From a point P on a level ground, the angle of elevation of the top of the tower is 30°. If the tower is 50 m high, the distance of point P from the foot of the tower is:
50\(\sqrt{3}\) m
\(\frac{50}{\sqrt{3}}\) m
25 m
\(\frac{50}{\sqrt{2}}\) m
This problem involves trigonometry, specifically the concept of the angle of elevation. We can model the situation as a right-angled triangle where the tower's height is the perpendicular side, the distance from point P to the foot of the tower is the base, and the angle of elevation is given. We will use the tangent function to relate these quantities.
Visualize the scenario: Let the top of the tower be T, the foot of the tower be F, and the point on the ground be P. The tower TF is perpendicular to the ground FP. This forms a right-angled triangle \(\triangle TFP\), with the right angle at F.
Identify given values:
Apply trigonometric ratio: In a right-angled triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. In \(\triangle TFP\):
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{TF}{FP} \]Substitute the values:
\[ \tan(30^\circ) = \frac{50}{x} \]Recall the value of \(\tan(30^\circ)\):
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]Solve for \(x\):
\[ \frac{1}{\sqrt{3}} = \frac{50}{x} \]\[ x = 50\sqrt{3} \text{ m} \]A) \(50\sqrt{3}\) m — This is the correct distance calculated using the tangent function.
The problem asks us to identify the letter-cluster pair that does not follow the same pattern as the others, based on the English alphabetical order. We need to find the relationship between the letters within each pair and then between the two clusters in the pair (e.g., WC and BY).
A) WC – BY does not follow the pattern. In this option, the first letter of the second cluster (B) is +5 from the first letter of the first cluster (W) when considering alphabetical wrap-around (W + 5 = B). However, the second letter of the second cluster (Y) is not -5 from the second letter of the first cluster (C). C (3) - 5 = X (24), not Y (25).
To find the product of 596 and 604, we can use the algebraic identity \((a-b)(a+b) = a^2 - b^2\). This method simplifies the multiplication of numbers that are equidistant from a common reference number.
D) 359984 is the correct value obtained by applying the algebraic identity.
The question asks to identify the revolt led by Bholanath Singh. This requires knowledge of various peasant and tribal uprisings in Indian history and their respective leaders.
Correct Option: C) Tamar Revolt
Bholanath Singh was indeed a leader associated with the Tamar Revolt. The Tamar Revolt was a series of uprisings by the Munda tribes in the Chota Nagpur region (present-day Jharkhand) against the British and local landlords. These revolts occurred periodically from the late 18th century to the mid-19th century, with Bholanath Singh being one of the notable leaders during its various phases.
This is a coding-decoding question based on letter positions in the English alphabet and arithmetic operations. We need to identify the pattern relating the given pairs and apply it to the third term.
C) XS -16 is the correct option because it follows the established pattern of the first letter advancing by 2, the second letter advancing by 3, and the number decreasing by 17.
To find Amit's income in 2021, we need to calculate the increment for each year starting from 2019. The income increases by 20% each year, which means it's a compound increase. We will calculate the income for 2020 first, and then use that value to calculate the income for 2021.
Alternatively, we can use the formula for compound growth:
\[ \text{Final Income} = \text{Initial Income} \times \left(1 + \frac{\text{Rate}}{100}\right)^{\text{Number of Years}} \]\[ \text{Income in 2021} = 22,000 \times \left(1 + \frac{20}{100}\right)^2 \]\[ \text{Income in 2021} = 22,000 \times \left(1 + 0.20\right)^2 \]\[ \text{Income in 2021} = 22,000 \times \left(1.20\right)^2 \]\[ \text{Income in 2021} = 22,000 \times 1.44 \]\[ \text{Income in 2021} = \text{₹}31,680 \]C) The income in 2021 is ₹31,680, as calculated by applying a 20% increment annually for two years (2020 and 2021).
The question asks about a major tax reform introduced by the 101st Amendment Act, 2016. This requires knowledge of significant constitutional amendments and their impact on India's taxation system.
Correct Option: A) Goods and Services Tax (GST)
The 101st Amendment Act, 2016, is a landmark constitutional amendment in India that paved the way for the introduction of the Goods and Services Tax (GST). This amendment inserted new articles into the Constitution, such as Article 246A, 269A, and 279A, to facilitate the implementation and administration of GST. It replaced multiple indirect taxes levied by the central and state governments with a single, unified tax.
The given series is a combination of arithmetic and multiplication operations. We need to identify the pattern by analyzing the relationship between consecutive numbers.
The first number is 13, and the second is 8. The difference is \(13 - 8 = 5\). So, the operation is \(-5\).
\[ 13 - 5 = 8 \]The second number is 8, and the third is 24. The relationship is multiplication by 3.
\[ 8 \times 3 = 24 \]The third number is 24, and the fourth is 19. The difference is \(24 - 19 = 5\). So, the operation is \(-5\).
\[ 24 - 5 = 19 \]The fourth number is 19, and the fifth is 57. The relationship is multiplication by 3.
\[ 19 \times 3 = 57 \]The fifth number is 57, and the sixth is 52. The difference is \(57 - 52 = 5\). So, the operation is \(-5\).
\[ 57 - 5 = 52 \]The sixth number is 52, and the seventh is 156. The relationship is multiplication by 3.
\[ 52 \times 3 = 156 \]The pattern observed is an alternating sequence of operations: subtract 5, then multiply by 3, then subtract 5, then multiply by 3, and so on.
The sequence of operations is: \(-5, \times 3, -5, \times 3, -5, \times 3, \ldots\)
The last operation was multiplication by 3 (from 52 to 156). Therefore, the next operation should be subtracting 5 from 156.
\[ 156 - 5 = 151 \]Correct Option: B) 151
The question asks for the number of rural households graded based on deprivation criteria as per the Socio Economic and Caste Census (SECC) concluded in March 2016 by the Ministry of Rural Development. This requires knowledge of the key findings of the SECC 2011 data pertaining to rural households.
C) 8.72 crore is the correct number of rural households graded based on deprivation criteria as per the SECC 2011 data, as concluded by the Ministry of Rural Development in March 2016.
The question asks about the constitutional body whose advice the President of India acts upon when appointing Governors of States and Lieutenant Governors of Union Territories. This falls under the executive powers of the President and the constitutional framework of India.
Correct Option: B) The Union Council of Ministers