2016 Archives

161. There are two boxes. Box I contains one white card and two black cards

There are two boxes. Box I contains one white card and two black cards and Box II contains one white card and a black card. Two persons P and Q play a game. P picks a card randomly from Box I. If P finds the white card, P wins and the game stops. If P finds the black card, Q draws a card randomly from Box II. If Q finds the white card, Q wins. The game stops whether Q draws the white card or the black card. Which one of the following is correct?

[amp_mcq option1=”If P loses, Q wins” option2=”If Q loses, P wins” option3=”Both P and Q may win” option4=”Both P and Q may lose” correct=”option4″]

This question was previously asked in

UPSC CAPF – 2016

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The correct option is D. The game setup allows for a scenario where neither P nor Q wins.

The game proceeds as follows:
1. P draws a card from Box I (1 White, 2 Black).
* If P draws White (probability 1/3), P wins and the game stops.
* If P draws Black (probability 2/3), P loses this step, and Q draws from Box II.
2. If P drew Black, Q draws a card from Box II (1 White, 1 Black).
* If Q draws White (probability 1/2), Q wins. The game stops.
* If Q draws Black (probability 1/2), Q loses. The game stops.

Let’s analyze the outcomes:
* Outcome 1: P draws White (Prob=1/3). P wins.
* Outcome 2: P draws Black (Prob=2/3) AND Q draws White (Prob=1/2). Q wins. Probability = (2/3)*(1/2) = 1/3.
* Outcome 3: P draws Black (Prob=2/3) AND Q draws Black (Prob=1/2). Neither P nor Q wins. Probability = (2/3)*(1/2) = 1/3.

Now let’s evaluate the options:
A) If P loses, Q wins: P loses if P draws Black. If P draws Black, Q draws from Box II. Q wins *only if* Q draws White. Q does *not* win if Q draws Black. So this statement is not always correct.
B) If Q loses, P wins: Q only plays if P loses (draws Black). If Q loses (draws Black), it means P already lost the first draw. P’s winning condition is drawing White in the *first* step. If Q gets to play and then loses, P cannot win *in that game instance*. So this statement is incorrect.
C) Both P and Q may win: In a single game instance, either P wins (game stops), or P loses and Q plays. If Q plays, either Q wins or neither wins. P and Q cannot both win in the same game. So this statement is incorrect.
D) Both P and Q may lose: This happens in Outcome 3, where P draws Black and Q draws Black. In this scenario, P did not win (as P drew Black) and Q did not win (as Q drew Black). This is a possible outcome with probability 1/3. So this statement is correct.

The total probability of winning for P is 1/3. The total probability of winning for Q is 1/3. The probability that neither wins is 1/3. The sum of probabilities is 1/3 + 1/3 + 1/3 = 1.

162. In a school there are three batches of players who play cricket, footb

In a school there are three batches of players who play cricket, football and hockey. An incomplete chart of number of students playing an individual sport belonging to an individual batch is displayed in the following table :

	Batch I	Batch III	Total
Cricket players		8	14
Football players		10	16
Hockey players	6	6	17
Total

Which one of the following is correct ?

[amp_mcq option1=”Batch II is empty” option2=”Batch I and Batch II do not have equal number of students” option3=”Batch I and Batch III can have equal number of students” option4=”Batch II and Batch III can have equal number of students” correct=”option2″]

This question was previously asked in

UPSC CAPF – 2016

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The correct option is B. By completing the table using the given row and column totals, we can determine the number of students in each batch and then evaluate the statements.

The given information is:
– Cricket: Batch I + Batch II + 8 = 14 => Batch I + Batch II = 6
– Football: Batch I + Batch II + 10 = 16 => Batch I + Batch II = 6
– Hockey: 6 + Batch II + 6 = 17 => Batch II = 17 – 12 = 5

Using Batch II = 5 in the first two equations:
– Batch I + 5 = 6 => Batch I = 1
– Batch I + 5 = 6 => Batch I = 1 (Consistent)

The number of students in each batch is the sum of players across sports:
– Batch I Total: Cricket (1) + Football (1) + Hockey (6) = 8
– Batch II Total: Cricket (5) + Football (5) + Hockey (5) = 15
– Batch III Total: Cricket (8) + Football (10) + Hockey (6) = 24
– Grand Total: 8 + 15 + 24 = 47 (Also 14 + 16 + 17 = 47)

Now, evaluate the options:
A) Batch II is empty (Batch II has 15 students) – False
B) Batch I and Batch II do not have equal number of students (Batch I = 8, Batch II = 15. 8 != 15) – True
C) Batch I and Batch III can have equal number of students (Batch I = 8, Batch III = 24. They are not equal) – False
D) Batch II and Batch III can have equal number of students (Batch II = 15, Batch III = 24. They are not equal) – False

The completed table is:

	Batch I	Batch II	Batch III	Total
Cricket players	1	5	8	14
Football players	1	5	10	16
Hockey players	6	5	6	17
Total	8	15	24	47

163. Which one of the following is the wrong number in the series 6, 14, 30

Which one of the following is the wrong number in the series 6, 14, 30, 64, 126 ?

[amp_mcq option1=”126″ option2=”64″ option3=”14″ option4=”6″ correct=”option2″]

This question was previously asked in

UPSC CAPF – 2016

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The correct option is B) 64.

We need to identify the pattern in the given series of numbers: 6, 14, 30, 64, 126.

Let the series be denoted by T(n), where n=1, 2, 3, …
T(1) = 6
T(2) = 14
T(3) = 30
T(4) = 64
T(5) = 126

Let’s examine the relationship between consecutive terms.
T(2) = 14. Is there a relation to T(1)=6? 6 * 2 + 2 = 12 + 2 = 14.
Let’s test this pattern for the next term: T(n+1) = 2 * T(n) + 2.
T(3) = 2 * T(2) + 2 = 2 * 14 + 2 = 28 + 2 = 30. This matches the given T(3).
Now let’s test this pattern for the fourth term:
Expected T(4) = 2 * T(3) + 2 = 2 * 30 + 2 = 60 + 2 = 62.
However, the given fourth term is 64. This indicates that 64 might be the wrong number.

Let’s assume the expected T(4) (which is 62) was correct and test the pattern for the fifth term:
Expected T(5) = 2 * (Expected T(4)) + 2 = 2 * 62 + 2 = 124 + 2 = 126.
This matches the given T(5).

So, the pattern T(n+1) = 2 * T(n) + 2 holds for all terms except for the fourth term, which should be 62 according to the pattern, but is given as 64. Therefore, 64 is the wrong number in the series.

164. A triangle is formed with vertices (0, 0), (0, 100) and (100, 100). Wh

A triangle is formed with vertices (0, 0), (0, 100) and (100, 100). What is the number of points inside the triangle with integer coordinates?

[amp_mcq option1=”5000″ option2=”4999″ option3=”4851″ option4=”4800″ correct=”option3″]

This question was previously asked in

UPSC CAPF – 2016

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The correct option is C) 4851.

The problem asks for the number of points with integer coordinates strictly *inside* the triangle formed by vertices (0, 0), (0, 100), and (100, 100).

The vertices of the triangle are A=(0, 0), B=(0, 100), and C=(100, 100).
The lines forming the boundaries of the triangle are:
1. The line segment AB is along the y-axis, from y=0 to y=100 (x=0).
2. The line segment BC is along the line y=100, from x=0 to x=100.
3. The line segment AC connects (0, 0) and (100, 100). The equation of this line is y = x.

A point (x, y) is strictly inside this triangle if it satisfies the following conditions:
1. It must be to the right of the line x=0: x > 0.
2. It must be below the line y=100: y < 100. 3. It must be above the line y=x: y > x.

Combining these, we are looking for integer coordinates (x, y) such that 0 < x < y < 100. Let's iterate through possible integer values for x. Since x > 0 and x < y < 100, the smallest possible integer value for x is 1. If x = 1, y must be an integer such that 1 < y < 100. Possible y values are 2, 3, ..., 99. The number of such y values is 99 - 2 + 1 = 98. If x = 2, y must be an integer such that 2 < y < 100. Possible y values are 3, 4, ..., 99. The number of such y values is 99 - 3 + 1 = 97. If x = 3, y must be an integer such that 3 < y < 100. Possible y values are 4, 5, ..., 99. The number of such y values is 99 - 4 + 1 = 96. ... What is the largest possible integer value for x? Since x < y < 100, the largest possible integer value for y is 99. This requires x to be at least 1 less than 99, i.e., x < 99. So, the largest possible integer value for x is 98. If x = 98, y must be an integer such that 98 < y < 100. The only possible y value is 99. The number of such y values is 1. (99 - 99 + 1 = 1) The total number of integer points inside the triangle is the sum of the number of possible y values for each x from 1 to 98. Total points = (99 - 1) + (99 - 2) + (99 - 3) + ... + (99 - 98) Total points = 98 + 97 + 96 + ... + 1 This is the sum of the first 98 positive integers. The formula for the sum of the first n positive integers is n(n+1)/2. Here, n = 98. Sum = 98 * (98 + 1) / 2 = 98 * 99 / 2 = 49 * 99. 49 * 99 = 49 * (100 - 1) = 4900 - 49 = 4851. The number of points inside the triangle with integer coordinates is 4851.

165. There are two concentric circles. The radii of the two circles are 100

There are two concentric circles. The radii of the two circles are 100 m and 110 m respectively. A wheel of radius 30 cm rolls on the smaller circle and another wheel rolls on the larger circle. After they have completed one revolution, it is found that the two wheels rolled equal number of times on their respective axes. What is the radius of the other wheel ?

[amp_mcq option1=”31 cm” option2=”32 cm” option3=”33 cm” option4=”34 cm” correct=”option3″]

This question was previously asked in

UPSC CAPF – 2016

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The correct option is C) 33 cm.

When a wheel rolls on the circumference of a circle, the distance covered by the wheel’s point of contact is equal to the circumference of the larger circle it’s rolling on. This distance is also equal to the number of revolutions the wheel makes on its own axis multiplied by its own circumference.

Radius of the smaller circle (R1) = 100 m = 10000 cm.
Radius of the larger circle (R2) = 110 m = 11000 cm.
Radius of the first wheel (r1) = 30 cm.
Let N be the number of revolutions made by each wheel on its axis.
The first wheel rolls on the smaller circle (R1). The distance covered by its point of contact is the circumference of the smaller circle, which is 2πR1. This distance is also equal to N times the circumference of the first wheel, which is N * 2πr1.
So, 2πR1 = N * 2πr1
R1 = N * r1
10000 = N * 30
N = 10000 / 30 = 1000 / 3

The second wheel rolls on the larger circle (R2). Let its radius be r2. The distance covered by its point of contact is the circumference of the larger circle, which is 2πR2. This distance is also equal to N times the circumference of the second wheel, which is N * 2πr2.
So, 2πR2 = N * 2πr2
R2 = N * r2

We know R2 = 11000 cm and N = 1000/3.
11000 = (1000/3) * r2
r2 = 11000 * (3 / 1000)
r2 = 11 * 3 = 33 cm.

The radius of the other wheel is 33 cm.

166. In the diagram given below, there is a circle, a square and a triangle

In the diagram given below, there is a circle, a square and a triangle dividing the region into five disjoint bounded areas. Each of these areas are labelled with number of players belonging to that area. The circle contains cricketers, the square contains football players and the triangle contains hockey players.
Which one of the following is not correct?

[amp_mcq option1=”Every hockey player plays football” option2=”Every cricket player plays either football or hockey” option3=”There are some hockey players who play both cricket and football” option4=”There are some football players who play neither cricket nor hockey” correct=”option3″]

This question was previously asked in

UPSC CAPF – 2016

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The correct option is C) There are some hockey players who play both cricket and football.

The question describes a diagram with a circle (cricketers, C), a square (football players, F), and a triangle (hockey players, H), divided into *five disjoint bounded areas*, each with a number of players. This implies that only these five specific areas within the sets contain players, and any intersection not represented by one of these five areas must contain zero players.

Let’s consider a plausible interpretation of the “five disjoint bounded areas” in a 3-set Venn diagram context. A standard 3-set Venn diagram has 7 possible regions inside the sets (C only, F only, H only, C&F only, C&H only, F&H only, C&F&H). If only five of these have players, two must be empty.
Statement C is “There are some hockey players who play both cricket and football”. This statement is true if the region representing players who play all three sports (C&F&H) has a number greater than zero. If the five disjoint areas described *do not include* the intersection of all three sets (C&F&H), then the number of players in that region is zero by implication. In this scenario, statement C would be false (“There are NO hockey players who play both cricket and football”).

Let’s test if this interpretation makes sense by assuming statement C is indeed the “not correct” one. This means C&F&H = 0. The five regions could then be, for example: C only, F only, H only, C&F only, C&H only, F&H only (this is 6, so one more must be 0). Or perhaps C only, F only, H only, C&F only, F&H only (omitting C&H only and C&F&H). Or C only, F only, H only, C&F only, C&H only (omitting F&H only and C&F&H).

If C&F&H = 0, then statement C is false. Let’s see if the other statements *could* be true under various configurations of 5 non-empty regions (where C&F&H=0):
A) “Every hockey player plays football”: Requires H only = 0 and C&H only = 0. Possible if the 5 regions are, for example, F only, C only, C&F only, F&H only, some other.
B) “Every cricket player plays either football or hockey”: Requires C only = 0. Possible.
D) “There are some football players who play neither cricket nor hockey”: Requires F only > 0. Possible.

Since statement C (C&F&H > 0) directly contradicts the implication that the region C&F&H is one of the *missing* regions among the five non-empty ones, statement C is the most likely to be the “not correct” one under a plausible interpretation of the five given areas. The wording “five disjoint bounded areas… labelled with number of players belonging to that area” strongly suggests these are the *only* regions within the shapes that contain players. If the C&F&H intersection is not one of these five, its count is 0.

167. If a place is affected by Tide at 6:00 AM Monday, normally at what tim

If a place is affected by Tide at 6:00 AM Monday, normally at what time the same place will be affected by Ebb ?

[amp_mcq option1=”6:26 PM Monday” option2=”6:52 AM Tuesday” option3=”12:13 PM Monday” option4=”00:26 AM Tuesday” correct=”option3″]

This question was previously asked in

UPSC CAPF – 2016

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The correct option is C) 12:13 PM Monday.

Tides follow a predictable pattern. In most places, there are two high tides and two low tides approximately every 24 hours and 50 minutes. The time between a high tide and the subsequent low tide (Ebb) is roughly half the period between two consecutive high tides, which is about 6 hours and 12.5 minutes. Ebb tide is the phase when the sea level is falling.

Given that high tide occurs at 6:00 AM Monday.
The low tide following this high tide will occur approximately 6 hours and 12.5 minutes later.
6:00 AM + 6 hours 12.5 minutes = 12:12:30 PM Monday.
The ebb tide is the period from high tide to low tide. The question asks “at what time the same place will be affected by Ebb?”. This is slightly ambiguous, as ebb is a period. However, among the options, 12:13 PM is the closest time to the predicted low tide, which marks the *end* of the ebb phase and the start of the flood phase. The other options represent subsequent high tide times. Therefore, 12:13 PM Monday is the most plausible answer representing the transition point following the ebb phase.

168. Which of the following statements relating to the solar system is/are

Which of the following statements relating to the solar system is/are correct?
1. Venus and Uranus rotate opposite to the direction of rotation of Earth
2. Venus is the nearest planet to Earth
Select the correct answer using the code given below:

[amp_mcq option1=”1 only” option2=”2 only” option3=”Both 1 and 2″ option4=”Neither 1 nor 2″ correct=”option3″]

This question was previously asked in

UPSC CAPF – 2016

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The correct option is C) Both 1 and 2.

Statement 1 refers to the rotation direction of Venus and Uranus compared to Earth. Statement 2 refers to the relative distance of Venus from Earth.

1. **Venus and Uranus rotate opposite to the direction of rotation of Earth:** Correct. Earth rotates counter-clockwise when viewed from above the North Pole (prograde rotation). Venus rotates clockwise (retrograde rotation). Uranus also has a unique rotation; it rotates on its side, but its rotation direction is also considered retrograde relative to its orbit.
2. **Venus is the nearest planet to Earth:** Correct, in the context of being Earth’s closest planetary neighbour at its point of closest approach. While Mercury is closer to the Sun and, on average, closer to Earth than Venus is, Venus makes the closest approach to Earth among all planets. In common parlance and many educational contexts, Venus is referred to as Earth’s nearest planet.
Both statements are considered correct in a general knowledge context like a UPSC exam.

169. Which of the following statements concerning the circum-pacific belt a

Which of the following statements concerning the circum-pacific belt are correct?
1. It is an active volcanic realm
2. It is an active seismic realm
3. It is a divergent plate boundary
4. It is prone to Tsunami
Select the correct answer using the code given below:

[amp_mcq option1=”1 and 2 only” option2=”3 and 4 only” option3=”1, 2 and 4 only” option4=”1, 2, 3 and 4″ correct=”option3″]

This question was previously asked in

UPSC CAPF – 2016

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The correct option is C) 1, 2 and 4 only.

The Circum-Pacific Belt, also known as the Ring of Fire, is a major area in the basin of the Pacific Ocean where a large number of earthquakes and volcanic eruptions occur. This high activity is due to the presence of numerous plate boundaries, primarily convergent ones, where tectonic plates are colliding or subducting.

Let’s evaluate each statement:
1. **It is an active volcanic realm:** Correct. The Ring of Fire contains a vast majority of the world’s active and dormant volcanoes.
2. **It is an active seismic realm:** Correct. It is the most seismically active region globally, experiencing frequent and powerful earthquakes.
3. **It is a divergent plate boundary:** Incorrect. While there are some segments of divergent boundaries (like the East Pacific Rise) and transform boundaries within or connected to the Ring of Fire, the dominant type of plate interaction responsible for the intense volcanism and seismicity in most parts of the belt is **convergent**, involving subduction zones. Divergent boundaries are typically associated with mid-ocean ridges.
4. **It is prone to Tsunami:** Correct. Tsunamis are often triggered by large underwater earthquakes, volcanic eruptions, or landslides, particularly those associated with subduction zones found along convergent plate boundaries like those prevalent in the Ring of Fire.
Therefore, statements 1, 2, and 4 are correct, while statement 3 is incorrect.