The problem asks us to identify the number that is different from the rest among the given four options. This type of problem usually involves finding a common pattern or property shared by three numbers, while one number does not follow that pattern. We will examine each number for various mathematical properties or digit relationships.
Let's re-evaluate the options for a common pattern. A common pattern in such questions often involves digit repetition, divisibility rules, or sums/products of digits.
Consider the pattern where the first digit is the same as the third digit:
This means that if this is the pattern, then D would be the odd one out. Let's re-check the question and options carefully. The question asks to select the number that is different from the rest. This implies three are alike and one is different.
Let's reconsider the options and look for another pattern, as the first pattern attempt (first digit = third digit) identified D as different, but the provided correct answer is A.
Let's look at the sum of the first two digits and the last two digits, or similar combinations.
Consider the relationship between the first digit and the last digit:
Based on this pattern, numbers B, C, and D are alike because their first and last digits are different. Number A is different because its first and last digits are the same.
A) 4174 — In this number, the first digit (4) is the same as the last digit (4).
Therefore, 4174 is the number that is different from the rest, as it is the only one where the first and last digits are identical.
The question asks to identify the entity to which the 'Panama Papers' belonged. This requires knowledge of a significant global event related to offshore finance and tax evasion.
Correct Option: C) Mossack Fonseca & Co
The 'Panama Papers' were a collection of 11.5 million leaked confidential documents that detailed financial and attorney-client information for more than 214,488 offshore entities. The documents belonged to Mossack Fonseca & Co., a Panamanian law firm and corporate service provider. The leak, which occurred in 2016, exposed how wealthy individuals and public officials worldwide used offshore tax havens to hide their wealth and avoid taxes.
If cos θ + sec θ = 2, then the value of sin6 θ + cos6 θ is:
The problem provides an equation involving \(\cos \theta\) and \(\sec \theta\) and asks for the value of an expression involving \(\sin 6 \theta\) and \(\cos 6 \theta\). The key to solving this is to first find the value of \(\theta\) from the given equation. Once \(\theta\) is known, substitute it into the expression \(\sin 6 \theta + \cos 6 \theta\) to find its value.
The given equation is:
\[ \cos \theta + \sec \theta = 2 \]We know that \(\sec \theta = \frac{1}{\cos \theta}\). Substitute this into the equation:
\[ \cos \theta + \frac{1}{\cos \theta} = 2 \]Multiply the entire equation by \(\cos \theta\) (assuming \(\cos \theta \neq 0\)):
\[ \cos^2 \theta + 1 = 2 \cos \theta \]Rearrange the terms to form a quadratic equation:
\[ \cos^2 \theta - 2 \cos \theta + 1 = 0 \]This is a perfect square trinomial, which can be factored as:
\[ (\cos \theta - 1)^2 = 0 \]Taking the square root of both sides:
\[ \cos \theta - 1 = 0 \]Therefore,
\[ \cos \theta = 1 \]If \(\cos \theta = 1\), then the principal value of \(\theta\) is \(0^\circ\) or \(0\) radians. (Other general solutions like \(2n\pi\) also exist, but \(0\) is sufficient for this problem.)
We need to find the value of \(\sin 6 \theta + \cos 6 \theta\). Substitute \(\theta = 0^\circ\) into the expression:
\[ \sin (6 \times 0^\circ) + \cos (6 \times 0^\circ) \]This simplifies to:
\[ \sin 0^\circ + \cos 0^\circ \]We know that \(\sin 0^\circ = 0\) and \(\cos 0^\circ = 1\). So, the expression becomes:
\[ 0 + 1 = 1 \]The value of \(\sin 6 \theta + \cos 6 \theta\) is 1.
B) 1 — As derived in the step-by-step solution, the value of the expression is 1.
The problem requires us to solve a trigonometric equation for the variable \(x\). We need to use the known value of \(\tan 240^\circ\), substitute it into the given equation, and then solve for \(x\).
The angle \(240^\circ\) is in the third quadrant. In the third quadrant, the tangent function is positive. We can write \(240^\circ\) as \(180^\circ + 60^\circ\).
\[ \tan 240^\circ = \tan (180^\circ + 60^\circ) \]\[ = \tan 60^\circ \]\[ = \sqrt{3} \]The given equation is \(2 \sin(3x - 3)^\circ = \tan 240^\circ\). Substituting \(\tan 240^\circ = \sqrt{3}\) gives:
\[ 2 \sin(3x - 3)^\circ = \sqrt{3} \]Divide both sides by 2:
\[ \sin(3x - 3)^\circ = \frac{\sqrt{3}}{2} \]We know that \(\sin 60^\circ = \frac{\sqrt{3}}{2}\). Therefore,
\[ 3x - 3 = 60 \]Add 3 to both sides:
\[ 3x = 60 + 3 \]\[ 3x = 63 \]Divide both sides by 3 to find \(x\):
\[ x = \frac{63}{3} \]\[ x = 21^\circ \]B) \(21^\circ\) is the correct value for \(x\) that satisfies the given equation.
Which Five year plan in India was based on P.C. Mahalanobis Model?
1st Five Year Plan
2nd Five Year Plan
4th Five Year Plan
3rd Five Year Plan
The question asks to identify which Five Year Plan in India was based on the P.C. Mahalanobis Model. This requires knowledge of the history and economic models associated with India's Five Year Plans.
Based on this historical context, the P.C. Mahalanobis Model is directly associated with the Second Five Year Plan.
B) 2nd Five Year Plan. The Second Five Year Plan (1956-1961) was indeed based on the P.C. Mahalanobis Model, which emphasized rapid industrialization and the development of heavy industries.
This question tests basic computer terminology related to data storage units. Understanding the fundamental units like bits, nibbles, and bytes is crucial for competitive exams.
Correct Option: A) Half a byte
The question asks to identify the modern country that was historically known as Abyssinia. This requires knowledge of historical geography and country names.
Correct Option: B) Ethiopia
Abyssinia is the ancient name for the country now known as Ethiopia. The name "Abyssinia" was derived from the Arabic "Habash," referring to the Habesha people who inhabited the region. The country officially adopted the name Ethiopia in the early 20th century, though it had been used interchangeably with Abyssinia for centuries.
The sum of the interior angles of any quadrilateral is always 360°. Given the ratio of the angles, we can represent each angle as a multiple of a common factor. By summing these multiples and equating them to 360°, we can find the value of the common factor and then determine each angle, including the smallest one.
Correct Option: D) 36°
The question asks to identify the Queen of Ahmednagar who resisted Emperor Akbar. This requires knowledge of prominent female rulers in Indian history and their interactions with the Mughal Empire, specifically during Akbar's reign.
Correct Option: A) Chand Bibi
Chand Bibi was a powerful regent of the Ahmednagar Sultanate. She is renowned for her valiant defense of Ahmednagar against the Mughal forces of Emperor Akbar in 1595. She negotiated a peace treaty with the Mughals, ceding Berar, but later fought again when the Mughals attacked. Her resistance is a significant event in Deccan history.
A and B together can dig a tank in 12 days, which A alone can dig in 24 days. How much of the tank can B dig in 1 day?
\( \frac{1}{4} \)
\( \frac{1}{36} \)
\( \frac{1}{24} \)
\( \frac{1}{12} \)
This problem can be solved using the concept of work and time. The total work is considered as 1 unit. We calculate the individual work rates (amount of work done in 1 day) for A and for A and B together. Then, we subtract A's work rate from the combined work rate to find B's work rate.
Calculate the combined work rate of A and B: If A and B together can dig the tank in 12 days, then in 1 day, they can dig \(\frac{1}{12}\) of the tank.
\[ (\text{A} + \text{B})'s \text{ 1-day work} = \frac{1}{12} \]Calculate the work rate of A alone: If A alone can dig the tank in 24 days, then in 1 day, A can dig \(\frac{1}{24}\) of the tank.
\[ \text{A's 1-day work} = \frac{1}{24} \]Calculate the work rate of B alone: To find how much B can dig in 1 day, subtract A's 1-day work from the combined 1-day work of A and B.
\[ \text{B's 1-day work} = (\text{A} + \text{B})'s \text{ 1-day work} - \text{A's 1-day work} \] \[ \text{B's 1-day work} = \frac{1}{12} - \frac{1}{24} \]Perform the subtraction: To subtract the fractions, find a common denominator, which is 24.
\[ \text{B's 1-day work} = \frac{2}{24} - \frac{1}{24} \] \[ \text{B's 1-day work} = \frac{2 - 1}{24} \] \[ \text{B's 1-day work} = \frac{1}{24} \]So, B can dig \(\frac{1}{24}\) of the tank in 1 day.
Correct Option: C) \(\frac{1}{24}\)