Find the value of cos 120° cos 240° cos 180° cos 60°.
\( \frac{-1}{8} \)
\( \frac{1}{2} \)
\( \frac{1}{4} \)
\( \frac{1}{16} \)
To find the value of the given trigonometric expression, we need to recall the values of cosine for the specific angles involved (60°, 120°, 180°, 240°). We will use the unit circle or trigonometric identities to determine these values and then multiply them together.
Recall the standard trigonometric values:
\( \cos 60^{\circ} = \frac{1}{2} \)
\( \cos 180^{\circ} = -1 \)
For \( \cos 120^{\circ} \), we can use the identity \( \cos (180^{\circ} - \theta) = -\cos \theta \):
\[ \cos 120^{\circ} = \cos (180^{\circ} - 60^{\circ}) = -\cos 60^{\circ} = -\frac{1}{2} \]For \( \cos 240^{\circ} \), we can use the identity \( \cos (180^{\circ} + \theta) = -\cos \theta \):
\[ \cos 240^{\circ} = \cos (180^{\circ} + 60^{\circ}) = -\cos 60^{\circ} = -\frac{1}{2} \]Now, substitute these values into the given expression:
\[ \cos 120^{\circ} \cos 240^{\circ} \cos 180^{\circ} \cos 60^{\circ} \]\[ = \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times (-1) \times \left(\frac{1}{2}\right) \]Perform the multiplication:
\[ = \left(\frac{1}{4}\right) \times (-1) \times \left(\frac{1}{2}\right) \]\[ = -\frac{1}{4} \times \frac{1}{2} \]\[ = -\frac{1}{8} \]A) \( \frac{-1}{8} \) is the correct value obtained after calculating the product of the cosine values for the given angles.
B) \( \frac{1}{2} \) — This would be incorrect as the product involves negative terms, resulting in a negative final value.
C) \( \frac{1}{4} \) — This value would be obtained if there were an even number of negative terms, or if \( \cos 180^{\circ} \) was positive, which is not the case.
D) \( \frac{1}{16} \) — This value is too small and positive. It would imply that all terms were positive and smaller in magnitude, or that there were more terms.
The question asks about the origin state of the Sabarmati River. To answer this, one needs to recall the geographical origin of major Indian rivers. The Sabarmati River is a prominent west-flowing river in India, and its source is located in the Aravalli Range.
Correct Option: D) Rajasthan
The Sabarmati River originates from the Aravalli Range, specifically from the Tepur village in the Udaipur District of Rajasthan. It then flows generally southwest through Rajasthan and Gujarat before emptying into the Gulf of Khambhat (Cambay) in the Arabian Sea.
To find the age of the eldest child, we can set up an algebraic equation. Since the children are born at intervals of four years, their ages form an arithmetic progression. We can define the age of the youngest child as 'x' and then express the ages of the other children in terms of 'x'. The sum of these ages is given as 80.
Let the age of the youngest child be \(x\) years.
Since the children are born at intervals of four years, their ages will be:
The sum of their ages is given as 80. So, we can write the equation:
\[ x + (x + 4) + (x + 8) + (x + 12) + (x + 16) = 80 \]Combine the like terms:
\[ (x + x + x + x + x) + (4 + 8 + 12 + 16) = 80 \]\[ 5x + 40 = 80 \]Solve for \(x\):
\[ 5x = 80 - 40 \]\[ 5x = 40 \]\[ x = \frac{40}{5} \]\[ x = 8 \]So, the age of the youngest child is 8 years.
The age of the eldest child is \(x + 16\).
\[ \text{Eldest child's age} = 8 + 16 = 24 \text{ years} \]Correct Option: B) 24
Plaster of Paris hardens by-
Giving off carbon dioxide
Changing into CaCO3
Giving out water
Combining with water
The question asks about the chemical process by which Plaster of Paris (POP) hardens. This involves understanding the chemical reaction that occurs when POP is mixed with water.
Correct Option: D) Combining with water
The question asks to identify the Mauryan emperor in whose court Chanakya served as a minister. This requires knowledge of the early Mauryan period and the key figures associated with its establishment and administration.
Correct Option: A) Chanakya was the chief minister and advisor in the court of Chandragupta Maurya. He was instrumental in Chandragupta's rise to power and the establishment of the Mauryan Empire.
The question asks to identify the physical quantity whose unit is Joule per Coulomb. This requires recalling the definitions and units of fundamental electrical quantities.
Correct Option: C) Volt (V)
The question asks to identify the scientist who discovered neutrons in the nucleus of an atom. This requires knowledge of the historical development of atomic theory and the contributions of key scientists.
D) J. Chadwick — James Chadwick discovered the neutron in 1932, a fundamental particle found in the nucleus of an atom.
The question asks to identify the Union Territory or state for which Kiran Bedi was the Lieutenant Governor as of 2019. This is a factual question related to current affairs and political appointments in India.
Correct Option: B) Puducherry
Kiran Bedi was indeed the Lieutenant Governor of Puducherry from May 29, 2016, to February 16, 2021. Therefore, as of 2019, she held this position.
This is a circular seating arrangement puzzle. We need to deduce the positions of eight captains around a circular table based on the given clues. We will start by fixing one person's position and then place others relative to them, using both direct and indirect clues. It's helpful to draw a circle and mark positions.
Clue 1: The Russian captain sits third to the left of the Indian captain.
Let's place the Indian captain (Ind) first. Then, the Russian captain (Rus) will be three positions to the left.
\[ \text{... Rus ... Ind ...} \]
Clue 2: Indian captain sits third to the left of the Mexico captain.
This means the Mexico captain (Mex) is third to the right of the Indian captain. If Ind is at position 1, then Mex is at position 4 (1 + 3 = 4).
\[ \text{... Ind ... Mex ...} \]
Clue 3: Only two people sit between the Russian captain and Spain captain.
From Clue 1, we have Rus and Ind. If we count two people between Rus and Spain (Spa), there are two possibilities for Spain's position relative to Russia. Let's consider the arrangement from Clue 1: Rus _ _ Ind. If Spain is two places away from Rus, then Spain could be to the immediate left or right of Ind, or further away. However, we also have Clue 4.
Clue 4: Neither the Russian captain nor Spain captain is an immediate neighbour of the Indian captain.
This is crucial. Since Rus is already three positions away from Ind, this clue confirms that Spain cannot be next to Ind. Combining this with Clue 3, if we place Rus, then count two places, and then place Spa, Spa cannot be next to Ind. Given Rus is 3rd left of Ind, and Mex is 3rd right of Ind, the arrangement so far is:
\[ \text{Position: 1 2 3 4 5 6 7 8} \\ \text{Captain: Ind _ _ Mex _ _ Rus _} \]
Now, let's place Spain. If we count two people between Rus and Spa, and Spa is not next to Ind, then Spa must be in the position that is two places to the right of Rus (or two places to the left, but that would put Spa next to Ind or Mex, which might be problematic). Let's try placing Spa two places to the right of Rus. This would put Spa in the position between Mex and Ind, which is not allowed. So, Spa must be two places to the left of Rus. This places Spa in the position between Ind and Rus, but not adjacent to Ind.
Let's re-evaluate with a circular diagram. Assume Ind is at position 1. Then Mex is at 4, and Rus is at 6. The positions are 1 (Ind), 2, 3, 4 (Mex), 5, 6 (Rus), 7, 8.
Now, two people between Rus (6) and Spa. If Spa is at 3, then 7, 8 are between Rus and Spa (clockwise). If Spa is at 8, then 7 is between Rus and Spa (clockwise). This is not two people. If Spa is at 3, then 7, 8, 1, 2 are between Rus and Spa (clockwise). This is too many. Let's count positions. Rus is at 6. Two people between Rus and Spa means Spa is at 3 or 9 (which is 1). If Spa is at 3, then positions 7, 8 are between Rus and Spa (2 people). This works. So, Spa is at position 3.
Current arrangement: Ind (1), _ (2), Spa (3), Mex (4), _ (5), Rus (6), _ (7), _ (8).
Check Clue 4: Neither Rus (6) nor Spa (3) is an immediate neighbour of Ind (1). This is true. Rus is at 6, Ind is at 1. Positions 7, 8 are between them. Spa is at 3, Ind is at 1. Position 2 is between them. This is correct.
Clue 5: The captain of Japan sits second to the right of the Peru captain.
This means Peru (Per) _ Japan (Jap).
Clue 6: Peru captain is not an immediate neighbour of the Indian captain.
Per cannot be at position 2 or 8.
Clue 7: Argentina captain sits third to the left of the Peru captain.
This means Per _ _ Arg.
Combining Clues 5, 6, 7:
We have positions 2, 5, 7, 8 remaining. Per cannot be at 2 or 8. So Per must be at 5 or 7.
Case A: If Per is at 5.
Then Jap is second to the right of Per (Clue 5). So Jap is at 7 (5 -> 6 -> 7).
Then Arg is third to the left of Per (Clue 7). So Arg is at 2 (5 -> 4 -> 3 -> 2).
This gives us: Ind (1), Arg (2), Spa (3), Mex (4), Per (5), Rus (6), Jap (7), _ (8).
The remaining team is Brazil (Bra), which must be at position 8.
Let's verify all conditions for this arrangement:
Current complete arrangement (clockwise starting from Ind):
\[ \text{Ind (1) - Arg (2) - Spa (3) - Mex (4) - Per (5) - Rus (6) - Jap (7) - Bra (8)} \]
Clue 8: Captains of Argentina and Brazil are not immediate neighbours.
In our arrangement, Arg is at 2 and Bra is at 8. They are not immediate neighbours (Ind is between them if we go clockwise from Arg to Bra, or Spa/Mex/Per/Rus/Jap if we go anti-clockwise). This condition is satisfied.
All conditions are met with the arrangement:
\[ \text{Ind - Arg - Spa - Mex - Per - Rus - Jap - Bra} \]
The question asks: Which of the following captains sits to the immediate right of the captain of Mexico?
Mexico (Mex) is followed by Peru (Per) in the clockwise direction (which is immediate right in a circular arrangement facing the center).
D) Peru — Based on the final arrangement, Peru is immediately to the right of Mexico.
The problem involves decoding a word based on a given coding pattern. We need to identify the rule applied to transform 'CASUAL' into 'GEWYEP' and then apply the same rule to 'PEOPLE'. This is a letter-coding type of question where each letter is shifted by a certain number of positions in the alphabet.
Correct Option: A) TISTPI is the correct code for 'PEOPLE' based on the identified pattern of shifting each letter forward by 4 positions.