The question asks to identify the landmark case that first established the principle of judicial review in constitutional law. This requires knowledge of significant legal precedents, particularly those from the United States and India, concerning the powers of the judiciary.
Correct Option: B) Marbury v. Madison (1803) — This landmark U.S. Supreme Court case, decided in 1803, is universally recognized for establishing the principle of judicial review in constitutional law. Chief Justice John Marshall's opinion asserted the Court's authority to declare an act of Congress unconstitutional.
The question pertains to a hypothetical Supreme Court ruling on 'dying declarations' in a specific case. In Indian law, dying declarations are admissible under Section 32(1) of the Indian Evidence Act, 1872. However, their evidentiary value is subject to certain judicial principles. The approach is to identify the established legal position regarding the reliability and corroboration requirements for dying declarations, especially when they appear suspicious.
Correct Option: A) Suspicious dying declarations require corroborative evidence.
To solve this number series problem, we need to identify the pattern or rule governing the sequence of numbers. We will look for differences between consecutive terms, and then examine if there's a pattern in these differences (a second-level difference pattern).
Calculate the first differences between consecutive terms:
The series of first differences is: 49, 44, 39, 34.
Identify the pattern in the first differences:
Observe the series 49, 44, 39, 34. Each subsequent number is 5 less than the previous one.
This indicates a consistent pattern where the difference decreases by 5 each time.
Determine the next difference in the sequence:
Following the pattern, the next difference should be \(34 - 5 = 29\).
Calculate the next term in the original series:
The last term given is 107. To find the next term, we subtract the calculated difference (29) from 107.
\[ 107 - 29 = 78 \]The next number in the series is 78, which corresponds to option B.
The question asks to identify which document is NOT mandatory for applying for Indian citizenship under the Citizenship (Amendment) Rules, 2024, framed under CAA 2019. We need to recall the key provisions and required documents as per these rules.
D) Birth Certificate — The Citizenship (Amendment) Rules, 2024, do not list a birth certificate as a mandatory document for applying for Indian citizenship under the CAA. The rules emphasize documents proving the applicant's country of origin (Afghanistan, Bangladesh, or Pakistan), their religious minority status, and their date of entry into India.
The question asks about the field for which Anant Nag received the Padma Bhushan in 2025. This requires knowledge of recent Padma Awards and the recipients' contributions.
Correct Option: A) Art
The problem asks us to find the principal amount invested given the difference between compound interest (CI) and simple interest (SI) for a specific period and rate. We can use the formula for the difference between CI and SI for 2 years, or calculate CI and SI separately and then find the principal.
The difference between compound interest and simple interest for 2 years is given by:
\[ \text{Difference} = P \left( \frac{R}{100} \right)^2 \]Let \(P\) be the principal amount. We have:
\[ 43 = P \left( \frac{4}{100} \right)^2 \]The sum invested was ₹26,875.
D) ₹26,875 — This is the principal amount calculated using the formula for the difference between CI and SI for 2 years.
This problem requires the application of the empirical formula relating mean, median, and mode for a moderately skewed distribution. The formula states that Mode \(= 3 \times\) Median \(- 2 \times\) Mean. We are given a relationship between the median and mode, and we need to find the difference between the median and the mean.
Recall the Empirical Formula:
The empirical relationship between mean, median, and mode for a moderately skewed distribution is:
Express the given information mathematically:
We are given that the median is 80.32 less than its mode. This can be written as:
Rearrange the given information to find Mode in terms of Median:
From the above, we get:
Substitute this into the empirical formula:
Substitute the expression for Mode from step 3 into the empirical formula from step 1:
Rearrange the equation to isolate the desired difference:
Our goal is to find the value of \( \text{Median} - \text{Mean} \). Let's rearrange the equation from step 4:
Factor out 2 and solve for the difference:
\[ 80.32 = 2 \times (\text{Median} - \text{Mean}) \]\[ \text{Median} - \text{Mean} = \frac{80.32}{2} \]\[ \text{Median} - \text{Mean} = 40.16 \]Thus, the median of the data exceeds its mean by 40.16.
C) 40.16 is the correct answer, as calculated by applying the empirical formula and the given relationship between median and mode.
The diameter of the base and slant height of a right circular cone are 30 cm and 113 cm, respectively. Find the volume (in cm³) of the given cone. (Use π = \( \frac{22}{7} \))
To find the volume of a right circular cone, we need its radius and height. The problem provides the diameter of the base and the slant height. We can calculate the radius from the diameter. Then, using the Pythagorean theorem, we can find the height of the cone from the radius and slant height. Finally, we apply the formula for the volume of a cone.
Calculate the radius (r):
Given diameter \(D = 30 \text{ cm}\).
The radius is half of the diameter:
\[ r = \frac{D}{2} = \frac{30}{2} = 15 \text{ cm} \]Calculate the height (h):
Given slant height \(l = 113 \text{ cm}\).
For a right circular cone, the radius, height, and slant height form a right-angled triangle. So, we can use the Pythagorean theorem:
\[ l^2 = r^2 + h^2 \] \[ 113^2 = 15^2 + h^2 \] \[ 12769 = 225 + h^2 \] \[ h^2 = 12769 - 225 \] \[ h^2 = 12544 \] \[ h = \sqrt{12544} \]To find the square root of 12544:
We can notice that \(110^2 = 12100\) and \(120^2 = 14400\). The number ends in 4, so its square root must end in 2 or 8. Let's try 112:
\[ 112 \times 112 = 12544 \]So, \(h = 112 \text{ cm}\).
Calculate the volume (V):
The formula for the volume of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]Substitute the values \(r = 15 \text{ cm}\), \(h = 112 \text{ cm}\), and \(\pi = \frac{22}{7}\):
\[ V = \frac{1}{3} \times \frac{22}{7} \times (15)^2 \times 112 \] \[ V = \frac{1}{3} \times \frac{22}{7} \times 225 \times 112 \]Simplify the terms:
Divide 225 by 3:
\[ \frac{225}{3} = 75 \]Divide 112 by 7:
\[ \frac{112}{7} = 16 \]Now, multiply the remaining terms:
\[ V = 22 \times 75 \times 16 \] \[ V = 22 \times (75 \times 16) \] \[ V = 22 \times 1200 \] \[ V = 26400 \text{ cm}^3 \]Correct Option: C) 26,400
The question asks to identify the type of combustion that occurs when a burning matchstick is brought near a gas stove's knob, and the gas is turned on. This scenario describes a common household event involving the rapid ignition of a fuel source (LPG gas) by an external ignition source (matchstick).
B) Rapid combustion — This is the correct answer because the gas from the stove burns quickly with the help of an external ignition source (the matchstick), producing heat and light. This is the characteristic definition of rapid combustion.
The question asks for the specific keyboard shortcut to open the Settings menu in Windows 11. This requires knowledge of common Windows operating system shortcuts.
The Windows Key + I combination is universally recognized in Windows operating systems, including Windows 11, as the shortcut to open the Settings application. This application allows users to configure various system settings, personalize their device, manage apps, and much more.
Correct Option: C) Windows Key + I