴ + y⁴)(x⁸ + y⁸) … how many such terms are there which will have only single x and rest ys?

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UPSC CAPF – 2018

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The product is (x + y)(x² + y²)(x⁴ + y⁴)(x⁸ + y⁸)… Let’s consider a product of n terms: P_n = (x^2^0 + y^2^0)(x^2^1 + y^2^1)…(x^2^(n-1) + y^2^(n-1)). When expanding this product, each term is formed by choosing either the first term (x raised to a power of 2) or the second term (y raised to the same power of 2) from each bracket. For example, from the i-th bracket (x^2^i + y^2^i), we pick either x^2^i or y^2^i. A term in the expansion is the product of one chosen term from each of the n brackets. We are looking for terms with a “single x”, which means the power of x in the term must be 1 (x¹). Let’s say we pick x^2^i from one bracket and y^2^j from all other brackets (j ≠ i). The resulting term is (x^2^i) * (product of y^2^j for all j ≠ i). The power of x in this term is 2^i. For this power to be 1 (x¹), we must have 2^i = 1. This occurs only when i = 0. So, the only way to get a term with x¹ is to choose x^2^0 = x from the first bracket (x+y) and y^2^j from all subsequent brackets (j = 1, 2, …, n-1). The resulting term is x * y² * y⁴ * y⁸ * … * y^(2^(n-1)) = x * y^(2¹ + 2² + … + 2^(n-1)). The power of y is the sum of a geometric series: 2(2^(n-1) – 1)/(2-1) = 2^n – 2. The term is x¹y^(2^n – 2). This is the only term in the expansion that has x raised to the power of 1. Any other combination will result in x raised to a power other than 1 (e.g., choosing x from multiple brackets, or choosing x^2^i where i > 0 results in a power 2^i > 1). Thus, there is only exactly one such term. The number of such terms is 1.

In the expansion of a product of factors of the form (x^a + y^a), terms are formed by selecting one element from each factor and multiplying them. To find terms with a specific power of x, analyze the possible combinations of selections from each factor.

This type of product (x+y)(x²+y²)(x⁴+y⁴)… is related to the identity (x-y)(x+y) = x²-y², (x²-y²)(x²+y²) = x⁴-y⁴, and generally (x^2^n – y^2^n)(x^2^n + y^2^n) = x^2^(n+1) – y^2^(n+1). If the product were multiplied by (x-y), it would telescope nicely to x^(2^n) – y^(2^n). However, the question asks about the terms in the expansion of the product itself.

23. In an examination, 53% students passed in Mathematics, 61% passed in P

In an examination, 53% students passed in Mathematics, 61% passed in Physics, 60% passed in Chemistry, 24% passed in Mathematics and Physics, 35% in Physics and Chemistry, 27% in Mathematics and Chemistry and 5% in none. The ratio of percentage of passes in Mathematics and Chemistry but not in Physics in relation to the percentage of passes in Physics and Chemistry but not in Mathematics is:

7:5

5:7

4:5

5:4

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UPSC CAPF – 2018

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Let M, P, and C represent the sets of students who passed in Mathematics, Physics, and Chemistry, respectively. We are given the following percentages: |M|=53, |P|=61, |C|=60, |M∩P|=24, |P∩C|=35, |M∩C|=27, |None|=5. The percentage of students who passed in at least one subject is |M∪P∪C| = 100 – |None| = 100 – 5 = 95%. Using the principle of inclusion-exclusion: |M∪P∪C| = |M| + |P| + |C| – |M∩P| – |M∩C| – |P∩C| + |M∩P∩C|. So, 95 = 53 + 61 + 60 – 24 – 27 – 35 + |M∩P∩C|. 95 = 174 – 86 + |M∩P∩C|. 95 = 88 + |M∩P∩C|. Thus, |M∩P∩C| = 95 – 88 = 7%.
Percentage of passes in Mathematics and Chemistry but not in Physics is the region (M∩C) excluding the triple intersection (M∩C∩P). This is |M∩C| – |M∩C∩P| = 27% – 7% = 20%.
Percentage of passes in Physics and Chemistry but not in Mathematics is the region (P∩C) excluding the triple intersection (M∩P∩C). This is |P∩C| – |M∩P∩C| = 35% – 7% = 28%.
The ratio of the percentage of passes in Mathematics and Chemistry but not in Physics to the percentage of passes in Physics and Chemistry but not in Mathematics is 20 : 28. Simplifying the ratio by dividing by 4, we get 5 : 7.

This problem requires using the principle of inclusion-exclusion for three sets and calculating the percentages corresponding to specific regions in a Venn diagram (intersections of two sets excluding the third).

Using a Venn diagram helps visualize the different regions. The percentage of students in M only is |M| – (|M∩P| + |M∩C|) + |M∩P∩C|. Similar calculations can be done for P only, C only, M∩P only, P∩C only, and M∩C only. The sum of percentages in all 7 regions plus the ‘none’ percentage should equal 100%.

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24. An equilateral triangle is inscribed in a circle of radius 1 unit. The

An equilateral triangle is inscribed in a circle of radius 1 unit. The area of the shaded region, in square unit, is:

π/3 - √3/4

π/3 - √3/2

π - 3

π - 3/4

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This question was previously asked in

UPSC CAPF – 2018

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An equilateral triangle inscribed in a circle of radius 1 unit has sides that subtend a central angle of 360°/3 = 120°. The area of a sector of the circle corresponding to this angle is (120°/360°) * π * r² = (1/3) * π * 1² = π/3. The area of the triangle formed by the center of the circle and one side of the equilateral triangle is (1/2) * r * r * sin(120°) = (1/2) * 1 * 1 * (√3/2) = √3/4. The region between the arc and the chord (one segment of the circle) has an area equal to the area of the sector minus the area of this triangle: π/3 – √3/4. Assuming the shaded region in the missing diagram refers to one such segment, this is the correct area. If the shaded area referred to the total area of the circle outside the triangle (three segments), the area would be 3 * (π/3 – √3/4) = π – 3√3/4, which is not among the options. Therefore, it is highly likely that the shaded region depicted was one segment.

For an equilateral triangle inscribed in a circle, each side subtends a central angle of 120°. The area of a circular segment formed by a chord and its corresponding arc is the area of the sector minus the area of the triangle formed by the radii and the chord.

The side length of an equilateral triangle inscribed in a circle of radius R is R√3. The area of an equilateral triangle with side ‘a’ is (√3/4)a². For R=1, side a=√3, Area of triangle = (√3/4)(√3)² = 3√3/4. Area of circle = πR² = π. Area outside triangle = π – 3√3/4. The question is solvable only by assuming the shaded region is one segment, which is a common type of shaded area in such diagrams.

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25. Which one of the following is the area of a sector of a circle of radi

Which one of the following is the area of a sector of a circle of radius 10 cm formed by an arc length of 15 cm?

10π cm²

15π cm²

75 cm²

150 cm²

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This question was previously asked in

UPSC CAPF – 2018

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The area of the sector is 75 cm².

The area of a sector of a circle can be calculated using the formula:
Area = (1/2) * r * L
where ‘r’ is the radius of the circle and ‘L’ is the length of the arc that forms the sector.
Given:
Radius (r) = 10 cm
Arc length (L) = 15 cm
Area = (1/2) * 10 cm * 15 cm
Area = 5 cm * 15 cm
Area = 75 cm²

Alternatively, the area of a sector can be calculated using the formula Area = (θ/360°) * πr², where θ is the central angle in degrees. To use this formula, one would first need to find the angle θ using the relationship L = rθ (where θ is in radians), or L = (θ/360°) * 2πr (where θ is in degrees). Using L = rθ, 15 cm = 10 cm * θ, so θ = 1.5 radians. Then, Area = (1/2) * r² * θ = (1/2) * (10 cm)² * 1.5 radians = (1/2) * 100 * 1.5 cm² = 50 * 1.5 cm² = 75 cm². Both methods yield the same result.

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26. Springs are common in which of the following areas? 1. Well jointed

Springs are common in which of the following areas?

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1. Well jointed rocks
2. Arid areas with underlying rocks
3. Karst topography
4. Tilted strata

Select the correct answer using the code given below.

1 and 3 only

1, 3 and 4

2 and 4

3 and 4 only

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UPSC CAPF – 2018

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Springs are common in areas with well-jointed rocks, karst topography, and tilted strata.

Springs occur where groundwater flows out onto the surface. This typically happens in areas where geological conditions facilitate the movement and emergence of groundwater.
1. **Well-jointed rocks:** Fractures and joints act as conduits for groundwater flow, allowing water to reach the surface. (Correct)
2. **Arid areas with underlying rocks:** Arid conditions generally mean low precipitation, low groundwater recharge, and deep water tables, making springs less common than in humid regions. (Incorrect)
3. **Karst topography:** Limestone or other soluble rocks are dissolved by groundwater, creating caves, sinkholes, and extensive underground drainage systems. Springs, often large ones (karst springs), are characteristic features of karst landscapes as groundwater emerges from these systems. (Correct)
4. **Tilted strata:** Layers of permeable rock (aquifers) between impermeable layers can become confined aquifers when tilted. If the impermeable upper layer is breached (e.g., by erosion or a fault), the pressurized water in the aquifer can rise to the surface as an artesian spring. (Correct)

The location and flow rate of springs are influenced by factors such as precipitation, aquifer properties, land use, and geological structures.

27. In mid-latitude regions, the diurnal variation in weather is due to

In mid-latitude regions, the diurnal variation in weather is due to

raising air/convection

sinking air/conduction

advection

radiation

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UPSC CAPF – 2018

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In mid-latitude regions, the primary driver of diurnal variation in weather is radiation.

Diurnal variation refers to the changes in weather elements (like temperature, humidity, wind, cloudiness) that occur over a 24-hour cycle. This cycle is fundamentally driven by the daily pattern of incoming solar radiation and outgoing terrestrial radiation. During the day, solar radiation heats the surface and atmosphere; at night, the Earth cools by emitting long-wave radiation. This daily heating and cooling cycle directly causes the diurnal temperature range and indirectly influences other diurnal patterns like convection (daytime heating leads to rising air and potential thunderstorms) or fog formation (nighttime cooling leads to condensation). While convection, conduction, and advection play roles in weather, radiation is the primary cause of the day-night difference that defines diurnal variation.

Advection, the horizontal movement of air masses, is a major cause of *weather changes* over periods longer than a day (synoptic scale), but the recurring daily cycle is primarily governed by the Earth’s rotation relative to the sun and the resulting radiative balance.

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28. Which one of the following processes is not a part of long-wave

Which one of the following processes is not a part of long-wave radiation?

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Conduction

Scattering

Convection

Radiation

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UPSC CAPF – 2018

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Scattering is not typically considered a part of the processes fundamentally defining or transferring energy as long-wave radiation in the Earth’s energy budget, unlike Conduction, Convection, and Radiation (emission/absorption).

Long-wave radiation refers to thermal radiation emitted by the Earth’s surface and atmosphere, primarily in the infrared spectrum. The transfer of this thermal energy within the Earth system involves processes like:
– **Radiation:** Emission and absorption of long-wave electromagnetic waves.
– **Conduction:** Heat transfer through direct contact, significant at the Earth-atmosphere interface.
– **Convection:** Heat transfer through the movement of air or water, driven by temperature differences resulting from radiative heating/cooling.
Scattering is a process where radiation is deflected in different directions by particles or molecules. While radiation of all wavelengths can be scattered, scattering is a dominant process affecting the path of incoming *short-wave* solar radiation (e.g., causing the blue sky). It is not a primary mechanism for the *transfer* or *distribution* of the heat energy associated with outgoing *long-wave* radiation in the same way as conduction, convection, and emission/absorption of radiation are.

The Earth’s energy balance involves the absorption of incoming short-wave solar radiation and the emission of outgoing long-wave terrestrial radiation. Conduction and convection are non-radiative heat transfer mechanisms that move heat absorbed from radiation within the atmosphere and surface.

29. Gersoppa Falls is formed by which one of the following rivers?

Gersoppa Falls is formed by which one of the following rivers?

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Lohit

Tungabhadra

Sharavati

Krishna

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Gersoppa Falls, also known as Jog Falls, is formed by the Sharavati river.

Jog Falls (Gersoppa Falls) is a major waterfall located in the Shimoga district of Karnataka, India. It is created by the Sharavati River dropping from a height in four distinct cascades.

Jog Falls is one of the highest plunge waterfalls in India. The Sharavati River is a westward-flowing river that originates and flows entirely within the state of Karnataka.

30. The Tropic of Cancer passes through which one of the following

The Tropic of Cancer passes through which one of the following countries?

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North Sudan

Chad

Mali

Yemen

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The Tropic of Cancer passes through Mali among the given options.

The Tropic of Cancer is a line of latitude approximately 23.5° North of the Earth’s Equator. It marks the northernmost latitude at which the sun can appear directly overhead. Among the countries listed, the Tropic of Cancer passes through the northern part of Mali. While it also passes through northern Sudan, the options require selecting only one country. Mali is a well-documented country crossed by this line.

Other countries the Tropic of Cancer passes through include Mexico, the Bahamas, Mauritania, Niger, Algeria, Libya, Egypt, Saudi Arabia, UAE, Oman, India, Bangladesh, Myanmar, China, and Taiwan.